TRANSIENT AND STEADY-STATE CREEP IN A Sn-Ag-Cu LEAD …

TRANSIENT AND STEADY-STATE CREEP IN A

Sn-Ag-Cu LEAD-FREE SOLDER ALLOY:

EXPERIMENTS AND MODELING

by

Dwayne R. Shirley

A thesis submitted in conformity with the requirements

for the degree of Doctor of Philosophy

Department of Materials Science & Engineering

University of Toronto

© Copyright by Dwayne R. Shirley 2009

ii

TRANSIENT AND STEADY-STATE CREEP IN A Sn-Ag-Cu LEAD-FREE SOLDER ALLOY:

EXPERIMENTS AND MODELING

Dwayne R. Shirley

Doctor of Philosophy

Department of Materials Science & Engineering University of Toronto

2009

Abstract

It has been conventional to simplify the thermo-mechanical modeling of solder joints by

omitting the primary (transient) contributions to total creep deformation, assuming that secondary

(steady-state) creep strain is dominant and primary creep is negligible. The error associated with this

assumption has been difficult to assess because it depends on the properties of the solder joint and

the temperature-time profile. This research examines the relative contributions of primary and

secondary creep in Sn3.8Ag0.7Cu solder using the constant load creep and stress relaxation

measurements for bulk tensile specimens and the finite element analysis of a chip resistor (trilayer)

solder joint structure that was thermally cycled under multiple temperature ranges and ramp rates. It

was found that neglect of primary creep can result in errors in the predicted stress and strain of the

solder joint. In turn, these discrepancies can lead to errors in the estimation of the solder thermal

fatigue life due to the changing proportion of primary creep strain to total inelastic strain under

different thermal profiles.

The constant-load creep and stress relaxation data for Sn3.8Ag0.7Cu span a range of strain

rates 8 1 4 110 10s sε− − − −< < , and temperatures 25°C, 75°C and 100°C. Creep and stress relaxation

measurements show that transient creep caused faster strain rates during stress relaxation for a

given stress compared to the corresponding minimum creep rate from constant-load creep tests. The

extent of strain hardening during primary creep was a function of temperature and strain rate.

iii

A constitutive creep model was presented for Sn3.8Ag0.7Cu that incorporates both transient

and steady-state creep to provide agreement for both creep and stress relaxation data with a single

set of eight coefficients. The model utilizes both temperature compensated time and strain rate to

normalize minimum strain rate and saturated transient creep strain, thereby establishing equivalence

between decreased temperature and increased strain rate. The apparent activation energy of steady-

state creep was indicative of both dislocation core and bulk lattice diffusion was the most sensitive

model parameter. A saturation threshold was defined that distinguishes whether primary or

secondary creep is dominant under either static or variable loading.

iv

ACKNOWLEDGEMENTS

The financial and in-kind contributions allocated to this project by the following entities are graciously

acknowledged. The University of Toronto’s School of Graduate Studies (SGS), the Department of

Materials Science & Engineering (MSE), and the Department of Mechanical & Industrial Engineering

(MIE). The research funding organizations the Ontario Centres of Excellence formerly entitled the

Centre for Microelectronics Assembly & Packaging (CMAP), and the National Sciences and

Engineering Research Council of Canada (NSERC). The commercial corporations Research In

Motion, Celestica, and Kester Solder.

On a personal note, I thank Prof. Jan K. Spelt and Dr. Laura J. Turbini for providing the opportunity to

pursue a doctoral dissertation and for graciously offering their expertise and support in pursuit of my

research and career objectives. I also thank Prof. Thomas H. North, Prof. Zhirui Wang and Dr. Polina

Snugovsky of Celestica for their time and contributions as doctoral dissertation committee members.

I especially thank my wife Dionne, my mother Joyce, my sister Michelle and all my family for standing

by my decision to pursue graduate studies while always supporting me in every way. I also thank my

friends and fellow lab members who through it all have supported, listened and kept me active

outside academia to help me truly enjoy this great journey.

This research is dedicated to the loving memory of my father Mr. Michael A. Shirley.

v

Table of Contents Abstract...........………………………………………..........................……………….... ii Acknowledgments...................................................................................................... iv List of Tables.............................................................................................................. viii List of Figures............................................................................................................. x CHAPTER 1 Introduction …..……………………………………………………........... 1 1.1 Background and Motivation……………………………………………………….... 1 1.2 Objectives……………..………..……………………………..……………………… 2 1.3 Literature review…………..……………………………………………………….... 3

1.3.1 Creep Mechanisms…………………………………………………………… 3 1.3.2 Differences between Bulk and Joint Type Test Specimens……………… 8

1.3.2.1 Cooling Rate Effect………………………………………………… 8 1.3.2.2 Aspect Ratio Effect………………………………………………… 9 1.3.2.3 Grain Size Coarsening.…………………………………………… 11

1.4 Creep Test Methodology…..………………………………………………………... 12 1.4.1 Constant Load Creep Testing……………………………………………… 13 1.4.2 Constant Constraint/Stress Relaxation Creep Testing……………..…… 14 1.4.3 Conversion of results between Creep Testing Methods…………..……. 15 1.4.4 Consideration of Primary Creep ……………………………………..…… 16

1.5 Overview of the Thesis…..………………………………………………………... 17 1.6 References…..……………………………………………………………………... 18 CHAPTER 2 Effect of Transient Creep and Plasticity in the Modeling of Thermal Fatigue………....……………………………………………….. 20 2.0 Introduction ..………………………………………..…………………………… 20 2.1 Finite difference trilayer model………………………………..………………… 21 2.2 Finite element modeling.................................................................................. 23 2.3 Constitutive modeling..……………………………………………………….... 26 2.4 Time hardening of solder during thermal cycling..…………………………… 31 2.5 Effect of plasticity……..………………………………………………………… 31 2.6 Thermo-mechanical cycling simulations……..……………………………..... 40

vi

2.7 Stabilized thermal cycling hysteresis loops……..…………………………... 41 2.8 Effect of temperature range on hysteresis loops……..…………………….. 45 2.9 Effect of ramp rate on hysteresis loops……..………………………………. 46 2.10 Effect of primary creep on damage metrics……..………………………….. 49 2.11 Neglect of primary creep in life prediction……..……………………………. 52 2.12 Conclusions……………………………….…..……………………………….. 60 2.13 References……………………………….…..……………………………….. 61 CHAPTER 3 Theory, Experiments and Data Analysis….…………………….. 63 3.0 Introduction ..………………………………………..………………………… 63 3.1 Experimental Procedures………………………………..…………………… 66

3.1.1 Bulk Tensile Specimen..................................................................... 66 3.2 Experimental Setup..………………………………………………………...... 68 3.3 Data Analysis………………………………...……………………………….… 69 3.3.1 Constant Load – Creep………………………………………………... 69 3.3.2 Constant Displacement – Stress Relaxation.………………………. 72

3.3.3 Machine Compliance………………………….……………………… 75 3.4 Results and Discussion…………………..………………………………..… 77

3.4.1 Constant Load Creep………………………………………………… 77 3.4.2 Stress Relaxation….………………………………………………….. 83 3.4.3 Microstructural Development………………………………………….. 88

3.5 Conclusions…………………..………………………………….……….…… 92 3.6 References…………………..………………………………………………… 93

CHAPTER 4 Constitutive Creep Model Development and Finite Element Analysis ……………………………………………………………….. 95 4.0 Introduction ..……………………………..………………………………………. 95 4.1 Constitutive Model Development………..………………………………………. 99

4.1.1 Steady-state Creep……………………………………………………….. 99 4.1.2 Normalized Steady-State Creep Rate…………………………………… 100 4.1.3 Normalized Transient Creep……………………………………………. 102 4.1.4 Modeling of Stress Relaxation Data……………………………………. 110

4.2 Discussion………..………………………………………………….……………. 115 4.2.1 Constitutive Model and Thermally-Activated Creep………………….. 114 4.2.2 Application to Thermal Cycling and Life Prediction Models………….. 115

4.3 Conclusions………..………………………………………………….……………. 117 4.4 References………..………………………………………………….……………. 118

vii

CHAPTER 5 Sensitivity of Creep-Fatigue Damage Metrics to Uncertainty in Creep Parameters ....……………………………………………….…… 120 5.0 Introduction ..……………………………..……………………..………………….. 120 5.1 Modeling Approach ..……………………………..……………………..………… 123

5.1.1 Creep Constitutive Model Equations…………………………………….. 123 5.1.2 Model Input Parameter Distributions…………………………………….. 124

5.2 Uncertainty Analysis Simulations………………..……………………..………… 126 5.2.1 Probabilistic Techniques………………………………………………….. 126 5.2.2 Bootstrap Methods……..………………………………………………….. 127

5.2.2.1 Residual Modification for Constant Variance…………….…….. 128 5.2.3 Monte Carlo Simulations with Latin Hypercube Sampling……..…….... 129 5.2.4 Kendall Tau Rank Correlation and Sensitivity Analysis………..…….... 131

5.2.4.1 Quantifying Sensitivity………………………..…………….…….. 132 5.3 Results………………..……………………………………………………………… 134

5.3.1 Creep Parameter Distributions…..……………………………………….. 134 5.3.2 Damage Metric Distributions…..………………………………………... 136 5.3.3 Sensitivity of damage metrics to creep parameter uncertainty……... 140

5.3.3.1 Substructure Affects on Stress Response….…………….…….. 145 5.3.3.2 Normalized Sensitivity Analysis……………..…………….…….. 148

5.4 Conclusions..………..……………………………………………………………… 153 5.5 References………….……………………………………………………………… 154 CHAPTER 6 Conclusions ....…………………………………………………….…… 156 6.0 Original Contributions.………………..……………………..………………….... 156 6.1 Practical Applications and Future Work....……..……………………..………… 157

viii

List of Tables CHAPTER 1 Introduction 1 CHAPTER 2 Effect of Transient Creep and Plasticity in the Modeling of 20

Thermal Fatigue Table 1. Trilayer dimensions and material properties 26 Table 2. Trilayer dimensions and material properties 29 Table 3. Constitutive models 32 Table 4. Comparison of strain energy density per cycle due to 35

rate-independent plastic deformation, Wplastic, and creep deformation, Wcreep, for stabilized von Mises stress-strain hysteresis loops with and without rate-independent plasticity.

Table 5. Comparison of accumulated equivalent plastic strain, εeqAcc.Plastic, and 36

accumulated equivalent creep strain, εeqAcc.Creep, for stabilized von Mises stress-strain hysteresis loops with and without rate-independent plasticity.

Table 6. Comparison of creep strain energy density, Wcreep, for stabilized 38

instantaneous von Mises stress-strain hysteresis loops using the secondary-only and primary plus secondary creep models.

Table 7. Comparison of accumulated equivalent creep strain, εeqAC for 39

stabilized instantaneous von Mises stress-strain hysteresis loops using the secondary-only and primary plus secondary creep models.

CHAPTER 3 Theory, Experiments and Data Reduction 63 Table 1. Test matrix for constant load creep. 68 Table 2. Stress relaxation experimental parameters 84

ix

CHAPTER 4 Constitutive Model Development 95

Table 1. Coefficients used to model steady-state creep and 99 normalize temperature compensated time and strain rate for Sn3.8 Ag0.7Cu using Eqs. 7, 8 and 9.

Table 2. Summary of coefficients used to model transient creep in 108

Sn3.8 Ag0.7Cu using Eqs. 10 and 12. CHAPTER 5 Sensitivity of Creep-Fatigue Damage Metrics to Uncertainty in 120 Creep Parameters

Table 1. Hyperbolic sine law parameters (Eq. (1)) as reported in the literature 123 for the Sn3.8 Ag0.7Cu solder alloy.

Table 2. Mean and uncertainty for each constitutive creep model parameter. 125 Table 3. Descriptive statistics for calculated damage metrics from steady-state 140

creep model, εss and transient plus steady-state creep model, εtr + εss. Table 4. εtr + εss (εss) constitutive creep model Kendall Tau correlation and 144

p-values. Table 5. Absolute sensitivity values of ∆ε ACS to uncertainty in creep parameters . 151

x

List of Figures CHAPTER 1 Introduction 1 Figure 1. Strain vs. Time behaviour during creep under constant load, Constant 5

engineering stress and the three states of creep (primary, secondary, tertiary).

Figure 2. Stress relaxation plot of the data points and fitted-line for Sn-3.6Ag-1.0 15 Cu at 25°C.

CHAPTER 2 Effect of Transient Creep and Plasticity in the Modeling of 20

Thermal Fatigue Figure 1. Free-body diagram of trilayer model. 22 Figure 2. Meshed finite element trilayer model (1672 nodes, 525 quadrilateral 25

PLANE183 elements). Figure 3. Thermal cycle with reference temperatures, 95°C/min and ΔT = 100°C. 32 Figure 4. SnAgCu von Mises stress vs. elapsed time over stabilized hysteresis 33

loop for the secondary creep model (Sec.); 95°C/min and ΔT = 165°C. Figure 5. SnAgCu von Mises stress vs. elapsed time over stabilized hysteresis 34

loop for the primary plus secondary creep model (Pri. + Sec.); 95°C/min and ΔT = 165°C.

Figure 6. SnPb von Mises stress vs. instantaneous von Mises creep strain 43

Hysteresis loops for the secondary creep model (Sec.) and the primary plus secondary creep model (Pri. + Sec.); 95 °C/min and ΔT=100°C.

Figure 7. SnAgCu von Mises stress vs. instantaneous von Mises creep strain 44

hysteresis loops for the secondary creep model (Sec.) and the primary plus secondary creep model (Pri. + Sec.); 95 °C/min and ΔT = 100°C.

Figure 8. Steady ratcheting of SnPb von Mises stress vs. instantaneous von 44

xi

Mises Creep strain hysteresis loops for the secondary creep model (Sec.) and the primary plus secondary creep model (Pri. + Sec.); 95°C/min and ΔT = 100°C.


hysteresis loops for the secondary creep model (Sec.) and the primary plus secondary creep model (Pri. + Sec.); ΔT = 100°C and ΔT = 165°C for 95°C/min.


hysteresis loops for the secondary creep model (Sec.) and the primary plus secondary creep model (Pri. + Sec.); ΔT=100°C and ΔT= 165°C for 95°C/min.


hysteresis loops for the secondary creep model (Sec.) and the primary plus secondary creep model (Pri. + Sec.); 14°C/min and 95° C/min for ΔT=100°C.


hysteresis loops for the secondary creep model (Sec.) and the primary plus secondary creep model (Pri. + Sec.); 14°C/min and 95° C/min for ΔT=100° C.

Figure 13. Accumulated creep strain vs. time for the secondary creep model 54

(Sec.) and the primary plus secondary creep model (Pri. + Sec.); (i) 95°C/min, ΔT=165°C; (ii) 95°C/min, ΔT=100°C; (iii) 14°C/min, ΔT=100°C.

Figure 14. Creep strain energy density vs. time for the secondary creep model 54

(Sec.) and the primary plus secondary creep model (Pri. + Sec.); (i) 95°C/min, ΔT=165°C; (ii) 95°C/min, ΔT=100° C, (iii) 14° C/min, ΔT = 100°C.

Figure 15. SnPb cycles to first failure vs. accumulated creep strain for the 55

secondary creep model (Sec.) and the primary plus secondary creep model (Pri. + Sec.); (i) 95°C/min, ΔT=165° C, (ii) 95° C/min, ΔT=100° C; (iii) 14° C/min, ΔT=100° C.

Figure 16. SnAgCu cycles to first failure vs. accumulated creep strain for the 57

xii

secondary creep model (Sec.) and the primary plus secondary creep model (Pri. + Sec.); (i) 95°C/min, ΔT = 165°C; (ii) 95° C/min, ΔT=100° C; (iii) 14°C/min, ΔT=100° C.

Figure 17. SnAgCu mean cycles for failure vs. accumulated creep strain for the 57

secondary creep model (Sec.) and the primary plus secondary creep model (Pri. + Sec.); (i) 95°C/min, ΔT=165° C; (ii) 95° C/min, ΔT=100°C, (iii) 14° C/min, ΔT= 100°C.

Figure 18. SnPb experimental cycles to first failure vs. calculated creep strain 58

energy density dissipation for the secondary creep model (Sec.) and the primary plus secondary creep model (Pri. + Sec.); (i) 95°C/min, ΔT=165°C; (ii) 95° C/min, ΔT=100° C; (iii) 14°C/min, ΔT= 100°C.

Figure 19. SnAgCu experimental cycles to first failure vs. calculated creep strain 58

energy density dissipation for the secondary creep model (Sec.) and the primary plus secondary creep model (Pri. + Sec.); (i) 95°C/min, ΔT=165° C; (ii) 95° C/min, ΔT=100°C; (iii) 14°C/min, ΔT=100° C.

Figure 20. SnAgCu experimental mean cycles to failure vs. calculated 59

creep strain energy density dissipation for the secondary creep model (Sec.) and the primary plus secondary creep model (Pri. + Sec.); (i) 95° C/min, ΔT=165°C; (ii) 95° C/min, ΔT=100°C; (iii) 14°C/min, ΔT=100° C.

CHAPTER 3 Theory, Experiments and Data Reduction 63 Figure 1. Creep Strain vs. Time, superposition of creep components that 65

determine the creep curve (T = 25° C, σ= 15 MPa). Figure 2. Figure 2. Tensile specimen (ruler units: mm). 67 Figure 3. Optical micrograph using cross-polarized light of aged as-cast 67

tensile specimen microstructure (no deformation): (a) low magnification (b) high magnification.

Figure 4. Sliding window schematic over a range of subwindows. 70 Figure 5. Stress Relaxation @ 100°C, FFT frequency-domain response, 73

xiii

fc = 0.0031 Hz. Figure 6. Stress Relaxation @ 100°C, raw and filtered time-domain response. 74 Figure 7. Nonlinear fit to measured creep strain versus time data at 25°C. 78 Figure 8. Nonlinear fit to measured creep strain versus time data at 75°C. 79 Figure 9. Nonlinear fit to measured creep strain versus time data at 100°C. 80 Figure 10. Figure 10. Saturation primary creep strain versus minimum creep rate 81

(25°C, 75°C, 100°C). Figure 11. Tensile stress versus minimum creep rate (25°C, 75°C, 100°C). 81 Figure 12. Extensometer measured total strain versus time (25°C, 75°C, 100°C). 85

Fast crosshead speed: 2.0/4.0/6.0 cm/min @ 25/75/100°C. Figure 13. Force versus time from stress relaxation tests (25°C, 75°C, 100°C). 85

Fast crosshead speed: 2.0/4.0/6.0 cm/min @ 25/75/100°C. Figure 14. Creep strain rate tensile stress from creep and stress relaxation 87

tests (25°C, 75°C, 100°C). Figure 15. Optical micrograph using cross-polarized light of aged and 89

crept specimen microstructure, σ= 8.6 MPa, T = 100°C: (a) low magnification (b) high magnification.

Figure 16. Optical micrograph using cross-polarized light of aged and crept 91

specimen microstructure, σ= 20 MPa, T = 25°C: (a) low magnification (b) high magnification.

CHAPTER 4 Constitutive Model Development 95 Figure 1. Minimum creep strain rate versus stress (T = 25°C, 75°C and 100°C). 100

Data points are from ref. [13] and the lines are Eq. (7) using the parameters of Table 1.

xiv

Figure 2. Normalized minimum secondary creep rate versus tensile stress term of 102 Eq. (7) (T=25°C, 75°C and 100°C) with optimal α corresponding to Table 1.

Figure 3. Saturation transient creep strain versus minimum creep rate 103

(T = 25°C, 75°C, 100°C). Figure 4. Normalization of saturation transient creep strain versus minimum 104

creep rate (T = 25°C, 75°C, 100°C). Figure 5. Creep Strain vs. Time, superposition of creep components that 104

determine the creep curve (T =°25 C, σ = 15 MPa). Figure 6. Stress dependence of the rate of exhaustion of transient creep strain 106

25°C, 75°C, 100°C). Figure 7. Normalization of the stress dependence of the rate of exhaustion of 107

Transient creep strain and minimum creep rate (25°C, 75°C, 100°C). Figure 8. Creep Strain response versus time, measured and modeled 109

(transient + steady-state creep), for σ=15 MPa at T = 25°C. Figure 9. Creep Strain response versus time, measured and modeled 109

(transient + steady-state creep), for σ=8 MPa at T = 75°C. Figure 10. Creep Strain response versus time, measured and modeled 110

(transient + steady-state creep), for σ=8.6 MPa at T = 100°C. Figure 11. Force response versus time, measured and modeled 111

(i. transient + steady-state; and ii. steady-state only), for stress relaxation tests (SRT) (25°C, 75°C, 100°C).

Figure 12. Normalized rate of transient creep exhaustion, rnorm versus 112

temperature compensated time, Ω (25°C, 75°C, 100°C) Figure 13. Creep strain versus tensile stress from creep 113

(minimum creep rate) and stress relaxation tests (SRT) (25°C, 75°C, 100°C).

xv

CHAPTER 5 Sensitivity of Creep-Fatigue Damage Metrics to Uncertainty in 120 Creep Parameters Figure 1. Flowchart describing role of creep parameters in each constitutive 122

creep model to estimate creep strain energy density dissipation and accumulated creep strain.

Figure 2. Probability density function for steady-state parameters: 135

C1, C3 and C4. Figure 3. Probability density function for transient creep saturation 135

Parameters: C5 and C6. Figure 4. Probability density function for normalized rate of transient 135

creep exhaustion parameters: C7 and C8. Figure 5. Histogram and empirical probability density of accumulated creep 138

strain damage metric. Figure 6. Histogram and empirical probability density of creep strain energy 139

density damage metric. Figure 7. Scatter plots for steady-state parameters ln(C1), C3 and C4 versus 142

damage metrics (εssmodel).

Figure 8. Scatter plots for steady-state parameters ln(C1), C3 and C4 versus 142 damage metrics (εtr + εss model).

Figure 9. Scatter plots for transient creep saturation parameters, ln (C5) and C6 143

versus damage metrics. Figure 10. Scatter plots for normalized rate of transient creep exhaustion 143

parameters, C7 and C8 versus damage metrics. Figure 11. Kendall Tau rank correlation for creep parameters versus accumulated 146

creep strain. Figure 12. Kendall Tau rank correlation for creep parameters versus creep strain 146

energy density. Figure 13. Normalized sensitivity coefficients for damage metrics based on a 1% 149

change in each creep parameter. Figure 14. Normalized sensitivity coefficients for accumulated creep strain based 150

on a 1% change in creep strain energy density.

Chapter 1. Introduction

1

Chapter 1

Introduction

1.1 Background and Motivation

Since 2006 several environmental legislations have become law that restrict the use of

hazardous materials in manufactured products, including electronic devices. The practical implication

has been that the industry standard Sn-Pb solder alloy, used to create electrical connections and

mechanical attachment, has been supplanted by a variety of lead-free solder alloy alternatives. The

most popular alternatives have been Sn-Ag-Cu alloys of varying compositions with melting

temperatures near 217°C – 227°C.

Successful implementation of lead-free soldering has required re-engineering of materials,

processes, designs, equipment as well as quality and reliability approaches. In regard to solder

reliability renewed concerns have emerged since creep-fatigue damage plays a dominant role in the

lifetime prediction of thermo-mechanical related ‘wear-out’ failures. Creep is a time-dependent and

thermally-activated inelastic (permanent) deformation that can occur at stress levels well below the

yield strength of a material. Since even at room temperature Sn-Ag-Cu is more than half its absolute

melting temperature, creep strain may readily occur.

Capturing the deformation of solder joint materials requires a constitutive creep model that

relates stress, strain, temperature and time to strain rate. Traditionally steady-state creep rate would

be used exclusively to describe the mechanical behaviour of Sn-Pb over a broad range of

temperatures and strain rates. However, this assumption may not hold with the transition to Sn-Ag-

Cu and other Sn-rich Pb-free solder alloys. The contribution of transient creep and its dominance

with respect to stress and strain conditions of interest to solder joints was an unresolved and largely

uncharacterized issue.


2

The work presented herein explicitly establishes the relevance of transient creep for

Sn3.8Ag0.7Cu through experimentation and constitutive creep model reformulation that incorporates

both transient and steady state creep. Further, the scatter of lifetime estimates are sensitive to creep-

fatigue damage metrics and the uncertainty in the constitutive creep model parameters. The extent of

this relationship was characterized through Monte Carlo simulations using a finite element model of a

solder joint subjected to accelerated thermal cycle conditions, which incorporates the affect transient

creep.

1.2 Objectives

Thermal fatigue in solder joints is a major factor in determining the reliability of many

microelectronic devices. Under manufacturing and service conditions, changes in temperature cause

thermal stresses on solder joints due to dissimilar thermal expansion characteristics of solder, circuit

board, chip carrier and other assembly materials. The thermal stresses cause a low strain rate, low

cycle fatigue of the solder joint. The resulting fatigue deformation, most often in shear, is a

consequence of the high homologous temperature (T/Tm) of the solder where creep is a major

deformation mode during the fatigue cycle. The process begins with the accumulation of cyclic creep

strains to some critical level leading to crack initiation and subsequent crack growth. Several

fundamental aspects of creep in solders are not well understood, making it difficult to construct

models of thermal fatigue failure. In addition, the implementation of lead-free solder materials into

electronic products has created a need to understand the creep and thermal fatigue behaviour of

these solder joints under service conditions.

The overarching objective of this work is to characterize and model creep deformation

behaviour, including the contributions of both transient and steady-state creep. This will be

accomplished by the measurement of primary and secondary creep in Sn3.8Ag0.7Cu solder alloy.

The mechanical deformation measurements acquired under a simple uni-axial stress state can be

related to the Von Mises or effective stress through the concept of equivalent distortion energy,

thereby making such results in this research applicable to the more complex stress states of solder

joints.


3

Primary and secondary creep data will be collected as a function of temperature, stress and

time using mechanical testing specimens designed through finite element modelling to provide

uniform stress states throughout the solder joint assuming isotropic properties. Strain rates below

10-4 s-1 are of interest in this work because thermal strain rates in actual joints fall within this range.

Fundamental relationships between creep deformation in laboratory tests and actual solder joints are

complicated by the state of stress (degree of tensile constraint), type of stress (tensile or shear) and

microstructure present under equivalent loading and isothermal temperature. In this research,

thermal cycling data for 2512 chip resistor solder joints previously obtained for both SnPb and

SnAgCu in the laboratory were used for drawing inference between thermal cycle modelling and

multi-axial creep. An understanding of primary and secondary creep in solder with similar

microstructure will improve the accuracy of thermal cycle modelling and reduce the need for

expensive and time-consuming accelerated thermal cycling tests.

1.3 Literature Review

This review is intended to highlight the rudimentary creep theory, materials issues and data

from published research relevant to solder joints. The content of this review is organized thematically

beginning with physical mechanisms of creep deformation. The second section addresses the

division within the literature of experimental data split among three test specimen types; namely, cast

bulk solder specimens, simplified model joints and actual joints. Thirdly, common creep test

specimens and the resulting states of stress created during experimental trials are explained as they

pertain to the interest of this project. Lastly, the effect of stress state on crack initiation is reviewed.

1.3.1 Creep Mechanisms

The physical mechanisms of creep deformation in crystalline materials including solder can

be broadly classified as diffusional flow, dislocation creep and grain boundary sliding. A general

equation for the steady-state creep rate is expressed as shown in Eq. (1) [1].

exp( )n

q

A Qd T RTσε −

= (1)


4

In this equation the strain rate (ε ) is a function of stress (σ), average grain diameter (d), and

absolute temperature (T). The stress exponent (n) and grain size (q) exponents, activation energy

(Q) and coefficient (A) depend on both the material and the active creep mechanism. Any changes in

the active creep mechanism are accompanied by changes in the activation energy and stress

exponent which are extracted from creep data. Steady state (secondary) creep as depicted in Fig.1,

has commonly been described in several solder materials using a constitutive relation similar to that

used in the Garofalo Model shown in Eq. (2) [2].

[ ]sinh( ) exp( )n QCRT

ε ασ −= (2)

The product of α and σ terms represents the transition of dislocation creep to viscous flow.

This relationship simplifies into a power law (Norton creep) relationship at low stresses ( )1.0ασ < ,

while at high stresses ( )1.2ασ > the hyperbolic sine law simplifies to an exponential creep

expression [3]. The C term is a constant and the other terms are the same as defined above. A

finite amount of primary creep takes place prior to achieving a steady state creep wherein a minimum

strain rate is observed. The material work hardens in primary creep. Additionally a time-independent

plastic strain component contributes to deformation at high stresses, so total inelastic strain is a sum

of creep and plastic strains.

total e p pc sc tcε ε ε ε ε ε= + + + + (3)

Total strain (εtotal) during creep testing can be resolved into several components as shown in

Eq. (3): elastic εe, time-independent plastic εp, primary creep εpc, secondary creep εsc and tertiary

creep εtc. As shown in Fig.1, elastic and time-independent plastic strains occur almost

instantaneously at the start of a creep test. Both elastic and plastic strains do not contribute to strain

during the remainder of creep testing and are subtracted from deformation measurements to isolate

the contributions of creep accordingly. The strain rate during primary creep gradually decreases as

primary creep strain contributions reach a saturation value when plotted with time, beyond this point

deformation is attributed to secondary creep [4]. An increase in strain rate from the minimum value


5

observed during secondary creep indicates crack initiation and tertiary creep leading to creep rupture.

In the proposed work some of the specimens will be tested to crack initiation and creep rupture. For

low cycle fatigue loadings with low temperature ramp velocities (low stress) solder creep is the most

important contributor to deformation [5].

The majority of solder creep data acquired from joint specimens has been tested under shear

loading conditions while most of the data on bulk samples has been obtained from tensile loading.

Equation (4) below is used to convert between shear (τultimate) and tensile (σultimate) strength

measurements while Eq. (5) allows for comparison of shear modulus (G) and tensile modulus (E) [6].

Shear and tensile creep measurements can be transformed using the Von Mises assumption with Eq.

(6) and (7). This criterion is widely used for materials with time-independent plastic flow. Although

widely accepted for use with eutectic SnPb solder, the applicability of the Von Mises assumption is

not fully established for SnPb at creep strain rates below 1x10-4 s-1 or at any strain rate for SnAgCu

solder alloys [7].

2(1 )ultimate

ultimateστ

ν=

+

(4)

3στ =

(6)

Figure 1. Strain vs. Time behaviour during creep under constant load, Constant engineering stress and the three states of creep (primary, secondary, tertiary) [1]


6

2(1 )EGν

=+

(5) 3γ ε= (7)

Igoshev [8] suggested that one of five possible creep mechanisms must dominate under a

given set of conditions: 1) dislocation glide, 2) non-conservative motion of dislocation, 3) grain

boundary sliding, 4) stress directed diffusion of vacancies, 5) intercrystalline void nucleation and

growth. Dislocation creep is assumed to dominate creep in SnPb solder, however a more thorough

assessment by Shi [9] clearly identifies creep strain rates and temperatures where each creep

mechanism is dominant. Igoshev [8] suggests that even a small change in absolute temperature will

considerably change the value of Young’s modulus. Even at normalized stresses as low as σ/E =

5x10-4, the low diffusion coefficient for Sn allows the diffusion creep mechanism to be rate controlling

over a wide temperature range up to its melting point [8]. However, the experimental creep results

cannot be explained by a dislocation creep mechanism alone. Additionally, the grain boundary sliding

mechanism with the presence of hard intermetallic inclusions at the grain boundaries can lead to

crack nucleation even at deformation as small as 0.2% [8].

Shi [9] found that creep flow of the SnPb solder may be dominated by more than one

mechanism over the temperature range from -40°C to 150°C and the applied tensile stress range of

0.75 MPa to 70 MPa. At a homologous temperature of 0.51Tm creep is usually controlled by the

dislocation glide process, while near 0.87Tm dislocation can both glide and climb. Hertzberg [10]

assumed that at sufficiently low stress levels, the creep process is controlled by diffusion. A unified

Nabarro-Coble creep model was shown to accurately predict the creep deformation for different

temperatures at very low stress levels. This motivates the present work to develop an awareness of

the strain rates and homologous temperatures where transitions occur between active creep

mechanisms in SnAgCu solder. Currently there is no published deformation mechanism map for

SnAgCu alloys and limited information available on creep of this alloy at shear stresses below 10

MPa.


7

A limited number of publications investigate creep deformation and the associated

constitutive equation parameters that describe SnAgCu solder creep in the low effective stress

regime (σ < 15 MPa) over a range of isothermal temperatures. Vianco [11] has gathered creep data

for as-cast 95.5Sn3.8Ag0.7Cu under compressive loading for over a wide range of stresses (2-34

MPa) and isothermal temperatures (-25°C-160°C). Wiese [12] performed creep testing in uni-axial

tension for multiple solders including 63SnPb37 and 95.5Sn3.8Ag0.7Cu for a similar stress range (2-

30 MPa) and smaller range of isothermal temperatures (20°C-150 °C). Other SnAgCu creep testing

publications at stresses above 10 MPa by Kariya [13] (96.5Sn3.0Ag0.5Cu) and Neu [14]

(96.5Sn2.5Ag0.8Cu0.5Sb) rely on power-law creep models similar to Eq.1. However, unlike the

hyperbolic sine creep model shown in Eq.2, these power-law models do not adequately represent

creep at low stresses [15].

The characteristic failure of metals at low temperatures is by fracture through the crystal while

failure occurs at crystal boundaries at high temperatures [16]. The temperature at which the manner

of fracture changes from intracrystalline to intercrystalline is called the equicohesive temperature.

Below the equicohesive temperature strain or work hardening tends to predominate and continuous

measurable creep will not occur unless the stresses are large enough to overcome the resistance

caused by strain hardening. The active strain mechanism in this region is associated with grain

elongation, an increase in dislocation density, and no grain boundary sliding [17]. At temperatures

above the equicohesive temperature the yield rate exceeds the strain hardening rate and creep will

proceed even under low stresses [16]. The active strain mechanism above the equicohesive

temperature is associated with grain boundary sliding, grain rotation but no grain elongation, and little

increase in dislocation density [17]. Near the equicohesive temperature there is a combination of

both mechanisms. Microstructural and creep strain rate analysis research conducted by Vianco [11]

and Clech [18] demonstrated that the equicohesive temperature of SnAgCu is 75°C, above which

matrix creep dominates and below grain boundary sliding dominates. Vianco [11] reported that a

separate set of model parameters for SnAgCu creep deformation above and below 75°C produces

more accurate results than a single model due to a transition of active creep mechanisms at this


8

temperature. Since the equicohesive temperature is strain rate sensitive [9], this breakpoint between

dominant dislocation creep mechanisms will likely occur at different temperatures throughout the

testing plan that covers strain rates between 10-9 and 10-3 s-1.

Shi [9] characterized creep in SnPb into two primary regimes, dislocation controlled and

diffusion controlled, and each of these groups were further subdivided. All strain rates smaller than

1x10-10 s-1 were considered elastic deformation as the stress was sufficiently low. A deformation map

was constructed for the eutectic SnPb solder alloy to identify the strain rates and temperature that

identify the dominance regime of each creep mechanism [9]. No similar deformation map for SnAgCu

solder alloy is currently available in the literature.

1.3.2 Differences between Bulk and Joint Type Test Specimens

There are many publications that study creep behaviour by using actual soldered joints or

simulated joint specimens. Results from simulated specimens are specific to the simulated

components and not easily applied to other cases. Consequently, models based on these specimens

represent creep behaviour of the test specimen rather than the solder material [9]. Additionally, as-

cast bulk solder test specimens are expected to exhibit different mechanical properties than joint type

specimens due to differences in microstructure, intermetallic compounds and constraint from the

assembly. The comparison between bulk and thin solder joints is presented with respect to cooling

rate, aspect ratio and grain coarsening.

1.3.2.1 Cooling Rate Effect

Mei [19] studied the effects of cooling rate during solidification on the shear creep behaviour

of eutectic SnPb solder joints and on bulk solder tensile properties. Mei [20] previously showed that

high-stress behaviour has the characteristic of dislocation (power law) creep, while the low-stress

behaviour is dominated by grain boundary sliding. It was shown that for quenched, air-cooled and

furnace cooled solder joints, very similar stress versus strain behaviour is observed above a strain

rate of 10-4 s-1, but below this strain rate there is a marked divergence between the different thermal


9

treatments. The microstructure of the quenched samples was finer than the air-cooled and furnace-

cooled specimens however, there was no difference in microstructure between air and furnace cooled

joints. The microstructure of the solder joints is not the typical lamella/colony structure of a slowly

cooled bulk solder, rather the Pb-rich phase is more globular consisting of short rod-like particles [20].

The agreement and differences in reported solder joint creep due to cooling rate is consistent

with results from recrystalized bulk specimens. In the dislocation creep region, the steady-state creep

rate is microstructure insensitive, while in the grain boundary sliding region, the creep rate is

microstructure sensitive. Bulk solders displayed similar shear creep behaviour as solder joints, where

dislocation gliding was dominant at shear strain rates above 10-4 s-1 where cold-worked, quenched

and furnace-cooled bulk specimens have the same strength despite differences in grain size and

grain morphology. At strain rates below 10-4 s-1 the reverse is true [19]. The SnAgCu microstructure

is distinct from SnPb, and the microstructure may be strengthened by a large number of fine

intermetallic particles. However, the Ag3Sn particles that precipitate during typical assembly

conditions are of a relatively large size and are incoherent with the matrix phase, thus limiting its

strengthening effect. Due to coarsening of intermetallics and active recovery processes at elevated

temperatures, SnAg and SnAgCu solder alloys exhibit decreasing tensile yield strength for

decreasing cooling rates [21].

1.3.2.2 Aspect Ratio Effect

Ranieri [22] investigated the relationship between joint aspect ratio (length/thickness) and the

fracture behaviour of eutectic SnPb solder joints. As the solder joint thickness decreases the aspect

ratio increases and the constrained thin layer of solder can develop a hydrostatic stress component

orders of magnitude greater than the yield stress of the bulk solder material. This aspect ratio effect

is referred to as the Friction Hill model [23]. It is caused by a tri-axial stress state induced in the thin

joint (25 µm-150 µm) upon the application of a load. Stress state tri-axiality is defined as the ratio

between macroscopic hydrostatic stress and macroscopic Von Mises effective stress [24]. Solder

joints geometrically constrained between an electronic package and a substrate board may develop


10

the hydrostatic stresses that are many times greater than stresses normally experienced by bulk

solder.

The average tensile strength of a constrained solder joint is dependent on the geometric

aspect ratio. The yield strength versus aspect ratio displayed a linear relationship for SnPb solder

joints with aspect ratios of 500 to 1000 resulting in average tensile strengths of 150 MPa to 200 MPa.

These tensile strength values are 3 to 6 times the average yield values of unconstrained bulk solder

[22]. The Friction Hill model correctly predicted that the tensile strength of high aspect ratio solder

joints exceeds that of bulk samples. However, the theoretical aspect ratios obtained from the Friction

Hill model are two orders of magnitude lower than the geometric ratios based on measured diameter

and thickness. This discrepancy was attributed to internal defects (oxide particles and microvoids)

and transverse grain boundaries that reduce the effective aspect ratio of the joint by creating

discontinuities across the solder bond plane. Thus the average tensile strength of the solder joint

was limited by these internal defects that reduce the constraint.

The presence of internal defects in thin solder joints may have an effect on creep

performance by weakening the joint and this result may become accentuated as cracks begin to

initiate causing an increase in the measured creep strain rate. Very little published work is available

on the topic of creep as a function of tensile constraint. Findley [25] considered the creep of 2618-

T61aluminum and 304 stainless steel under combined tensile and shear forces. Gooch [26] edited a

collection of papers that characterize creep and creep rupture under multi-axial stress conditions that

rely primarily on tension-torsion, cruciform, notched specimens and pressurized tubes. Yousefiani

[27] considered creep rupture times of 7075 aluminum as a function of tensile constraint (i.e. uni-

axial, bi-axial, tri-axial), however this work did not characterize creep deformation in the bi-axial and

tri-axial specimens due to the complexity introduced by the notched specimens tested. Bao [28]

investigated ductile fracture formation in tensile tests of 2024-T351 aluminum as a function of stress

tri-axiality, stress and strain ratios, however, this work considered time-independent plastic

deformation and did not include the contribution of time-dependent creep deformation.


11

1.3.2.3 Grain Size Coarsening

Frear [29] studied the microstructure mechanisms that lead to eutectic SnPb solder joint

failures in thermal fatigue for a shear configuration. The fundamental study of microstructure

evolution in an actual solder joint is very complex, thus a simplified solder joint was designed that

experiences simple shear upon thermal cycling. The joint design consists of an aluminum plate

sandwiched between two copper plates.

The microstructure evolution versus the number of thermal cycles was evaluated for the

simplified Cu/Al/Cu joint, unconstrained eutectic SnPb bulk solder and Cu/Solder/Cu joints between -

55°C and +125°C. In the Cu/Al/Cu simplified joint after 625 cycles microstructure in the 250 µm thick

SnPb joint heterogeneously coarsened parallel to the direction of shear in the regions of high shear

strain at the ends of the specimen. When a 500 µm thick SnPb joint was used the shear strain was

reduced from 19% to 10% near the ends. Despite the reduction in strain the joints fail after 1000

cycles through the coarsened region. In both cases heterogeneous coarsened regions developed

throughout the solder parallel to the shear direction [29].

The unconstrained bulk eutectic solder was increasingly coarsened as the number of cycles

increased from zero to 2000. The coarsening was homogeneous in both the Sn and Pb regions

throughout the solder, wherein the Pb-rich regions contained Sn precipitate. In the heterogeneous

coarsening observed in the simplified joint the Pb-rich phases did not contain Sn precipitates. In the

Cu/Solder/Cu joints heterogeneous coarsening parallel to the shear direction occurred after 2000

cycles. This contrasts the simplified Cu/Al/Cu joint where heterogeneous coarsening was observed

after 625 cycles when the joint is bounded by copper and aluminum plates.

Yenawine [30] directly observed heterogeneous coarsening of the solder microstructure in

electronic packages during thermal fatigue. The coarsening in the study was complicated because

the solder joints tested did not experience simple shear deformation. Nonetheless, the most

extensive coarsening was found to occur in the regions of the joint that encountered the highest shear

strains. This coarsening effect in solder joints caused by thermally induced strains may relate to work

by Mei [19] with eutectic SnPb bulk tensile bar specimens. Mei [19] found that the grain size in the


12

deformed necked region of the bulk tensile bar was larger than the less deformed material that

closely resembled the original bulk microstructure.

The implication of grain coarsening on creep testing is important because solder joint fatigue

failures and creep failures are the result of similar metallurgical mechanisms [31]. The development

of an accurate creep response for solder material under multi-axial stress states is enhanced by

cataloguing the grain size coarsening effect in relation to stress, strain rate, temperature and joint

thickness parameters, particularly when accounting for similarities and differences in creep and crack

initiation between bulk and joint specimens.

1.4 Creep Test Methodology

The simplest and lowest cost method of constant load creep testing is the direct creep testing

machine. During the creep process an extensometer measures strain along a gauge length as a

function of time while fixed loads apply the stress needed to cause elongation. The linear portion of

the resultant strain versus time plot gives the minimum or steady-state strain rate associated with

secondary creep. It is common to use constant load testing instead of constant true stress testing

because it is easier to control applied load than to account for variations in the instantaneous cross

sectional area of the specimen. The converse of creep testing is stress relaxation where a constant

constraint (i.e. constant total strain) is applied and the reduction of stress is measured with a load cell

as a function of time. Although the total strain is constant throughout the stress relaxation test, the

creep strain within the material increases at the expense of elastic strain. A strength testing method

applies a constant strain rate to determine the ultimate tensile strength of the specimen at a given

strain rate. Once the ultimate tensile strength is reached the typical response of solder is to further

elongate at a constant stress level.

The general process for acquiring creep plots begins with an experimental setup that includes

an extensometer or similar device capable of resolving small changes in length. The extensometer is

attached to a gauge length section along the test sample and polled periodically by a data acquisition

system that includes software and hardware capable of converting output voltage readings into strain

measurements recorded at a given instant. It is also common to record instantaneous stress and/or


13

sample surface temperature with the data acquisition system as appropriate to the creep testing

approach. For tests conducted at elevated temperatures the temperature reading is used along with

a heating element and appropriate enclosure to control the nominal test temperature. Strain

measurements are recorded for subsequent analysis.

1.4.1 Constant Load Creep Testing

The constant load (constant engineering stress) creep test typically is conducted under

isothermal conditions, as shown in Fig.1 is a means of determining deformation through strain

accumulation as a function of stress, temperature and time. When the test is prolonged until the

onset of fracture, the test is considered a creep rupture test. Through careful laboratory testing as

described above, this method is a simple means of collecting reliable data that covers a wide range of

materials and temperatures. The experimental data are obtained in terms of engineering stress

where the stress of the specimen is taken to be constant at a value equal to the applied load divided

by the original cross-sectional area and the engineering strain is taken as the change in length

divided by the initial specimen length. The resultant stress-strain curves and strain-time curves may

be converted to true stress and true strain [10, 32]. Since this is the easiest way to gather relevant

data, it is desirable that other testing approaches be related to this test if possible [33]. The primary

advantage of this method is its low cost since it requires only direct dead weight loading or perhaps a

simple lever system to apply a constant tensile uni-axial load. Also, strain accumulation data may be

readily acquired through all stages of creep including primary, secondary and tertiary leading to

fracture. The major drawbacks include relatively long test durations per test specimen particularly at

lower temperatures.

During the secondary creep phase a minimum creep rate is achieved where the strain rate

becomes constant. The applied stress that corresponds to this minimum creep rate is termed the

saturation stress [32]. The saturation stress and corresponding strain rate are unique and may be

determined through alternate creep test methods. Since the constant load creep test corresponds to

only one applied stress, only one pair of saturation load and strain rate may be determined per test,

making it a relatively inefficient testing method since a large number of specimens must be tested to


14

acquire saturation stress as a function of strain rate and temperature. Additionally, a test specimen

cannot be reused due to the history effect of non-recoverable deformation caused during the test.

Therefore a large number of samples and long overall testing timeframe is required to complete a

design of experiment that includes a range of several applied loads and isothermal temperatures.

Constant load testing is an important part of the experimental design that allows for the monitoring of

primary creep, steady-state creep and the onset of crack initiation with tertiary creep.

1.4.2 Constant Constraint/Stress Relaxation Creep Testing

Stress relaxation tests (constant engineering strain) have been proposed as an alternative

test method to constant load and constant stress creep tests for studying the high temperature

deformation of metals and alloys. With the stress relaxation approach the creep strain rate versus

stress relationship is extracted from the experimental results through manipulation of the constitutive

creep equation. The appeal of a stress relaxation approach is that a single specimen covers a wide

range of creep strain rates and stress magnitudes [34]. Additionally, as shown if Fig. 2, stress

relaxation permits creep test completion in much less time than a conventional constant load creep

test [35]. This is advantageous when a large experimental range of stress and temperature are to be

examined in a minimum amount of time without incurring significant time-independent plastic strain in

the sample [36]. It has been shown that there is no history-effect observed in stress relaxation tests,

so similar results will be observed at a given stress level independent of the initial stress applied to

achieve a fixed constraint [37]. Further, it has been asserted that stress relaxation data used to

develop constitutive relations for modelling are more likely to reflect the actual deformation process

than steady state creep data because stress relaxation tests mimic the same deformation process

that takes place in solder joints during temperature hold periods in thermal cycling tests [35].

The determination of the effective shear modulus of a solder joint (i.e. solder, machine

stiffness, etc.) is non-trivial due to microstructure variations and small joint width. This issue

prompted a sensitivity analysis of lead free solder alloys including SnAgCu that showed the shear

strain rate versus shear stress curve is not very sensitive within a reasonable choice of shear

modulus ranging between 10 and 30 MPa [34]. Comparatively, the effective Young’s modulus of a


15

tensile test can calculated in a straightforward manner [35]. Elongation data is not obtained with this

testing approach and it is not possible to follow creep through to fracture.

Figure 2. Stress relaxation plot of the data points and fitted-line for Sn-3.6Ag-1.0Cu at 25°C [34]

Stress relaxation testing is core to the experimental design because it provides a large

number of data points for the steady state strain rate versus stress curve over a relatively

compressed time scale compared with the equivalent experimental time required with constant load

creep testing.

1.4.3 Conversion of results between Creep Testing Methods

Common to all creep test methods are the four variables stress, strain, time and temperature.

Creep tests may be conducted on individual specimens at several combinations of loads and

temperature to acquire data on the material response. The applicability of results from one creep test

method to another is an important consideration when selecting an experimental testing method.

Although microstructure, specimen size and thermal history of a sample may differ between

investigations, it is possible to compare the results obtained from constant load and stress relaxation

methods conducted at similar temperatures and strain rates. Such comparisons are made by using

multiple variable linear regression analysis to compute parameters of constitutive creep equations of


16

a form similar to Eq. (2). These model curves for particular alloys indicate trends within the data and

identify outliers.

1.4.4 Consideration of Primary Creep

Throughout the literature many different combinations of x-axis and y-axis quantities are used

to plot creep data measurements. Some of the variations on stress include true stress, normalized

stress, effective stress, average stress, force and load. Similarly variations on strain and strain rate

include true strain, principal strain, equivalent strain, average strain and displacement. The two most

common ways of expressing creep graphically within the solder reliability literature are: (i) Tensile

Strain rate (s-1) vs. Tensile Stress (MPa), and (ii) Shear Strain rate (s-1) vs. Shear Stress (MPa).

These curves are most often used to describe steady state creep strain rate. The trend in the

literature is to limit the scope of investigation to steady state creep due to the basic assumption that

damage in solder joints during temperature cycling is primarily due to steady state creep strain

accumulation, where the effects of primary creep and time-independent plasticity are minor [38]. For

a given temperature, steady state creep tends to dominate deformation under relatively higher stress

conditions for SnAgCu. However, pending temperature, at lower stress levels deformation may

include non-negligible primary creep as with Sn-3.5Ag [39].

In the low stress regime accurate estimation of primary creep becomes important in the

determination of total creep strain incurred during service. Excluding the contribution of primary

creep to total strain may result in an underestimation of total creep at constant stress/load or an

overestimation of total creep strain under constant strain (stress relaxation) conditions as noted for

SnPb solder alloy [40]. The primary creep strain can be obtained from constant load creep test data

by subtracting the elastic, plastic, and steady state creep strain from the total strain. Primary creep

strain reaches a limiting or saturation value over time that shows power-law dependence on stress for

both SnPb and SnAgCu solder alloys [41].


17

1.5 Organization of Thesis

Each of the chapters that follow represents research articles that have been published or are

currently under peer-review for publication in the Journal of Electronic Materials and Microelectronics

Reliability.

Chapter 2, Literature Effect of Transient Creep and Plasticity in the Modeling of Thermal

Fatigue, examines the relative contributions of plasticity, primary and secondary creep in Sn40Pb and

Sn3.8Ag0.7Cu solders using the analysis of a trilayer solder joint structure with finite elements and a

finite difference technique based on measurements of a 2512 chip resistor. The influences of

temperature amplitude and ramp rate were quantified.

Chapter 3, Theory, Experiments and Data Analysis, follows-up on the prominent contributions

of transient creep for established in Chapter 2 by characterizing a bulk tensile specimen. This

chapter presents constant-load creep and stress relaxation test methodology and results for

Sn3.8Ag0.7Cu spanning a range of strain rates 8 1 4 110 10s sε− − − −< < , and temperatures 25°C, 75°C

and 100°C.

Chapter 4, Constitutive Creep Model Development and Finite Element Analysis, explains a

constitutive creep model presented for Sn3.8Ag0.7Cu that incorporates both transient and steady-

state creep to provide agreement for both creep and stress relaxation data with a single set of eight

coefficients. The model utilizes both temperature compensated time and strain rate to normalize

minimum strain rate and saturated transient creep strain, thereby establishing equivalence between

decreased temperature and increased strain rate. The model is readily applied to solve for creep-

fatigue damage metrics.

Chapter 5, Sensitivity of Creep-Fatigue Damage Metrics to Uncertainty in Creep Parameters,

investigated the sensitivity of these damage metrics to uncertainty in the parameters of two creep

constitutive models using a Latin Hypercube sampling scheme. The two damage metrics,

accumulated creep strain and creep strain energy density dissipation, were calculated for the thermo-

mechanical cycling of a trilayer solder joint using finite element analyses encompassing both primary

and secondary creep in the constitutive modelling of Sn3.8Ag0.7Cu solder.


18

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19

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[29] Frear, D., Grivas, D., and Morris, J.W., A Microstructural Study of the Thermal Fatigue Failures of 60Sn-40Pb Solder Joints. Journal of Electronic Materials, 1998. 17(2): p. 171-80.

[30] Yenawine, R., Wolverton, M., Burkett, A., Waller, B., Russel, B., and Spritz, D. Today and Tomorrow in Soldering. in Naval Weapons Electronics Manufacturing Seminar. 1987. China Lake, CA, USA.

[31] Tribula, D. and Morris, J.W., Creep in Shear of Experimental Solder Joints. Journal of Electronic Packaging, 1990. 112(2): p. 87-93.

[32] Wilde, J., Becker, K., Thoben, M., Blum, W., Jupitz, T., Wang, G.Z., and Cheng, Z.N.N., Rate dependent constitutive relations based on Anand model for 92.5Pb5Sn2.5Ag solder. IEEE Transactions on Advanced Packaging, 2000. 23(3): p. 408-14.

[33] Penny, R.K. and Marriott, D.L., Design for creep. 2nd ed. 1995, London: Chapman & Hall. x, 430 p.

[34] Unal, O., Anderson, I.E., Harringa, J.L., Terpstra, R.L., Cook, B.A., and Foley, J.C., Application of an asymmetrical four point bend shear test to solder joints. Journal of Electronic Materials, 2001. 30(9): p. 1206-13.

[35] Hare, E.W. and Stang, R.G., Stress-Relaxation Behavior of Eutectic Tin-Lead Solder. Journal of Electronic Materials, 1995. 24(10): p. 1473-84.

[36] Mavoori, H., Chin, J., Vaynman, S., Moran, B., Keer, L., and Fine, M., Creep, stress relaxation, and plastic deformation in Sn-Ag and Sn-Zn eutectic solders. Journal of Electronic Materials, 1997. 26(7): p. 783-90.

[37] Baker, E. and Kessler, T.J., Influence of Temperature on Stress Relaxation in a Chill-Cast, Tin-Lead Solder. IEEE Transactions on Parts Hybrids and Packaging, 1973. PHP9(4): p. 243-6.

[38] Syed, A. Accumulated Creep Strain and Energy Density Based Thermal Fatigue Life Prediction Modls for SnAgCu Solder Joints. in ECTC. 2004: IEEE.

[39] Yang, H., Deane, P., Magill, P., and Murty, K.L. Creep Deformation of 96.5Sn-3.5Ag Solder Joints In A Flip Chip Package. in ECTC. 1996: IEEE.

[40] Schroeder, S.A., Morris, W.L., Mitchell, M.R., and James, M.R., eds. A Model for Primary Creep of 63Sn-37Pb Solder. Fatigue of Electronic Materials, ed. ASTM. 1994, STP 1153. 82-94.

[41] Zhang, Q., Isothermal Mechanical and Thermomechanical Durability Characterization of Selected Pb-Free Solders, in Mechanical Engineering. 2004, University of Maryland/CALCE.

Chapter 2. Effect of Transient Creep and Plasticity in the Modeling of Thermal Fatigue 20

Chapter 2

Effect of Transient Creep and Plasticity in the Modeling of Thermal Fatigue

2.0 Introduction

Damage metrics such as accumulated creep strain and creep strain energy density are used

frequently to estimate the thermal fatigue lives of solder joints. Constitutive models that define a

relationship between strain rate, stress, temperature and material properties are required to evaluate

such damage metrics. In the electronics packaging literature, it is commonplace to assume that

secondary creep dominates creep deformation in both SnPb and lead-free solder joints, leading to the

neglect of primary creep in solder constitutive models [1-4]. Although absolute strain estimation may

be inaccurate in this case, it is possible to obtain reasonable thermal fatigue life predictions in certain

circumstances [4]; for example, when the error is insensitive to changes in the solder thermal strain

due to variations in the applied temperature.

Some lead-free solders such as SnAgCu exhibit more primary creep than SnPb solder [5-7].

The neglect of primary creep in SnAgCu can therefore be of particular concern, leading to inaccurate

predictions of the amount of deformation in the solder and in the calculated fatigue damage metrics

such as creep strain energy density and accumulated creep strain per stabilized hysteresis loop.

Moreover, in some applications such as opto-electonics, the accurate estimation of absolute creep

deformation rather than a fatigue damage metric is necessary [8].

Constitutive models that include the effects of primary and secondary creep for SnPb and

SnAgCu solder have been investigated under conditions of constant stress and temperature [9, 10].

In these studies it was found that the saturated (ultimate) primary creep strain was approximately

constant. This value was obtained from a graph of the total (including primary) creep strain versus

time, extrapolating the secondary creep slope back to zero time. An extension of these studies


considered primary creep during stepwise increases in stress at constant temperature, and found that

the saturation primary creep strain was a function of the secondary creep strain rate [6]. The

constitutive relation used in the present modeling builds on that in references [6, 9, 10] by applying

time hardening and temperature compensated time to account for changes in temperature, stress and

strain rate inherent in thermo-mechanical cycling.

This chapter considers the effect of neglecting rate-independent plasticity and primary creep

in a thermo-mechanical analysis of a generic solder joint that has been simplified to a fundamental

trilayer assembly comprised of a silicon chip, solder interlayer and an FR-4 substrate. The finite

element model incorporated four types of strain: elastic, plastic, primary creep and secondary creep.

In addition, a two-dimensional finite difference model for calculating thermal stresses of trilayer

assemblies, originally developed in [11-13] with experimental accelerated reliability thermal-cycling

data (IPC SM-785) for a 2512 chip resistor solder joint previously measured in this laboratory, was

extended here to include the contributions of primary creep and secondary creep. This finite

difference solution was used here as an alternate approach to finite elements that validates an elasto-

creep finite element model. This basis of comparison was useful, since the finite element model was

relatively complex, involving plasticity and nonlinear rate and temperature-dependent creep

deformation.

2.1 Finite difference trilayer model

Figure 1 shows a free-body diagram of the trilayer model where the elastic and creep

rigidities of each layer are assumed to be temperature-dependent. The layers undergo rate-

independent elastic and rate-dependent creep deformation as temperature changes. A uniform time-

varying temperature is assumed throughout the joint, because the effect of transient temperature

gradients on the stress and strain fields has proved insignificant for both SnPb and SnAgCu solders

under the thermal profiles of interest [14]. The interfacial shear stress, τxym(x,t), and peel stresses,

σym (x,t), vary with position along the two solder interfaces; where subscript m refers to the FR4/solder

(m=1) and Si/Solder (m=2) interface. The axial, transverse shear and bending loads acting at the


midplane of each layer are given, respectively, by Tj(x,t), Vj(x,t) and Mj(x,t); where j refers to the

material with FR4 (j=1), solder (j=2), and silicon (j=3). The axial and transverse shear forces are due

to the interlaminar shear and peel stresses. The transverse shear, bending loads and the shear

stresses created by the mismatch of the coefficients of thermal expansion (CTE) vanish at the free

ends of the finite difference solution to the trilayer.

Figure 1. Free-body diagram of trilayer model.

This trilayer model incorporates the essential characteristics of many solder joints, thereby

serving as a useful generic sandwich model that illustrates general trends attributable to the inclusion

of plasticity and primary creep in an elasto-creep model. The model is applicable to trilayers with any

aspect ratio (i.e. both long and short) under either plane stress or plane strain conditions [11]. The

comparison of constitutive models using the same generic sandwich, both with and without the

inclusion of plasticity and primary creep, illustrates the general trends attributable to primary creep.

During the finite difference solution, the interfacial shear and peel stress distributions along

the two interfaces were computed with respect to time, and hence temperature in thermal cycling.

The upper interface of Fig. 1 was of interest in the present work, because the large CTE, elastic

(stiffness) and flexural rigidity mismatches between the silicon chip and solder generated the greatest


interfacial stresses. In ref. [11], the flexural creep rigidity relating the creep compliance of the solder

layer is derived for a shear loaded strip. From these results, the other components of stress and

strain were defined for each time interval.

A limitation of the present finite difference solver was the lack of dynamic memory allocation

which limited the number of attainable hysteresis loops to less than that required to reach complete

stabilization. Nevertheless, the finite difference results could be used to verify the damage metric

trends for each elasto-creep constitutive model computed in the finite element analysis.

2.2 Finite element modeling

The finite element solution experiences a singularity where it produces stresses at the free

ends (x=l) that are both unstable, being highly dependent on the local mesh density, and unrealistic

since they predict a singular stress field at the free ends of interfaces even when solved with an

inelastic analysis [11, 15]. One method used to deal with this singularity is termed volume weighted

averaging in which the stress is averaged over the interfacial elements in the vicinity of the free edge

at x=l (Figure 1) [16, 17]. This approach reduces mesh sensitivity in the critical area near the singular

stress field making the analysis more robust, but there is still some dependency on the thickness of

the interface elements. A second approach, used in the present work, expresses model results in

terms of a “calibration element” at a location close to the free edge, but distant enough to give

stresses that are independent of the mesh density. It was found that x=0.8*l was a suitable location

[11, 12]. The latter approach was adopted in this work because it adequately and simply represents

the calculated results over the majority of the solder joint. This approach is more straightforward to

calculate and interpret than the volume averaged result. Although the stress level at the free end was

always higher than at the calibration element, the trend of creep and plasticity damage was the same

at both locations [11].


The trilayer was discretized into a system of 2D 8-node plane strain structural elements

(Ansys™ 8.0 PLANE183) with two degrees of freedom at each node (x and y displacement). The

finite element model of the trilayer (Fig. 2) utilized a symmetry displacement boundary at x=0,

displacement in the x-direction (UX=0) and rotation in the z-plane (ROTZ=0) are constrained at all

nodes along the boundary line. A single node at the bottom of the symmetry plane was constrained

in the y-direction (UY=0) to prevent rigid body motion. The finite element mesh was incrementally

refined locally in areas of high stress concentration until the results at the calibration point did not

change with increasing mesh density. Figure 2 shows the element meshing with 1,672 nodes and

525 quadrilateral PLANE183 elements that produced mesh size independent results at the calibration

element. Near the singular region, elements were refined to a width of 2.6 µm and a height of 4.5

µm. Table 1 gives the dimensions and material properties of the solder joint trilayer used in the

present work for both Sn40Pb and Sn3.8Ag0.7Cu.

To obtain interfacial stress and strain fields within the solder layer, "element solutions" were

employed that report component values at the nodes of the selected element. This is in contrast to

an alternative "nodal solution" that averages data components such as strain and stress among

adjacent elements [18], thereby obscuring the interfacial creep strain. The strain and stress results

are discontinuous at the interface except for interfacial shear stress and peel stress components.


Figure 2. Meshed finite element trilayer model

(1672 nodes, 525 quadrilateral PLANE183 elements).


Table 1. Trilayer dimensions and material properties [30] Solder Half

Length, l

(mm)

Layers Height,

h (mm)

Young’s

Modulus, E

(GPa)

Poisson’s

Ratio, ν

CTE, α

(PPM/K)

Yield

Strength

(MPa)

SnPb 0.380

Si 0.65 131 0.30 2.8

SnPb 0.12 75.940 - 0.152 x

θ 0.40 22

0°C: 21.1

50°C: 13.8

100°C: 6.1

FR-4 1.23 22 0.28 19

SnAgCu 0.380

Si 0.65 131 0.30 2.8

SnAgCu 0.12

25°C: 46

50°C: 44

100°C: 41

0.30 17

25°C: 31.8

75°C: 21.0

125°C: 13.6

FR-4 1.23 22 0.28 19

2.3 Constitutive modeling

Contributions to the creep strain of metals under constant stress and temperature can be

partitioned into three components: 1. primary (transient) creep having a decelerating rate as the metal

hardens; 2. secondary (steady-state) creep where the hardening and softening processes are in

equilibrium and the metal deforms approximately linearly with time; and 3. tertiary (transient)

creep where voids form in the metal and the rate of creep accelerates leading to rupture. When a

solder joint is exposed to thermal cycling, creep becomes a function of time-varying stress and

temperature because of the mismatch of thermal expansion coefficients and differences in the

elasticity and creep rigidities amongst the layers of the joint.

As mentioned previously, it is common to simplify the creep modeling of solder joints in

microelectronics packaging by considering only contributions from secondary creep, under the

assumption that primary creep is negligible. This assumption simplifies modeling and is facilitated by

the many secondary creep equations available in FE software, such as the Garofalo hyperbolic sine


creep expression. By contrast, modeling both primary and secondary creep with the equations

outlined below required a user defined implicit creep function and additional FORTRAN compiler

software [19]. Additionally, the inclusion of primary creep increases computation time. For some

engineering purposes, the neglect of primary creep in the modeling of SnPb solder deformation is

acceptable [1]. However, it is recognized that fluctuating temperature and stress during thermal

cycling produces primary creep contributions prior to the onset of secondary creep [20].

As temperature fluctuates during a thermal cycle, creep deformation is accompanied by

elastic and possibly, plastic strain (rate-independent) at stress levels above the yield strength.

Tertiary creep was neglected from constitutive models in the present work, because it only represents

deformation contributions at the onset of imminent creep rupture. The contribution of plastic strain

was modelled with a von Mises isotropic hardening constitutive model, where 10% strain hardening

was assumed after plastic yield [21]. Hence, total strain deformation in the present model had

elastic, plastic, and creep components:

, ,total elastic plastic creep primary creep secondaryε ε ε ε ε= + + + (1)

Two established empirical constitutive creep models were considered: (i) creep strain rate

contributions from secondary creep (εsc) given by the constitutive relation of Eq.(2) [7, 22]; and (ii)

contributions from secondary creep plus primary creep (εpc) given by the constitutive relations of

Eqs.(2), (3) and (4) [6, 23]:

( )sinh exp3Csc 41 2

CC Ctε σ

θ∂ − = ∂

(2)

1 exp8C

sat scpc pc 7C t

tεε ε

∂ = − − ∂ (3)


6Csat scpc 5C

tεε ∂ = ∂

(4)

where, C1 - C8 are experimentally determined material constants with C3 known as the stress

exponent; C4 is the ratio of the activation energy (Qa) over the ideal gas constant (R); θ is the

absolute temperature; t is elapsed time, and satpcε is the saturated primary creep strain. Equation (4)

describes the dependence of the saturated primary creep strain on the steady state creep rate, and

hence temperature [6, 20, 23]. Table 2 gives the values of the coefficients from the literature [6, 24]

for the constitutive models of the two solders used in this chapter. The values differ between the two

solder materials with SnPb showing a temperature dependence for 1C and 2C . All the coefficients

for SnAgCu are constants derived from regression over a broad temperature range.

Throughout the electronics packaging literature there is scatter among reported creep

coefficients based on differences in test methods, stress state, specimen size and aging history of the

solder used in monotonic creep testing. The secondary creep constants from monotonic creep

testing are used to calculate the minimum creep rate during cyclic thermal fatigue, while time

hardening is used to account for primary creep. The significance of primary creep as reported herein

is independent of scatter in creep constants. Although the absolute magnitude of stresses and

strains may change due to scatter in creep constants, the trends caused by primary creep are

insensitive to this for two principal reasons. Firstly, primary creep strain is a function of the secondary

creep strain rate, and hence they are related for a given set of creep constants. Secondly, Eq. (7)

shows that the inclusion of primary creep will always reduce the creep rigidity thereby increasing the

accumulated equivalent creep strain regardless of the particular constants used in the creep models.


Table 2. Constitutive equation constants [6, 35]

Sn40Pb Sn3.8Ag0.7Cu

C1 (1/s) ( )926 × 508-θθ

441000

C2 (1/Pa) ( )6

1

37.78×10 - 74414×θ -95×10

C3 3.3 4.2

C4 (K) 6360 5412

C5 (s) 1.613 1.710

C6 0.467 0.400

C7 111 109

C8 0.79 0.60

Equation (2) was implemented in Ansys™ 8.0 through the implicit creep algorithm termed the

“Generalized Garofalo” model [18]. Since the primary creep constitutive relations in Eqs. (3) and (4)

were not available in the standard software release, they were implemented with a user-defined

implicit creep model through “usercreep.f”. In addition to Ansys™ 8.0, Compaq Visual Fortran 6.6A

was required to compile and link the custom Ansys™ executable [19].

The damage metrics of interest in this work are “accumulated equivalent strain” and “strain

energy density” as dissipated per stabilized stress versus strain hysteresis loop. The accumulated

equivalent strain ( .eqAccε , Eq.(6)), due to both creep and plastic components, is a monotonically

increasing quantity computed as the summation of equivalent (or von Mises) incremental strains

( eqε∆ , Eq.(5)) over all time steps (λ) in terms of principal strains. The accumulated equivalent strain

is effectively the integral of instantaneous equivalent stress divided by instantaneous creep rigidity,

over the length of the thermal cycle ( eqε∆ , Eq.(7)). This interpretation of accumulated creep strain is

illustrative, because the accumulated equivalent creep strain increases as the creep rigidity (β)

decreases, with the inclusion of primary creep strain. The accumulated equivalent strain is an ever-


increasing function, independent of the sense of the instantaneous equivalent strain direction; hence

a useful quantity for cyclic applications with load reversals. Inelastic deformation is irreversible and

the solder is assumed incompressible (effective Poisson’s ratio = 0.5) during both creep and plasticity

[25]. However, the elastic calculations used the Poisson’s ratio from Table 1. If the equivalent von

Mises stress computed using elastic, plastic and creep properties exceeds the material yield strength,

then plastic strains occur. Plastic strains were based on the assumed von Mises isotropic hardening

rule.

( ) ( ) ( )1

2 2 2 22 3 3 1 1 2

23eqε ε ε ε ε ε ε ∆ = ∆ −∆ + ∆ −∆ + ∆ −∆ (5)

( ).1

eqAcc eqi

λ

ε ε=

= ∆∑ (6)

.0

teq

eqAcc dsσ

εβ

= ∫ (7)

In Ansys™ 8.0, accumulated equivalent creep strain ( .,eqAcc creepε ) and accumulated equivalent

plastic strain ( .,eqAcc plasticε ) are output options of the PLANE183 element. The strain energy density

dissipation during creep deformation ( creepW ) of solder at the interfacial calibration element was

calculated by integrating the product of instantaneous equivalent (von Mises) stress and accumulated

equivalent (von Mises) strain (Eq.(8)) [26]. In Ansys™ 8.0 the accumulated strain energy density is

output using the time-history results post processor. The integration function of the post processor is

used to compute Eq.(8) for the instantaneous equivalent stress and accumulated equivalent creep

strain. Similarly, the plastic strain energy density is computed using the accumulated equivalent

plastic strain. The accumulated strain energy density (or strain energy dissipation) is also an ever-

increasing function that increases monotonically during thermal cycling, despite load reversals and

changes in the sense of instantaneous equivalent strains.

.

0

eqAcc

creep eq ACSW dε

σ ε= ∫ (8)


2.4 Time hardening of solder during thermal cycling

Primary creep modeling was accomplished using the time hardening approach in which the

creep strain rate is a function of the time and stress at a particular temperature, independent of the

stress history [27]. Conceptually, whenever the stress or temperature change, the deformation is

assumed to proceed according to the creep curve for the new stress and temperature, starting at the

point on this curve corresponding to the value of total elapsed time [25]. Modeling changes in stress

that result from changes in temperature was done by substituting the Sherby-Orr-Dorn temperature-

compensated time parameter (Ω), Eq.(9), for the elapsed time (t) in Eq.(3) [28]. Substituting for t

and satpcε , the primary creep strain rate is given by Eq.(10).

Creep data for pure metals and simple alloys at homologous absolute temperatures (θ/θmelt

with temperature in K) above 0.45, obtained at the same stress but different temperatures,

superimpose when plotted against Ω [25, 29]. Equation (9) is applicable where the activation energy

is constant. Equation (10) is satisfied over the temperature ranges of interest in the present work,

because the activation energy defined in Eq.(2), is constant for SnPb and SnAgCu solders at both low

and high stresses.

4exp CtΩθ− =

(9)

6 8

5 71 expC C

sc scpc C C

t tε εε Ω

∂ ∂ = − − ∂ ∂ (10)

2.5 Effect of plasticity

The total strain consists of elastic and inelastic components due to rate-independent plastic

strain and primary and secondary creep (rate-dependent) strain (Eq.(1)). The temperature-

dependent Young’s modulus for SnAgCu is larger than that of SnPb solder. This relatively larger

stiffness means SnAgCu is more prone to experience plastic strains.


Figure 3. Thermal cycle with reference temperatures, 95°C/min and ∆T = 100°C.

In Fig. 3, the 95°C/min-∆100°C (5 min upper and lower dwell) thermal cycle is shown for 10

loops. According to Fig. 4, when this thermal profile was applied to the elastic-plastic-creep

(secondary-only creep) model for SnAgCu, the majority of the stabilized effective (von Mises) stress

versus time plot was above the temperature dependent yield stress. Table 3 gives the four creep

models considered, along with the defining constitutive equation and stress-free reference

temperature used in each simulation.

Table 3. Constitutive models Case # Material, Creep Contributions Constitutive Equation # Reference Temperature (°C)

1 Sn40Pb, Secondary (2) 183

2 Sn40Pb, Primary + Secondary (10) 183

3 Sn3.8Ag0.7Cu, Secondary (2) 217

4 Sn3.8Ag0.7Cu, Primary + Secondary (10) 217


Figure 4. SnAgCu von Mises stress vs. elapsed time over stabilized hysteresis loop for the secondary

creep model (Sec.); 95°C/min and ∆T = 165°C.

In contrast, Fig.5 shows that for the same thermal cycle, when the elasto-plasto-creep

(primary plus secondary creep) model was considered for SnAgCu, the entire stabilized effective (von

Mises) stress versus time plot was below the yield stress. This suggests that for SnAgCu, when

primary creep is neglected, plastic strain contributes towards total inelastic deformation, but plastic

strain is zero when primary creep is considered.


Figure 5. SnAgCu von Mises stress vs. elapsed time over stabilized hysteresis loop for the primary

plus secondary creep model (Pri. + Sec.); 95°C/min and ∆T = 165°C.

Tables 4 and 5 show that the plastic components of strain energy density and accumulated

equivalent strain are negligible for both SnPb and SnAgCu compared with the total inelastic

deformation. These damage metrics were dominated by creep deformation regardless of whether

primary creep was considered. As mentioned previously, when primary creep was included, plastic

deformation for SnAgCu was zero for the three thermal profiles of interest because the stress level

throughout the hysteresis loop was below the temperature dependent yield strength. Similarly, for

SnPb the plastic contribution for either damage metric was orders of magnitude smaller than the

creep contribution to the total inelastic damage.


Table 4. Comparison of strain energy density per cycle due to rate-independent plastic deformation, Wplastic, and creep deformation, Wcreep, for stabilized von Mises stress-strain hysteresis loops with and

without rate-independent plasticity Elastic-Creep Model

Sec.=10 loops; Pri.+Sec=15

loops

Elastic-Plastic-Creep Model

Sec.=15 loops; Pri.+Sec=30 loops

creepW (MPa) plasticW (MPa) creepW (MPa) totalW (MPa)

SnPb, 14°C/min, ∆T=100°C

Sec. 1.69x10-1 5.62x10-4 1.68x10-1 1.69x10-1

Pri. + Sec. 1.19x10-1 1.42x10-4 1.21x10-1 1.21x10-1


Sec. 2.26x10-1 8.51x10-4 2.25x10-1 2.26x10-1

Pri. + Sec. 1.63x10-1 1.92x10-4 1.65x10-1 1.65x10-1


Sec. 4.39x10-1 2.60x10-3 4.36x10-1 4.38x10-1

Pri. + Sec. 3.18x10-1 4.18x10-4 3.21x10-1 3.21x10-1

SnAgCu, 14°C/min, ∆T=100°C

Sec. 1.21x10-1 8.33x10-4 1.20x10-1 1.21x10-1

Pri. + Sec. 6.62x10-2 0 6.88x10-2 6.88x10-2


Sec. 1.49x10-1 2.84x10-3 1.44x10-1 1.47x10-1

Pri. + Sec. 9.31x10-2 0 9.58x10-2 9.58x10-2


Sec. 2.48x10-1 1.05x10-2 2.41x10-1 2.52x10-1

Pri. + Sec. 1.84x10-1 0 1.91x10-1 1.91x10-1


Table 5. Comparison of accumulated equivalent plastic strain, εeqAcc.Plastic, and accumulated equivalent creep strain, εeqAcc.Creep, for stabilized von Mises stress-strain hysteresis loops with and

without rate-independent plasticity. Elastic-Creep Model


Elastic-Plastic-Creep Model


.eqAcc Creepε .eqAcc Plasticε .eqAcc Creepε .eqAcc Totalε


Sec. 9.12x10-3 1.46x10-5 9.10x10-3 9.12x10-3

Pri. + Sec. 1.17x10-2 6.32x10-6 1.17x10-2 1.17x10-2


Sec. 9.08x10-3 1.79x10-5 9.06x10-3 9.09x10-3

Pri. + Sec. 1.17x10-2 6.62x10-6 1.17x10-2 1.17x10-2


Sec. 1.55x10-2 3.38x10-5 1.55x10-2 1.55x10-2

Pri. + Sec. 1.91x10-2 8.22x10-6 1.91x10-2 1.91x10-2


Sec. 4.40x10-3 1.67x10-5 4.39x10-3 4.40x10-3

Pri. + Sec. 1.05x10-2 0 1.04x10-2 1.04x10-2


Sec. 3.97x10-3 4.70x10-5 3.93x10-3 3.98x10-3

Pri. + Sec. 1.03x10-2 0 1.02x10-2 1.02x10-2


Sec. 5.83x10-3 1.00x10-4 5.81x10-3 5.91x10-3

Pri. + Sec. 1.76x10-2 0 1.74x10-2 1.74x10-2


In the secondary-only creep model for the 95°C/min-∆165°C thermal profile, where the

thermal stresses are the largest of the three profiles considered, the plastic strain component for the

SnPb and SnAgCu solders accounted for only 0.22% and 1.70%, respectively, of the total inelastic

accumulated equivalent strain. A similar trend was noted for the strain energy density dissipation for

the 95°C/min-∆165°C thermal profile where the plastic strain contributions were 0.59% and 4.16%, for

the SnPb and SnAgCu solders, respectively. Thus, the inclusion of plastic strain had minimal impact

on the calculated damage metrics used for fatigue life prediction. Although the relative contribution of

rate-independent plastic strain increases proportionally with the difference between stress and yield

strength, both damage metrics remain dominated by creep. When stress levels are very high in

solder the corresponding creep rate is fast, thereby accommodating stress relaxation, and resulting in

creep deformation vastly exceeding plastic deformation over any given stabilized thermal cycle. This

applies equally for this trilayer geometry and other variants that may generate larger stress levels.

For this reason, subsequent analyses conducted with the finite difference and finite element

approaches omit plastic strain by using the elasto-creep constitutive model (Eq.(11)). The results of

these analyses are given in Tables 6 and 7 and Figs. 6-16.

, ,total elastic creep primary creep secondaryε ε ε ε= + + (11)


Table 6. Comparison of creep strain energy density, Wcreep, for stabilized instantaneous von Mises stress-strain hysteresis loops using the secondary-only and primary plus secondary creep models.

Finite Difference (FORTRAN) Finite Element (ANSYS™)

creepW

(MPa)

creep,sec

creep, pri+ sec

WW

% Diff. creepW

(MPa)

creep,sec

creep, pri+ sec

WW

% Diff.

SnPb, 14°C/min, ∆T=100°C 4 loops Sec.=10 loops; Pri.+Sec=15 loops

Sec. 1.21x10-1 1.3 29

1.69x10-1 1.4 35

Pri. + Sec. 9.04x10-2 1.19x10-1


Sec. 1.59x10-1 1.3 25

2.26x10-1 1.4 32

Pri. + Sec. 1.23x10-1 1.63x10-1


Sec. 3.12x10-1 1.3 26

4.39x10-1 1.4 32

Pri. + Sec. 2.41x10-1 3.18x10-1

SnAgCu, 14°C/min, ∆T=100°C 4 loops Sec.=10 loops; Pri.+Sec=15 loops

Sec. 8.25x10-2 2.1 71

1.21x10-1 1.8 58

Pri. + Sec. 3.93x10-2 6.62x10-2


Sec. 1.02x10-1 1.8 54

1.49x10-1 1.6 46

Pri. + Sec. 5.88x10-2 9.31x10-2


Sec. 1.69x10-1 1.5 39

2.48x10-1 1.4 29

Pri. + Sec. 1.13x10-1 1.84x10-1


Table 7. Comparison of accumulated equivalent creep strain, εeqAC for stabilized instantaneous von Mises stress-strain hysteresis loops using the secondary-only and primary plus secondary creep

models. Finite Difference (FORTRAN) Finite Element (ANSYS™)

eqAcc.ε

eqAcc.,sec

eqAcc., pri+ sec

εε

% Diff. eqAcc.ε

eqAcc.,sec

eqAcc., pri+ sec

εε

% Diff.


Sec. 6.91x10-3 0.72 33

9.12x10-3 0.78 25

Pri. + Sec. 9.65x10-3 1.17x10-2


Sec. 6.73x10-3 0.71 34

9.08x10-3 0.78 25

Pri. + Sec. 9.47x10-3 1.17x10-2


Sec. 1.16X10-2 0.75 29

1.55x10-2 0.81 21

Pri. + Sec. 1.55X10-2 1.91x10-2


Sec. 3.20x10-3 0.44 78

4.40x10-3 0.42 82

Pri. + Sec. 7.33x10-3 1.05x10-2


Sec. 2.89x10-3 0.40 87

3.97x10-3 0.39 89

Pri. + Sec. 7.31x10-3 1.03x10-2


Sec. 4.25x10-3 0.35 97

5.83x10-3 0.33 100

Pri. + Sec. 1.22x10-2 1.76x10-2


2.6 Thermo-mechanical cycling simulations

The finite difference and finite element models were executed to simulate thermal cycling for

the trilayer assembly with SnPb and SnAgCu solders, each with two constitutive models: secondary-

only creep (Eq.(2)) and secondary plus primary creep (Eqs.(2) and (10)). For each of these four

cases outlined in Table 3, the following thermal profiles were modeled: i) 14°C/min ramp rate between

0°C and 100°C; ii) 95°C/min ramp rate between 0°C and 100°C; iii) 95°C/min ramp rate between -

40°C and 125°C. In all cases, 5 minute dwell periods were used at both temperature extremes. The

solder melt temperature was considered the stress free reference temperature for the analysis (Table

3). In order to simulate a hypothetical thermal history following assembly, each finite element

simulation was initiated by assuming a cooling phase from the solder melt temperature to 25°C over a

one hour period, followed by a one hour dwell at room temperature, before starting the thermal

cycling [30]. An example of such an accelerated thermal profile is shown in Fig. 3 for the 95°C/min

ramp rate and ∆T=100°C.

These conditions, taken from JEDEC accelerated thermal cycling standards [31], were used

previously to test resistor joints [32] made with Sn40Pb and Sn3.8Ag0.7Cu solders. The ramp rate of

14°C/min is typical of many accelerated thermal cycling tests, while the ramp rate of 95°C/min

represents a thermal shock condition. The material input parameters for the trilayers are given in

Tables 1 and 2.

The damage metrics, namely accumulated equivalent strain and strain energy density per

cycle, were obtained from the stabilized instantaneous equivalent (von Mises) stress versus

instantaneous equivalent (von Mises) strain hysteresis loops of each of the 12 cases that were

modeled (2 constitutive models, 2 solders, 3 thermal profiles). Hysteresis loops were considered

stable once the difference in the damage metric between subsequent loops was less than 1%. The

SnPb and SnAgCu secondary-only creep models were run for 10 loops, while the combined primary

and secondary creep models required 15 loops to reach the stability criterion. Hysteresis loops

shown in the enclosed figures are always given for the final stabilized loop.


Additionally, a stability limit was required for the time step size to ensure accuracy and

convergence [33]. The nonlinear material response of the solder is non-conservative, because

energy is dissipated by creep deformation. An analysis of a non-conservative system is path-

dependent; therefore, the actual load-response history of the system must be followed closely to

obtain accurate results. The path-dependent creep solution requires slow load application with many

sub-steps to reach the final value [34]. For an accurate yet efficient solution, the maximum increment

of creep strain was limited to 10% of the maximum total accumulated equivalent creep strain at all

integration points of the system (implicit creep ratio) [33]. Whenever this stability limit was exceeded

during model execution, the current time step size was reduced and the iteration step repeated.

2.7 Stabilized thermal cycling hysteresis loops

The stabilized von Mises stress versus instantaneous von Mises creep strain hysteresis loops

shown in Figs. 6 and 7 for SnPb and SnAgCu respectively, are the result of thermal cycling as shown

in the inset figures. Each thermal cycle is depicted as beginning and ending at 25°C, and the

stresses and strains correspond to the values after 10 cycles (secondary-only creep) and 15 cycles

(combined primary and secondary creep).

Figure 6 illustrates that the inclusion of primary creep in the SnPb trilayer model caused a

large decrease in the von Mises stress range and an increase in the strain range per cycle because

creep deformation relaxes stress. The slope of the AB segment from the secondary-only model is

slightly larger than the corresponding AB segment from the model with primary creep, suggesting that

secondary creep is dominant during the high temperature dwell. At the beginning of the cooling

phase (B), the stresses and strains in the two models are approximately equal because the von Mises

stress is almost fully relaxed by creep deformation. There is a shear stress and peel stress reversal

at the end of the high dwell (B), and the subsequent cool-down leads to the maximum stress at the

onset of the lower temperature dwell (C). The inclusion of primary creep increases the joint

compliance, causing the maximum von Mises stress at C to be one half as large as that of the

secondary-only model, and the strain to be approximately 15% larger. During the low temperature


dwell (CD), both curves have a similar slope as the stress relaxes, showing again that secondary

creep is dominant during the dwell periods. The temperature increases at D, causing the stress to

decrease rapidly to a local minimum before rising once again and then falling to the value at A. This

happens at the end of the low dwell (D) because, as at (B), there is a shear stress reversal causing

the von Mises stress to be a local minimum.

In Figs. 6 and 7, for SnPb and SnAgCu respectively, the magnitude of stress relaxation that

occurs during the cold dwell (CD) is larger than that occurring during the high temperature dwell (AB).

This was observed for all three thermal profiles both when primary creep was included and with the

secondary-only creep model. This is caused by the relatively fast creep rates attained at the high

stress levels and low temperatures, particularly as residual stress builds during the temperature

ramp-down where the flexural creep rigidity of the solder is relatively high. Although the flexural

creep rigidity is lower at high temperatures, the much smaller residual stress levels result in less

creep strain during the high temperature dwell. Following the high temperature dwell (BC), the

temperature decreases and the stress level increases with higher creep rigidity for both SnPb and

SnAgCu. Note that after the cold dwell (DA) a sizeable residual stress exists because creep is less

activated at lower temperatures [11]. The subsequent temperature rise accompanies a decrease in

stress with lower creep rigidity. Stress relaxation served to diminish the residual stress levels during

the temperature ramp-up preceding the hot dwell, thereby resulting in a partially relieved residual

stress at its onset. This causes the creep strain during the high temperature dwell to be seemingly

dwarfed by that during the low temperature dwell, where stress relaxation occurs only once the cold

dwell has commenced at much higher stress levels.

In Fig. 6 the line segment CD, at the start of the hysteresis loop and line segment DA at the

end of the hysteresis loop are not contiguous. This is caused by cyclic creep ratcheting where the

mean strain increases with the number of cycles as shown in Fig. 8 for multiple stabilized hysteresis

loops. When a nonlinear creep model is used for solder, the cyclic increase in the mean creep strain

usually establishes an approximately constant value after a few cycles (steady ratcheting). In the

stabilized hysteresis loops with steady ratcheting the difference between the maximum and minimum


von Mises creep strain for one cycle does not change in subsequent cycles. The typical steady cyclic

creep ratcheting shown in Fig. 8 for SnPb, also occurs in SnAgCu, but to a lesser extent [35].

The trends of Fig. 6 for SnPb solder are similar but amplified in Fig. 7 for the SnAgCu alloy.

The inclusion of primary creep causes a fivefold decrease in the overall stress range and a threefold

increase in the strain range. Over the high temperature dwell period defined by AB, the stress-strain

slope decreased markedly with the inclusion of primary creep, in contrast to Fig. 6. A similar

observation is noted at the low temperature dwell period (CD). Over the cooling phase (BC), the

contribution of primary creep to the joint compliance produces a large decrease in stress and

increase in strain. As with the SnPb solder, during the heating period (DA), both models show a local

minimum that is followed by a sudden increase in stress. The inclusion of primary creep causes the

creep strains during the heating and cooling phases to be much larger than those produced by the

secondary-only model, similar to Fig. 6. Comparing Figs. 6 and 7, it is evident that primary creep is

more significant for SnAgCu than SnPb solder.

Figure 6. SnPb von Mises stress vs. instantaneous von Mises creep strain hysteresis loops for the secondary creep model (Sec.) and the primary plus secondary creep model (Pri. + Sec.); 95 °C/min

and ∆T=100°C.


Figure 7. SnAgCu von Mises stress vs. instantaneous von Mises creep strain hysteresis loops for the secondary creep model (Sec.) and the primary plus secondary creep model (Pri. + Sec.); 95 °C/min

and ∆T = 100°C.

Figure 8. Steady ratcheting of SnPb von Mises stress vs. instantaneous von Mises creep strain hysteresis loops for the secondary creep model (Sec.) and the primary plus secondary creep model

(Pri. + Sec.); 95°C/min and T = 100°C.


2.8 Effect of temperature range on hysteresis loops

The effect of temperature range on the relative influence of primary creep was examined by

comparing the simulations for ramp rates of 95°C/min and temperature ranges of 100°C (0°C ≤ T ≤

100°C) and 165°C (-40°C ≤ T ≤ 125°C), shown in Figs. 9 and 10 for SnPb and SnAgCu, respectively.

As expected, the larger temperature range produces greater stress and strain ranges for both

constitutive models in both solders. Since the creep rigidity decreases exponentially with

temperature, the portion of the temperature range above 100°C for the ∆T=165°C cycle causes a

much higher creep strain than experienced when ∆T=100°C. Similarly, at temperatures below 0°C,

the creep rigidity is much greater in the ∆T=165°C cycle than experienced during the ∆T=100°C

cycle, thereby accounting for the much higher stress range in the ∆T=165°C thermal cycle.

Figure 7. SnPb von Mises stress vs. instantaneous von Mises creep strain hysteresis loops for the secondary creep model (Sec.) and the primary plus secondary creep model (Pri. + Sec.);

∆T = 100°C and ∆T = 165°C for 95°C/min.


Figure 8. SnAgCu von Mises stress vs. instantaneous von Mises creep strain hysteresis loops for the secondary creep model (Sec.) and the primary plus secondary creep model (Pri. + Sec.);

∆T=100°C and ∆T= 165°C for 95°C/min.

The temperature range increased by a factor of 1.65 from ∆T=100°C to ∆T=165°C and this

same ratio of 1.65 is observed between the ranges of stress or strain, for these two thermal cycles,

for both SnPb and SnAgCu using either secondary-only or primary plus secondary creep models.

Details of the relative significance of primary creep in each of these cases are quantified in Section 11

in terms of two damage metrics: (i) accumulated equivalent creep strain; and (ii) creep strain energy

density.

2.9 Effect of ramp rate on hysteresis loops

The effect of ramp rate was examined by simulating typical thermal cycling (14°C/min) and

thermal shock (95°C/min) conditions for a temperature range of 100°C (0°C ≤ T ≤ 100°C). Figures 11

and 12, for the SnPb and SnAgCu solders, respectively, show a general trend of increasing stress

range at the higher ramp rate and a strain range that is relatively insensitive to ramp rate.

The shorter elapsed time for temperature change during the 95°C/min ramp does not provide

sufficient time for thermal stresses to relax through creep deformation to the same extent observed in


the 14°C/min case. However, since the temperature range is the same in both cases, they have the

same creep rigidity. Consequently, the strain range is very similar between the two cooling rates for

both materials when either secondary-only or primary plus secondary creep models are considered.

The shapes of the loops are consistent with those of Figs. 6-10. The following section quantifies the

differences between these curves in terms of the two aforementioned damage metrics.


Figure 9. SnPb von Mises stress vs. instantaneous von Mises creep strain hysteresis loops for the secondary creep model (Sec.) and the primary plus secondary creep model (Pri. + Sec.); 14°C/min

and 95° C/min for ∆T=100°C.

Figure 10. SnAgCu von Mises stress vs. instantaneous von Mises creep strain hysteresis loops for the secondary creep model (Sec.) and the primary plus secondary creep model (Pri. + Sec.);

14°C/min and 95° C/min for ∆T=100° C.


2.10 Effect of primary creep on damage metrics

In Figs. 13 and 14, the accumulated equivalent creep strain and the creep strain energy

density damage metrics are shown for time elapsed during a single stabilized thermal cycle beginning

at t=0. Results are shown for the three thermal profiles. It is apparent that the inclusion of primary

creep in the constitutive model can have a large effect on either damage metric for both solder

materials. Interestingly, the damage metrics are affected differently by primary creep.

In Fig. 13, the accumulated equivalent creep strains of the primary plus secondary creep

models are greater for both solders than those of the secondary-only models. The opposite trend is

observed in Fig. 14 for creep strain energy density, where the primary plus secondary creep models

predict less damage than the secondary-only creep models.

Tables 6 and 7, for the strain energy density and accumulated equivalent strain respectively,

summarize the changes in damage metric as a function of solder material, thermal profile and

constitutive model.

The greater accumulated equivalent creep strain predicted by the primary plus secondary

creep model is due to a decrease in creep rigidity relative to the secondary-only model. The primary

plus secondary models, however, predict a reduction in stress which is proportionately larger than the

increase in strain, so that the strain energy density as defined in Eq. (8) decreases relative to the

prediction of the secondary-only models.


Figure 11. Accumulated creep strain vs. time for the secondary creep model (Sec.) and the primary plus secondary creep model (Pri. + Sec.); (i) 95°C/min, ∆T=165°C; (ii) 95°C/min, ∆T=100°C; (iii)

14°C/min, ∆T=100°C.

The trends for damage accumulation differ not only among the damage metrics, but also

among the SnPb and SnAgCu solders. Among the temperature profiles considered in this work, each

solder has a distinct proportion of primary creep to total creep strain that is the combined result of the

difference in creep rigidity at different temperatures, the time available for stress relaxation under

varying ramp rates, and differences in the time hardening behaviour. Since SnPb displays less

primary creep, the relative differences between the primary plus secondary and the secondary-only

creep models were similar among all thermal profiles for both accumulated equivalent creep strain

(21%-25% difference) and creep strain energy density dissipation (32%-35%) damage metrics.

SnAgCu has a more pronounced primary creep response as reflected by the higher percentage


difference between the primary plus secondary and secondary-only creep models; 82%-100% for the

accumulated equivalent creep strain and 29%-58% for the creep strain energy density dissipation.

For SnAgCu, the proportion of primary creep in each of the three thermal profiles varies significantly

for both damage metrics. As discussed in Section 12, this has an important implication for the

establishment of reliable thermal fatigue-life empirical correlations using these two damage metrics.

Figure 12. Creep strain energy density vs. time for the secondary creep model (Sec.) and the primary plus secondary creep model (Pri. + Sec.); (i) 95°C/min, ∆T=165°C; (ii) 95°C/min, ∆T=100° C, (iii) 14°

C/min, ∆T = 100°C.

The finite difference solution of the trilayer confirms the damage metric trends obtained with

the finite element method for both accumulated equivalent creep strain and creep strain energy

density dissipation per cycle. Tables 6 and 7 show that the finite difference and finite element

approaches produce different estimates of the damage metrics; an anticipated result because the

finite difference solution did not run for sufficient loops to reach a stabilized solution. The results

obtained with the Ansys™ finite element method are considered more accurate than the finite


difference results for two principal reasons: (1) Ansys™ dynamic memory allocation allows for a

larger number of thermal loops to satisfy the stability criteria; and (2) optimized time stepping in

Ansys™ reduced the solution sub-step time intervals and more accurately accounted for primary

creep as the proportion of primary creep changed with temperature. Nevertheless, the trends

displayed by the finite difference model and the finite element model are very similar. In all cases, the

secondary-only creep model predicted a much larger creepW and smaller .eqAcc Creepε than did the

primary plus secondary creep model. The percentage differences in the damage metrics (secondary-

only compared with primary plus secondary creep) are comparable for both the finite element and

finite difference models, and both show greater profile-to-profile variability between the two creep

models for SnAgCu than for SnPb solder.

2.11 Neglect of primary creep in life prediction

Empirical thermal fatigue life correlations are based on damage metrics calculated for solder

joints that are thermally cycled until some fraction of the initial sample size population has failed. For

example, Eqs.(12) and (13) show the modified Coffin-Manson relations for the prediction of cycles to

first failure, Nf,1, expressed in terms of accumulated equivalent creep strain, .eqAcc Creepε , and creep

strain energy density, creepW .

( ),1 .

m

f eqAcc CreepN A ε−

= ∆ (12)

( ),1f creepN B Wϕ−

= ∆ (13)

The selection of cycles to first failure is arbitrary, as similar trends are also evident when

considering the mean cycles to failure (Nf,50%). Typically, fatigue life correlations are established by

thermally cycling a large number of solder joints with specific ramp rates and dwell periods leading to

creep-fatigue failure. A stress-strain analysis of the failed solder joints is then used to calculate the

value of a particular damage metric per stabilized hysteresis loop. This damage metric is then

correlated to the number of cycles required to cause first failure or the failure of a certain fraction of


the tested sample. The regression constants , , ,A B m ϕ in Eqs.(12) and (13) are then determined by

a best fit through data points obtained under multiple thermal cycle profiles with different temperature

ranges and ramp rates.

As an illustration, Figures 15 and 16, for SnPb and SnAgCu respectively, use

experimentally measured published data of cycles to first failure of resistor joints [32] to correlate

computed accumulated equivalent creep strain for both elasto-creep models (with and without

primary creep).

For comparison, Fig.17 for SnAgCu, correlates mean cycles to failure with the accumulated

equivalent creep strain. In this case, the present trilayer model was taken to be an approximation of

the solder joints from the resistor 2512 assembly test vehicle described in [32], subject to the

14°C/min-∆100°C, 95°C/min-∆100°C and 95°C/min-∆165°C thermal fatigue profiles. The cycles to

first failure and mean cycles to failure of the resistor joints, for each of the three thermal fatigue

profiles, were reported in [36] and are used here to demonstrate a correlation between the elasto-

creep models and physical data.


Figure 13. SnPb cycles to first failure vs. accumulated creep strain for the secondary creep model (Sec.) and the primary plus secondary creep model (Pri. + Sec.); (i) 95°C/min, ∆T=165° C, (ii) 95°

C/min, ∆T=100° C; (iii) 14° C/min, ∆T=100° C.

Figure 14. SnAgCu cycles to first failure vs. accumulated creep strain for the secondary creep model (Sec.) and the primary plus secondary creep model (Pri. + Sec.); (i) 95°C/min,

∆T = 165°C; (ii) 95° C/min, ∆T=100° C; (iii) 14°C/min, ∆T=100° C.


The accumulated equivalent creep strain per stabilized hysteresis loop was calculated using

both the secondary-only and primary plus secondary creep models. As discussed below, if primary

creep is neglected, the approximated damage metric-life correlations will have an undetermined and

variable error dependent on the thermal profile; therefore, the life predicted by the correlation may

differ widely from the actual life for any thermal profile not specifically used in the test program

leading to the correlation. In the previous section it was shown that for SnAgCu the accumulated

equivalent creep strain damage metric may differ by 100% for the 95°C/min-∆165°C (5 min high and

low dwell) thermal profile with the exclusion of primary creep, thereby vastly under-predicting the true

damage per cycle and over-predicting service lifetime in the field.

Figure 15. SnAgCu mean cycles for failure vs. accumulated creep strain for the secondary creep model (Sec.) and the primary plus secondary creep model (Pri. + Sec.); (i) 95°C/min, ∆T=165° C; (ii)

95° C/min, ∆T=100°C, (iii) 14° C/min, ∆T= 100°C.


The only exception would be if the approximate damage metric differed from the true value by

a constant multiple for all possible thermal profiles of interest – then the approximate and true

damage metrics would vary with thermal profile parameters in exactly the same way. This is the case

for SnPb in Fig. 15, where the secondary-only and primary plus secondary creep models for

accumulated equivalent creep strain differ by approximately a constant multiple of approximately 0.8

(Table 7) for the three thermal profiles. Therefore, the error in the approximate secondary-only

creep model is a constant multiple that is independent of the thermal profile, and so the correlation

will likely be applicable to thermal profiles other than those specifically used to develop the

correlation.

Figures 16 and 17 for SnAgCu show the cycles to first failure (Nf,1) and mean cycles to failure

(Nf,50%) respectively, and give a correlation for the SnAgCu solder among the three thermal profiles

tested. However, the accumulated equivalent creep strain calculated using the secondary-only model

differs from that of the primary plus secondary model by a varying ratio of between 0.33 and 0.42

(Table 7). Therefore, the error in the secondary-only accumulated equivalent creep strain damage

metric is an unknown function of the parameters of the thermal profiles. Consequently, the

approximate secondary-only correlation of Figs.16 and 17 may not satisfy thermal profiles other than

those three used to develop the correlation.

Accumulated equivalent creep strain and creep strain energy density dissipation have

differing sensitivities to the inclusion of primary creep. Figures 18 and 19 show the creep strain

energy density dissipation per stabilized hysteresis loop correlated with cycles to first failure for both

SnPb and SnAgCu, respectively measured from solder joints of 2512 chip resistors under accelerated

thermal cycling. Figure 20 for SnAgCu, shows a similar trend among both creep models correlated to

mean cycles to failure. For SnPb, the creep strain energy density calculated using the secondary-

only model is 1.4 times larger than that from the primary plus secondary model for all three thermal

profiles, while for SnAgCu the ratio ranged between 1.4 and 1.8 (Table 6).


Figure 16. SnPb experimental cycles to first failure vs. calculated creep strain energy density

dissipation for the secondary creep model (Sec.) and the primary plus secondary creep model (Pri. + Sec.); (i) 95°C/min, ∆T=165°C; (ii) 95° C/min, ∆T=100° C; (iii) 14°C/min, ∆T= 100°C.

Figure 17. SnAgCu experimental cycles to first failure vs. calculated creep strain energy density

dissipation for the secondary creep model (Sec.) and the primary plus secondary creep model (Pri. + Sec.); (i) 95°C/min, ∆T=165° C; (ii) 95° C/min, ∆T=100°C; (iii) 14°C/min, ∆T=100° C.


Figure 18. SnPb experimental cycles to first failure vs. calculated creep strain energy density

dissipation for the secondary creep model (Sec.) and the primary plus secondary creep model (Pri. + Sec.); (i) 95°C/min, ∆T=165°C; (ii) 95° C/min, ∆T=100° C; (iii) 14°C/min, ∆T= 100°C.

Figure 19. SnAgCu experimental cycles to first failure vs. calculated creep strain energy density

dissipation for the secondary creep model (Sec.) and the primary plus secondary creep model (Pri. + Sec.); (i) 95°C/min, ∆T=165° C; (ii) 95° C/min, ∆T=100°C; (iii) 14°C/min, ∆T=100° C.


Figure 20. SnAgCu experimental mean cycles to failure vs. calculated creep strain energy density

dissipation for the secondary creep model (Sec.) and the primary plus secondary creep model (Pri. + Sec.); (i) 95° C/min, ∆T=165°C; (ii) 95° C/min, ∆T=100°C; (iii) 14°C/min, ∆T=100° C.

This suggests that for SnAgCu, the approximate creep strain energy density dissipation,

obtained with the secondary-only creep model, will also have an undetermined and variable error.

This unknown error, also a function of the parameters of the thermal profiles, may differ from that of

the accumulated creep strain damage metric.

These results illustrate the limitations of using empirical fatigue life correlations based on

damage metrics calculated using a secondary-only model. Significant errors may be encountered

when extrapolating such a correlation to predict the life of a component under a thermal profile other

than those used to develop the correlation. Similarly, any changes to the component that alter the

thermal stress distribution may also result in a different proportion of primary creep to total inelastic

deformation and hence a different error in the secondary-only correlation.


2.12 Conclusions

Three trends emerge regarding the accumulated creep strain and creep strain energy density

dissipation damage metrics: (1) SnAgCu is much more sensitive to primary creep contributions than

SnPb in all three thermal profiles; (2) the inclusion of primary creep strain increases accumulated

equivalent creep strain and decreases the creep strain energy density; and (3) the differences

between the damage predictions of the secondary-only and primary plus secondary creep constitutive

models varies with the thermal profile, particularly in the case of SnAgCu.

The contributions of rate-independent plastic strains were shown to be a negligible fraction of

the total inelastic damage as creep dominates both accumulated equivalent inelastic strain and

inelastic strain energy density dissipation per cycle. The inclusion of primary creep in constitutive

models changes the computed stress and strain fields significantly in both SnPb and SnAgCu solder

joints. When primary creep is neglected, high stress levels may be obtained for large temperature

amplitudes, particularly for SnAgCu, as demonstrated with a ∆T=165°C and 95°C/min thermal profile

with 5 minute dwells. The neglect of primary creep strain over-predicts the creep strain energy

density and under-predicts the accumulated creep strain thermal-fatigue damage metric. The effect

of neglecting primary creep is more pronounced for accumulated creep strain than for creep strain

energy density dissipation. The SnAgCu solder is more sensitive than SnPb solder to the inclusion of

primary creep contributions over the three thermal cycling profiles that were examined.

The proportion of primary creep strain to total creep strain changes with different thermal

profiles; therefore, the unknown and variable error in neglecting primary creep is a function of thermal

profile parameters. Thus, empirical thermal fatigue life correlations (damage metric versus cycles to

failure) based on secondary-only creep constitutive models, may be inaccurate when used to predict

lifetime for thermal profiles (ramp rate, temperature range, dwell period, etc.) other than those actually

used in the thermal cycling experiments leading to the correlation.


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Engineering. 1986, New York: McGraw-Hill. [29] Sherby, O.D., Orr, R.L., and Dorn, J.E., Creep Correlations of Metals at Elevated

Temperatures. Transactions of the AIME - Journal of Metals, 1954. 200: p. 71-80. [30] Pucha, R.V., Tunga, K., Pyland, J., and Sitaraman, S.K., Accelerated Thermal Cycling

Guidelines for Electronic Packages in Military Avionics Thermal Environment. Transactions of the ASME - Journal of Electronic Packaging, 2004. 126(2): p. 256-64.

[31] JEDEC, Temperature Cycling - JESD22-A104C (Revision of JESD22-104B). 2005: Arlington, VA, USA. p. 16.

[32] Qi, Y., Ghorbani, H.R., Snugovsky, P., and Spelt, J.K., Temperature profile effects in accelerated thermal cycling of SnPb and Pb-free solder joints. Microelectronics Reliability, 2006. 46(2-4): p. 574-88.

[33] Zienkiewicz, O.C. and Cormeau, I.C., Visco-plasticity: Plasticity and creep in elastic solids - A unified numerical solution approach. International Journal for Numerical Methods in Engineering., 1974. 8(4): p. 821-45.

[34] ANSYS, ANSYS Theory Reference. 2004: Canonsburg, PA, USA. p. 129-30. [35] Ghorbani, H.R., Analytical Modeling of Thermal Stresses in Solder Joints - Appication to

Reliability Prediction, in Mechanical Engineering. 2005, University of Toronto: Toronto. p. 256.

[36] Qi, Y., Ghorbani, H.R., and Spelt, J.K., Thermal fatigue of SnPb and SAC resistor joints: Analysis of stress-strain as a function of cycle parameters. IEEE Transactions on Advanced Packaging, 2006. 29(4): p. 690-700.

Chapter 3. Theory, Experiments and Data Analysis 63

Chapter 3

Theory, Experiments and Data Reduction

3.0 Introduction

The creep of SnAgCu solder alloy is an important factor governing the reliability of solder

joints. The solidus temperature ( sT ) of Sn3.8Ag0.7Cu is 217°C (490K), so the homologous

temperature ( h sT T T= ) defined in absolute temperature terms is relatively high ( 0.6hT > ) even at

room temperature. Consequently, solder joints at typical service temperatures and during thermal

cycling tests are subject to significant creep deformation even at low stresses. Creep is time-

dependent plastic deformation that that may readily occur due to thermal or applied stresses below

the proof stress of a metal. Creep strain is a function of temperature, strain rate and stress.

Historically, creep in SnAgCu has been studied at relatively high strain rates and stresses that are

often not representative of conditions in actual solder joints.

The total strain exhibited during creep testing can be partitioned into elastic and inelastic

deformation (Eq.(1)). The inelastic deformation is comprised of time-independent (plastic) and time-

dependent (creep) deformation. There is no plastic deformation, so the inelastic deformation is solely

creep.

total elastic inelasticε ε ε= + (1)


Creep strain contributions in metals under constant load and temperature are partitioned into

three stages: (1) primary creep with a decelerating strain rate as the metal strain hardens; (2)

secondary creep where the strain hardening and dynamic recovery (softening) processes are in

equilibrium and the metal deforms approximately linearly with time at an averaged minimum creep

rate; and (3) tertiary creep where voids form and the strain rate accelerates to final rupture. Tertiary

creep is usually omitted in solder joint deformation modeling because it only represents deformation

contributions at the onset of imminent creep rupture. Given that true strains rates are additive, the

total true strain rate is given by Eq.(2), where elasticε is the true elastic strain rate, trε is the true

transient creep strain rate and minε is the true minimum (steady-state) creep strain rate.

Total creep strain is a superposition of transient and constant-rate steady state creep

components in both primary and secondary stages. Although both contribute to total creep

deformation as shown in Fig. 1, transient creep contributions are dominant during the primary creep

stage and steady-state creep contributions are dominant during the secondary creep stage [1].

Although transient and steady-state creep are considered distinct material responses to applied

stress and temperature, both are strongly interconnected at high homologous temperatures [2]

mintotal elastic trε ε ε ε= + + (2)


Figure 1. Creep Strain vs. Time, superposition of creep components that determine the creep curve (T = 25° C, σ= 15 MPa)

During isothermal creep testing at a given stress level, the contributions of transient creep

gradually saturate during primary creep and subsequent creep contributions are dominated by

steady-state creep (secondary creep). Whenever a stress change occurs within a metal, some

transient creep is produced prior to the onset of steady-state creep [3]. Stress fluctuations and hence

transient creep are present in thermally cycled solder joints due to mismatch in the thermal expansion

coefficients of adjacent materials. The isothermal dwell period of a thermal cycle is analogous to a

constant displacement loading where stress relaxes over time. Although transient (primary) creep is

often ignored in solder joint modeling and within the electronics packaging literature [4-6], it has been

shown to make a significant contribution to total deformation during thermal cycling of SnAgCu solder

joints [7].


This chapter describes primary and secondary creep measurements made with

Sn3.8Ag0.7Cu solder alloy under both constant-load and constant-displacement (stress relaxation)

conditions at three temperatures. Stress and strain rates were selected to extend down to ranges

representative of actual solder joints in service ( 4 110 sε − −< and 20MPaσ < ). Several non-

standard data reduction techniques were used to interpret the results and characterize both the

primary and secondary creep regimes. The data are used in Chapter 4 to develop a creep

constitutive model incorporating both transient and steady-state creep strain.

3.1 Experimental Procedures

3.1.1 Bulk Tensile Specimen

The alloy investigated was the near-eutectic ternary alloy Sn3.8Ag0.7Cu: Ag 3.73, Cu 0.68

and balance Sn (in wt. %), as per the manufacturer certificate of analysis. The bulk solder tensile

specimen (Fig. 2) was designed according to international standard guidelines[8, 9] to achieve a

uniform uniaxial stress in the reduced radius gauge section with a 40 mm gauge length and 8 mm

diameter. Bulk bar stock of Kester Sn3.8Ag0.7Cu was melted, cast into 14 mm diameter copper pipe

molds and air cooled at 2-3°C/s, representative of cooling during reflow processing in microelectronic

assembly. Each casting was scanned using non-destructive C-mode scanning acoustic microscopy

(C-SAM) to inspect for voids, cracks and foreign material inclusions [10]. The gage section was

machined with a CNC lathe to give a 6 mm shoulder fillet radius that transitioned to a 12.7 mm

diameter straight cylindrical cross-section along the balance of the tensile specimen. The ends were

machined flat and soldered to copper rods (60 mm long and 12.7 mm diameter). The rods were

threaded along half the length and subsequently inserted into steel sockets that connected the

specimen to the load train of the constant-load creep testing frame. This prevented thread failure that

otherwise occurred at elevated temperatures, when threads made directly in bulk solder deformed

excessively.


Once machined, each specimen was aged at 2

3* solidusT (54°C) for 16 h to relax residual

stresses and provide a common starting microstructure [11]. Fig.3 (a, b) shows that the aged and

untested specimen had large Sn-rich regions containing ternary eutectic Ag3Sn and Cu6Sn5 rod-like

particles (some appear as small dark dots) that define the edge of β -Sn grain boundaries. The

samples were mounted in epoxy resin and prepared using standard grinding and polishing

techniques. The polishing was finished with alumina of abrasive size 0.05 µm followed by colloidal

silica suspension of abrasive size less than 0.05 µm. Lastly, the samples were etched in a silica

suspension for 90 sec for improved resolution of the sample cross section.

Figure 2. Tensile specimen (ruler units: mm)

Figure 3. Optical micrograph using 1/4 wavelength lambda plate of aged as-cast tensile specimen microstructure (no deformation): (a) low magnification (b) high magnification.


Table 1. Test matrix for constant load creep

Temperature (°C) Engineering Tensile Stress (MPa)

25 6.5, 10.0, 15.0 (duplicated), 20.0

75 2.6, 7.8 (duplicated), 11.3, 14.8

100 2.6, 3.9, 5.2 (duplicated), 8.6, 12.0

3.2 Experimental Setup

The isothermal constant-load creep test matrix (Table 1) included tests at several stress

levels at each of 25°C, 75°C and 100°C. The constant-displacement test matrix included duplicate

isothermal tests each run at 25°C, 75°C and 100°C. The specimens in all tests were placed within an

aluminum sheet enclosure, through which an opening was made to accommodate a pair of T-type

thermocouples and an extensometer. Thermocouples were attached to the specimen surface near

the top and bottom portions of the reduced diameter gage section. The uniform strain within this

section was measured with a clip gauge extensometer (10 mm gage length) affixed with springs.

Heating was provided by a set of rope heaters attached directly to both steel sockets that connected

with the threaded copper ends of the tensile specimen. The heaters were PID-algorithm controlled

with a set of solid state relays using a temperature feedback loop to keep the temperature within ±1°C

of the set point.

The data acquisition program recorded temperature, strain and force at a sampling rate of 60

and 600 samples/min, for creep and stress relaxation tests, respectively. Electrical noise was filtered

using a low-pass finite impulse response (FIR) filter with a cutoff frequency equal to one-tenth the

sampling frequency.


Most of the constant-load creep tests were conducted with the dead-weight load frames by

attaching calibrated weights to the load train and gradually lowering a supporting platform from below

until the weights were freely suspended from the specimen. Some additional creep tests were made

using the screw-driven actuator load frame in load control mode (±2N). This load frame was also

used in stroke-control mode for all constant-displacement stress relaxation tests.

3.3 Data Analysis

3.3.1 Constant Load - Creep

The start of the creep test (the “zero moment”) was the moment when the test load was fully

applied, corresponding to the maximum measured strain rate [8]. During the finite loading time

required to strain the specimen the stress is described by the Boltzmann integral [12]. The zero

moment was readily determined by fitting the loading portion of the strain ( totalε ) versus time ( t )

curve to a sigmoidal Boltzmann function (Eq.(3)) and solving for the maximum slope; where 1A ,

2A , 0t and dt are fitting parameters.

1 22

01 exptotal

A A At t

dt

ε −= +

− +

(3)

Throughout the creep test, particularly at low strain rates, the extensometer measurement

was subject to residual signal noise. A smooth curve was obtained using an adjacent averaging

approach for particularly noisy strain-time data. Following the zero moment, the creep strain rate

decelerated to an approximately linear strain rate, denoting the transition between primary and

secondary creep regimes. This transition occurs gradually and is difficult to determine precisely by

inspection for many creep curves; however, the definition of the saturation transient creep strain and


minimum creep rate relies on the accurate determination of the transition time satt , the elapsed time

to transient strain saturation. An algorithm was created to define this transition point over time, based

on the partial memory sliding-window method that used a statistical hypothesis test to detect change

[13, 14]. A schematic of the sliding window approach is shown if Fig. 4 where subsequent windows

span adjacent ranges of sub-windows.

Figure 4. Sliding window schematic over a range of subwindows

The initial reference time window was taken from the end of the creep curve, in the linear

portion of the secondary creep regime with subsequent windows sliding progressively backwards in

time. The width of this window was selected as described below. The curves were truncated if the

full data set extended into the tertiary creep regime. In such cases, a similar technique was used to

determine the transition from secondary to tertiary, with the initial reference window within the linear

region and subsequent windows sliding forwards in time. From the central limit theorem an


appropriate sample size for the statistical test was obtained using a time window subdivided into 30

sub-windows where the slope of each was determined by linear regression [13]. The mean of these

first 30 slopes became the reference window slope. The algorithm then advanced to the next time

window closer to the start of the creep curve ( t = 0 s) by an increment of two sub-windows, adding

two new recent slopes and removing the oldest two slopes from the current window. The slope of

each of the 30 examples within the current window was taken by linear regression and the mean

compared with the mean of the reference window using a two-tailed t-test with an alpha level of 0.05.

If the mean of the slopes were statistically equal, two new sub-windows were added and two removed

from the end of the window and the process repeated. When the mean slope of the current window

was not statistically equal to that of the reference window, the transition time, satt , was considered

the middle of this window. Since the gradual nature of change in the primary and secondary creep

regime transition was similar between specimens, a fixed size time window was implemented. In

cases where the data was noisy a larger window size was useful for smoothing, although a window

either too large or too small resulted in decreased prediction accuracy. Since the calculated transition

time was sensitive to the window size an iterative approach was used to find the best estimate. The

initial transition time solution was calculated with a window size of one-half of the total elapsed time

( elapsedt ) of the measured creep strain versus time data set. Subsequent solutions were re-run with

progressively smaller windows down to one-eighth the total elapsed time, where the iterative window

size was defined by elapsedt n , where 2,3,...,7,8n = . The estimate of transition time ( satt ) was

then taken as the median of the array of all calculated times computed from each of the several

window sizes.


The linear portion of the creep curve beyond the transition time was fitted using linear

regression to solve for the slope and y-axis intercept that correspond to the minimum creep rate and

saturated transient creep strain, respectively.

3.3.2 Constant Displacement - Stress Relaxation

The stress relaxation tests were loaded by moving the screw-driven actuator crosshead at a

constant speed to a fixed displacement. However, since loading requires a finite time, during and

immediately after the loading period some relaxation occurred, mostly of high frequency components

(considering the strain-time response as a Fourier series) [12]. Consequently, pure relaxation under

constant displacement starts from values less than the maximum stress at a time beyond the end of

the loading period. If the tensile specimen has a relaxation time that is shorter than the loading time,

then the amplitude associated with loading will dissipate faster than it is applied, and decrease

considerably from its true value by the time loading is complete.

The main limitation of stress relaxation is the collection of high frequency data components.

The limiting factor is not the sampling data rate, but the finite time taken to strain the specimen.

Accordingly this finite loading time was minimized with relatively fast constant crosshead speeds (2

cm/min at 25°C; 4 cm/min at 75°C; and 6 cm/min at 100°C). The simultaneous relaxation and

application of stress during the initial loading period means that all frequency components will be at

least partially present at the end of loading, regardless of crosshead speed [12].

The identification of the start of pure relaxation, somewhat later than the final crosshead

movement, was adapted from [12] using a fast Fourier transform (FFT) analysis and an FFT low-pass

filter. The FFT analysis approximately transforms a periodic time-domain waveform, like force versus

time response during stress relaxation, into the frequency domain as a finite summation of sinusoids

of different frequencies and magnitudes [15]. Although the stress relaxation is a continuous analog


function of time, the data was acquired at a fixed sampling rate making the force measurement a

discrete function of time and thereby introducing a distortion termed aliasing, most acute at the

highest frequencies. Aliasing was mitigated by setting the measurement rate well above the Nyquist

sampling rate [15]. Similarly, the truncation of an infinite series to a finite representation of the

Fourier series introduced a distortion termed leakage that affected the frequency resolution of the

FFT. The leakage of the FFT was significantly reduced by use of the Hanning window function in the

signal processing tools [16].

Figure 5. Stress Relaxation @ 100°C, FFT frequency-domain response, fc = 0.0031 Hz.

The amplitude-frequency single-sided power spectrum (Fig. 5) was used to determine the

start of pure relaxation using the fact that the high frequency components have relaxation periods

much shorter than the loading period. Consequently, at the end of loading the lower frequency

components will have higher magnitudes and tend towards their true relaxation values, whereas the

higher frequency components, due to the finite time required for crosshead movement, would have

decreased considerably [12]. For example, Fig.5 shows the FFT frequency response for stress

relaxation at 100°C, where the pure signal and incomplete high frequency regimes are indicated. The


point at which the amplitude decreased sharply from its maximum value, beyond the initial lobe, was

used to determine the cutoff frequency at the start of pure relaxation. Once the cutoff frequency was

identified, an FFT low-pass filter was applied to the load cell output to yield a stress versus time

response without the higher frequency components. Subtraction of the highest frequency

components can also effectively mitigate a large part of the signal degradation due to aliasing [12].

Figure 6 shows the original and filtered time-domain representation of this stress relaxation response

where the high frequency components have been removed. The approximation of the finite Fourier

series is unable to exactly represent the abrupt end of the relaxation curve resulting in the second-

order “end effect” of the filtered load-time response [17]. This oscillatory end effect was an artifact of

the FFT approach and was omitted from the pure relaxation response.

Figure 6. Stress Relaxation @ 100°C, raw and filtered time-domain response.


3.3.3 Machine Compliance

The interpretation of the constant-displacement stress relaxation measurements required

consideration of the compliance of the tensile specimen and the screw-driven actuator test machine

[18, 19]. Crosshead displacement was the sum of the elastic and plastic elongation of both the

tensile specimen and machine elements including the crosshead, linkages and load cell. Assuming

that all elements of the machine are linearly elastic and that all inelastic deformation is isolated to the

tensile specimen gage length, (Eq.(4)) defines the total crosshead displacement over time ( t ).

0 00

0

t

PL AXdt LE K

σ σ ε= + +∫ (4)

where X = crosshead speed, t = time, σ = gage section stress, K = machine modulus, E = Young’s

modulus of solder, A0 = initial cross-sectional area, Pε = true plastic strain and L0 = initial gage length.

In a tensile specimen with a large shoulder fillet diameter some inelastic deformation may in

fact occur in the shoulder transition beyond the parallel gage section. This may be incorporated as

an increase in the initial gage length. This effective gage length was calculated with Eq.(5) by

measuring the uniform total true strain ( totalε ) and the final shoulder-shoulder separation ( ,f shoulderL )

with dial calipers.

,0

f shoulder

total

LL

ε= (5)

In the case of a constant-strain (strain control) test the total true strain rate ( totalε = 0) is zero

resulting in effectively infinite machine stiffness. However, in the case of constant displacement

(stroke control) the crosshead is fixed ( X = 0), thus machine stiffness ( K ) does contribute to true


plastic strain rate and the interpretation of stress relaxation results. The coupling relationship

between the tensile specimen and machine is given by taking the derivative of Eq.(4) and dividing by

the gage length. Simplifying the resultant coupling expression further by substituting the machine

modulus ( M ), where 0

0

KLMA

= , gives Eq.(6). Thus the combined specimen-machine elastic

modulus ( C ) is given by Eq.(7), where the coupled elastic response acts as two springs in series.

0

1 1P

XL E M

σ ε = + +

(6)

1 1 1C E M= + (7)

Eq.(6) is exact for load-independent machine stiffness which is a good approximation over

the stress relaxation test [19]. The machine stiffness is known to vary between tests, even for

identical test conditions, hence K must be calculated for each mechanical test performed rather than

assuming a universal value for the machine [18].

The machine stiffness was determined using the total deformation method. During initial

loading the crosshead speed ( X ) was constant and the linear portion of the loading rate ( P ) and the

corresponding total strain rate ( totalε ) were each calculated by linear regression. Machine stiffness

was calculated using Eq.(8) which was derived from Eq.(6) for the elastic portion of initial loading with

constant crosshead speed.

( )1

0

0

* elastic plasticL XKP L

ε ε−

= − +

(8)


3.4 Results and Discussion

3.4.1 Constant Load Creep

The constant load creep strain versus time data for 25°C, 75°C and 100°C are given in Figs.

7-9, respectively (omitting the duplicate tests indicated in Table 1 for clarity). The figures show the

transitions from primary creep to secondary creep regimes. The transient saturation creep strain,

,tr satε , minimum creep rate, min

ddtε

, and saturation transition time, satt , calculated as explained

above are shown in each figure. The total creep strain in these curves is a superposition of two creep

processes, transient and steady-state creep components [1]. Although transient and steady state

creep are sometimes considered distinct material responses to applied stress and temperature, they

occur concurrently and are strongly interconnected at high homologous temperatures [2]. Transient

creep was the dominant component of total creep strain at the beginning of the constant load creep

tests increasing gradually until it reached a maximum value ( ,tr satε ) at the onset of saturation ( satt ).

Beyond satt additional increases in total creep strain were due solely to the constant-rate creep

component that gradually dominates total creep strain over time.

Consider a set of creep results for similar minimum strain rates ranging from 3x10-6 s-1 to

5x10-6 s-1 measured at different temperatures and applied stresses: Fig.7 ( 25°C, σ = 20 MPa); Fig.8

(75°C, σ = 11.3 MPa); and Fig.9 (100°C, σ = 8.6 MPa). Note that as the temperature increases a

similar strain rate is obtained at lower stress levels. In Fig. 7 there is a large initial slope at 25°C that

corresponds to strain (also termed work) hardening during primary creep. In Fig. 8 and Fig. 9 for

75°C and 100°C respectively, the initial slope for the same 3x10-6 s-1 to 5x10-6 s-1 strain rate range is

progressively smaller as is the absolute value of creep strain measured. This is the consequence of


decreased strain hardening at higher temperatures as a result of increased thermally-activated

recovery

processes [20]. This trend is consistent with Fig. 10 for saturated transient creep strain versus

minimum creep rate. Strain hardening lead to larger absolute values of saturation transient creep

strain at 25°C and progressively smaller amounts at 75°C and 100°C. In Fig. 11 tensile stress was

plotted versus minimum creep rate. Note that the repeated creep results in Fig. 10 and 11 for each

test temperature (Table 1) give similar results indicating experimental repeatability.

Figure 7. Nonlinear fit to measured creep strain versus time data at 25°C.






Figure 10. Saturation primary creep strain versus minimum creep rate (25°C, 75°C, 100°C).

Figure 11. Tensile stress versus minimum creep rate (25°C, 75°C, 100°C).


As widely reported in the literature, the minimum constant creep rate is a function of applied

stress and temperature (Fig. 11), where faster minimum creep rates are achieved at a given load for

higher temperatures, due to thermally-activated creep mechanisms. The observation that the curves

of tensile stress versus minimum creep rate are nearly parallel at the three temperatures is indicative

of a common apparent activation energy and underlying creep mechanism, or set of creep

mechanisms, that controls deformation at each of these temperatures and strain rates.

During the strain hardening process, the dislocation density gradually increases leading to

a decreasing creep strain rate as dislocation mobility is impeded. Eventually during secondary creep

a steady-state is achieved between the competing processes of recovery (dislocation annihilation)

and strain hardening (dislocation generation) where the ratio of the rates of these two competing

processes yields the minimum creep strain rate [21]. However, the treatment of strain hardening and

consequently transient creep strain is not consistent in the literature. The saturated transient creep

strain has been assumed a constant over all temperatures for Sn3.8Ag0.7Cu. It has also been

considered solely a function of minimum strain rate for Sn4.0Ag0.5Cu [22]. In both cases, the strain

hardening was assumed not to be thermally-activated. In contrast to these two assumptions, Fig.11

shows that transient saturation creep strain, ,tr satε , for Sn3.8Ag0.7Cu was a function of both the

minimum creep strain rate and temperature. Larger values of saturated transient creep strain were

measured at lower temperatures, while there was a gradual increase in saturated transient creep

strain at higher minimum creep strain rates for a given temperature.

Strain hardening is commonly thought to result from dislocation pile-up at barriers (planar

glide), occurring over a relatively large number of interatomic distances and consequently being a

function of strain rate but independent of temperature [23]. However, strain hardening may also

occur through the intersection of dislocations, termed dislocation forests (wavy glide), where


dislocations moving in the slip plane cut through other dislocations intersecting the active slip plane

[1, 23]. This hardening mechanism is enhanced by cross-slip that accelerates the dislocation

rearrangement process in high stacking fault energy materials. Strain hardening by this cutting

process arises from short-range forces occurring over distances less than 5 to 10 interatomic

distances, and therefore it is subject to thermal recovery and is both strain rate and temperature

dependent. Therefore, the results of Fig.11 suggest that strain hardening through dislocation climb

and the cutting of intersecting dislocation forests is active in Sn3.8Ag0.7Cu. Generally metals with

high stacking fault energy like tin show only a very small propensity to harden by dislocation pile-up

because they can deform easily by cross-slip. Since Sn-rich lead-free solders are principally

comprised of β -Sn matrix, they exhibit strain hardening deformation characteristic of high stacking

fault energy pure metals [20].

3.4.2 Stress Relaxation

Knowledge of the elastic modulus of the tensile specimen and the machine compliance of the

loading frame were essential to the correct interpretation of the stress relaxation test. Young’s

modulus measurements can vary widely depending on the rate of loading and the scale of

deformation. For example, the dynamic modulus measured using the propagation of an acoustic

wave through the material is associated with purely elastic deformation [24], and may be significantly

greater than the static modulus measured from the stress-strain response of the solder loaded in

tension at the start of the stress relaxation test. In both cases, the Young’s modulus decreases with

increasing temperature, because at higher temperatures atoms in the lattice move farther apart due

to thermal expansion lowering inter-atomic bonding forces [23]. The static modulus values measured

at the start of the stress relaxation tests (Table 2) are similar to those reported in [25] for

Sn3.8Ag0.7Cu bulk solder tensile specimens.


Table 2. Stress relaxation experimental parameters

Temperature, T (°C) 25 75 100

Crosshead speed, X (cm/min) 2 4 6

Static Young’s modulus, E (GPa) 46 30 19

Specimen-machine elastic

modulus,C (GPa)

1.33 3.13 12.43

The combined specimen-machine elastic modulus was calculated with Eqs. 5-8 where total

strain rate and load rate inputs were also calculated. The total strain rate ( elastic plasticε ε+ ) was

computed from the total strain versus time response in Fig. 12. Similarly, the load rate ( P ) was

computed from the force versus time response in Fig. 13. The noisy experimental data for total strain

and load responses versus time, were each fit with a 3-parameter logarithmic expression given by

Eq.(9) [26], where y is either total strain or load; X is time, and α , β and γ are fitting parameters.

Accordingly, the creep strain rate was subsequently calculated by subtracting the elastic component

from the total strain rate.

*log( )y Xα β γ= − + (9)


Figure 12. Extensometer measured total strain versus time (25°C, 75°C, 100°C). Fast crosshead speed: 2.0/4.0/6.0 cm/min @ 25/75/100°C.

Figure 13. Force versus time from stress relaxation tests (25°C, 75°C, 100°C). Fast crosshead speed: 2.0/4.0/6.0 cm/min @ 25/75/100°C.


As shown in Fig. 12, the total strain in the gage section was not constant during these fixed

displacement stress relaxation tests, but rather increased with time over all test temperatures. Due to

faster creep rates at higher temperatures it took significantly less time for the stress relaxation to

approach the practical resolution limit of the test (1x10-8 s-1), hence the discrepancy in test durations

at 25°C, 75°C and 100°C. As anticipated, more strain was measured at higher temperatures as a

result of more elastic strain during crosshead movement and higher creep rates. This was also

evident in Fig. 13 where the larger strains at higher temperatures coincided with periods of faster

relaxation.

Figure 14 shows the creep strain rate versus stress from the stress relaxation tests together

with the minimum creep strain rate versus stress from the constant load creep tests. It is evident that

the creep strain rate for stress relaxation was faster than the minimum creep strain rate for the

corresponding stress measured from constant load creep tests. This was simply a result of the

steady-state and transient creep both contributing to deformation during stress relaxation. The

magnitude of the strain rate offset between stress relaxation and constant load creep results is

indicative of the relative importance of transient creep, and Fig. 14 shows that the transient

contributions were not uniform over all temperature and strain rates; i.e. the strain rate difference

increased as the temperature fell and, to a lesser extent, as the stress decreased. This is consistent

with Fig. 11 which showed that the largest saturation transient creep strain among all constant load

creep tests occurred at 25°C, while the smallest values were measured at 100°C. For all three test

temperatures the amount of saturation transient creep strain increased with increasing minimum

creep rate.


Figure 14. Creep strain rate versus tensile stress from creep and stress relaxation (25°C, 75°C, 100°C).

During stress relaxation, the stress decreases over time. During the finite moments between

incremental load drops, there may be insufficient time to approach transient creep saturation;

therefore the strain rate is higher than it would be under purely steady-state creep. The extent to

which transient creep saturation is approached during the stress relaxation test changes as a function

of temperature and strain rate. In Fig. 14 for 25°C, the entire stress relaxation response has a

significant transient component and does not appear to approach saturation at any point. However,

at 75°C at the highest strain rates (1x10-5 s-1) the strain rate offset between the stress relaxation and

constant load curves is relatively small and at incremental load drops, transient creep saturation may

be approached. This changes for the same temperature at lower strain rates (1x10-7 s-1) where

transient saturation is now much further away at each incremental load drop, hence the larger strain


rate offset. Considering 100°C curves, the strain rate offsets are indistinguishable at the highest

strain rates (1x10-5 s-1) through to nearly (1x10-7 s-1), and only at strain rates below this point does the

strain rate offset become notable. This implies that during a large portion of the 100°C stress

relaxation response, saturation transient creep strain was approached at each incremental load drop.

These observations suggest that the proportion of transient and steady-state creep changes

as a function of temperature and strain rate. Thus a constitutive model that considers only steady-

state creep when fit to a stress relaxation response for SnAgCu will have some unknown and

variable error when applied to temperatures and strain rates beyond its original data set, because it

does not account for the effect of the changing proportions of transient and steady-state contributions.

Such a finding was reported in [27] where a steady state creep model was fit to creep and stress

relaxation data for 3.5Sn Ag eutectic solder. Different sets of parameters, including stress exponent

and activation energy, were required to fit the creep and stress relaxation measurements. The

discrepancy was attributed to a complex dependency on stress and strain rate beyond the capability

of the steady state creep model.

3.4.3 Microstructural Development

The microstructure shown in Fig.15 resulted from solder that was cast and cooled at 2°C/s

prior to aging at 23 mT (54°C) for 16 h - the same thermal history used for all creep and stress

relaxation specimens. Figure 3 shows that the pre-test aging period purged the β -Sn matrix of most

fine dispersions of 3Ag Sn and 6 5Cu Sn as the intermetallics coarsened forming rod like particle

groupings, binary and eutectic solid solution regions. After isothermal creep testing at 100°C and an

applied engineering stress of 8.6 MPa, the microstructure became more complex (Fig.15). From the


corresponding creep curve (Fig.9), sufficient time had elapsed for the microstructure to reflect a

saturation of transient creep and a complete transition to a steady-state creep rate.

Figure 15. Optical micrograph using 1/4 wavelength lambda plate of aged and crept specimen microstructure, σ= 8.6 MPa, T = 100°C: (a) low magnification (b) high magnification.

One notable difference between the microstructure of specimens before and after creep is

the appearance of substructure development in individual grains. During primary creep, intragranular

creep deformation of the β -Sn matrix by the dislocation climb mechanism causes a rearrangement

of dislocations into a lower-energy configuration (less local lattice strain) of what appears to be low-

angle boundaries consisting of dislocation arrays that ultimately form the polygon-like network of

subgrains seen in Fig.16; Electron Backscatter Diffraction (EBSD) characterization can be used to

confirm orientation mismatch. This polygonization process is prominent in high stacking fault energy

materials like Sn and Sn-rich lead free solders when the temperature and extent of deformation are

low enough to prevent the spontaneous formation of new grains by recrystallization [1, 28]. Subgrain

formation and recrystallization are two competitive processes where the occurrence of the former

reduces the tendency of the latter [29]. Recrystallization was not evident in Fig.16 and probably had


minimal impact during the creep and stress relaxation tests given the low creep strain deformation

measured.

In Fig.15, the absence of void coalescence at boundaries, and the lack of trans-granular or

intra-granular cracking, is an indication that grain boundary sliding has not contributed significantly to

creep deformation in these experiments. Furthermore, the subgrain contains β -Sn with a distribution

of Cu6Sn5 particles that also cross the low-angle boundaries. In a few instances the Cu6Sn5

particles appeared discontinuous at the boundary, but each portion of the particle remained aligned

across the boundary indicating that grain boundary sliding was absent. More commonly, the Cu6Sn5

particles that spanned the low-angle boundaries remained clearly continuous across the boundary.

The presence of primary phase Cu6Sn5 and Ag3Sn particles serves to resist defect movement,

especially when located along the grain boundary [2]. However, the effect of Ag3Sn and Cu6Sn5

primary phases in improving creep resistance is limited for diffusion based mechanisms because of

the contribution of thermally-activated vacancy diffusion to mass transport, whether alone or in

assisting dislocation climb. In general, diffusion based creep mechanisms (i.e. self-diffusion and core

diffusion), either alone or in combination, are expected to dominate in the present experiments

because of the high homologous temperature ( )0.6 mT T> .

Figure 16 shows the microstructure of a specimen tested under constant load creep at 25°C

with an applied engineering stress of 20 MPa. In contrast to the creep test at 100°C (Fig.15), where

the total creep strain was less than 2%, the microstructure of Fig.16 corresponds to a total creep

strain of about 6% at room temperature. The lower temperature resulted in a higher dislocation

density with more subgrains being created during primary creep. At room temperature additional

strain hardening occurred during primary creep because the thermally-activated recovery processes


played a smaller role. This substructure development is the reason a larger saturation transient creep

strain was measured at lower temperatures and at higher strain rates.

Figure 16. Optical micrograph using 1/4 wavelength lambda plate of aged and crept specimen microstructure, σ= 20 MPa, T = 25°C: (a) low magnification (b) high magnification.

Strain hardening is caused by the penetration of dislocations into subgrain boundaries that

increase the local dislocation density and form a barrier to dislocation motion. The competing

recovery process is caused by a lowering of the dislocation density through mutual climb and

annihilation of edge dislocations [30]. The dislocation density increases directly as a function of

increasing stress and the creep rate is proportional to the dislocation density [20]. This effect was

observed for constant load creep tests where faster minimum creep rates were measured with higher

dislocation densities at higher stresses. At the start of the stress relaxation test a high stress was

obtained creating greater dislocation density than would be obtained for a constant load creep test at

lower stress levels. This elevated initial dislocation density accounts for the faster measured creep

rates at a given stress in the stress relaxation tests compared to the corresponding minimum creep

rate from the constant load tests. The smaller relative offset in creep rates between these two tests


at higher temperatures can be accounted by the increased thermal recovery that decreases the

dislocation density and reduces the creep strain rate during stress relaxation.

The role of thermal activation in the polygonization that underlies strain hardening during

primary creep in SnAgCu has been included in a constitutive creep model presented in Chapter 4.

3.5 Conclusions

Both transient and steady-state creep play a significant role in creep deformation over a

broad range of strain rates (1x10-8 < ssε < 1x10-4) and temperatures (25°C, 75°C and 100°C). A

combination of matrix creep and dislocation core diffusion, either alone or in combination, dominate

over these strain rates and temperatures.

The extent of strain hardening during primary creep and hence the magnitude of the

saturation transient creep strain was a function of temperature and strain rate.

The measured strain rates from stress relaxation were consistently higher than the minimum

creep rates measured during constant-load creep testing, with the difference being a strong function

of temperature and, to a lesser extent, strain rate.

Transient creep strain can be a significant portion of total creep strain in Sn-rich lead-free

solders such as Sn3.8Ag0.7Cu. The relative amount of transient creep is strongly dependent on the

temperature and strain rate. Cyclic loading applications, like accelerated thermal cycling, are affected

by primary creep because relatively rapid temperature and/or stress changes occur throughout a

given cycle.


3.6 References

[1] Dieter, G.E., Mechanical Metallurgy. 3rd. ed. McGraw-Hill Series in Materials Science and Engineering. 1986, New York: McGraw-Hill.

[2] Vianco, P., Fatigue and Creep of Lead-Free Solder Alloys: Fundamental Properties, in Lead-free solder interconnect reliability, D. Shangguan, Editor. 2005, ASM: Materials Park, OH. p. 67-106.

[3] Kennedy, A.J., Processes of creep and fatigue in metals. Wiley Series on the Science and Technology of Materials. 1963, New York: Wiley.

[4] Ladani, L.J. Reliability Estimation for Large-Area Solder Joints Using Explicit Modeling of Damage. in IEEE T Device Mat Re. 2008.

[5] Nie, X., Bhate, D., Chan, D., Chen, W., Subbarayan, G., and Dutta, I. Rate-dependent behavior of Sn3.8Ag0.7Cu solder over strain rates of 10−6 to 102 s−1. in ITHERM. 2008. Orlando: IEEE.

[6] Zhang, Q., Dasgupta, A., Nelson, D., and Pallavicini, H., Systematic Study on Thermo-Mechanical Durability of Pb-Free Assemblies: Experiments and FE Analysis. Journal of Electronic Packaging, 2005. 127: p. 415-29.

[7] Cuddalorepatta, G. and Dasgupta, A. Effect of primary creep behavior on fatigue damage accumulation rates in accelerated thermal cycling of Sn3.0Ag0.5Cu Pb-free interconnects. in EuroSimE. 2008. Germany: IEEE.

[8] ISO 204:1997 Metallic materials - Uninterrupted uniaxial creep testing in tension - Method of test. 1997, ISO: Geneva, Switzerland.

[9] ASTM. 2000, ASTM E139-00. p. 285-96. [10] Ma, L., Bao, S., Lv, D., Du, Z., and Li, S. Application of C-mode Scanning Acoustic

Microscopy in Packaging. in Int'l Conf Elec Pack Tech. 2007. Shanghai: IEEE. [11] Plumbridge, W.J., Matela, R.J., and Westwater, A., Structural integrity and reliability in

electronics : enhancing performance in a lead-free environment. 2003, Boston: Kluwer. [12] Holmes, A.D. and Jing, F., The experimental nature of stress relaxation and recovery of high-

frequency information. Journal of Physics D: Applied Physics, 1994. 27: p. 2475-9. [13] Koychev, I. and Lothian, R. Tracking Drifting Concepts by Time Window Optimisation. in Res

Dev in Intel Sys XXII. 2005. Cambridge, UK: Springer. [14] Kukar, M. Drifting Concepts as Hidden Factors in Clinical Studies. in Art Intel Med Eur. 2003.

Protaras, Cyprus: Springer. [15] Brigham, E.O., The fast Fourier transform and its applications. Prentice-Hall signal

processing series. 1988, Englewood Cliffs, N.J.: Prentice Hall. xvi, 448 p. [16] Elliott, D.F. and Rao, K.R., Fast transforms : algorithms, analyses, applications. 1982, New

York: Academic Press. xxii, 488 p. [17] Pan, C. Gibbs Phenomenon Removal and Digital Filtering Directly through the Fast Fourier

Transform. in IEEE T Signal Proces. 2001. [18] Hockett, J.E. and Gillis, P.P., Mechanical Testing Machine Stiffness: Part I - Theory and

Calculations. International Journal of Mechanical Sciences, 1971. 13: p. 251-64. [19] Holbrook, J.H., Swearengen, J.C., and Rohde, R.W. Specimen-Test Machine Coupling and

Its Implications for Plastic Deformation Models. in ASTM STP. 1982. [20] Sherby, O.D. and Burke, P.M., Mechanical Behavior of Crystalline Solids at Elevated

Temperature. Progress in Materials Science, 1968. 13: p. 323-90. [21] Ojediran, S.O. and Ajaja, O., The Bailey-Orowan Equation. Journal of Materials Science,

1988. 23: p. 4037-40.


[22] Schubert, A., Dudek, R., Auerswald, E., Gollhardt, A., Michel, B., and Reichl, H. Fatigue life models for SnAgCu and SnPb solder joints evaluated by experiments and simulation. in ECTC. 2003. New Orleans: IEEE.

[23] Hertzberg, R.W., Deformation and fracture mechanics of engineering materials. 1996, New York: Wiley.

[24] Nakamura, Y. and Ono, T., Temperature dependence of anelastic properties in Sn–Ag–Cu system and Sn-3.5Ag alloys of lead-free solders Journal of Materials Science, 2005. 40(12): p. 3267-9.

[25] Pang, J.H.L., Xiong, B.S., Neo, C.C., Zhang, X.R., and Low, T.H. Bulk Solder and Solder Joint Properties for Lead Free 95.5Sn-3.8Ag-0.7Cu Solder Alloy. in ECTC. 2003. New Orleans.

[26] Ratkowsky, D.A., Handbook of nonlinear regression models. 1990, New York: Marcel Dekker. [27] Mavoori, H., Chin, J., Vaynman, S., Moran, B., Keer, L., and Fine, M., Creep, stress

relaxation, and plastic deformation in Sn-Ag and Sn-Zn eutectic solders. Journal of Electronic Materials, 1997. 26(7): p. 783-90.

[28] Gay, P. and Kelly, A., X-ray Studies of Polycrstalline Metals Deformed by Rolling. II. Examination of the Softer Metals, Tin, Zinc, Lead and Cadmium. Acta Crystallographica, 1953. 6: p. 172-7.

[29] Hardwick, D., Sellars, C.M., and Tegart, W.J., The Occurrence of Recrystallization During High-Temperature Creep. Journal of the Institute of Metals, 1961. 90: p. 21-2.

[30] Thorpe, W.R. and Smith, I.O., A model for High-Temperature Creep Incorporating Both Recovery and Thermally Activated Glide. Physica Status Solidi, 1979. 52: p. 487-97.

Chapter 4. Constitutive Creep Model Development and Finite Element Analysis 95

Chapter 4

Constitutive Creep Model Development and Finite Element Analysis

4.0 Introduction

The total deformation of a solder joint can be represented as the sum of quasi-static elastic and

plastic deformation plus transient and steady-state creep. Changing temperatures during service often

cause fluctuating stresses that usually do not exceed yield; however, under long-term repetitive usage

solder joints can fail in creep-fatigue [1]. Creep strain contributions in metals under constant load and

temperature are partitioned into three stages: (1) primary; (2) secondary; and (3) tertiary. Tertiary creep

is omitted from most constitutive models because it only represents deformation contributions at the

onset of imminent creep rupture. Hence, in the present discussion, total deformation is viewed as

comprising elastic, transient and steady-state creep contributions (Eq.(1)).

, ,total elastic creep transient creep steadystateε ε ε ε= + + (1)

Fundamentally, time-dependent plasticity (creep) and time-independent plasticity are both the

result of dislocation motion. Upon the application of load, the dislocation density increases causing

strain hardening and deceleration of the strain rate during primary creep. During secondary creep, the

transient creep strain has saturated at a value defined as ,tr satε , and the rate of strain hardening and the

competing dynamic recovery process have reached a steady-state, resulting in a constant minimum

creep rate, minε . Whenever the stress or temperature change, new primary creep contributions occur

as the solder alloy takes a finite time to reach a new steady state between strain hardening and dynamic


recovery [2]. The magnitude of strain hardening during primary creep is a function of temperature and

strain rate for Sn3.8Ag0.7Cu, as controlled by the subgrain formation and associated thermally-activated

dynamic recovery.

The creep deformation of SnPb eutectic solder has usually been modeled solely in terms of

secondary creep, whereby a steady-state is assumed to occur instantly with a change in load. In the

case of SnAgCu solder alloys, this may not be appropriate since they have a relatively high stacking

fault energy that increases the contribution of primary creep strain to the total creep deformation over a

broad range of temperatures and strain rates [3]. Consequently, a constitutive creep model for

deformation in SnAgCu should account for both primary (transient) and secondary (steady-state) creep.

Omission of transient creep strain from the solder constitutive model was shown to have an

appreciable impact on damage metric estimation used for strain and energy-based modified Coffin-

Mason life prediction estimates in Chapter 2. Since the relative contribution of transient creep to total

creep changes as a function of the time-temperature profile in service, the calculation of creep-fatigue

damage metrics, can have an undetermined and variable error if transient creep is neglected.

At high homologous temperature, secondary creep is often characterized by a power law

relationship at low stresses. As the stress increases ( 31 10Eσ −> × ), a power law breakdown occurs

where the minimum creep rate increases more rapidly than expected based on extrapolation from lower

stresses, resulting in larger values of the stress exponent in the constitutive equation [4]. The traditional

means of circumventing the effect of power law breakdown on secondary creep modeling has been to

use either the double power law creep equation or the hyperbolic sine constitutive creep equation [5, 6].

The double power law equation given in Eq. (2) has one term to represent low stress creep,

where dislocation climb is dominant, and a second term to account for high stress creep where a

combination of grain boundary sliding and dislocation climb are present. The coefficient A , activation

energy Q , and stress exponent n for each power law term are found by applying regression methods


to experimental creep data. nσ is assumed equal to one, T is the absolute temperature, and R is the

universal gas constant.

( )( ) ( )( )min exp explow highn nlow low n high high nA Q RT A Q RTε σ σ σ σ= − + − (2)

The hyperbolic sine law for steady-state creep rate given in Eq.(3) also spans the power law

breakdown and represents both low and high stress creep. The creep coefficients are again obtained

through regression of experimental creep data. Equation (3) does not explicitly partition the low and

high stress creep mechanisms, but rather treats steady-state creep with an effective or apparent

activation energy that may represent multiple creep mechanisms. This relationship simplifies into a

power law (Norton creep) relationship at low stresses ( )1.0ασ < as in Eq.(4), while at high stresses

( )1.2ασ > the hyperbolic sine law simplifies to an exponential creep expression as in Eq.(5) [7]. When

the hyperbolic sine law is followed, the experimental creep data measured at different temperatures

appears as a series of parallel straight lines when plotted as log minε versus log( ( )sinh ασ ).

min sinh( ) expn QART

ε ασ − =

(3)

( )min 1 expnA Q RTε σ= − ( )1.0ασ < (4)

( ) ( )min 2 exp expA n Q RTε α σ= − ( )1.2ασ > (5)

At high homologous temperatures, where the temperature is greater than half the absolute

melting temperature ( )/ 0.5mT T > , the dislocation mechanisms in hot metal working

( )2 1 2 110 ( ) 10 ( )s sε− − + −< < and creep ( )4 110 ( )sε − −< are related as they both feature strain

hardening and dynamic recovery processes, even though the strain rates are usually quite different.

Therefore, the constitutive modeling of large plastic strains in hot metal working is applicable to


modeling the much smaller creep strains experienced by solder joints. Complementary to steady-state

creep laws, is the time hardening transient creep expression, Eq.(6) [8], originally published to represent

creep in type-316 austenitic stainless steel at temperatures near 700°C at flow stresses below yield.

The time hardening relation of Eq.(6), along with variants where the ( )rt− term inside the exponential

argument is given in terms of minimum strain rate [9] and raised to a power [10], have similar

applicability to creep in Sn3.8Ag0.7Cu and other Pb-free solder alloys under isothermal constant load

creep deformation. In Eq.(6) t is elapsed time, r is the rate of exhaustion of transient creep, and minε

is the minimum creep rate. The parameters ,tr satε , r and minε yield accurate creep predictions only for

a given applied load and test temperature. Changes in these conditions produce a different material

strain hardening response which must be modeled using another set of parameters.

, min(1 exp( ))creep tr sat rt tε ε ε= − − + (6)

The constitutive model developed below includes both transient and steady-state creep

contributions to total inelastic deformation using a single set of creep coefficients generated from

constant load creep testing using tensile specimens over a range of temperatures, as described in

Chapter 3. The increased capability to predict both primary and secondary creep under a diverse set of

temperatures and stress levels lends itself to predicting deformation under dynamic conditions where

the load and/or temperature are varying. For verification purposes, experimental measurements of

stress relaxation from Chapter 3 are modeled using the coefficients from the constant load creep tests.


4.1 Constitutive Model Development

4.1.1 Steady-state Creep

The four steady-state creep coefficients in Eq. (3) were obtained for Sn3.8Ag0.7Cu using the

constant load creep measurements from Chapter 3 for isothermal conditions at 25°C, 75°C and 100°C

over a range of stresses and minimum strain rates. An iterative approach was used in which α was

incremented from an initial value of 100 1 10α −= × using a constant step size 12

1 1 10i iα α −+ = + × . At

each step, a set of best fit parameters (α , A , n , Q ) was found using a multiple-variable nonlinear

regression (Matlab™ Statistics Toolbox™, nlinfit command [11]). This continued until a local minimum

was detected in the mean square error for a particular set of the four parameters. The corresponding

optimal α was then confirmed with a second iterative solution where 0α was set just below the

previously determined optimal value and a finer step size 161 1 10i iα α −+ = + × was used to again find the

minimum mean square error.

Table 1. Coefficients used to model steady-state creep and normalize temperature compensated time and strain rate for Sn3.8Ag0.7Cu using Eqs. 7, 8 and 9.

1C (1/s) 2C (1/Pa) 3C 4C (K)

3.8 0.7Sn Ag Cu 8.09 x 105 1.15 x 10-7 5.02 1.01 x 104

95% C.I. (lower) 1.07 x 105 --- 4.87 9.70 x 103

95% C.I. (upper) 6.10 x 106 --- 5.17 1.04 x 104

Distribution type lognormal constant normal normal

Table 1 gives the optimal set of parameters, along with the 95% confidence interval and

distribution type, for Eq. (3) in terms of the nomenclature used by the Ansys™ finite element analysis


software (generalized Garofalo implicit creep option [12]) where the creep coefficients A , α , n and

Q R are equivalent to 1C , 2C , 3C and 4C , respectively, as in Eq. (7).

( ) 3 4min 1 2sinh exp

C CC CT

ε σ − = (7)

Fig.1 shows good agreement between the steady-state creep model (Eq. (7)) and the measured

minimum creep rates as a function of the tensile stress, expressed as ( )( )sinh ασ , at each of the three

test temperatures over a strain rate range extending from 8 11 10 s− −× to 4 11 10 s− −× .

Figure 1. Minimum creep strain rate versus stress (T = 25°C, 75°C and 100°C). Data points are from creep experiments and the lines are Eq. (7) using the parameters of Table 1.

4.1.2 Normalized Steady-state Creep Rate

The Sherby-Orr-Dorn parameter, Ω , defined by Eq.(8) is useful for correlating creep data

under different test temperatures onto a single master creep curve for a given applied stress [13]. The

principal assumption of such normalization is that the apparent activation energy is independent of

temperature.


( )4exp /t C TΩ = − (8)

where t is the elapsed time and T is the absolute temperature. Ω is also termed the temperature

compensated time, because it relates the time scale of creep deformation to the temperature.

The temperature compensated strain rate ( )d dε Ω is then given by Eq.(9), which is also

termed the Zener-Hollomon parameter [7, 14]. This parameter enables a normalization of the minimum

creep strain rate when a constant apparent activation energy is assumed over a broad range of strain

rates. As previously shown in Fig.1, this assumption is valid for the experimental creep data

of 3.8 0.7Sn Ag Cu .

( ) ( )min 4minexpd d C Tε εΩ = (9)

As seen in Fig.2, the use of temperature compensated time produces a single relationship

between the temperature compensated minimum creep rate ( )mind dε Ω and the tensile stress

( )( )sinh ασ for all test temperatures, stresses and strain rates. In practice, this reduces the number of

required experimental measurements since accurate interpolation can be made between measured

minimum creep strain rates at different temperatures.


Figure 2. Normalized minimum secondary creep rate versus tensile stress term of Eq. (7) (T=25°C, 75°C and 100°C) with optimal α corresponding to Table 1.

4.1.3 Normalized Transient Creep

The transient creep model of Eq.(6) [8], can be written in a normalized form using temperature

compensated time, Ω , as

( )( ) ( )8

, 7 min1 exp

Ccreep tr sat normC r d dε ε ε= − − Ω + Ω Ω (10)

In this newly modified form of the constitutive equation ( )mind dε Ω is the temperature

compensated minimum creep rate (Eq.(9)) based on the hyperbolic sine law (Eq.(7)). The constitutive

model was implemented in Ansys 11 as a user-defined implicit creep model (usercreep.f) using Intel

Fortran Compiler 9.1 to compile and link the custom Ansys executable [15].

The saturated primary creep strain, ,tr satε , in Eq.(10) is a function of temperature and the

minimum creep rate as seen in Fig. 3 for the Sn3.8Ag0.7Cu solder using the constant load creep data.

This relationship is greatly simplified when considered in terms of temperature compensated strain rate

expressed as Eq. (11)) and shown in Fig.4, where a single relationship spans all test temperatures,


applied loads and strain rates. As with the minimum creep rate, the strain hardening in transient creep

exhibits equivalence between decreased temperature and increased strain rate. For instance, in Fig.4

the saturation transient creep strain corresponding to the slowest temperature compensated minimum

creep rate at 25°C ( 64 10d dε Ω = × 1s− , 81 10d dtε −= × 1s− ) nearly overlaps the corresponding

data for the second fastest creep rate for 100°C ( 62 10d dε Ω = × 1s− , 65 10d dtε −= × 1s− ), despite

the fact that the corresponding minimum creep rates are more than two decades apart. This

equivalence applies to both the steady-state creep rate (Fig.2) and strain hardening (Fig.4), because at

faster minimum creep rates a greater dislocation density occurs within the microstructure, and similarly,

at lower temperatures thermally-activated dynamic recovery is less effective, thereby also resulting in a

greater dislocation density at the onset of transient creep saturation. Hence, the experimental

measurements show that thermal activation plays a notable role in the motion of dislocations and

vacancies that underlies the competitive strain hardening and dynamic recovery processes.

( )( ) 6

, 5 min

C

tr sat C d dε ε= Ω (11)

Figure 3. Saturation transient creep strain versus minimum creep rate (T = 25°C, 75°C, 100°C).


Figure 4. Normalization of saturation transient creep strain versus minimum creep rate (T = 25°C, 75°C, 100°C).

Figure 5. Creep Strain vs. Time, superposition of creep components that determine the creep curve (T =°25 C, σ = 15 MPa); data are from ref. [13].


In Eq. (10), the normalized rate of transient creep exhaustion, normr , is defined as

( ) ,minnorm tr satr d dε ε= Ω (12)

In the original formulation of Eq. (6), the rate constant r was thought to be a function of stress

with a form similar to that of the minimum strain rate, minε [8]. The normr relationship presented in this

work originates from recent experimental observations of the constant load creep curves detailed in

Chapter 3 near the transition between primary and secondary creep at satt , where the transient

contributions to total creep strain are near saturation. Figure 5 shows schematically the contributions of

both transient and steady-state creep contributions towards total creep strain for a constant load creep

test at 25°C and an applied stress of 15 MPa, where the transient creep contribution clearly attains a

plateau value after the finite time required for transient saturation. The saturation time satt , the

saturation creep strain ,tr satε and the total creep strain creepε were defined for a constant load creep test,

where it was possible to quantify the ratio of steady-state creep to transient creep at the onset of

transient creep saturation as ( ) ,min sat tr satd dt tε ε× .

Figure 6 shows the dependence of r , in its non-normalized form min ,tr satr ε ε= , on stress and

temperature, indicating that the rate of exhaustion of transient creep decreases at lower temperatures

for a given stress; therefore, satt will be much larger at 25°C than at 100°C as was experimentally

observed.


Figure 6. Stress dependence of the rate of exhaustion of transient creep strain (25°C, 75°C, 100°C).

Figure 7 shows the effect of normalization using temperature compensated time on both the

normalized minimum strain rate ( )mind dε Ω , and normalized rate of exhaustion of transient creep

normr , as a function of tensile stress ( )( )sinh ασ . As in [8], both normr and ( )mind dε Ω share the

same form of stress dependence. However, replacing time with temperature compensated time makes

the functional form of the stress dependence of normr and ( )mind dε Ω applicable over a broad range of

temperatures.


Figure 7. Normalization of the stress dependence of the rate of exhaustion of transient creep strain and minimum creep rate (25°C, 75°C, 100°C).

In summary, Eq. (10) allows a single set of creep coefficients to represent transient and steady-

state creep behavior under different temperatures, stresses and strain rates. Table 2 shows the creep

coefficients 5C , 6C , 7C and 8C , along with the 95% confidence interval and distribution type, solved

with a multiple-variable nonlinear regression for the Sn3.8Ag0.7Cu constant-load creep data. In

particular, the coefficients 5C and 6C were solved using the measured saturated transient creep strain,

,tr satε , and normalized minimum strain rate data, ( )mind dε Ω , where the previously determined 4C

( )Q R was used to compute the temperature compensated minimum strain rate. The solution of 7C

and 8C used the measured total creep strain and normalized time data, where similarly 4C ( )Q R was

used to compute the temperature compensated time.


Table 2. Summary of coefficients used to model transient creep in Sn3.8Ag0.7Cu using Eqs. 10 and 12

5C 6C 7C 8C

3.8 0.7Sn Ag Cu 8.36 x 10-5 2.46 x 10-1 4.02 7.12 x 10-1

95% C.I. (lower) 4.89 x 10-5 2.28 x 10-1 4.00 7.10 x 10-1

95% C.I. (upper) 1.45 x 10-4 2.63 x 10-1 4.04 7.15 x 10-1

Distribution type lognormal normal normal normal

Figures 8-10, compare the predictions of the current model (Eqs. (7)-(10)), using the coefficients

1 8C C− from Tables 1 and 2, with the measured isothermal creep strain measurements conducted at

25°C, 75°C and 100°C, respectively. The agreement is generally good over both the transient and

steady-state portions of the creep curves. The initial portion of the model of Fig.10 strain hardens

somewhat faster than the measured curve at 100°C and consequently reaches transient creep

saturation sooner. Overall, it is evident that a single set of coefficients can fit creep data over a broad

range of temperatures, stresses and strain rates, while incorporating the contributions of both transient

and steady-state creep.


Figure 8. Creep Strain response versus time, measured and modeled (transient + steady-state creep), for σ=15 MPa at T = 25°C; data are from ref. [13].

Figure 9. Creep Strain response versus time, measured and modeled (transient + steady-state creep), for σ=8 MPa at T = 75°C, data are from ref. [13].


Figure 10. Creep Strain response versus time, measured and modeled (transient + steady-state creep), for σ=8.6 MPa at T = 100°C, data are from ref. [13].

4.1.4 Modeling of Stress Relaxation Data

As a further test of the creep model, Eqs. (7)-(10) were used to simulate the stress relaxation

data in Chapter 3 at 25°C, 75°C and 100°C, using the single set of coefficients 1 8C C− obtained from

the constant load creep tests at these temperatures (Tables 1 and 2). Figure 11 shows that the model

agreed well with the measured data. This test of the model is particularly suited to the requirements of

thermal cycling since the stress and strain are continually changing, thereby preventing primary creep

from saturating and making it significant throughout the test. This is illustrated in Fig. 11 by comparing

the full model with the predictions of a purely steady-state creep model (Eq.(7) using only the

coefficients 1 4C C− ).


Figure 11. Force response versus time, measured and modeled (i. transient + steady-state; and ii. steady-state only), for stress relaxation tests (SRT) (25°C, 75°C, 100°C); data are from ref. [13].

Since the steady-state creep model does not accommodate the relatively faster creep rates of

transient creep, the force is not relaxed as quickly and the predictions overestimate the measured

values. The error is larger at lower temperatures because proportionately more strain hardening occurs

at lower temperatures owing to the relatively smaller role of thermally-activated dynamic recovery. The

inability of a secondary-only creep model to simultaneously fit both constant-load creep data and stress

relaxation data at different temperatures was also observed in ref. [16] for a Sn3.5Ag eutectic solder.

The contributions of steady-state and transient creep to the total stress relaxation response of

the solder may be seen in Fig. 12 which graphs normr (Eq.(12)) versus temperature compensated time

Ω . A single data point corresponds to each of the isothermal constant-load creep curves at the onset

of transient creep saturation satt . These points lie on a common power law curve denoted as the

,tr satε threshold (creep fit) line. Points above the ,tr satε threshold correspond to strain rates and


temperatures where the rate constant of transient creep exhaustion is high enough that as incremental

changes in load occur, the minimum creep rate is approached, and transient creep is exhausted nearly

immediately (secondary creep). Below the ,tr satε threshold, the rate constant of transient creep

exhaustion is low enough that transient creep plays a pronounced role during load change, as steady-

state creep does not ensue immediately. The relative importance of primary and secondary creep

throughout the stress relaxation test becomes apparent when each of the isothermal stress relaxation

curves measured at 25°C, 75°C and 100°C are superimposed on the ,tr satε (creep fit) line (Fig.12).

Figure 12. Normalized rate of transient creep exhaustion, rnorm versus temperature compensated time, Ω (25°C, 75°C, 100°C); data are from creep and stress relaxation experiments

The normr versus Ω solutions for the stress relaxation curves were obtained by substituting the

measured normalized creep rate ( )d dε Ω for the minimum normalized creep rate ( )mind dε Ω in

Eqs.(11) and (12). The stress relaxation curve for 25°C lies entirely below the ,tr satε threshold in the trε

non-saturation region (primary creep). The 75°C and 100°C stress relaxation curves each have the


initial portions of the response above the ,tr satε threshold, but as time elapses the curves gradually move

below the saturation threshold. This observation corresponds well with a comparison of the strain rates

measured in stress relaxation for a given stress and the minimum creep rate in a constant load creep

test (Fig.13). At 25°C the large offset between the two tests was caused by the dominant role of primary

(transient) creep as indicated in Fig.12. As the temperature increases to 75°C and 100°C, thermally-

activated recovery processes increasingly curtail the extent of strain hardening, thereby a smaller offset

results between creep and stress relaxation, particularly at the highest strain rates where the initial

portion of the stress relaxation line lies above the saturation threshold of Fig.12. Application of this

approach can similarly identify the relevance of primary creep in any given isothermal or thermal cycling

simulation.

Figure 13. Creep strain versus tensile stress from creep (minimum creep rate) and stress relaxation tests (SRT) (25°C, 75°C, 100°C); data are from creep and stress relaxation experiments


4.2 Discussion

4.2.1 Constitutive Model and Thermally-Activated Creep

The apparent activation energy Q ( )4C R of 83.6 kJ/mol for steady-state creep over the three

measurement temperatures (25°C, 75°C, 100°C) found in the preceding section is comparable to values

reported for Sn-rich lead-free solder alloys, including Sn3.9Ag0.6Cu [17] and Sn3.5Ag [18], under

similar temperature, strain rate and stress testing conditions. This apparent activation energy is smaller

than that of self-diffusion in β -Sn (90-110 kJ/mol) and larger than the activation energy of dislocation

core diffusion (50-60 kJ/mol) [19]. These two interdependent diffusion processes occur concurrently at

high homologous temperatures, and the apparent activation energy is an effective value representing

the contributions of both types of diffusion over the three temperatures. The apparent activation energy

value is also similar to that of grain boundary sliding (GBS) in Sn. However, the microstructure does not

suggest that GBS was active under the experimental test conditions [19].

In metals, a homologous transition temperatureϕ , defines the homologous temperature where

the contributions of lattice self-diffusion through the crystal bulk and diffusion through the core of

dislocations contribute approximately equally to creep deformation. For mT T ϕ> the activation energy

of creep equals the activation energy of self-diffusion; whereas, for mT T ϕ< , core diffusion creep has

increased importance. Since 0.8ϕ = for β -Sn [20], and mT T = 0.61, 0.71, 0.76 at 25°C, 75°C and

100°C, respectively for Sn3.8Ag0.7Cu, it is expected that dislocation core diffusion is the dominant

creep deformation mechanism. Furthermore, the activation energy for core diffusion, although a

function of temperature, has been experimentally shown to be independent of crystal structure or

cooling rate [20, 21]. Therefore, it may be expected that the present model parameters, particularly 4C ,

are representative of Sn-rich solders in the range of 25°C to 100°C where creep deformation is similarly

rate controlled by core diffusion.


Dislocations and vacancies in a pure metal or a solid solution such as a Sn-rich lead-free solder

alloy, may diffuse via dislocation glide or dislocation creep. The stress exponent n (Eq.(3)) or 3C

(Eq.(7)) is indicative of which method of dislocation movement is rate controlling. The experimental

results of [11] suggest that 3 5.0C = (Table 1) is indicative of dislocation climb. Values of n between 4

and 6 have been experimentally observed for pure metals and some solid solutions including Sn-rich

Sn3.5Ag [22] and a number of near-eutectic SnAgCu [17, 23, 24] ternary alloy compositions. The climb

of dislocations and vacancies can occur via the aforementioned core diffusion [25]. When the stress

exponent is near the third power ( )3.0 3.5n< < , dislocation movement is controlled by dislocation

glide and transient creep is generally minimal, as in the case of eutectic 37Sn Pb where n equals 3.3.

4.2.2 Application to Thermal Cycling and Life Prediction Models

Since steady-state creep is not achieved immediately upon change of stress or temperature,

thermal cycling of solder joints comprises both transient and steady-state creep as a function of

temperature, strain rate and stress. This was demonstrated in Fig.11 where steady-state creep

coefficients for Eq.(7) did not fit constant-displacement stress relaxation data. Moreover, the relative

amounts of primary and secondary creep are a function of the temperature and strain rate, as seen in

Fig. 12. As mentioned previously, the neglect of primary creep will produce damage metrics (e.g. creep

strain energy or creep strain) that have an error that is dependent on the thermal cycling profile.

Similar results were seen in ref[26] with the measured thermomechanical fatigue response of

Sn4.0Ag0.5Cu using a joint specimen cycled between 40°C and 140°C, with 10 min dwell and ramp

periods. A stabilized hysteresis curve characteristic of the crack initiation period was achieved within

three cycles, while crack propagation associated with the weakening of the hysteresis curve occurred as

additional damage accumulated well past 15 cycles. A finite element model prediction using a steady-

state creep equation fit with deformation data from the dwell periods, overestimated the stress amplitude

and underestimated the total strain range. Moreover, a significant difference between the measured


and the steady-state creep model was observed during the ramp period. The inclusion of transient

creep in the model would have decreased the stress amplitude and increased the strain amplitude in

keeping with the measured response.

The implementation of temperature compensated time and temperature compensated strain

rate in this chapter, unlike a typical time hardening approach (Eq.(6)), is suitable for thermal cycling

applications because it satisfies the principle of objectivity, whereby strain rates are invariant under a

time translation [27].

Thermal fatigue damage metrics based on a creep model such as the present one, is only

meaningful up to the point of significant crack propagation. Generally, this crack nucleation phase

corresponds to the period during which the stress-strain hysteresis loop is stable, as seen in ref[26].


4.3 Conclusions

A novel constitutive creep model was demonstrated that incorporates both transient and steady-

state creep for Sn3.8Ag0.7Cu. The functional form is believed to be equally applicable to other Sn-rich

lead-free solder alloys under diffusion-dominated dislocation creep at high homologous temperatures.

A temperature-normalizing procedure used temperature compensated time and temperature

compensated strain rate to characterize the stress dependence of the minimum (steady-state) creep

strain rate and the saturated transient creep strain of Sn3.8Ag0.7Cu. This approach is based on the

equivalence of decreased temperature and increased strain rate. It facilitates the use of the model in

thermal cycling simulations. A single set of eight regression coefficients fit from isothermal constant-

load creep data ( 1 4C C− steady-state creep, 5 8C C− transient creep), produced good predictions of

both isothermal creep and isothermal stress relaxation measurements over a range of strain rates

( )8 4 110 10 sε− − −< < at 25°C, 75°C, 100°C.

A transient creep saturation threshold was established that enables the distinction between

primary and secondary creep dominant regimes under variable loading conditions.

The presented creep model improves stress/strain prediction within a solder joint by accounting

for the fluctuation of transient and steady-state creep during cyclic loading. This can improve the

prediction of strain and energy-based damage metrics used in creep-fatigue failure theory to compute

the number of thermal cycles to failure.


4.4 References

[1] Rodriguez, P. and Rao, K.B.S., Nucleation and growth of cracks and cavities under creep-fatigue interaction. Progress in Materials Science, 1993. 37(5): p. 403-80.

[2] Kennedy, A.J., Processes of creep and fatigue in metals. Wiley Series on the Science and Technology of Materials. 1963, New York: Wiley.

[3] Sherby, O.D. and Burke, P.M., Mechanical Behavior of Crystalline Solids at Elevated Temperature. Progress in Materials Science, 1968. 13: p. 323-90.

[4] Sinha, N.K., Power-law breakdown—An examination using strain relaxation and recovery tests on a nickel-base superalloy, IN-738LC. Journal of Materials Science Letters, 2002. 21(11): p. 871-5.

[5] Clech, J.P. An Obstacle-Controlled Creep Model For Sn-Pb and Sn-Based Lead-Free Solders. in SMTAI. 2004. Chicago: SMTA.

[6] Syed, A. Accumulated Creep Strain and Energy Density Based Thermal Fatigue Life Prediction Modls for SnAgCu Solder Joints. in ECTC. 2004: IEEE.

[7] Dieter, G.E., Mechanical Metallurgy. 3rd. ed. McGraw-Hill Series in Materials Science and Engineering. 1986, New York: McGraw-Hill.

[8] Garofalo, F., Resistance to creep deformation and fracture in metals and alloys, in ASTM STP 283. 1961. p. 82-98.

[9] Darveaux, R. and Banerji, K. Constitutive relations for tin-based solder joints. in T Compon Hybr. 1992: IEEE.

[10] Mayuzumi, M. and Onchi, T., Creep Deformation of an Unirradiated Zircaloy Nuclear Fuel Cladding Tube Under Dry Storage Conditions. Journal of Nuclear Materials, 1990. 171(2-3): p. 381-8.

[11] MATLAB, Statistics Toolbox 7: User's Guide. 2008: Natick, MA, USA. p. 360. [12] ANSYS, ANSYS Elements Reference. 2007. p. 55. [13] Dowling, N.E., Mechanical behavior of materials : Engineering methods for deformation,

fracture, and fatigue. 2nd ed. 1999, Upper Saddle River, N.J.: Prentice Hall. [14] Zener, C. and Hollomon, J.H., Effect of Strain Rate Upon Plastic Flow of Steel. Journal of

Applied Physics, 1944. 15(1): p. 22-32. [15] ANSYS, Programmer's Manual for ANSYS. 2007: Canonsburg, PA. [16] Mavoori, H., Chin, J., Vaynman, S., Moran, B., Keer, L., and Fine, M., Creep, stress relaxation,

and plastic deformation in Sn-Ag and Sn-Zn eutectic solders. Journal of Electronic Materials, 1997. 26(7): p. 783-90.

[17] Vianco, P., Creep behavior of the ternary 95.5Sn-3.9Ag-0.6Cu older: Part I - As-Cast Condition. Journal of Electronic Materials, 2004. 33(11): p. 1389-400.

[18] Mooris, J., Song, H.G., and Hua, F. Creep Properties of Sn-Rich Solder. in ECTC. 2003. New Orleans: IEEE.

[19] Vianco, P., Fatigue and Creep of Lead-Free Solder Alloys: Fundamental Properties, in Lead-free solder interconnect reliability, D. Shangguan, Editor. 2005, ASM: Materials Park, OH. p. 67-106.

[20] Sherby, O.D. and Weertman, J., Diffusion-Controlled Dislocation Creep: A Defense. Acta Metallurgica, 1979. 27(3): p. 387-400.

[21] Ochoa, F., Deng, X., and Chawla, N., Effects of cooling rate on creep behavior of a Sn-3.5Ag alloy. Journal of Electronic Materials, 2004. 33(12): p. 1596-607.

[22] Kerr, M. and Chawla, N., Creep deformation behavior of Sn–3.5Ag solder/Cu couple at small length scales. Acta Materialia, 2004. 52(15): p. 4527-35.

[23] Clech, J.P., Review and analysis of lead-free solder material properties, in NEMI Lead-Free Project, NIST, Editor. 2003, IEEE/Wiley.

[24] Pang, J., Xiong, B., and Low, T. Comprehensive Mechanics Characterization of Lead-free 95.5Sn-3.8Ag-0.7Cu Solder. in Micromaterials and Nanomaterials. 2004.


[25] Nabarro, F.R.N., Steady-state Diffusional Creep. Philosophical Magazine, 1967. 16(140): p. 231-7.

[26] Pao, Y.H., Badgley, S., Govila, R., and Jih, E. Thermomechanical and Fatigue Behavior of High-Temperature Lead and Lead-Free Solder Joints. in ASTM STP. 1994.

[27] Stouffer, D.C. and Dame, L.T., Inelastic deformation of metals. 1996, New York: Wiley.

Chapter 5. Sensitivity of Creep-Fatigue Damage Metrics to Uncertainty in Creep Parameters 120

Chapter 5

Sensitivity of Creep-Fatigue Damage Metrics to Uncertainty

in Creep Parameters

5.0 Introduction

Solder joint reliability is typically estimated by linking actual service conditions to an

accelerated thermal cycling test. An approximation of the mean time to failure of the accelerated test

allows for an estimation of the distribution of lifetimes an actual solder joint may experience in the

field. The reliability of electronic components is generally characterized by the instantaneous failure

rate or hazard rate resembling a ‘bathtub curve.’ At the beginning of service life the failure rate may

be high due to infant mortality, then the defect rate levels off to a relatively constant level when parts

fail by wear out, until the defect rate gradually increases as damage accumulates.

During the final portion of this curve, creep-fatigue processes dominate the accumulation of

damage that ultimately leads to solder joint failure [1-3]. Therefore, the characterization of the time-

dependent thermo-mechanical reliability of a solder joint depends on the constitutive creep model

used to calculate the creep rate as a function of stress and temperature during thermal cycling.

A practical difficulty in such modeling is the scatter reported in the literature for constitutive

creep model parameters for lead-free solder alloys. Table 1 shows the creep model parameters as

reported by several authors for Sn3.8Ag0.7Cu solder alloy for the hyperbolic sine steady state creep

expression (Eq.(1)),


( ) ( )minsinhε ασ −∆ =

n aHd dt ART

(1)

where ∆ aH is the apparent activation energy. It is seen that a broad range of values were able to

provide ‘best fits’ to Sn3.8Ag0.7Cu and Sn3.9Ag0.6Cu steady state creep rate data. Most notable is

the lack of agreement on values for the constant A that spans five orders of magnitude and the pre-

stress multiplier α that spans two orders of magnitude. This problem is symptomatic of the

variability due to differences in specimen preparation, thermal aging, solidification, heterogeneity of

microstructure, divergence from creep testing standards [4], and possible errors such as

misalignment in joint loading causing undesired bending moments [5]. In addition, damage

accumulation including nucleation and growth of internal defects, pores and microcracks during creep

is intrinsically probabilistic, leading to uncertainty that may be substantially greater than that due to

experimental variability [6].

Therefore, the calculation of creep-fatigue damage metrics must inevitably utilize creep

constitutive models with uncertain parameters; however, the sensitivity of the calculation to variations

in each specific parameter is unknown. This makes it impossible to judge the accuracy required of

each parameter in a creep constitutive model.

The present work is an assessment of sensitivity in the calculated damage metrics due to the

propagation of uncertainty from the creep model input parameters. The creep strain energy density

and accumulated creep strain damage metrics were computed using a finite element solution, as

explained in detail in Chapter 2 for a model trilayer joint assembly of Si and FR4 with Sn3.8Ag0.7Cu

solder. Three complete thermal cycles (0°C ≤ T ≤ 100°C, 95°C/min ramp rate, and 5 min symmetric

dwell) were simulated to obtain a stabilized hysteresis strain-strain loop. The creep behavior of the

Sn3.8Ag0.7Cu solder was modeled using two different constitutive creep models: (1) a steady state

creep (hyperbolic sine) model; and (2) a transient plus steady-state creep model, developed in the


preceding chapters. Figure 1 relates the creep parameters of each constitutive model to the

calculation of the two creep-fatigue damage metrics. The two models have different scopes and

underlying assumptions. Principally, the steady-state only model ignores hardening and recovery

processes that result in transient creep whenever there is a change in stress, as in thermal cycling.

Figure 1. Flowchart describing role of creep parameters in each constitutive creep model to estimate creep strain energy density dissipation and accumulated creep strain.


Table 1. Hyperbolic sine law parameters (Eq. (1)) as reported in the literature for the Sn3.8Ag0.7Cu and Sn3.9Ag0.6Cu solder alloys

A ( -1s ) α ( -1MPa ) n aH∆ ( kJ mol )

3.8 0.7Sn Ag Cu [7] 52.78 10× 0.02447 6.41 54

3.9 0.6Sn Ag Cu [8] 248.4 0.188 3.79 63

3.8 0.7Sn Ag Cu [9] 52.6 10−× N/A 3.69 36

3.9 0.6Sn Ag Cu [10]

as-cast

54.41 10× 0.005 4.2 45

3.9 0.6Sn Ag Cu [10]

Aged 150°C/24h

73.46 10× 0.005 5.3 49

5.1 Modeling Approach

5.1.1 Creep Constitutive Model Equations

The eight creep parameters shown in Fig. 1 correspond to the constitutive creep model

developed in Chapter 4, where: 1 4C C− define the minimum steady state creep rate ( )mind dtε

(Eq. (2)) where t is time; 5 6C C− pertain to the saturated transient creep strain ,tr satε (Eq. (4)); and

7 8C C− give the normalized rate of transient creep exhaustion normr (Eq. (5)). In Chapter 3, the

model parameters were fit to a body of experimental tensile creep data collected from bulk

Sn3.8Ag0.7Cu solder alloy. The wide range of measured strain rates were accommodated using the

logarithmic transformation of Eqs. (2) and (4), into the regression functions specified in Eqs. (6) and


(7), respectively. In Eqs. (6) and (7), temperature T (K), tensile stressσ (Pa), and minimum

temperature compensated strain rate ( )mind dε Ω ( 1s− ) are independent variables; where Ω is the

temperature compensated time (Eq. (3)).

( ) ( ) 3 41 2min

sinh expC Cd dt C C

Tε σ − =

(2)

( )4exp /t C TΩ = − (3)

( )( ) 6

, 5 min

C

tr sat C d dε ε= Ω (4)

( )( ) ( )8

, 7 min1 exp

Ccreep tr sat normC r d dε ε ε= − − Ω + Ω Ω (5)

( )( ) ( ) ( )41 3 2min

ln ln ln sinhCd dt C C CT

ε σ= − + (6)

( ) ( ) ( )( ), 5 6 minln ln lntr sat C C d dε ε= + Ω (7)

5.1.2 Model Input Parameter Distributions

The second column of Table 2 gives the values of the creep parameters resulting from the

nonlinear regression to the measured transient tensile creep data. The other columns give an

estimation of the uncertainty in each creep parameter treated as a random variable ( jC ) for

1, ,8j = , as discussed below. The log-normally distributed parameters 1C and 5C are given in

terms of the geometric mean making the confidence interval limit estimates robust to sampling error

caused by the highly skewed distributions.


Table 2. Mean and uncertainty for each constitutive creep model parameter

Mean

Standard Error of

Mean

Lower 95%

Confidence Limit

Upper 95%

Confidence Limit

Parametric

Distribution

1C ( 1s− ) 58.091 10× N A 52.852 10× 62.295 10× Lognormal

( )1ln C 11.360 10× 1 2.801 10−± × 11.256 10× 11.465 10× Normal

2C ( 1Pa− ) -71.148 10× N A N A N A Constant

3C 5.020 2 4.200 10−± × 4.682 5.357 Normal

4C ( K ) 41.006 10× 1 9.833 10± × 39.265 10× 41.085 10× Normal

aH∆

( kJ mol )

18.359 10× N A 17.703 10× 19.017 10× Normal

5C ( 6Cs ) 58.360 10−× N A 56.360 10−× 41.114 10−× Lognormal

( )5ln C -9.390 -2 7.535 10± × -9.980 -8.785 Normal

6C 12.460 10−× 3 4.795 10−± × 12.074 10−× -12.836 10× Normal

7C 4.020 -3 4.271 10± × 3.990 4.050 Normal

8C -17.124 10× 4 6.256 10−± × -17.080 10× 17.168 10−× Normal

* N A = not applicable to constant or lognormal distributions; recall 4 aC H R= ∆


In a conventional sensitivity analysis, a damage metric is tested for change due to variation in

a single creep model parameter, while all other model parameters remain unchanged. This approach

may be suitable for first order approximations to understand the most significant contributors to model

predictions; however, it cannot identify the effect of the simultaneous change of multiple creep

parameters. For example, a damage metric may be relatively insensitive to error in 1C for a

particular set of 2C - 8C , but become much more sensitive to 1C variations when one or more of the

other parameters are at a different level. Given that all the creep model parameters jC have

uncertainty about their means, a more accurate approach is to gauge the sensitivity of the damage

metric calculation to the simultaneous variation of multiple jC terms.

5.2 Uncertainty Analysis Simulations

5.2.1 Probabilistic Techniques

A Monte Carlo simulation allows for the simultaneous variation of all creep parameters

defined in the model as random variables with a given distribution. It is expected that the damage

metrics will also have a distribution. Hence, both the creep parameters and damage metrics can be

described by cumulative density functions ( )F x that describe the probability that a variate takes on

a value less than or equal to a value of x . The cumulative density function is related to the

probability function ( )P x by Eq. (8); where ( )P x is simply the derivative of ( )F x (Eq. (9)).

( ) ( )iF x P C x= ≤ (8)

( ) ( )F x P x′ = (9)


The simultaneous change of creep parameters in the Monte Carlo simulation uses a joint

multi-variate distribution (Eq. (10)) that allows sensitivity testing of a damage metric due to variation in

a single creep parameter, while all other creep model parameters are also changing.

( ) ( )1 8 1 1 8 8, , , ,F x x P C x C x= ≤ ≤ (10)

The present work used the following approaches to determine the statistical distributions of

the creep constitutive model parameters jC , and the resulting uncertainty in the two damage metrics:

(1) The residuals of the nonlinear regression fit to the experimental constant-load creep data, were

used in a distribution-free bootstrap simulation to obtain the best parametric distribution for each

creep parameter; the distribution, mean and variance used to define input of Monte Carlo simulations.

(2) The relation between damage metric sensitivity and each jC was evaluated using Latin

Hypercube sampling (sampling without replacement) to reduce the total sets of jC required from the

Monte Carlo simulation. (3) A rank correlation was used to measure the strength of association

between the damage metrics and creep parameters.

5.2.2 Bootstrap Methods

The distribution-free (non-parametric) bootstrap simulation allows for an estimate of the

distribution type (i.e. normal, lognormal, etc.), sample mean, sample variance and confidence interval

of a creep parameter without assuming a priori the parametric distribution [11]. The bootstrapping

procedure involves two steps: In the first step the residuals (difference between the measured and

predicted creep strains) from the original nonlinear regression fit for each of Eqs. (5), (6), and (7)

(yielded the creep parameters of the second column of Table 2) . In the second, the residuals from

the original nonlinear regression solution were randomly re-sampled 1000 times to obtain replicate

sets of estimates and the associated creep model parameters for total creep strain totalε , minimum


creep strain rate ( )mind dtε , and saturated transient creep strain ,tr satε as given by Eqs. (5), (6) and

(7), respectively.

Estimates of totalε , ( )mind dtε and ,tr satε are nonlinear functions of multi-variate

distributions, so results from a non-parametric bootstrap give a true representation of the distribution

of each creep parameter jC . The distribution of each of these three functions and their associated

jC were replicated through separate bootstrap simulations where the solution for each iteration is

given by (Eq. (11); where ( )iE Y gives the expected value of each iteration step i [ 1, ,1000i = ],

ˆiµ gives the function estimate for either Eq. (5), (6), or (7) , and iδ gives the error term sampled from

the residual error distribution.

( ) ˆi i j jE Y X x µ δ= = + (11)

5.2.2.1 Residual Modification for Constant Variance

Prior to the sampling from the raw residuals of the least squares nonlinear regression (step 1

in the bootstrap simulation), the residuals were modified using Eq. (12), where jr is the modified

residual, ˆj jy µ− is the raw residual (difference between data point and estimated value), and jh are

the individual leverages of the hat matrix H where ˆ j jHyµ = [12]. The modified residuals ensure

that the error distribution from which jδ is drawn has a constant variance. This is necessary

because the bootstrapped residuals are assumed both mutually independent and identically

distributed (i.i.d.) [11].

( )1 2

ˆ

1j j

j

j

yr

h

µ−=

− (12)


The Matlab Statistics Toolbox version 7.0 (R2008b) was used for the bootstrap simulation

and to generate smooth empirical probability density (PDF) plots to establish the most appropriate

distribution type for the random variable of each creep parameter [12, 13]. The distribution type,

mean and variance defined the probability density of the creep parameters in the Monte Carlo

simulation.

5.2.3 Monte Carlo Simulations with Latin Hypercube Sampling

The Monte Carlo analysis ran multiple instances of the finite element model simulating three

thermal cycles of the 0-100°C thermal profile described in Chapter 2, Fig. 3. As described in Chapter

2, a calibration solder element that bounds the Si/Solder interface was used to extract stress and

strain information to calculate the creep strain energy density dissipation and accumulated creep

strain damage metrics.

Each simulation instance populated a sampling space defined by the multi-variate distribution

of all creep parameters. In populating this sampling space with traditional direct Monte Carlo

sampling, a cluster of points may result very close together that would provide no new information or

insight into the behavior of the trilayer solder joint while some portions of the distribution may remain

underrepresented. This clustering effect becomes less significant as the simulation size n

approaches infinity. However, it is relatively inefficient to obtain the desired accuracy with a finite

sample size. Therefore, a sampling without replacement strategy termed Latin Hypercube sampling

(LHS) was used to prevent such clustering of data points or distribution underrepresentation with a

simulation of finite size n .

The LHS strategy initially split the probability distribution for each creep parameter into n

non-overlapping intervals of equal probability. Once selected, a given interval was not reselected for

any of the remaining n simulations, thereby ensuring representation in all portions of the distribution.


Within each of these intervals a value was selected at random using a uniform random

number generator, where the selection of any value within the interval was equally likely [14]. The

random probability value selected within the interval domain was converted into a value of the creep

parameter using the respective inverse cumulative distribution function [15].

A sample size of 100n = was used in the present work. However, to increase the

representation of creep coefficients within each interval, the simulations were repeated 10 times.

With each simulation sample size of n , unique solutions were obtained, yielding a sample size grand

total equal to 1000 ( 1000=N ). Repeating the simulation set multiple times increased the

randomness of the sampling process. Further, the large sample size increased the statistical power

of hypothesis testing used to determine the significance of correlation between variates; where the

hypothesis of mutual independence was accepted for p -values more than 0.05.

The probabilistic design suite in Ansys R11 (SP1) was used to control the Monte Carlo

analysis with Latin Hypercube sampling, execute the trilayer thermal simulation, and post-process the

results to calculate creep strain energy density dissipation and creep strain energy density for each of

the 1000 simulations. The creep strain energy density and accumulated creep strain damage metrics

were computed from a stabilized stress versus strain hysteresis loop following three complete thermal

cycles (0°C ≤ T ≤ 100°C, 95°C/min ramp rate, and 5 min symmetric dwell) for the model trilayer joint

assembly of Si and FR4 joined with Sn3.8Ag0.7Cu solder. Details of the implementation of the

transient plus steady-state creep constitutive model were given elsewhere [16]. The user-defined

implicit time integration algorithm provided an unconditionally stable and efficient solution of the

nonlinear constitutive creep model for any time step size [17-19].


5.2.4 Kendall Tau Rank Correlation and Sensitivity Analysis

The sensitivity of the two creep-fatigue damage metrics to uncertainty in the constitutive

creep model parameters was assessed using the Kendall Tau (τ ) rank correlation, where

[ ]1 1τ− ≤ ≤ + . This provided a normalized comparison of the relative impact of uncertainty in the

creep parameters. The magnitude of the Kendall Tau ranges from -1 for a perfect negative

correlation, zero for a perfectly random correlation, and +1 for a perfect positive correlation.

The Kendall Tau is defined by Eq.(13), where CN is the number of concordant pairs, DN is

the number of discordant pairs, and n is the number of observations. For example, a bi-variate

comparison of two instances of the creep model parameter 3C and the corresponding creep strain

energy dissipation per cycle ∆ creepW would yield a concordant pair if Eq.(14) were true, and a

discordant pair were Eq.(15) true.

( )1 2C DN N

n nτ −=

− (13)

( ) ( )2 1 2 13, 3, , ,sgn sgn creep creepC C W W− = ∆ −∆ (14)

( ) ( )2 1 2 13, 3, , ,sgn sgn creep creepC C W W− = − ∆ −∆ (15)

Since the distribution of the Kendall Tau quickly approaches the normal distribution [20], a

p -value significance level can be calculated to test the null hypothesis that correlation does not

exceed that of random error (no correlation). When the null hypothesis is rejected, correlation is

statistically significant at a level of 0.05α = .


5.2.4.1 Quantifying Sensitivity

The Kendall Tau provides a measure of significance of a correlation between changes in a

given creep parameter and the damage metric that is invariant to transformation of either. This is

distinct from the sensitivity of the damage metric to changes in the creep parameter. For example, a

high Kendall Tau might exist for a creep parameter to which the damage metric is not very sensitive

in absolute terms; i.e. large incremental changes in the creep parameter may only produce small

changes in the damage metric. However, normalized sensitivities provide a common basis for

comparison of absolute sensitivities among creep parameters. The sensitivity of each damage metric

to an incremental change in the creep parameters was quantified in a three step process.

Firstly, from the joint bi-variate normal distribution obtained from scatter plots of either

creepW∆ versus jC or ACSε∆ versus jC , a confidence ellipse was defined at the 0.05α = level as

described in [21, 22]. The Hotelling T2 statistic is a measure of the bi-variate distance of each

observation from the center of the data set ( 1x , 1y ). Accordingly for confidence level α , the statistic

2Tα can be used to calculate the perimeter values of a confidence ellipse ( Eq. (16)) that encloses

( )100 1 α− % of the bi-variate distribution; where 1x is the horizontal axis, 1y the vertical axis, 1x is

the horizontal axis mean, 2x the vertical axis mean, 21s is the horizontal axis variance, 2

2S is the

vertical axis variance, and 212s is the bi-variate co-variance.

( )( ) ( ) ( )( )2 22 2

21 1 2 2 1 21 1 2 21 22 2 2 22 2 21 2 1 21 2 12

2x x x x s x x x xs ss s s ss s s T α

− − − −+ − =

− (16)

Secondly, principal components analysis of the confidence ellipse data was used to calculate

the slope of its major axis from the loadings npv of the principal components using Eq. (17); where


n degrees of freedom for 1,2n = and p is the dimensionality parameter for 1,2p = . The

orthogonal regression line, equal to the slope of the major axis, minimizes the sum of squares of

deviations perpendicular to the line itself.

21

11or

vbv

= (17)

Thirdly, the sensitivity iju of output damage metrics to the uncertainty in input creep

parameters (Eq. (18)) was quantified directly from the slope of the orthogonal regression line. As is

common practice when performing a sensitivity analysis, these sensitivity values were normalized

with a function ijN (Eq. (19)) that represents the approximate percentage change in the output

damage metric iX ( creepW∆ , ACSε∆ ) for a 1% change in each input creep parameter jC

( 1 8, ,C C ) [23]; where sensitivities were calculated for n =2 creep damage metrics and m creep

parameters. The most significant correlations as identified by the Kendall Tau were evaluated for

sensitivity, while the correlations approaching mutual independence were omitted from the sensitivity

analysis.

iij

j

XuC∂

=∂

1, ,i n= 1, ,j m= (18)

jij ij

i

CN u

X= ⋅ 1, ,i n= 1, ,j m= (19)

The Matlab Statistics Toolbox version 7.0 (R2008b) was used to compute the

aforementioned requisite calculations to quantify the sensitivity of creep-fatigue damage metrics to

variations in the constitutive creep model parameters.


5.3 Results

5.3.1 Creep Parameter Distributions

Figures 2-4 compare the probability distributions (PDF) of jC ( 1 8, ,C C ) obtained in several

ways: (i) the distribution of jC in the replicate data set of the 1000 iterations of the bootstrap

simulation; (ii) the mean and standard deviation from the bootstrap results were used to generate a

theoretical PDF assuming a log normal or normal distribution; (iii) the distribution of jC obtained from

the 1000 iterations of the Monte Carlo simulation obtained by Latin Hypercube sampling. Probability

density plots for Monte Carlo results from both the steady-state creep model and the steady-state

plus transient creep model were shown to verify that the same distributions were used for 1C , 2C ,

3C and 4C in both sets of analysis.

Figure 2 shows a marked positive skewness 1γ ( 1 1.34γ = + ) in the PDF for 1C

characteristic of a lognormal distribution. In contrast, the PDF plots for 3C and 4C were both

normally distributed. Very little difference existed between these different ways of estimating the

distributions of 1C , 2C , 3C and 4C .

Figure 3 shows the PDFs for the creep model parameters describing the transient creep

saturation; 5C and 6C , respectively. Figure 5 clearly shows positive skewness ( 1 0.97γ = + )

indicative of a lognormal distribution for 5C , while 6C (Fig. 6) was normally distributed. The

normalized rate of transient creep exhaustion is characterized by 7C and 8C , both of which were

normally distributed (Fig. 4). In Fig. 4 for 7C and 8C , the mean was nearly the same for all three

curves; however, the bootstrap simulation showed a higher probability density peak resulting in a

smaller confidence interval.


Figure 2. Probability density function for steady-state parameters: 1C , 3C and 4C .

Figure 3. Probability density function for transient creep saturation parameters: 5C and 6C .

Figure 4. Probability density function for normalized rate of

transient creep exhaustion parameters: 7C and 8C .


This was expected and caused by the non-parametric bootstrap giving an inference specific to the

bounds of the large number of creep strain data points; i.e. a sample size n of nearly 700,000 points,

spanning all constant load creep tests conducted in Chapter 2. In contrast, the parametric

distributions of the theoretical and LHS PDFs are unbounded, resulting in a broader confidence

interval [24]. It is noteworthy that the steady-state creep parameters 1C , 3C and 4C were correlated

with 2C (Eq. (6), thus a constant value of 2C that minimized the mean square error of regression

was used throughout [9]. Further, logarithmic transformations were applied to 1C and 5C , so that all

creep parameters implemented in the constitutive model solution were normally distributed. Kendall

Tau rank correlation results reflecting comparisons between creep parameters showed no statistically

significant correlations.

5.3.2 Damage Metric Distributions

During accelerated thermal cycling the solder was subject to displacement controlled

deformation as it was bounded by layers of materials with different coefficients of thermal expansion.

As the temperature fluctuates thermal stress and strain result as calculated by either the ssε or

tr ssε ε+ constitutive creep models. The distribution of damage accumulated per thermal cycle as

calculated with either the accumulated creep strain ( ε∆ ACS ) and creep strain energy density

(∆ creepW ) are multi-variate statistical functions of the jC discussed in the preceding section.

Inelastic deformation is irreversible and the solder is assumed incompressible (effective Poisson’s

ratio = 0.5) during creep calculations.

The accumulated equivalent strain ACSε in terms of principal strains (Eqs. (20)-(21)) is an

ever-increasing function, independent of the sense of the instantaneous equivalent strain direction;

hence a useful quantity for cyclic applications with load reversals calculated over all thermal cycles


λ . The strain energy density dissipation during creep deformation ( creepW ) of solder calculated at the

interfacial calibration element was calculated by integrating the product of instantaneous equivalent

(von Mises) stress and accumulated equivalent (von Mises) strain (Eq. (22)) [25]. Similarly, the creep

strain energy density dissipation is also an ever-increasing function that increases monotonically

during thermal cycling for all ACSε , despite load reversals and changes in the sense of instantaneous

equivalent strains.

( ) ( ) ( )1

2 2 2 22 3 3 1 1 2

23eqε ε ε ε ε ε ε ∆ = ∆ −∆ + ∆ −∆ + ∆ −∆ (20)

( )1

ACS eqi

λ

ε ε=

= ∆∑ (21)

.

0

eqAcc

creep eq ACSW dε

σ ε= ∫ (22)

The histograms and smooth distribution-free empirical PDF plots for both ε∆ ACS and

∆ creepW were given in Figs. 5 and 6, respectively. Table 3 summarized the statistical parameters for

these distributions including Shapiro-Wilk p -values for testing the hypothesis that a normal

distribution well represents the empirical distributions of each damage metric. The results indicate

that normality both ε∆ ACS distributions are well modeled by a normal distribution, while both ∆ creepW

distributions cannot be well modeled by a normal distribution at the 0.05α = level.


Figure 5. Histogram and empirical probability density of accumulated creep strain damage metric.


Figure 6. Histogram and empirical probability density of creep strain energy density damage metric.

The inclusion of transient creep led to large accumulated creep strain over the thermal cycles

as well as increased creep strain energy dissipation per cycle compared to the steady state omitting

transient creep contributions. Although additional creep contributions from transient creep had a

direct and effect on ε∆ ACS (Eq. (21)), the transient creep strain effect on ∆ creepW was impacted by

the joint proportionally of strain and stress (Eq. (22)). The inclusion of transient creep resulting in a

smaller net effect on ∆ creepW compared to ε∆ ACS that is affects sensitivity of some jC more

accurately than others, as discussed in following section.


Table 3. Descriptive statistics for calculated damage metrics from steady-state creep model, ε ss and

transient plus steady-state creep model, ε ε+tr ss . Distributions shown in Figs. 9-10.

Mean

Standard

Error Low 95% CI High 95% CI

Shapiro-Wilk

p -value

∆ creepW

(MPa)

ε ss 11.00 10−× 42.45 10−× 28.65 10−× 11.12 10−× 31 10−<< ×

ε ε+tr ss 11.18 10−× 42.79 10−× 11.01 10−× 11.30 10−× 31 10−<< ×

ε∆ ACS

ε ss 32.87 10−× 51.26 10−× 32.22 10−× 33.53 10−× 0.65

ε ε+tr ss 37.68 10−× 51.91 10−× 36.69 10−× 38.69 10−× 0.42

5.3.3 Sensitivity of damage metrics to creep parameter uncertainty

The sensitivity of ε∆ ACS and ∆ creepW to uncertainty in each creep parameter jC is revealed

in scatter plots obtained from the Monte Carlo simulations for the steady-state and transient plus

steady-state creep models (Figs. 7-10). Each figure includes an ellipse encompassing 95% of the

points in the joint distribution of the damage metric and the specific jC .

The data points in the scatter plots were either randomly distributed forming a near circle-

shaped confidence ellipse, or were more strongly correlated producing a joint distribution with a

narrower confidence ellipse. The Kendall Tau rank correlation (τ ) quantifies the strength of this

association and was plotted for ε∆ ACS (Fig. 11) and ∆ creepW (Fig. 12), as a series of horizontal bar

graphs relating the magnitude and direction of τ for each creep parameter in descending order.

Table 4 presents the Kendall Tau τ and the statistical significance ( p -value) that describe the


strength of association between each jC versus either damage metric ∆ creepW and ε∆ ACS . The

hypothesis of independence between jC and either damage metric is rejected when the p -value is

less than 0.05. The dependence structure of the rank correlation τ and p -value are invariant to

transformations of the creep parameters. This permits the logarithm of 1C and 5C to share the same

damage metric correlation as the untransformed values. However, the transformed values are

presented in the scatter plots to satisfy the bi-variate normal distribution pre-condition of the

confidence ellipse; the slope of the major axis defines sensitivity in absolute terms.


Figure 7. Scatter plots for steady-state parameters ln(C1), C3 and C4 versus damage metrics ( ssε model).

Figure 8. Scatter plots for steady-state parameters ln(C1), C3 and C4 versus damage metrics ( tr ssε ε+ model).


Figure

Figure 9. Scatter plots for transient creep saturation parameters, ln (C5) and C6

versus damage metrics.

Figure 10. Scatter plots for normalized rate of transient creep exhaustion parameters, C7 and C8 versus damage metrics.


Table 4. tr ssε ε+ ( ssε ) constitutive creep model Kendall Tau correlation and p -values

1C 3C 4C 5C 6C 7C 8C

∆ creepW

tr ssε ε+ , τ

( p -value)

-0.46

(<0.05)

-0.07

(<0.05)

+0.55

(<0.05)

+0.04

(0.10)

+0.09

(<0.05)

-0.02

(0.43)

+0.02

(0.28)

ssε , τ

( p -value)

+0.03

(0.20)

+0.24

(<0.05)

-0.77

(<0.05)

ε∆ ACS

tr ssε ε+ , τ

( p -value)

+0.49

(<0.05)

+0.10

(<0.05)

-0.51

(<0.05)

+0.07

(<0.05)

+0.05

(<0.05)

+0.02

(0.40)

-0.02

(0.32)

ssε , τ

( p -value)

+0.03

(0.21)

+0.25

(<0.05)

-0.76

(<0.05)

The apparent activation energy term 4C (Eq.(2)) corresponds to the largest absolute τ

values for ACSε∆ and creepW∆ for both the ssε and tr ssε ε+ creep models. Therefore, uncertainty in

4C causes the greatest sensitivity in both calculated damage metrics using either creep model. An

increase in the apparent activation energy for steady-state creep reduces the accumulated creep

strain as shown for ACSε∆ vs 4C in both Figs. 7 ( ssε model) and 8 ( trε + ssε model). Similarly,

creepW∆ decreases with increasing 4C using the ssε model (Fig. 7), but creepW∆ increases with

increasing 4C with the tr ssε ε+ model, because the decrease in creep strain is more than offset by

an increase in stress.


The second largest τ with the ssε model occurs with 3C , the stress exponent in the steady-

state constitutive equation. The term 3C is mathematically defined in Eq. (23) by taking the logarithm

and subsequently differentiating Eq. (2). From Fig. 7 examining both ACSε∆ and creepW∆ for 3C we

see that creep strain increased with increasing 3C values. From Eq. (23), for ssε this increase in

creep strain was accompanied by an increase in strain rate and stress for 0.24τ = + . Similarly for

Fig. 8, from ACSε∆ and creepW∆ we determine that creep strain increases with increasing 3C .

However, contrarily from Eq. (23), the strain rate and stress must decrease to accommodate

0.07τ = − . This example illuminates the principle limitation of the steady-state creep model, its

inability to compensate for changes in strain hardening and recovery processes. This difference

uniquely distinguishes the scatter plots of 3C between the two models for creepW∆ .

( )3ln

ln sinhC ε

ασ∂

=∂

(23)

5.3.3.1 Substructure Affects on Stress Response

In [26], the yield stress yσ was used as a proxy to represent the affects of substructure

features such as the width of partial dislocations, dislocation density and subgrain size. The stress

term was normalized as ( )yσ σ in a Brown-Ashby [27] relation of the form given for Eq. (24), where

lower creep rates resulted in lower yσ values and lower dislocation density because the slower

creep rate allowed more time for microstructural coarsening (recovery processes) to lower dislocation

density.

( ) ( ) ( )min

n

yd dt A f Tε σ σ= (24)


The link with subgrain structure is analogous to the finding in Chapter 3 where at the highest

temperature compensated creep rate ( 910 1s− ) the subgrains were densely packed leading to a high

dislocation density, while a much lower temperature compensated creep rate ( 610 1s− ) the subgrains

Figure 11. Kendall Tau rank correlation for creep parameters versus accumulated creep strain.

Figure 12. Kendall Tau rank correlation for creep parameters versus creep strain energy density.


were more sparsely packed and a lower dislocation density resulted. Similarly, this finding was also

evident in compression creep tests of Sn3.9Ag0.67Cu conducted in [10, 28], where for lower

temperatures (-25°C to +75°C), relatively high stresses and high dislocation density, regression of

experimental data yielded 4.4 0.7n = ± . While for tests at elevated temperatures (75°C to 160°C),

relatively lower stress levels and lower dislocation density regression yielded 5.2 0.8n = ± .

The dimensionless pre-multiplier constant 1C (Eq. (2)) is an accumulation of frequency

factors, Burgers vectors, dislocation volumes, etc., depending upon the specific theory used to create

ssε model. This physical meaning is simply neglected in mechanism analyses of creep data. The

( )1ln C has a very small positive correlation for both ACSε∆ and creepW∆ that is not discernable from

random sampling variation for 0.05α = (Table 4). However, for the tr ssε ε+ model the correlation

with ( )1ln C is the second most significant, and as creep rate and stress decrease the ( )1ln C value

increases for both ACSε∆ and creepW∆ . This suggests that ( )1ln C may be sensitive to changes in

sub-structure or that the nonlinear structure of the transient portion of the model may amplify the

uncertainty in the 1C creep parameter as uncertainty is propagated to the calculated damage metrics

[24].

In considering the creep parameters corresponding to saturation of transient creep, Figs. 11

and 12 shows that 5C and 6C have small positive correlations for both ACSε∆ and creepW∆ , where a

decrease in stress and creep rate accompany an increase in both creep parameters. This follows

from Eq. (4) where an increase in the minimum temperature compensated strain rate ( )mind dε Ω

accompanies an increase in the saturated transient creep strain and hence creep strain more

generally. Since an increase in stress also relates directly to an increase in ( )mind dε Ω , the


correlation between 5C and 6C is always positive for both ACSε∆ and creepW∆ . However, the small

correlation indicates that uncertainty of these creep parameters does not drive uncertainty

propagation in the calculated damage metrics.

Lastly, the creep parameters describing the normalized rate of transient creep exhaustion are

given in Fig. 10 for 7C and 8C . As shown in Figs. 11 and 12, 7C and 8C have the weakest

correlation of all creep parameters with either ACSε∆ or creepW∆ . The p -values (Table 3) indicate

these weak correlation cannot be discerned from random sampling variability at a significance level of

0.05α = . However, the correlation direction observed from Fig. 11 for ACSε∆ follows from Eq. (5).

When 8C is increased the transient creep term decreases, while when 7C is increased the transient

term increases. The extension to creepW∆ follows as suggested by Fig. 12, because lower strain

accompanies increased stress and strain rate leading to opposite directions of correlation between

7C and 8C for both ACSε∆ and creepW∆ . Clearly the creep parameters describing the normalized

rate of transient creep exhaustion do not drive uncertainty propagation in the calculated damage

metrics.

5.3.3.2 Normalized Sensitivity Analysis

Further insight may be obtained from Fig. 13 where the normalized sensitivity ijN (Eq. (19))

are given as a function of ACSε∆ and creepW∆ . The absolute sensitivities iju used to calculate the

normalized sensitivities are given in the Table 5. Note that ( )1ln C does not appear in normalized

sensitivity results for the ssε model because it did not have a statistically significant Kendall Tau τ at

the 0.05α = level; similarly other creep parameters jC with no significant correlation (Table 4) were


also omitted. For the tr ssε ε+ model the normalized sensitivity for ( )1ln C remains nearly constant

across the range of ACSε∆ values. However, for increases in creepW∆ there is a notable decrease in

ijN as the damage metric increases. By contrast, while 3C and 4C have reversed directions of

sensitivity between ACSε∆ and creepW∆ , the magnitudes are similar as reported from Figs. 11-12.

Figure 13. Normalized sensitivity coefficients for damage metrics based

on a 1% change in each creep parameter


Figure 14. Normalized sensitivity coefficients for accumulated creep strain based on a 1% change in creep strain energy density.

This sharply contrasts with Fig. 13 results for the ssε model that is notably more sensitive to

3C and 4C for ACSε∆ than to creepW∆ . In this case it appears that the seemingly moderating effect

of the creepW∆ dependence on strain and stress, makes ACSε∆ twice as sensitive to 3C compared

with creepW∆ over the complete range of damage metrics. For 4C the ACSε∆ was more than twice

as sensitive compared to creepW∆ for lower values of damage metric, and approached near parity of

normalized sensitivity as the damage metric increased.

This firmly demonstrates the point raised in Chapter 2 of a variable and unknown error that

affects damage metric results from ssε and tr ssε ε+ models. Although the standard deviation of the

damage metric distributions is smaller the ssε model compared with the tr ssε ε+ results, the

normalized sensitivity is larger for the creep parameters 3C and 4C . When the contribution of

steady-state creep is largest, the sensitivity to damage metrics is most severe, raising the importance

of accurate values for steady-state creep parameter estimates.


Table 5. Absolute sensitivity values of ACSε∆ to uncertainty in creep parameters

jC iX ( )i jss

X Cε

∂∆ ∂ ( )i jsstr

X Cε ε+

∂∆ ∂

( )1ln C ACSε∆ N A 44.50 10−×

creepW∆ N A 36.14 10−− ×

3C ACSε∆ 31.15 10−× 47.13 10−×

creepW∆ 22.13 10−× 36.80 10−− ×

4C ACSε∆ 61.06 10−− × 61.30 10−− ×

creepW∆ 52.07 10−− × 51.98 10−×

( )5ln C ACSε∆ N A 42.82 10−×

creepW∆ N A N A

6C ACSε∆ N A 33.28 10−×

creepW∆ N A 11.17 10−×

* N A = Not applicable for steady-state creep model

The transient creep parameters ( )5ln C and 6C both showed nearly constant normalized

sensitivity over the range of damage metrics that was much lower than the steady-state creep

parameters. Notably all creep parameters become less sensitive to additional incremental increases

as the damage metrics increase in magnitude.


Lastly, to better understand the nature of the correlation between creepW∆ and ACSε∆

damage metrics, a normalized sensitivity plot (Fig. 14) was created for both correlated model outputs.

As with all the ijN results the sensitivity of creepW∆ to incremental increases decreases as the

ACSε∆ increases. The ssε model is remarkable more sensitive than the than the tr ssε ε+ model. As

reflected in Fig. 5, the magnitude of ACSε∆ values is less than half that compared to the strain that

occurs when transient creep is included. These issues capture the main difference between the

stress response between the constitutive creep models, as ε and σ are in near lock-step with a

0.98τ = + for the ssε model. The tr ssε ε+ model with 0.82τ = − is less sensitive and more

accurately reflects the response of solder with transient creep contributions.

In summary, uncertainty in the damage metrics ACSε∆ and creepW∆ calculated using the

transient plus steady-state creep model is mainly attributable to uncertainty in the parameters 4C and

1C . Uncertainty in these damage metrics calculated using the steady-state creep model is due

principally to 4C and 3C .


5.4 Conclusions

Practically the sensitivity analysis to creep parameter uncertainty provides a benchmark of

comparison upon which to assess the broad array of scatter in the solder literature. The rank

correlation provided a means of gauging which creep parameters are most impactful, while the

sensitivity analyses on the Monte Carlo simulation results permit for greater clarity into how sensitivity

may be affected as parameters change.

Considering the scatter from the literature (Table 1), some trends can now be taken in better

perspective. For instance, the 1C parameter (equivalent to A , Eq. (1)) was shown to be mutually

independent of either ACSε∆ or creepW∆ damage metrics for the ssε creep model. Thus the scatter

over 5 orders of magnitude recorded in the literature does not pose an impediment to finite element

modeling. Recall, the 2C parameter (equivalent to α , Eq. (1)) ideally is selected to minimize the

mean square error. From a pragmatic standpoint, focus can be addressed on the activation energy

term 4 aC H R= ∆ , since it has the highest magnitude of Kendall Tau and the largest normalized

sensitivity of all creep parameters. However, the analyst can quickly determine from the absolute

sensitivity (Table 5) if the difference in alternative values results in a significant change in expected

damage metric for a given application.

There is a clear trend of decreased sensitivity with increased creep strain energy density

dissipation and accumulated creep strain. Since the inclusion of transient creep contributions enlarge

the mean values and increase the variance of the damage metrics, models that incorporate transient

creep have creep parameters with less sensitivity compared with those from the steady-state model.

The analyst is recommended, especially for profiles with significant steady-state creep contributions,

to focus on obtaining the best estimates of steady-state creep parameters for the best damage

metrics and reliability predictions.


5.5 References

[1] Gong, J., Liu, C., Conway, P.P., and Silberschmidt, V.V., Micromechanical modelling of SnAgCu solder joint under cyclic loading: Effect of grain orientation. Computational Materials Science, 2007. 39(1): p. 187-97.

[2] Plumbridge, W.J., Matela, R.J., and Westwater, A., Structural integrity and reliability in electronics : enhancing performance in a lead-free environment. 2003, Boston: Kluwer.

[3] Stipan, P.M., Beihoff, B.C., and Shaw, M.C., Electronics Package Reliability and Failure Analysis: A Micromechanics-Based Approach, in The Electronic Packaging Handbook, G.R. Blackwell, Editor. 2000, CRC Press: Boca Raton.

[4] Wilshire, B. and Evans, R.W., Acquisition and Analysis of Creep Data. Journal of Strain Analysis, 1994. 29(3): p. 159-65.

[5] Pierce, D.M., Sheppard, S.D., Fossum, A.F., Vianco, P.T., and Neilsen, M.K., Development of the Damage State Variable for a Unified Creep Plasticity Damage Constitutive Model of the 95.5Sn–3.9Ag–0.6Cu Lead-Free Solder. Journal of Electronic Packaging, 2008. 130.

[6] Farris, J.P., Lee, J.D., Harlow, D.G., and Delph, T.J., On the Scatter in Creep Rupture Times. METALL TRANS A, 1990. 21A: p. 345-52.

[7] Schubert, A., Dudek, R., Auerswald, E., Gollhardt, A., Michel, B., and Reichl, H. Fatigue life models for SnAgCu and SnPb solder joints evaluated by experiments and simulation. in ECTC. 2003. New Orleans: IEEE.

[8] Zhang, Q., Dasgupta, A., and Haswell, P. Viscoplastic constitutive properties and energy-partitioning model of lead-free Sn3.9Ag0.6Cu solder alloy. in ECTC. 2003: IEEE.

[9] Frear, D.R., Jang, J.W., Lin, J.K., and Zhang, C., Pb-free solders for flip-chip interconnects. JOM, 2001. 53(6): p. 28-33.

[10] Vianco, P.T., Rejent, J.A., and Kilgo, A.C., Creep behavior of the ternary 95.5Sn-3.9Ag-0.6Cu solder: Part II—Aged condition Journal of Electronic Materials, 2004. 33(12): p. 1473-84.

[11] Efron, B. and Tibshirani, R.J., An Introduction to the Bootstrap. 1993, New York: Chapman & Hall.

[12] Davison, A.C. and Hinkley, D.V., Bootstrap Methods and Their Application. 1997: Cambridge University Press.

[13] Wand, M.P. and Jones, M.C., Kernel smoothing. 1995, New York: Chapman & Hall. [14] ANSYS, Commands Reference. 2007: Canonsburg. p. 1059-66. [15] Robert, C.P. and Casella, G., Monte Carlo statistical methods. 1999, New York: Springer. [16] ANSYS, Programmer's Manual for ANSYS. 2007: Canonsburg, PA. [17] ANSYS, Theory Reference for ANSYS and ANSYS Workbench. 2007: Canonsburg, PA. [18] Bathe, K.-J., Finite element procedures. 1996, Englewood Cliffs, N.J.: Prentice Hall. [19] Hughes, T.J.R., The finite element method: linear static and dynamic finite element analysis.

2000, Mineola, NY: Dover. [20] Conover, W.J., Practical nonparametric statistics. 2d ed. 1980, New York: Wiley. [21] Jackson, J.E., A user's guide to principal components. 1991, New York: Wiley. [22] Paradowski, L.R., Uncertainty ellipses and their application to interval estimation of emitter

position. Aerospace and Electronic Systems, IEEE Transactions on, 1997. 33(1): p. 126-33. [23] Hearne, J.W., Sensitivity analysis of parameter combinations. Applied Mathematical

Modelling, 1985. 9(2): p. 106-8. [24] Cullen, A.C. and Frey, H.C., Probabilistic Techniques in Exposure Assessment. 1998, New

York: Plenum. [25] Zyczkowski, M., Combined loadings in the theory of plasticity. 1981: Kluwer Academic

Publishers. [26] Walser, B. and Sherby, O.D., The Structure dependence of power law creep. Scripta

Metallurgica, 1982. 16(2): p. 213-9.


[27] Brown, A.M. and Ashby, M.F., On the power-law creep equation. Scripta Metallurgica, 1980. 14(12): p. 1297-302.

[28] Vianco, P., Creep behavior of the ternary 95.5Sn-3.9Ag-0.6Cu older: Part I - As-Cast Condition. Journal of Electronic Materials, 2004. 33(11): p. 1389-400.

Chapter 6. Conclusions 156

Chapter 6

Conclusions

6.0 Original Contributions

(1) The research presented in this dissertation characterized the transient creep behavior of

Sn3.8Ag0.7Cu lead-free solder alloy over a range of temperatures and strain rates of interest to

the electronics packaging industry ( 20effσ < MPa). The transient and steady-state creep

contributions were found to be functions of both temperature and strain rate.

(2) Through use of the temperature compensated time and temperature compensated strain rate

equivalence was established between decreased temperature and increased temperature

compensated strain rate for Sn3.8Ag0.7Cu under both creep and stress relaxation testing.

(3) A newly presented constitutive model based on the time-hardening approach permits the fitting

of creep data from across a wide range of conditions, and also predicted stress relaxation data

set using a common set of creep parameters.

(4) A transient creep threshold was defined that related the dominant creep contribution, whether

primary or secondary creep, for a given inelastic deformation temperature and response.

(5) A sensitivity analysis of two damage metrics as a function of the uncertainty in creep model

parameters demonstrated that transient creep contributions reduce sensitivity, and that the

largest effects are caused by steady-state creep parameters.

Chapter 6. Conclusions 157

6.1 Practical Applications and Future Work

Creep-fatigue life estimates based on crack initiation can be estimated through damage metrics

such as, the accumulated creep strain and the creep strain energy density. The constitutive creep model

presented better captures the changing influence of transient creep through a thermal cycle, thereby

improving the accuracy of damage metric calculations.

Further work should be done to incorporate this modeling approach into a framework that

encompasses both crack initiation and propagation.

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TRANSIENT AND STEADY-STATE CREEP IN A Sn-Ag-Cu LEAD …

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