Cumulative Damage Analysis

I

FATIGUE STUDIES AND

CUMULATIVE DAMAGE ANALYSIS

OF THE SURGICAL ALLOY

VITALLIUM

by

Kieran J. Claffey

9529349

A report submitted in partial fulfillment of the

requirements for the undergraduate degree of

Mechanical Engineering

University of Limerick, Ireland.

1997

Approved by Dr. Tim Mcgloughlin Project Supervisor

II

Abstract

This study is based on the fatigue failure of the trade-marked, cobalt-chrome, surgical alloy,

Vitalliumtm which is used in the manufacture of prosthetic hip implants. Different aspects of

fatigue failure are explored within a completely reversed bending, fatigue test programme.

The effect of mean stress on alternating stress is considered in the form of a fatigue strength

diagram. An exploratory S-N curve is determined for cast vitalliumtm. A notch sensitivity

analysis is conducted for two different notch types. A cumulative damage analysis is also

conducted to determine the most suitable life prediction theory for vitalliumtm. The effects of

machining and hot isostatic pressing on fatigue resistance are examined. The nature of the

fatigued microstructure and macrostructure is inspected. It was found that Ben-Amoz’s

theory and the Unified theory were the best cumulative damage life prediction theories for

application to vitalliumtm.

III

Dedication

To my parents for being ever supportive and making all this possible.

Table of Contents

IV

Table of Contents

Chapter Description Page

1 Introduction .................................................................................................................... 1

2 Objectives ....................................................................................................................... 4

3 Literature Review ......................................................................................................... 5

3.1.1 Introduction to Fatigue Behaviour ................................................................................. 5

3.1.2 The Three Stages of Fatigue........................................................................................... 5

3.1.3 Fatigued Surface Characteristics .................................................................................... 6

3.1.4 The S-N Curve ............................................................................................................... 6

3.1.5 Determination of S-Log N Curve ................................................................................... 7

3.2.1 Fatigue Tests .................................................................................................................. 8

3.2.2 R.R. Moore Fatigue Test ................................................................................................ 9

3.2.3 Strain Controlled Tests ................................................................................................... 9

3.3.1 Mathematical Relationship between Strength and Number of Cycles......................... 10

3.3.2 The Basquin Relation ................................................................................................... 10

3.4.1 Combined Alternating and Mean Stress ...................................................................... 11

3.4.2 Constant Life Diagram ................................................................................................. 11

3.5.1 Notch Effects ................................................................................................................ 12

3.5.2 Notch Sensitivity, Stress Concentrations and Fatigue Notch Factors .......................... 13

Table of Contents

V

3.5.3 Tensile Notch Sensitivity ............................................................................................. 16

3.5.4 Multiple Notches in a Plate .......................................................................................... 16

3.6.1 Cumulative Damage Analysis ...................................................................................... 17

3.6.2 Cumulative Damage Theories ...................................................................................... 17

3.6.3 Two Level Step Tests ................................................................................................... 17

3.7.1 Damage Tolerant Design .............................................................................................. 18

3.7.2 Linear Elastic Fracture Mechanics (LEFM) ................................................................ 18

3.7.3 Thickness Considerations for LEFM ........................................................................... 19

3.7.4 Determination of Fracture Toughness .......................................................................... 21

3.7.5 Life Prediction using LEFM ........................................................................................ 23

3.7.6 Randomly Distributed Small Crack Data Method of Fatigue Evaluation ................... 24

3.8.1 The Finite Element Method.......................................................................................... 24

3.9.1 Information Required for Damage Assessment ........................................................... 26

3.9.2 Damage Assessment ..................................................................................................... 26

3.10.1 Ductile Materials .......................................................................................................... 30

3.10.2 Brittle Materials............................................................................................................ 30

3.11.1 Investment Casting of Hip Implants and Fatigue Test Specimens .............................. 31

3.11.2 Hot Isostatic Pressing ................................................................................................... 32

Table of Contents

VI

3.11.3 HIPing Investment Castings ......................................................................................... 34

3.11.4 HIPing – The End Result ............................................................................................. 35

3.11.5 Fatigue Resistance due to HIP ..................................................................................... 36

3.12.1 Specimen Design .......................................................................................................... 38

3.12.2 Specimen Preparation ................................................................................................... 38

3.12.3 Machining ..................................................................................................................... 39

4 Life Prediction Theory ............................................................................................... 40

4.1 Miner’s Cumulative Damage Theory ........................................................................... 40

4.2 Subramanyan’s Cumulative Damage Theory ............................................................. 42

4.3 Ben-Amoz’s Cumulative Damage Theory .................................................................. 44

4.4 Corten-Dolan’s Cumulative Damage Theory ............................................................. 47

4.5 Marin’s Cumulative Damage Theory .......................................................................... 48

4.6 Manson’s Double Linear Cumulative Rule ................................................................. 50

4.7 Henry’s Cumulative Damage Theory ......................................................................... 50

4.8 Henry’s Modified Cumulative Damage Theory ......................................................... 51

4.9 Gatt’s Cumulative Damage Theory ............................................................................. 52

4.10 Unified Theory of Cumulative Damage ...................................................................... 53

4.11 Marco-Starkey’s Cumulative Damage Theory ........................................................... 55

5 Experimentation ......................................................................................................... 57

5.1 Experimental Programme ............................................................................................. 57

5.2 Specimen Preparation ................................................................................................... 57

5.3 Preparation of Notched Specimens .............................................................................. 58

5.4 Apparatus ..................................................................................................................... 59

5.5 Standard Fatigue Test Precautions ............................................................................... 61

Table of Contents

VII

5.6 Experimental Procedure .............................................................................................. 61

5.7 Test Problems and Solutions ........................................................................................ 63

6 Experimental Results ................................................................................................. 65

7 Analysis of Data .......................................................................................................... 73

7.1 Determination of S-N Curve ........................................................................................ 73

7.2 Mean Stress Effects on Machined Vitalliumtm ............................................................. 75

7.3 Constant Life Diagram for Machined Vitalliumtm ....................................................... 76

7.4 Constant Life Diagram with Mean Stress Correction for Machined Vitalliumtm ........ 77

7.5 Fatigue Strength Diagram for Cast Vitalliumtm ........................................................... 78

7.6 Cumulative Damage ..................................................................................................... 81

7.7 Notched Vitalliumtm ..................................................................................................... 83

7.8 HIPed Vitalliumtm ......................................................................................................... 85

8 Inspection .................................................................................................................... 87

8.1 Microscopic Examination ............................................................................................ 87

8.2 Procedure for Microstructure Analysis ........................................................................ 87

8.3 Results of Micro-Structural Inspection ........................................................................ 88

8.4 Macroscopic Inspection................................................................................................ 90

9 Discussion .................................................................................................................... 92

9.1 Machining Effects on Fatigue Resistance .................................................................... 92

9.2 Notch Analysis ............................................................................................................. 92

9.3 Damage Assessment ..................................................................................................... 93

9.4 Cumulative Damage Theories ...................................................................................... 94

9.4.1 Marco-Starkey’s Theory .............................................................................................. 94

9.4.2 Miner’s, Subramanyan’s and Ben-Amoz’s Theories ................................................... 95

Table of Contents

VIII

9.4.3 Corten-Dolan’s and Marin’s Theories.......................................................................... 96

9.4.4 Manson’s Theory.......................................................................................................... 96

9.4.5 Henry’s, Henry’s Modified and Gatt’s Theories .......................................................... 97

9.4.6 Unified Theory ............................................................................................................. 98

9.5 HIPed Specimens ......................................................................................................... 99

9.6 Mean Stress Effects .................................................................................................... 100

9.7 Inspection ................................................................................................................... 100

9.8 The Fatigue Testing Machine ..................................................................................... 101

10 Conclusions ............................................................................................................... 102

11 Recommendations .................................................................................................... 103

References ................................................................................................................. 104

Appendix .........................................................................................................................

Derivation of Equation 5.1 ........................................................................................ A.1

Specimen Specifications ........................................................................................... A.2

Machined Vitalliumtm S-N Curve .............................................................................. A.3

S-N Curve for Cast Vitalliumtm ................................................................................. A.4

Constant Life Diagram for Machined Vitalliumtm .................................................... A.5

Sample Life Prediction Calculations based upon Cumulative Damage Theories..... A.6

Nomenclature

IX

Nomenclature

Symbol

Description Units

B

C

C

CR

D

D

D1

De

E0

E1

Kf

Kt

Ktf

K1C

M

N

N1,2

Nd

NI

NK

R

S

Se

Seo

Specimen width

Material constant for LEFM

Material constant used in Gatt’s cumulative damage theory

Strength reduction factor for reliability

Damage fraction

Specimen thickness

Damage fraction after first stress level application

Equivalent damage fraction

Original endurance limit

Endurance limit after first stress application

Fatigue notch factor

Stress concentration factor

Estimated fatigue notch factor for design purposes

Fracture toughness (1st mode)

Bending moment

Number of cycles to failure

Number of cycles to failure at first or second applied stress levels

Number of delay cycles

Number of cycles required to initiate a crack to the propagation stage

Number of cycles to knee-point of S-N curve (Subramanyan’s theory)

Stress ratio

Alternating stress amplitude

Current value of endurance limit

Original value of endurance limit

mm

-

-

-

-

mm

-

-

MPa

MPa

-

-

-

-

Nm

-

-

-

-

-

MPa

MPa

MPa

Nomenclature

X

SN

SU

a

a

d

m

mi

n

n

n1

nf

n

nf

n12

ni2

np2

q

q

r

w1

ek

ε

a

f

m

N

TS

Stress amplitude for a safe life of N cycles

Ultimate tensile strength

Material constant for notch sensitivity

Crack length used in LEFM

Material constant used in Corten-Dolan’s theory

Material constant used in the Unified theory

Exponent used in Marco-Starkey’s cumulative damage theory

Number of applied cycles

Material constant for LEFM

Number of cycles applied at S1

Number of remaining cycles to failure

Factor of safety

Number of remaining cycles to failure

Equivalent no. of cycles for 2nd stress level after application 1st stress level (Subramanyan).

Number of cycles which cause crack initiation at stress level 2

Number of cycles which cause crack propagation at stress level 2

Notch sensitivity

Material constant used in Marin’s cumulative damage theory

Notch radius

Work done at n1 cycles

Constant used in Subramanyan’s and Corten-Dolan’s cumulative damage theory

Equivalent cycle ratio after k levels of stress

Strain

Stress

Alternating stress

Stress for static failure/fracture

Mean stress

Nominal stress

Ultimate tensile strength

MPa

MPa

-

mm

-

-

-

-

-

-

-

-

-

-

-

-

mm

KJ

-

-

-

MPa

MPa

MPa

MPa

MPa

MPa

Nomenclature

XI

Y

α

µ

e

u

da/dN

dD/dn

de/dn

K

Yield strength

Reliable stress for a design life of N cycles

Weighting coefficient used in the Unified theory

Non-dimensional maximum cyclic stress

Instantaneous non-dimensional endurance limit

Non-dimensional original ultimate tensile strength

Rate of crack growth (Paris’s Law)

Rate of damage growth

Rate of reduction in the non-dimensional endurance limit (Unified theory)

Change in stress intensity factor

MPa

-

-

-

-

-

-

-

Introduction Cumulative Damage Analysis – Kieran J. Claffey

Chapter 1

Introduction

The study of fatigue failure began more than a century ago. The English engineer, Sir

William Fairbairn, carried out the first recorded fatigue tests with wrought iron girders. He

discovered that a girder which could withstand a static load of twelve tons for an indefinite

period would fail if a load of three tons was applied cyclically about three million times. His

explanation was that the metal had become tired. Thus was born the concept of metal

fatigue.

Fatigue accounts for approximately eighty per cent of all metal failures. Therefore, the more

information that can be obtained on the subject, the better it is for society. The general

public first became aware of fatigue failure because of the comet airline disasters in the

1960’s and from the fatigue of fuselage and propellers in ageing civilian and military aircraft.

Fatigue affects us in ways other than aeroplane disasters. An example is of fatigue failure of

the pinion teeth in a rack and pinion automotive steering system. If this were to suddenly fail

when coming down a winding mountain road, the consequences would be obvious. Another

practical example is the small pin which connects the gear shift of a car to the transmission.

This pin is subjected to high stresses every time a gear is changed and cannot fail in a fatigue

event.


2

It is obvious that these are extreme situations. Fatigue can cause a lot of trouble in industry.

For example a crack may occur in a notched key way on a motor shaft and as a result, fail

due to fatigue. The damage would necessitate repair and the motor may need to be replaced.

Either of which is expensive. Any part of a machine that vibrates around a point of stress

concentration is liable to be subjected to fatigue and possible failure. It is therefore,

important to design machines to operate at speeds which avoid resonant frequencies, thus

eliminating strong vibrations and unnecessary fatigue failures.

A large area which warrants fatigue information is that of the prosthetic bone industry.

Cobalt chrome alloys have been used as a bone prosthesis material for decades. Vitalliumtm

is one such alloy used by Howmedica to manufacture knee implants and fasteners for

insertion into the human body. Some of the early prosthesis in the 1960’s and ‘70’s failed

during service, due to fatigue. The Charnley-Muller type femoral component in hip

prosthesis was one of these. Today, implants do not fail. This can be accredited to the

design engineers who relentlessly made improvements to get to this stage.

However, the failure of a metal component in a person cannot be over emphasised because of

the excruciating pain involved. It is dangerous to become complacent especially with the

design of new implant products. Surgeons are still reluctant to do hip replacement operations

on people who are under sixty years of age. The implant cannot be one hundred percent

guaranteed against failure under very active conditions. This is a materials problem that

needs to be addressed. This report concentrates on the selection of the best cumulative

damage theory to apply to vitalliumtm , in order to predict when failure is likely to occur

when the material is in a pre-stressed condition.


3

The study examines certain areas of the fatigue failure of heat treated cast vitalliumtm, heat

treated machined-cast vitalliumtm and hot isostatically pressed (HIPed) heat treated cast

vitalliumtm.

These areas include

1. Determination of S-N curves.

2. Notch sensitivity analysis.

3. Cumulative damage analysis.

4. Effects of mean stress (Fatigue Strength Diagram).

5. Microscopic and macroscopic examination.

The literature review explains each aspect of fatigue, which was considered relevant to

vitalliumtm. The theory section breaks down each cumulative damage theory that was used

and shows how to apply the theory for life prediction purposes. The chapter entitled

‘Analysis of Data’ is basically a summary of how the results were analysed to produce

design charts and design data for vitalliumtm.

The investment casting, hot isostatic pressing and X-raying was conducted by Howmedica

Limerick (Pfizer Corp.). The machining, microscopic analysis and fatigue testing were

conducted by the author in the University of Limerick materials laboratory.

Objectives Cumulative Damage Analysis – Kieran J. Claffey

4

Chapter 2

2.1 Objectives

The main objective of this report is to develop an extensive understanding and empirical

knowledge about the fatigue behaviour of the surgical alloy, Vitalliumtm. In particular, to

determine a suitable cumulative damage theory for application to this material. The previous

researcher [29] provided a basis for the study by producing an S-N curve and cumulative

damage results for machined Vitalliumtm.

The primary objectives are shown below.

1. To plot a reliable S-N curve for cast vitalliumtm by conducting experimental fatigue tests.

2. To conduct a cumulative damage analysis so as to determine the best life prediction

theory for vitalliumtm.

3. To conduct a notch sensitivity analysis.

4. To produce an S-N diagram for notched cast vitalliumtm and to evaluate the affects of

geometric discontinuities on fatigue behaviour.

5. To produce a fatigue strength diagram for cast vitalliumtm.

6. To determine and quantify whether hot isostatic pressing (HIP) treatment affects the

fatigue resistance of cast vitalliumtm.

7. To evaluate the affect of machining on the fatigue behaviour of cast vitalliumtm.

8. To conduct a microscopic and macroscopic examination of the fatigued surface.

Literature Review Cumulative Damage Analysis – Kieran J. Claffey

5

Chapter 3

3.1.1 Introduction to Fatigue Behaviour

Fatigue results from repeated plastic deformation on a microscopic level. Without

this repeated plastic yielding, fatigue failure could not occur [1]. This is illustrated

below in figure 3.1. The area around the notch tip behaves in a plastic manner while

the rest of the material behaves elastically. It is better for the engineer to design a

component, which is subjected to cyclic stresses, with fatigue data, as opposed to

tensile data because failure may occur well below the yield strength of the material.

Figure 3.1 Illustration of the plastic behavior at a fatigued area.

Local plastic yielding may strain harden a material and prevent the growth of a crack.

If yielding is any more than this it causes a loss of local ductility. The resulting cyclic

strain causes failure.

3.1.2 The Three Stages of Fatigue

Fatigue is widely believed to occur in three separate stages. The first stage is known

as crack initiation or crack nucleation. Crack initiation is caused by a phenomenon

known as dislocation. A dislocation is a fault where half-planes of atoms are missing

within a crystal [2]. These dislocations travel along crystallographic planes until they

reach the surface where they form a minute step. This microscopic step acts as a stress

raiser and a crack initiates. The second stage is known as crack propagation. This

basically is the development of a crack across a surface caused by repeated loading.


6

A loss of ductility is associated with this stage. The third stage is a catastrophic

failure. This occurs when the decreasing cross-sectional area, can no longer support

the applied load.

3.1.3 Fatigued Surface Characteristics

A failed surface can be examined, to determine whether it failed due to fatigue or not.

There are three distinct areas to look out for. The first is a small smooth area where

the crack begins to initiate, the second is characterised by fibrous beach marks caused

during crack propagation, as shown below, and the third is a crystalline (bumpy) area

where final fast failure occurs.

Figure 3.2 Beach marking transitions to a suddenly failed crystalline section.

3.1.4 The S-N Curve

This is a graphical relationship between strength and number of cycles to failure. It

can be a logarithmic or semi-logarithmic plot. The semi-log plot is more often used

because it is much easier to read stresses on the ordinate compared to a logarithmic

plot.

The fatigue limit is the threshold stress at which failure will no longer occur no matter

how many bending cycles are applied. It is also commonly known as the endurance

limit.

The fatigue test is a simple tensile test in which a load, less than the ultimate, is

applied and released cyclically. A large magnitude of specimens are subjected to this

test with a different load for each case. The data from these tests are used to provide


7

data points on the S-N plot. The curve of best fit is drawn, which gives 50%

reliability.

Figure 3.3 Experimental S-N plots, illustrating different levels of reliability.

The scatter is due to the statistical nature of fatigue, and remains despite taking care to

make all specimens identical. At 106 cycles the statistical spread can be taken as

Gaussian with a standard deviation of 8%. If the design stress is reduced by 8%, the

probability for survival for 106 cycles is 0.841. However, if the design stress is

reduced by three standard deviations (24%), a theoretical reliability of 99.9% can be

achieved.

This means the strength reduction factor for reliability ‘CR’ has a value of 0.76.

Therefore, the reliable stress for a design that is to have a safe life of N cycles is

σα = CR SN Equation 3.1

3.1.5 Determination of S-log N Curve

It is desirable to apply forces in the initial tests which result in a final life (fatigue

cracked specimen). These forces are selected on the basis of past experience of the

ratio between fatigue strength and tensile strength of the material. For wrought

aluminum, copper and nickel alloys, the ratio of the fatigue strength at 107 cycles to


8

the ultimate tensile strength is typically 0.35 to 0.5 in a completely reversed bending

test with stress ratio R = -1.

Figure 3.4 Typical S-Log N curve (Wohler curve).

Refer to figure 3.4. The scatter in fatigue strength corresponding to a given life, is

small. The scatter in fatigue life corresponding to a given stress level, is large. It is

therefore important to pick the correct stress level for each test. Frequency has a

negligible effect on the fatigue life of most metallic materials, except at frequencies

greater than 1000 Hz or at temperatures where significant creep occurs during each

cycle. This is useful since accelerated testing can be used to explore failure

conditions [3].

3.2.1 Fatigue Tests

There are four different types of fatigue tests which may be carried out on a material.

1. Rotating bending (R.R Moore test).

2. Reversed bending.

3. Reversed axial loading (push-pull).

4. Reversed torsional loading.


9

A completely reversed bending test is carried out in this analysis.

3.2.2 R.R Moore Fatigue Test

This is a strict standardised fatigue test which is used to determine the S-N curve

(Wohler curve) for a material under rotating bending conditions. In rotating bending,

the maximum stress acts all around the circumference. Fatigue failure will occur at

the weakest point of the circumference. In reversed bending the maximum stress acts

at the top and the bottom of the test specimen. There is a statistical probability that

cracks will not initiate at the place of least cross-sectional area. This means that the

fatigue strength in reversed bending is slightly greater than in rotating bending.

Reversed axial loading subjects the entire cross-sectional area to the maximum stress,

therefore giving approximately 10% lower fatigue strengths than the R.R Moore test.

These tests can serve as a strong basis for the fatigue knowledge of a material.

3.2.3 Strain Controlled Tests

Some structures such as aeroplane cabins and pressure vessels are subjected to

infrequent large strains which may exceed the elastic limit. The materials S-N curve

is often invalid for this type of loading because the large strains are not taken into

consideration. Specialised hydraulic fatigue testing machines are used to conduct

strain controlled tests to produce a relationship between strain and the number of

cycles to failure. There are changes to the hysteresis loop in the stress-strain plot

under cyclic loading. For example, an annealed metal will increase in stress and

undergo cyclic softening, whereas a hardened cold worked metal will decrease in

stress and undergo cyclic hardening [4].


10

3.3.1 Mathematical Relationship between Strength and Number of

Cycles

To save testing time, a mathematical relationship between stress and cycles to failure

can be used instead of the S - N curve. When the number of cycles is greater than 106

the endurance limit is significant. Between 103 and 106 cycles the straight line

formula can be used to find the endurance stress for a given life.

S SS

SN

LogN

10

10

10

6

36

3

6

Equation 3.2

Or if the safe number of cycles for a given stress is required the formula transposes to

N

S N

S

S

S

10

6 3106

103

106

log

log

Equation 3.3

These formulae can only be used when the endurance stresses are known at 103 and

106 cycles and only apply to materials which show an asymptotic relationship on a

S –log N curve at 106 cycles.

It is common practice to use static design data between 1 and 10 cycles because the

strength reduction is considerably small when compared to cycles above 103 cycles.

A safety factor is introduced to compensate for the cyclic nature of the load.

3.3.2 The Basquin Relation

This is a method of determining the life of a material when subjected to a high stress

in low cycle fatigue. That is, from 102 to 105 cycles. The Basquin relation is

expressed as follows.


11

F

c

fNS )2( Equation 3.4

Where σF = The true stress for fracture in tension

c = exponent lying in the range 0.05 to 0.12.

Once the exponent ‘c’ is known the number of cycles to failure ‘Nf’ can be

determined.

3.4.1 Combined Alternating and Mean Stress

The majority of strength problems involve a combination of steady and alternating

stresses. This means that components are subjected to fluctuating stresses as

illustrated below.

Figure 3.5 Illustration of how stress alternates with time for zero mean stress and mean stress conditions.

3.4.2 Constant Life Diagram

A constant life diagram allows one to determine the new safe value for the cyclic

component of alternating stress when a mean stress is applied. The application of a

mean stress, results in a lesser allowable alternating stress. This stress needs to be

known. The constant life diagram is then superimposed on another plot, on which

maximum stress is the ordinate and the minimum stress is the abscissa to produce


12

what is known as a fatigue strength diagram. An example of one such diagram is

shown below [5]. The chart can be used if the application of a mean stress is

unavoidable or if it is the desired effect.

Figure 3.6 Fatigue strength diagram for 7075-t6 aluminum alloy, Su = 82 Ksi, Sy = 75 Ksi [5]

If a known mean stress is applied, and if the maximum stress and the required life

cycle of a component are also known, then the maximum allowable alternating stress

can be determined using the fatigue strength diagram. Having mean stress correction

has the effect of increasing the materials chances of yielding before actual failure. It

reduces the possibility of catastrophic failure due to mean stress.

3.5.1 Notch Effects

Most components contain necessary discontinuities in their structure such as holes

and edges. Collectively, these are known as notches and reduce the fatigue resistance

of components. It is necessary, therefore to account for this effect. The study of this

is called a notch analysis. A sharp notch is considered to have a radius smaller than

0.25 mm whilst a blunt notch has a radius of approximately 2 mm. Sharp notches can


13

initiate cracks at low energy whereas blunt notches need more energy to initiate

cracks.

Two types of test exist

To assess notch sensitivity as a material property.

To generate information for design purposes.

3.5.2 Notch Sensitivity, Stress Concentration and Fatigue Notch

Factors

The introduction of a notch causes a stress concentration in one local area. The stress

at the notch is higher than the nominal stress throughout the component. For

example, consider the S-N curve below.

Figure 3.7 S-N curve illustrating how the introduction of a notch can affect the fatigue life.

The fatigue life of the plain material with no notch is given by ‘N1’. The material

would not last as long as long as N1 if a notch were to be introduced. Therefore, the

life of the notched material is given by ‘N2’ at the new higher local stress. In reality,

this is not the new fatigue life. The introduction of the new local higher stress does

not reduce the fatigue limit as much as it was thought it would. This is where the

fatigue notch factor and notch sensitivity come into play. The notch, in actual fact

has an equivalent stress level at ‘N3’.


14

The notch sensitivity is defined as

q = Kf - 1 Equation 3.5

Kt - 1

Kf = Fatigue notch factor

Kt = Stress concentration factor

Note: When q = 0, implies no notch effect.

q = 1, implies full theoretical notch effect.

nf

f

fK

Equation 3.6

f = Fatigue limit of unnotched material

nf = Fatigue limit of notched material

Note: It is desirable that notch effects do not adversely change the fatigue properties.

Therefore, one wants to keep the fatigue notch factor ‘Kf’ as close to unity as

possible. The fatigue notch factor is always lower than the stress concentration factor.

The estimated fatigue notch factor ‘Ktf’is used for design purposes, only when the true

notch factor is unknown. It represents a calculated estimate of the actual fatigue

notch factor ‘Kf’.

Ktf = q(Kt - 1) + 1 Equation 3.7

Note: In high strength steels the effect of small holes or scratches is more

pronounced than in steels of lesser strength. Vitalliumtm is a high strength material

which is susceptible to notch effects of small holes. This is the reason why such

small notch radii were chosen for the notch analysis. It can be seen from the graph q

vs. r, (figure 3.8) that materials (especially brittle materials) are more notch sensitive

at small radii.


15

Figure 3.8 Notch sensitivity curves for steels of different hardness and an aluminum alloy [1].

It is difficult to find ‘q’ because the notch sensitivity factor ‘Kf’ must be known first.

However the approximate nature of ‘q’ as shown in the above graph, can be found

using the following formula.

r

aq

1

1 Equation 3.8

Where a = material constant

r = notch radius

Note: The softer the material is; the higher the value of ‘a’.

a = 0.02 for aluminium alloy.

The notch sensitivity ‘q’ and the material constant ‘a’ are unknown for vitalliumtm.

One of the main purposes of doing a notch sensitivity analysis is to determine the

proper ‘Kf’ fatigue notch factor for vitalliumtm as opposed to using stress


16

concentration factors. This allows us to accurately quantify the effects of notches on

material properties.

3.5.3 Tensile Notch Sensitivity

There exists another definition of notch sensitivity which is obtained from material

tensile data. It is defined by the reduction in ductility of an area that has a tri-axial

stress field [6].

Notch strength ratio (NSR) = -

Equation 3.9

Where SULT-n = ultimate tensile strength in notched specimen.

SULT = ultimate tensile strength in plain material (unnotched).

If NSR < 1 => notch sensitive material

> 1 => notch insensitive material

3.5.4 Multiple Notches in a Plate

It is known that a single notch represents a higher degree of stress concentration than

a series of closely spaced notches of the same kind.

Single notch Multiple notches

Figure 3.9 Stress contours associated with multiple notches.

Sometimes it is preferable to introduce multiple notches into the design of a

component so as to avoid having high stress concentrations with single notches.

When doing a notch sensitivity analysis single notches are preferred because they

produce higher localised stress concentrations.


17

3.6.1 Cumulative Damage Analysis

Cumulative damage is the damage that accumulates in a material after it has been

subjected to various stresses, for an arbitrary number of cycles. It involves the

prediction of the actual damage that has occurred, in parts stressed above the

endurance limit at different stress levels and the remaining life of the material at a

particular stress level. It is, at best, a rough procedure due to the large scatter band of

failed specimens. Several cumulative damage theories have been developed, most of

these in the 1950’s and 60’s. There is no absolute correct theory. It is up to the

engineer or scientist to determine whether any theory may apply and to determine the

most suitable theory for a particular material.

3.6.2 Cumulative Damage Theories

There are several damage theories available, some of which are more suitable to

particular materials than others. Miner’s rule is the most simple cumulative damage

theory. It suggests that when the sum of the cycle ratios equals unity, complete

damage, which is failure, will occur.

n/N = 100% Damage ≈ 1 Equation 3.10

3.6.3 Two Level Step Tests

This is a test method used to model the action of a material which is subjected to

varying loads above the fatigue limit. The specimen can be stressed high, then

stressed low until failure occurs (high-low test) or it can be stressed low, then stressed

high until failure (low-high test). The cumulative damage theories are then used to

predict the number of cycles to failure at the second stress level. The Avery-Denison

fatigue testing machine in UL does not have the facility to apply fluctuating loads that

can be repeated for sustained periods. This method of testing acts only as a model to

the real situation.


18

3.7.1 Damage Tolerant Design

Damage tolerant fatigue design ensures that cracks will not propagate to failure either

within the design life or between inspection periods. It was introduced for the design

of military aircraft where it was impractical to adopt the fail-safe procedures used for

commercial aircraft. The concept was that an assumed pre-existing defect would not

propagate to failure between two inspections [7], and it depends on the application of

fracture mechanics.

Fatigue strength calculation is based on the determination of the fatigue or threshold

stress which corresponds to a particular life (S-N curve) whereas linear elastic

fracture mechanics (LEFM) is based on the determination of fracture stress which

corresponds to a particular crack length. LEFM has an advantage over fatigue

strength calculation which is that it is more useful to a maintenance engineer. A crack

may develop, due to unusual loading circumstances, which could possibly fail at a

fracture stress below the fatigue threshold stress. Fatigue strength calculation will not

take this into account but fracture mechanics will.

3.7.2 Linear Elastic Fracture Mechanics (LEFM)

The durability of a component is governed by the rate of degradation of the load

bearing capacity as a result of sub-critical crack growth. The sub-critical crack is

defined as a crack that is smaller than the critical crack length for a particular stress.

LEFM describes the useful life of a component as a function of the materials

subcritical crack growth resistance [8]. This mathematical relationship is known as

Paris's Law.

da

dNC K

n Equation 3.11

K is the change in the stress intensity factor and can be measured experimentally.


19

It can be expressed in terms of geometry, stress and crack size as

aYKf Equation 3.12

Where Y is a function of the geometry of the instantaneous crack size ‘a’ and the

specimen thickness ‘B’, [Y= f (a/B)].

For a given crack size, a specific critical failure stress is defined by its fracture

toughness ‘Kc’or failure locus (figure 3.10 below). Conversely, for a given stress, a

critical crack size exists.

Figure 3.10 Schematic representation of failure locus.

3.7.3 Thickness Considerations for LEFM

The failure locus is different for different materials of varying thicknesses. For

example, Harrison [9] applied an LEFM analysis to the fatigue behaviour of

transverse non-load carrying fillet welds and discovered that the fatigue strength tends

to decrease with increasing thickness (B). Refer to figure 3.11


20

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

1 10 100

Thickness, mm

Re

lati

ve

Fa

tigu

e S

tre

ngth

Square butt w elds

Fillet w elds

Figure 3.11 Influence of plate thickness on fatigue strength of weldments [9].

The reason the thickness effects the failure locus is because ‘Y’ is a function of (a/B).

Thus, the high stress region extends further for a thick plate than for a thin one.

Therefore, for two weld joints with the same initial crack size, but with different plate

thicknesses, the ‘Y’ factor and hence the stress intensity factor (K) will be greater for

the thicker plate, causing the crack to propagate faster in the thicker plate. The same

situation exists for pressure vessels in that fatigue strength will decrease with

increasing thickness. However, in bridge girders, (I - beams, T - beams, J - beams),

the fatigue strength will increase with flange thickness.

LEFM thickness predictions have been investigated experimentally and have

correlated reasonably with conventional fatigue test data.

The mode one stress intensity factor ‘K1’ varies with thickness ‘B’ but the critical

stress intensity factor ‘K1c’ is a material property that has a critical value according to

equation 3.13. Refer to figure 3.13 below.

BK c

Y

2 5

1

2

.

Equation 3.13


21

Figure 3.12 Illustration of how material thickness affects the stress intensity factor.

The critical stress intensity factor ‘K1c’ is more commonly known as the fracture

toughness parameter. To perform a valid test to find K1c the thickness B must be in

agreement with equation 3.13 above.

3.7.4 Determination of Fracture Toughness (K1C)

The use of alloys with high fracture toughness can act as a safeguard against

catastrophic failure towards the end of the design life of a component. The

determination of the fracture toughness provides an insight into the materials ability

to resist failure after crack nucleation and propagation. The figure below shows an

example of typical test procedure (Figure 3.13). A specimen is cyclically excited

with constant amplitude load. Two such loads () yield two different fatigue crack

growth curves for a titanium alloy.


22

Figure 3.13 Constant load amplitude fatigue crack growth curves for a mill annealed Ti-6A1-4V alloy tested in

vacuum at room temperature.

Paris's Law relates the slope (da/dN) of figure 3.13 to the stress intensity range (K).

K can be found from the equation

K a1 . Equation 3.14

and da/dN can be obtained from figure 3.13. A plot with da/dN as ordinate and K1

as abscissa (figure 3.14) yields the fracture toughness value, K1C.

Figure 3.14 Relationship between the rate of crack growth and fracture toughness.


23

The value of the fracture toughness on the K1 (fracture toughness) axis occurs when

the slope of figure 3.14 goes to infinity.

3.7.5 Life Prediction using LEFM

Paris's Law, [da / dN = C (K )n] can be solved to yield a life prediction, once the

initial crack length ai and the final crack length af are known. Through integration by

separation of variables equation 3.11 becomes

dN

C Kda

N

N

na

af

i

f

0

1

Equation 3.15

But if equation 3.14 is substituted into equation 3.15, the equation then becomes

dN

Ca da

N

N

nn

n

a

af

i

f

0

1

2

2

Equation 3.16

The constants C and n can be found in the following manner

Take the logarithm of Paris's equation to yield a linear equation

log (da/dN) = log C + n log K Equation 3.17

The values of C and n can be obtained from the plot of log (da/dN) vs. log K. as

shown in figure 3.15.

Figure 3.15 Demonstration of method used to determine the constants ‘C’ and ‘n’.

The equation which allows the calculation of the number of cycles to failure is

simplified to give this final equation.

NC

a

nf n n

n

a

a

i

f

1

1 22

1 2

Equation 3.18


24

It should be noted that LEFM can only be utilised in materials which exhibit a

relatively linear elastic relationship according to Hooke's Law ( Elastic-plastic

fracture mechanics [EPFM] has been developed to deal with materials that act in a

plastic nature around the crack zone. EPFM is the preferred fracture mechanics

method when dealing with materials that tend to work-harden during cyclic loading.

3.7.6 Randomly distributed small crack data method of fatigue

evaluation

Conventional fatigue testing using S-N curves and fracture mechanics are the better

known methods of fatigue damage measurement. Another method, developed by

Kitagawa et als, utilises small crack data, which are randomly distributed throughout

test specimens [10]. Their method incorporates the probabilistic laws that cracks

follow and a quantitative method for the fatigue damage evaluation and fatigue life

prediction of materials. It is based on the growth rate of distributed small cracks and

on the crack length distribution function. This function is dependent on geometry,

initial crack size and the type of loading involved.

The first step is the preparation of test specimens having small cracks randomly

distributed, in a deliberate manner, over their surfaces. The second step is the

sampling of statistical data on the cracks from these specimens and from real

structures. Finally, the third step is fatigue damage evaluation and life prediction

based on various statistical treatments of the obtained data. This method of fatigue

damage evaluation has yielded good results for some steels. It is not used very much

in practice because of the lack of experimental data available.

3.8.1 The Finite Element Method (FEM)

Fatigue testing can be quite long and expensive. Any method capable of predicting

the outcome of these tests in the early stage of the design process would be useful.

Viceconti et als created three different finite element computer models, of increasing

complexity, to predict the stress field in a prosthetic hip implant.


25

The first model was a linear model with all the material assumed to have an elastic

behaviour. In the second model the non-linear behaviour of the stem interface was

taken into account by incorporating simple gap elements. In the third model the

bonding cement was modelled as an elastic-perfectly plastic material. Their study

highlighted the important point that FEM cannot be blindly used to predict failure

because the first two FE models failed completely to predict the actual stress field

with only the most complex model achieving an acceptable degree of accuracy.

Figure 3.16 Finite element model of a hip prosthesis [11].

McNamara et als investigated the load transfer (stress shielding) of press fitting and

full bonding of hip prosthesis by comparing a finite element model with an

experimental model [11]. They discovered that press-fitting a stem does not provide

the same stress shielding effect as obtained by gluing. The FE method predicted

strains which correlated well with the experimental strains. In this case the FE

method was accurate.

The problem associated with FE is not the actual computer package but the wisdom of

the engineer who makes the simplifying assumptions. Often it is impossible to be

able to preconceive actual material behaviour. It is always recommended to conduct

experiments and not to rely on FEM alone.


26

3.9.1 Information Required for Damage Assessment

Before the fatigue life of an actual component can be obtained, the working stress and

the cycles of operation must be known. This may often be difficult as it involves

precise instrumentation and experimental work, regarding the operation. S-N curves

are required for the particular part in question. An example of which occurs when

conducting a cumulative damage analysis of a hip prostheses. This involves

producing an S-N curve for the actual component type and applying the damage

theories. It is necessary to take component geometry into consideration in this

instance because there is a critical stress concentration approximately 5 mm below the

level of fixing cement in the human femur. However, if the critical region has a

negligible stress concentration, S-N Curves of the basic material can be used.

3.9.2 Damage Assessment

Cumulative damage theories are at best a rough approximation of the damage that

actually accumulates during the working life of a component, because the nature of

fatigue, in itself, is random but more importantly the definitions and evaluation

methods for damage are a much debated question [12]. Cumulative damage theories

provide a means for life prediction based upon accumulated damage over time.

The damage methods may be divided into two categories:

1) Damage is considered as the presence of defects.

2) Damage is considered as the alteration of a property.

The first method presents difficulties in that several kinds of defects exist at the same

time ( geometry, size, and inclusion defects for example ) The problem of adding the

effects of different kinds of defects to obtain ‘total damage’ has not yet been solved

[12] although LEFM methods have produced some promising results for variable

loading conditions.


27

Attempts have been made to predict fatigue life under variable - amplitude loading

using LEFM methods. For randomised load spectra, LEFM models are reasonably

successful in predicting trends in the rate of fatigue crack growth and fatigue life.

The rate of crack growth may be characterised reasonably well by using the root-

mean-square value of the stress intensity range (Krms) [9]. Life prediction for

random load spectra may then be possible by replacing K by Krms in equations 3.11

and 3.16.

Failure conditions must be checked along the way by comparing the crack size ‘a’

against the critical crack size ‘af’ for the current maximum load.

For ordered loading spectra, such as in two level step tests, an iterative procedure

developed by Wei and Shih [8] appears to work. The general form of Paris's law is

preserved, but is modified by adding numbers of cycles of delay ‘ND’, where no crack

growth takes place. These delay cycles are illustrated in figure 3.17 below and they

are determined by experimentation.

Figure 3.17 Schematic illustration of delay in fatigue crack growth (definition of delay cycles, Nd).

Their results [8] suggest the assumption, that damage is directly related to defects, is

correct. This method is long winded however as the number of delay cycles must be

found experimentally for the magnitude of each loading level, whether it is high- low

stressing, or low-high stressing. The second method damage being considered is the

alteration of a material property. Some properties which have been correlated with


28

damage evolution include resistivity, ductility, fatigue limit, Young's modulus,

mechanical energy dissipation and plastic strain.

When the second method is used, the damage correlated with various properties

shows a non-linear characteristic with respect to cycle life ‘β’, (Figure 3.18 below).

Figure 3.18a. Figure 3.18b.

Figure 3.18 Damage accumulation curves for two level tests. (a) Without interaction

(b) With interaction

The forms of the curves are dependent on the loading level; the higher levels

correspond to a higher damage level.

Consider a two level step test. Proposed test models may be divided into two cases:

[12]

1.) There is no interaction of the first level upon the second level. (Figure 3.18a).

2.) There is interaction of the first level upon the rate of damage accumulation of

second level. (Figure. 3.18b).

For the first case, at S1, damage accumulates along path OAC until point A is reached,

where the stress level is changed to S2. The damage then begins to accumulate along

path OBC, which is the path along which damage would have accumulated if the

stress had been equal to S2 all along. However, for the second case (Fig. 3.18b);

damage accumulates along path BD after the stress change. This case is what actually

happens in reality, especially when the first stress is greater than the second stress

(high-low) in a two level step test.


29

Pluvinage and Raguet [12] investigated the affects of eight different damage

properties for low cycle fatigue. They defined damage as

DX X

X X

n

R

0

0

Equation 3.19

Where X0 is the value of the considered property for the virgin material, and XR is the

value of the considered property after N cycles of strain.

Their results suggested that the measurement of fatigue damage depends on the

properties that are used as damage indicators (Figure 3.19 below). Clearly the

damage indicator affects the cycle ratio ‘β’ and the overall life prediction.

Figure 3.19 Damage evolution curves measured by 8 different methods.

From figure 3.19 it can be seen that the eight properties can be put into three

categories.


30

1) Those which are sensitive to the formation of macrocracks ( Elasticity modulus

and electrical resistivity )

2) Those which are sensitive to the migration of dislocations ( density and

Brinell hardness ).

3) Those which are sensitive to both of the above (Ductility, Limit of

Reversibility, load drop during strain controlled tests).

Pluvinage et al [12] suggested that the examination of the evolution of a property

seems to be easier than the defect method for a quantitative measurement of fatigue

damage.

3.10.1 Ductile Materials

Ductile materials can be deformed considerably by tension before fracture will occur.

As a general rule, ductile materials do not suffer loss of strength due to steady

increasing uniaxial load. In completely reversed bending fatigue tests they tend to

form pockets or indentations at the surface. Ductile materials are more fatigue

resistant by nature than brittle materials [2]. Cast vitalliumtm has a 15% elongation

before ultimate failure and is considered to be a ductile material.

3.10.2 Brittle Materials

Brittle materials are materials that are typically hard but not tough. Experience has

shown that it is wise to apply the full stress concentration factor ‘Kt’in the design of

brittle material components, despite the fact that experiments show that the full effect

is not usually obtained. The full ‘Kt’factor is used to compensate for poor shock

resistance and tends to give conservative estimates for failure criteria. Conservative

estimates are quite satisfactory when dealing with notches in brittle materials due to

their inherent ability to fail in a catastrophic manner.


31

The Mohr theory is suggested for design purposes for brittle materials subjected to

alternating stress [13]. The factor of safety for bending loading (normal stress) is

given as,

n = f / Kt a Equation 3.20

3.11.1 Investment Casting of Hip Implants and Fatigue Test Specimens

Vitalliumtm is an expensive material which warrants ‘casting to size’ to eliminate

machining waste. When large quantities of cast products, under 2 kg in weight, are

required, investment casting is ideal. The fatigue test specimens were cast and then

machined and ground to size.

A master mould, in two separate halves, is produced to manufacture the wax patterns.

This metal mould can be machined or cast using a low melting point alloy around a

pattern. A typical example of a wax mould making process can be broken into five

steps below.

1. Molten wax is injected into the clamped master mould at a pressure of about

3.5 MPa.

2. When the wax has solidified it is removed from the metal mould and its gate is

trimmed.

3. The wax pattern is attached to a central wax runner. This is repeated until the

required number of castings is reached and produces a ‘wax tree’ which is then

attached to a bottom plate.

4. An open metal flask is placed over the assembly and a liquid investment

material is poured into the flask. The bottom plate vibrates to bring entrapped

air bubbles to the surface during investment solidification. A typical

investment material is a mixture of fine ‘sillimanite’ sand and ethyl silicate.

5. It takes about eight hours for the investment to dry. The bottom plate is

removed, the flask is inverted and is passed through an oven at approximately

150°C. The wax melts and leaves a cavity which is then used as the mould for

the original required parts.


32

Figure 3.20 The investment mould making process [2].

3.11.2 Hot Isostatic Pressing

Hot isostatic pressing is a process of bonding materials at high temperatures with high

pressures acting on all surfaces simultaneously and equally. It is used to heal internal

casting defects and to reduce scrappage. The work piece is placed in a furnace which

is contained within a pressure vessel as shown below [14].


33

Figure 3.21 HIP vessel with work piece [14].

The work piece is then pressurised with an inert gas and heated to a softened state.

The gas pressure exerts a large force equally distributed over the part to attain

absolute density. When healing defects in castings, temperatures generally range from

500°C to 1300 °C and pressures from vacuum to 200 MPa.

Companies, such as Howmedica, who conduct high integrity casting, can accrue the

following advantages utilising HIP.

1. Repair of subsurface defects.

2. Improved mechanical properties which reduce rejection rates and inspection

frequencies.

3. Elimination of unwanted porosity to significantly reduce premature failure.

Repair of used turbine blades, such as the one in figure 3.22, by HIP has been

successful on both stationary and aircraft turbine engines. The cost to HIP is much

less than the cost to manufacture new turbine blades.


34

Figure 3.22 Turbine blade repaired by HIP [14].

3.11.3 HIPing Investment Castings

Precision casting of cobalt- base superalloys to near net shape is well established as a

means of producing fasteners for insertion into the human body. The mechanical

properties and microstructures are expected to be those of the perfect as-cast material.

The fabrication of investment castings is such that they are prone to developing

residual casting defects which effect mechanical properties [14]. An example is that

of microporosity which is readily formed from shrinkage, during cooling, in castings.

The application of HIP has changed the integrity of investment castings which can

now actively compete in net shape technology of high performance forged parts. It is

used extensively for full densification of investment castings.

For the production of prosthetic bone implants and fasteners, it is absolutely critical

that material with casting defects does not pass through the system and break in the

patient’s body. Casting defects are inevitable. This entails a large amount of

scrappage which is rather costly. HIPing is expensive, yet it offsets scrappage and re-

manufacturing costs in most cases.


35

In HIP cycles for castings, restrictions on the temperature range are influenced by the

strength of the material (minimum temperature) and the occurrence of melting

(maximum temperature). The range may be as little as 30 °C. Temperature control is

more critical than pressure or dwell time control. Frequent sampling of gas purity is

necessary during a typical cycle to prevent gas inclusions.

A typical HIPing process is as follows.

1. Castings are loaded into a container which is then transferred into a pressure

vessel, surrounded by insulation.

2. The cold system is closed, evacuated, flushed with argon and re-evacuated.

3. The vessel is then pressurised and the furnace is started. The operating

conditions in terms of temperature and pressure are controlled until they reach

the desired values simultaneously.

4. The dwell time at peak pressure and temperature typically varies between two

to four hours. The power is shut off and the system is allowed to cool by itself

[14].

3.11.4 HIPing - The End Result

The application of HIP to castings has the effect of healing the microstructure. This

has a knock on effect of improving the mechanical properties of castings.


36

Figure 3.23 Illustration of the microshrinkage that can appear in titanium castings which can be closed

by HIP [14].

HIP has been proven to increase the stress rupture life under creep tests, the fatigue

life under fatigue tests and tensile strength for most alloys. Also, mechanical

properties are improved in castings that have no detectable defects.

3.11.5 Fatigue Resistance due to HIP

Fatigue data under high cycle (30 Hz) and low cycle fatigue (0.17 Hz) is illustrated

below for a titanium alloy.

Figure 3.24 High cycle fatigue properties of titanium alloy before and after HIP [14]


37

Figure 3.25 Low cycle fatigue properties of titanium alloy before and after HIP [14].

It can be seen that the HIPed specimens give fatigue lives to the right and above the

mean S-N curve.

A typical set of fatigue test results is shown in table 3.1 for the alloy RENE 120. In

this case HIPing improves the fatigue resistance.

As-cast HIP 4h @ 103 MPa

1177 0C 1204

0C

2487 8,543 24,919

Cycles to failure 2300 10,933 13,052

773 13,685 5,723

1853 11,053 12,077

Table 3.1 Effect of HIP on RENE 120 Fatigue Properties [14].


38

3.12.1 Specimen Design

Bending specimens may be round or flat and maybe tested under plane bending or

rotating bending conditions. Plane bending specimens are usually flat and may be

tested as beams or cantilevers. Examples of flat test specimens [15], according to BS.

3518, are shown in figure 3.26 below. The test specimen may be subjected to a

uniform or varying moment.

Figure 3.26 Typical bending test pieces of rectangular cross-section, outlined by BS. 3518 [15].

It is desirable to keep the continuous radii ,‘r’, as large as practicable in order to

minimise stress concentration effects. The continuous radii are necessary in order to

ensure fatigue cracking close to the centre of the specimen. The design of the

gripping portions is dictated by the clamping arrangement of the particular machine.

3.12.2 Specimen Preparation

It is essential that uniform specimen preparation procedures are practiced. The

procedures should be carefully specified for the various stages of rough machining,

finish machining and polishing [16]. Machining should be longitudinal on flat

specimens. To obtain the required thickness of cast specimens it is necessary to

perform gradual grinding operations. Sharp edges can be removed by breaking them

by hand to an approximate radius of 0.125mm, then using a fine emery paper (no.

500), moving in the longitudinal direction in order to leave no harmful scratches.

Notched specimens can be prepared by rough machining or grinding of notches,

depending upon the strength of the material and cutting tools available. It should be

noted that notched specimens are designed so that failure will occur at the notched


39

cross-section. The surface preparation of the remainder of the test section is therefore

not as important as in the case of unnotched specimens [16].

Polishing is a cutting process used to remove scratches. It is not a smearing operation

which merely smudges scratches into themselves. If a polishing process smears

scratches, then they still exist. So even, if there is a perfect mirror finish it is not

guaranteed to be scratch free. For most non-ferrous specimens a machined surface is

often more desirable than a polished surface because it is easily produced and it

represents a practical condition. Experience has shown that complicated polishing

procedures are no more effective in obtaining fatigue values, than in a uniform

standard finish. However, it is sometimes necessary to polish specimens, so that the

surface finish is similar to that of the desired finished product. Polishing methods in

use have roughness limits as low as 0.05µm and as high as 0.8µm. Notched

specimens may be polished by means of a rotating member immersed in an abrasive.

Two successively finer grades of abrasive slurry are used, for example, 280 grit in

SAE 30 oil and then 3F grit in SAE 30 oil.

3.12.3 Machining

The tolerance on the thickness of the test piece is X + 0.5mm. Test pieces should be

ground to size in the following manner.

0.025mm depth of cut to 0.1mm oversize.

0.005mm depth of cut to 0.03mm oversize.

0.0025mm depth of cut to size.

The sequence of polishing is arranged so that the last paper used is 600 grade

waterproof silicon carbide paper.

Life Prediction Theory Cumulative Damage Analysis – Kieran J. Claffey

40

Chapter 4

4.1 Miner's Cumulative Damage Theory

This theory adopts two major assumptions [17]:

1.) The loading cycle is sinusoidal, (figure 4.1).

2.) The total amount of work that can be absorbed produces failure (i.e. no work

hardening occurs).

Figure 4.1 Assumed sinusoidal loading cycle.

W = work absorbed at failure.

W1 = work done at n1cycles.

N1 = number of cycles to failure at stress S1.

n1 = number of cycles applied at S1.

Figure 4.2 Typical S-N curve illustrating the effects of cumulative damage for two

different levels of stress [17].


41

If W represents the net work absorbed at failure, then

w

W

n

N

w

W

n

N

1 1 2 2 ; ………….etc. Equation 4.1

and w1+ w2+w3+…..wn=W (at failure)

Hence w

W

w

W

w

W

w

W

n1 2 31 ...... Equation 4.2

Substituting values of equation 4.1 into equation 4.2.

n

N

n

N

n

N

n

N

n

n

1

1

2

2

3

3

1 ...... Equation 4.3

Or n/N = 1 Equation 4.4

Miner experimented with 245 - T Alcad aluminium at two or more stress levels, with

constant stress ratio, and found that the average test value for n/N was 0.98.

He discovered that all loading cycles were significant in the eventual failure of

materials. To prove this he based experiments on the assumption that only the final

stress cycling caused failure. The average value of n/N was 0.37 in this

circumstance, which indicated that the damage from the other loading cycles cannot

be ignored.

Miner also experimented with aluminium specimens, at different stress levels, and

also with variance in the stress ratio R (Smin/Smax). He did this to determine the effect,

the ratio had in particular cyclic loading patterns. He discovered that the average

value of n/N was 1.05 for these experiments. Fatigue data by Johnson and Oberg

[18] when readjusted gave an average value of n/N equal to 1.05. Miner published

his paper on cumulative damage in 1945. He did not know whether his rule would


42

apply to materials other than 24 S-T Alcad aluminium. Subsequent fatigue testing of

various materials has proven him correct. More complicated cumulative damage

theories have been developed since but none is more widely used than Miners' theory

because of its sheer simplicity. His rule has been developed by other authors to

increase its accuracy. It has been discovered [19] that for a two level stress test, in

which one stress is applied for a number of cycles and then run to failure at a second

stress, that if S1 < S2, then n/N > 1

and for S1 > S2, then n/N < 1

In addition, the variation from unity is greater for larger differences between S1 and S2

stress levels.

4.2 Subramanyan’s Cumulative Damage Theory

Subramanyan's theory states that lines of constant damage (isodamage lines) exist

between applied stress levels and that all these lines converge to a kneepoint [20],

(Figure 4). This differs to Miner's rule which assumes the constant damage line

between stress levels lie parallel to the S-N curve.

Figure 4.3 Comparison of Subramanyan’s constant damage line approach to Miner’s

approach for remaining life prediction theory [20].


43

The isodamage line is drawn from the point (S1, n1) to the point ( Se, Ne ) which is

known as the knee point. The equivalent damage D at ( S2, n12 ), caused by stress 1 at

n1 is found from

DN N

N n

N N

N n

k

k

k

k

log log

log log

log log

log log

1

1

2

12

Equation 4.5

This equation is transposed to give

log log

log log

n N

n N

12 2

1 1

Equation 4.6

From figure 4.3, n2 = N2 - n12. Cycle ratio may be defined as C = n/N then equation

2 may be written as follows.

log logC C1 21 Equation 4.7

But

log n12 <=> - Se

log n1 <=> - Se

log N2 <=> - S2

log N1 <=> - S1

From equation 4.6 this implies that

S S

S S

e

e

2

1

log log

log

N N

LogN N

k

k

2

1

Equation 4.8

When the number of remaining cycles to failure in a two level step test is required,

equation 4.7 transposes to the following form.

n Nn

N2 2

1

1

1

Equation 4.9


44

4.3 Ben-Amoz’s Cumulative Damage Theory

Ben-Amoz’s theory is a development of both Miner’s and Subramanyan’s theories.

He proposed the idea of introducing upper and lower bounds on remaining fatigue life

in two-stage cycling [21]. Subramanyan assumed that all damage curves converge to

a knee-point. Ben-Amoz considered this to be too restrictive and so relaxed the idea

to produce a lower bound on damage S-N curves. Refer to figure 4.4 below. Miner’s

rule states that damage curves run parallel to the original S-N curve. According to

Ben-Amoz this provides an upper bound on remaining life. The principle of this

theory is to use both Subramanyan’s and Miner’s rules to give two values which

inbound the true value of remaining life.

Figure 4.4 Upper and lower bounding damage curves used in Ben-Amoz’s cumulative damage theory

[21].

To prove his theory he made two assumptions.

1. Damage curves constitute a family of curves of which the base S-N curve is a

member.

2. Damage curves form a non-intersecting family of curves.


45

The damage curves A’ and B’ in figure 4.4 above violate the second assumption and

so are considered to be invalid.

Miner’s theory states that n

N

n

N

2

2

1

1

1

Subramanyan’s theory states that n

N

n

N

2

2

1

1

1

For a two-level, high-low stress test, Ben-Amoz suggests that

1 11

1

2

2

1

1

n

N

n

N

n

N

Equation 4.10

And for a low-high test, that

1 12

2

1

1

2

2

1

n

N

n

N

n

N

Equation 4.11

Ben-Amoz proved that the bounds apply equally aswell to linear as well as non-linear

S-N curves. The bounds can become narrower if crack initiation information is

available. The fatigue process is broken into two phases, crack initiation and crack

propagation. The remaining life is considered to be of the form

n2 = ni2 + np2 Equation 4.12

where ni2 = number of cycles which cause crack initiation at stress level 2.

np2 = number of cycles which cause crack propagation at stress level 2.

Two cases arise.

Case 1 For n1 Ni1, the bounds are considered to be

1 12

1

1

1

2

2

2

1

1

1

N

N

nN

n

NN

nN

N

i

i

i

i

Equation 4.13


46

where Ni1 = number of cycles to initiate crack at stress 1.

N i2 = Ni/N = crack initiation life fraction at Stress 2.

n1 = number of cycles applied at stress 1.

N1 = number of cycles to failure at stress 1.

Case 2 For n1 > Ni1, the bounds are considered to be

N

nN N

N

n

NN

nN N

Np

i

p

p

i

p

2

1

11

1

2

2

2

1

11

1

1 1

Equation 4.14

Where N p2 = Np/N = crack propagation life fraction at stress 2.


47

4.4 Corten-Dolan’s Cumulative Damage Theory

Corten and Dolan [22] took a different approach to other cumulative damage

researchers. They reduced the study of damage accumulation down to the atomic

level as opposed to treating damage on a continuum basis. They also determined a

method of dealing with periodic variations in stress amplitude, such as those shown in

the figure 4.5 below [23].

Figure 4.5 Periodic variations in stress amplitude [23].

nr = total number of cycles in each repeated block.

= fraction of nr cycles that are incurred at the higher stress level S1.

In this instance = 2/5

Their theory allows a direct method of determining the number of remaining cycles to

failure ‘nf’.

nN

S

S

S

S

S

S

f d d

i

i

d

1 2

2

1

3

3

1 1

..........

Equation 4.15


48

where N = cycles to failure at the highest stress amplitude S1.

1,2,….i = the fraction of cycles imposed at stresses S1, S2….,Si,

respectively.

d = material constant.

Corten and Dolan experimentally obtained the mean value of ‘d’ which was 6.57 for

an alloy steel and 6.0 for an aluminium alloy.

If a large amount of periodic stress variation occurs in a component, the theory is

quite suitable because equation 4.15 is a geometric progression and can be easily

inserted into a computer program.

For a two-level-stress test equation 4.15 can be modified to yield ‘nf’.

nN

SS

f d

2

1

Equation 4.16

4.5 Marin’s Cumulative Damage Theory

Marin’s theory states that damage is a function of cycle ratio, as shown below in

figure 4.6 [24].

Figure 4.6 Damage as a function of cycle ratio for seven different stress levels [24].


49

Each stress level has its own damage curve. Lines of constant damage are plotted on

an S-N curve. For example, a damage line of D = 0.4 is shown in figure 4.7 which is

extrapolated from points 1 to 7 in figure 4.6.

Figure 4.7 S-N plot showing lines of constant damage developed by Marin’s theory [24].

He approximated the shape of the S-N curve to be

SxN = K Equation 4.17

where x is an exponent describing the shape of the curve. He also defined

q = y - x Equation 4.18

where y is a material constant (same as ‘d’ in Corten and Dolan’s theory).

The number of cycles to failure is given by

n Nn

N

S

S

n

N

S

S

n

N

S

Sf i

i

q

i

q

i

i

i

i

1

1

1

1 2

2

2 1

1

1........ Equation 4.19

where Ni = number of cycles to failure at the last stress level Si.

For a two-level-stress test equation 4.19 simplifies to give

n Nn

N

S

Sf

q

2

1

1

1

2

1 Equation 4.20


50

4.6 Manson’s Double Linear Cumulative Damage Rule

This rule can be considered as Miner’s rule applied to both the crack initiation and

crack propagation stages of fatigue. Through an empirical method he discovered that

a special point ‘P’ was common to all S-N curves regardless of damage level [25].

Refer to figure 4.8 below.

Figure 4.8 Manson’s cumulative damage rule applied to a two-level-stress test [25].

The number of cycles to failure for a two-level-stress test is given by

n Nn

Nf

NN

NN

P

P

2

1

1

1

2

1

log

log

Equation 4.21

4.7 Henry’s Cumulative Damage Theory

This theory is based on the concept that the fatigue limit decreases after each load.

Fatigue damage is defined as the ratio of the reduction in fatigue limit to the original

fatigue limit of virgin material, that is

DE E

E1

0 1

0

Equation 4.22

where E0 = original fatigue limit

E1 = fatigue limit after damage


51

This may be solved for E1 to give

E

S nN

S EE

nN

10

0

1

1

Equation 4.23

where S = applied stress

n/N = cycle ratio.

If equation 4.23 is substituted into equation 4.22 a damage equation can be

formulated, to give

D

nN

ES E

nN

10

01 1

Equation 4.24

The value of E0 must be updated after the application of each stress level, using

equation 4.23 above, to yield E0, E1, E2…., where E1 is the fatigue limit after applying

n1 cycles at stress level S1 and so on. Failure will occur when the sum of damage

fractions equals unity [26].

D = 1 failure.

4.8 Henry’s Modified Cumulative Damage Theory

Henry hypothesised that damage accumulation is a function of the change in the

endurance limit after each stress application. This theory can be applied to a semi-log

S-N curve. Equation 4.23 can be used to find the new fatigue limit after each stress

level. This new endurance limit is plotted at 106 cycles and connected to the point

‘Su’ at one cycle by a straight line [23]. Refer to figure 4.9 below.


52

Figure 4.9 Illustration of the set of new S-N curves according to Henry’s modified cumulative damage

theory.

This method yields a new S-N curve after each new load which can then be used to

predict failure. For example, in figure 4.9 above, the application of a third stress level

‘S3’ will produce failure at n = 103 cycles.

4.9 Gatts’s Cumulative Damage Theory

Gatts postulated that the fatigue strength and the fatigue limit change continuously

with the application of each stress cycle [27] [23]. This is different to Henry’s theory

in that the fatigue limit changes with each stress cycle as opposed to each stress level

application. He developed a non-dimensionalised version of the S-N curve and used

this to formulate the damage expression shown below.

e

C

11

1 11

Equation 4.25

where e = Se/Seo = fatigue limit ratio (ratio of current value of fatigue limit to the

original value of fatigue limit).

= S/Seo = stress amplitude ratio (ratio of the stress amplitude to the original

value of the fatigue limit).


53

= n/N = cycle ratio (ratio of number of cycles applied to number of cycles to

failure at stress amplitude S).

C = material constant.

Gatts’s theory calculates the new endurance limit ‘Se’ after stressing. This new value

can be substituted into equation 4.24 from Henry’s theory, to evaluate the damage

fraction ‘D1’. The same failure criteria applies to Gatts’s and Henry’s theory, which

is

D = 1 failure.

For two-level-stress test the number of cycles to failure is given by

n

DD S

S S N

D SS S

e

e

e

e

2

22

22

2

21

Equation 4.26

where D2 = 1 - D1

4.10 Unified Theory of Cumulative Damage

This theory is based on the relationship between the rate of change of damage and the

rate of change of the non-dimensional endurance limit. The theory was developed by

Dubuc et als [28], who managed to combine work from previous researchers such as

Sheh and Shanley, and Valluri and Gatts. Damage growth may be expressed in

differential form as follows.

dD

dn

d

dn

e

Equation 4.27

Where µ is a weighting coefficient used to yield a value of unity to the

damage function at failure.


54

dD

dn = damage growth.

e = instantaneous non-dimensional endurance limit.

e eo/

The determination of the rate of change of the non-dimensional endurance limit is the

backbone of the theory. Once this was found an expression relating the damage

fraction ‘D’ to cycle ratio ‘’, to maximum cyclic stress and to original ultimate

tensile strength, was derived.

D

u

m

1

1

Equation 4.28

β = Cycle ratio.

= non-dimensional maximum cyclic stress = / eo

u = non-dimensional original ultimate tensile strength = o

/

m = constant greater than 1.

A recurring theme in cumulative damage prediction, is the search for an equivalent

cycle ratio at the new stress level, that would have caused the same damage to occur

at the original stress level. The equivalent cycle ratio ‘βek’ corresponding to the last

stress level, which would cause the same amount of damage De may be evaluated by

means of equation 4.28.

ek

e

kk

uk

m

k

ek

uk

m

k

D

D

1

1

1

1

Equation 4.29


55

The theoretical number of cycles that the specimen will sustain at the last level k is

n Nk K ek 1 Equation 4.30

For a two-level step test, the number of remaining cycles can be found by

n N e2 2 21 Equation 4.31

4.11 Marco-Starkey’s Cumulative Damage Theory

This theory is based on the following assumptions [23].

1) Damage curves for each level of sinusoidal stress amplitude may be defined by the

relationship,

Dn

N

mi

Equation 4.32

Where mi is a function of the stress level. This approximates the non-linear

relationship between damage and cycle ratio.

2) Failure will occur when ‘D’ reaches unity and when (n/N) reaches a critical

value.

The critical value ‘ (n/N)’ can be determined from the damage history on a damage

vs. cycle ratio plot. Consider figure 4.10.


56

Figure 4.10 Fatigue damage as a function of cycle ratio for low-high stressing, illustrating Marco-

Starkey cumulative damage theory.

For a low-high, two-level-stress test, S1 is applied for a cycle ratio (n/N) of 0.5,

followed by S3 until failure occurs at D = 1. Path 0-A-B-C represents the damage

history. The critical value of n/N may be computed from the curve as

n

N

n

N

n

NS S

1 3

Equation 4.33

From figure 4.10, for example, it can be observed that the critical value is

n/N = 0.5 + (1 - 0.1) = 1.4

Experimentation Cumulative Damage Analysis – Kieran J. Claffey

57

Chapter 5

5.1 Experimental Programme

All tests were carried out using the Avery-Denison Fatigue testing machine. The

experimental programme consisted of five parts, each with its own purpose.

1. Determination of the S-N curve for cast vitalliumtm.

2. Determination of the S-N curve for notched vitalliumtm with a notch radius of

1.5mm

3. Determination of the S-N curve for notched vitalliumtm with a notch radius of

2.5mm

4. Determination of the S-N curve for HIPed vitalliumtm.

5. Two-level-stress tests for a cumulative damage analysis.

5.2 Specimen Preparation

The vitalliumtm test specimens were cast using the investment casting process, as

described earlier in chapter three. Howmedica were requested to cast sixty test

specimens with 3 mm thickness. The casting dye had to be modified to produce the

required thickness. The process proved to be troublesome and produced test

specimens that included shrinkage voids. From the sixty specimens, twenty two

failed. The remaining thirty eight specimens were then X-rayed to determine the

existence of internal voids. About half of these contained minute shrinkage voids. A

decision had to be made as to whether they should be scrapped or not. The outcome

of this was to keep all thirty eight specimens. The decision was based on the fact that

the micro-voids were all contained in areas of low stress, well away from the centre of

the specimen where the maximum stress was designed to occur. Also, the majority of

the specimens (sixteen in all), which contained the shrinkage voids were hot

isostatically pressed. HIPing is a special process that uses high temperature and


58

pressure to densify cast components. This process eliminated all traces of the micro-

voids. This meant that thirty eight specimens were available for testing.

All specimens were ground in a longitudinal direction according to BS 3518

specifications. Grinding proved to be slow because a cut, no larger than 0.002 mm

could be machined at one pass. Notched fatigue test specimens should have their

notches machined in the same direction that they were cast. That is, across flow as

opposed to along flow, during casting.

5.3 Preparation of Notched Specimens

A sufficient number of notched test pieces should be tested to allow for scatter in

notch condition due to manufacturing variables. A total of ten notched test pieces

were chosen to allow for an exploratory analysis. The notch sensitivity ‘q’ does not

change very much above a notch radius of 3mm [13]. In general, materials are more

notch sensitive at small radii. The author therefore chose round notch radii below

3mm.

Two types of notch were chosen for the analysis.

1) Opposite single U-shaped notches in a finite width plate.

2) Opposite single semi-circular notches in a finite width plate.

Vitalliumtm is an extremely hard material to machine. It was decided to machine semi-

circular notches as opposed to elliptical or V-notches due to the machining difficulties

of resilient vitalliumtm. Blunt notches were chosen above sharp notches for

machining practicality.


59

A rough guide for obtaining maximum stress concentration in a test specimen is to

make the smaller width about 3/4 of the larger width, assuming the radius ‘r’ and the

thickness ‘D’ are given. Specimen dimensions are shown in Appendix A.2.

5.4 Apparatus

Figure 5.1 Front view of the Avery-Denison Fatigue testing machine.

Figure 5.2


60

Figure 5.3

1 Motor 9 Dial gauge

2 Rev-counter 10 Adjustable thimble

3 Start-stop switches 11 Plunger

4 Indicator lights 12 Torsion bar arm

5 Dashpot switches 13 Test specimen

6 Torsion bar housing 14 Nuts

7 Crank 15 Oscillating spindle

8 Eccentric wheel


61

5.5 Standard Fatigue Test Precautions

1. In attaching the specimen to the grips of the fatigue machine, one should be

careful not to prestrain the specimen excessively, since large strains can

influence the results [30].

2. When the test is running, the test engineer should study the behaviour of the

specimen to determine whether there is excessive vibration or heat build-up

[30].

3. When a test is started or stopped without specimen failure, there may be

spurious loads or strains applied to the specimen that may influence the test.

To take account of this load, strain-time traces should be recorded for

inclusion in the test log.

5.6 Experimental Procedure

1. When all grinding processes were finished the width ‘B’ and thickness ‘d’ of each

specimen was measured. The required bending moment for that specimen at the

desired stress was found using

MB D

. . 2

6 Equation 5.1

(Refer to Appendix A.1 for derivation).

2. The dial gauge deflection which corresponds to the bending moment was

determined from the calibration chart. The 30 Nm torsion bar was used for high

stresses and the 10 Nm torsion bar was used for stresses that required less than 10

Nm of bending moment.

3. The eccentric wheel was set to zero.

4. The test specimen was clamped to the machine.

Note: The four allen head bolts were tightened evenly to prevent uneven

stress levels across the specimen.


62

5. The top and bottom dial gauges were both set to zero when contact was made with

the torsion bar arm.

6. The eccentric wheel was offset by four wheel divisions.

7. The upper and lower deflections of the torsion bar arm were adjusted until they

were the same. This ensured that the neutral axis was in the centre of the test

specimen. The adjusting was done using the two allen head bolts (adjusters) on

the machine head.

Note: One bolt was loose while the other was being tightened and both bolts

were tight when the eccentric was rotated by hand.

8. Both dial gauges were offset by the value of deflection obtained from the

calibration chart.

9. The eccentricity of the connecting rod was increased until both dial gauges

registered a tiny deflection. This indicated that the deflection in the torsion bar

delivered the required stress to the specimen and that it occurred at the centre of

the specimen.

10. The number of revolutions on the rev-counter was recorded, the dial gauges were

removed from the machine and the adjusters were both tightened.

11. The upper contact on the switch was lowered until the red light illuminated. The

stop button was pressed and the switch lowered again until the red light

illuminated. The stop button was pressed again. This was repeated until the red

light remained on. The switch connection was broken with the use of a phase-

tester and the machine was started. The switch was lowered until the shortest

spark length was obtained.

12. The steps, one to eleven, were repeated for each application of stress.

13. The same procedure was carried out for the cumulative damage tests, with the

exception that the machine was manually turned off after the application of the

first stress level and reset for the application of the second stress level.


63

5.7 Test Problems and Solutions

Various problems were encountered during the test programme, most of which were

associated with the switching mechanism. These problems were either solved or

compromised. They are discussed below.

1. The dashpot switch proved to be a problem, especially with high cycle tests.

Through experimentation it was found that the switch was optimal when the spark

between the switching contacts was at a minimum length. When the spark length

was allowed to increase, the machine cut out. During high cycle tests the point of

the switch flattened due to the high level of repeated contact. This meant that the

spark length was allowed to increase; resulting in machine cut out when the

specimen had not yet fractured. To overcome this problem the point of the lower

contact had to be sharpened with rough emery paper between each test. Also, for

high cycle tests, the upper contact had to be slightly lowered approximately every

four hours. This ensured a short spark length until the specimen fractured.

2. The Avery-Denison fatigue testing machine contains two dashpot switches. The

lower switch was missing parts and the remaining parts were broken. The upper

switch was capable of doing the job on its own so it was decided not to use the

lower switch at all.

3. During low cycle testing the machine was not cutting out when the specimen

fractured. Through an investigation the problem was discovered. The oil in the

dashpot switch contained wear particles. These particles made the oil less

viscous. This meant the dashpot piston was too slow to come down the necessary

distance to allow full contact across the switch. The problem was solved by

changing the oil every four or five tests.


64

4. Some oil splashing occurred. This seeped down the thread of the adjustable upper

contact and across the switch, cutting out the machine. The solution was to put

the bare minimum of oil into the dashpot to eliminate splashing.

5. The pivot screws in the dashpot switch tended to loosen and needed to be

tightened throughout the test programme. This was also the case for the two

tapered bolts which secured the torsion bar.

6. Two adjusters were used to centre the position of the load in the test specimen.

The internal and external threads on one of the adjusters completely wore out.

The reason for this must be that the torque exertion at the adjuster was excessive.

However, it was necessary to apply such a high torque, to bend the test specimen

with the torsion bar. The problem was rectified by taking the machine apart, re-

boring and re-tapping the internal thread to a larger size (M10).

7. The front damping plate, which supports the machine head, cracked at the end of

testing program. A crack initiated at a point of high stress concentration and

appeared to propagate across the plate. This problem can be solved by making

another plate and replacing the damaged one.

8. The screw which bounces continuously off the torsion bar arm has worn it down.

Therefore when high stresses were applied the screw bounced higher than usual

causing the switch to close. This had the random effect of switching off the

machine when the piece had not broken. The problem was solved by damping the

screw. The worn area of the torsion bar arm was covered with paper and wrapped

with insulating tape. Even though it was a temporary job it improved the switch

performance immensely.

Results Cumulative Damage Analysis – Kieran J. Claffey

65

Chapter 6

Experimental Results

Stress

Amplitude

MPa

Required Bending

Moment

Nm

Cycles to

Failure

Cycles to

Run-out

Comments

650 23.54 2x104 - valid

580 19.58 2.9x104 - valid

400 11.22 1.47x105 - valid

370 9.05 1.46x105 - questionable

validity

300 9.13 1.13x106 - valid

250 6.55 - 1.51x106 valid

Table 6.1 S-N curve results for cast vitalliumtm

.

Notch

Radius

mm

Stress

Amplitude

MPa

Required

Bending Moment

Nm

Cycles to

Failure

Cycles to

Run-out

Comments

2.5 580 11.57 2.3x104 - valid

2.5 510 12.01 8.6x104 - valid

2.5 300 7.06 5.81x105 - valid

2.5 200 5.05 1.15x106 - valid

1.5 580 15.87 9.5x103 - valid

1.5 510 10.86 9.2x104 - valid

1.5 400 10.24 1.91x105 - valid

1.5 300 8.1 7.78x105 - valid

1.5 200 4.71 - 2.34x106 valid

Table 6.2 S-N curve results for notched cast vitalliumtm

.


66

Stress

Amplitude

MPa

Required Bending

Moment

Nm

Cycles to

Failure

Cycles to

Run-out

Comments

650 22.18 1.4x104 - valid

580 15.92 5.7x104 - valid

500 12.89 7.8x104 - valid

400 11.78 3.86x105 - valid

300 9.99 1.34x106 - valid. Switch

problem, therefore

may be as much

as .25x106 higher.

Table 6.3 S-N curve results for HIPed cast vitalliumtm

.

Cumulative Damage Results

1 N1 n1 2 N2 n2 = nf

Experimental

n2 = nf

Miner

Comments

(Conservative /

Non-conservative)

High- 650 1.0x104 8,000 580 1.85x10

4 2,500 3,700 non-con

Low 600 1.45x104 6,000 500 1.15x10

5 92,000 67,414 con

Tests 600 9x103 6,000 500 1.6x10

4 62,000 53,333 notched

specimen

con

Low- 580 1.85x104 2,500 650 1.0x10

4 14,000 8,649 con

High 550 1.05x105 60,000 580 1.85x10

4 8,500 7,928 con

Tests 550 1.05x105 20,000 580 1.85x10

4 33,500 14,976 con

450 1.24x105 100,000 550 1.05x10

5 25,000 20,322 con

Table 6.4 Miner’s prediction for two level-stress-tests.


67

1 N1 n1 2 N2 n2 = nf

Experimental

n2 = nf

Subramanyan

Comments

(Conservative /

Non-

conservative)

High- 650 1.0x104 8,000 580 1.85x10

4 .811 2,500 3,062 non-con

Low 600 1.45x104 6,000 500 1.15x10

5 .687 92,000 52,304 con

Tests 600 9x103 6,000 500 1.6x10

4 .687 62,000 38,924 con

Low- 580 1.85x104 2,500 650 1.0x10

4 1.233 14,000 9,152 con

High 550 1.05x105 60,000 580 1.85x10

4 1.111 8,500 8,565 non-con

Tests 550 1.05x105 20,000 580 1.85x10

4 1.111 33,500 15,568 con

450 1.24x105 100,000 550 1.05x10

5 1.588 25,000 30,383 non-con

Table 6.5 Subramanyan’s prediction for two level-stress-tests.

1 N1 n1 2 N2 n2 = nf

Experimental

n2 = nf

Ben-Amoz

Comments

(Conservative /

Non-

conservative)

High- 650 1.0x104 8,000 580 1.85x10

4 2,500 3,700 n2

3,062

non-con

Low 600 1.45x104 6,000 500 1.15x10

5 92,000 67,414 n2

52,304

con

Tests 600 9x103 6,000 500 1.6x10

4 62,000 53,333 n2

38,924

con

Low- 580 1.85x104 2,500 650 1.0x10

4 14,000 8,649 n2

9,152

con

High 550 1.05x105 60,000 580 1.85x10

4 8,500 7,928 n2

8,565

correct

Tests 550 1.05x105 20,000 580 1.85x10

4 33,500 14,976 n2

15,568

con

450 1.24x105 100,000 550 1.05x10

5 25,000 20,322 n2

30,383

correct

Table 6.6 Ben-Amoz’s prediction for two level-stress-tests.


68

1 N1 n1 2 N2 n2 = nf

Experimental

n2 = nf

Corten-Dolan

d = 6.3

Comments

(Conservative /

Non-conservative)

High- 650 1.0x104 8,000 580 1.85x10

4 2,500 20,500 non-con

Low 600 1.45x104 6,000 500 1.15x10

5 92,000 45,731 con

Tests 600 9x103 6,000 500 1.6x10

4 62,000 28,384 con

Low- 580 1.85x104 2,500 650 1.0x10

4 14,000 4,878 con

High 550 1.05x105 60,000 580 1.85x10

4 8,500 13,239 non-con

Tests 550 1.05x105 20,000 580 1.85x10

4 33,500 13,239 con

Does not take

n1 into a/c for

2 level test.

450 1.24x105 100,000 550 1.05x10

5 25,000 29,658 non-con

Table 6.7 Corten-Dolan’s prediction for two level-stress-tests (d=6.30).

1 N1 n1 2 N2 n2 = nf

Experimental

n2 = nf

Corten-Dolan

d = 9.82

Comments

(Conservative /

Non-conservative)

High- 650 1.0x104 8,000 580 1.85x10

4 2,500 30,616 very non-con

Low 600 1.45x104 6,000 500 1.15x10

5 92,000 86,881 con

Tests 600 9x103 6,000 500 1.6x10

4 62,000 53,926 con

Low- 580 1.85x104 2,500 650 1.0x10

4 14,000 3,266 con

High 550 1.05x105 60,000 580 1.85x10

4 8,500 10,981 non-con

Tests 550 1.05x105 20,000 580 1.85x10

4 33,500 10,981 con

Does not take

n1 into a/c for

2 level test.

450 1.24x105 100,000 550 1.05x10

5 25,000 14,634 con

Table 6.8 Corten-Dolan’s prediction for two level-stress-tests (d=9.8 ).


69

1 N1 n1 2 N2 q n2 = nf

Experimental

n2 = nf

Marin

Comments

(Conservative / Non-

conservative)

High- 650 1.0x104 8,000 580 1.85x10

4 4.42 2,500 6,122 non-con

Low 600 1.45x104 6,000 500 1.15x10

5 -1.53 92,000 51,003 con

Tests 600 9x103 6,000 500 1.6x10

4 -5.96 62,000 17,992 very con

Low- 580 1.85x104 2,500 650 1.0x10

4 4.42 14,000 5,227 con

High 550 1.05x105 60,000 580 1.85x10

4 -22.8 8,500 26,711 non-con

Tests 550 1.05x105 20,000 580 1.85x10

4 -22.8 33,500 50,455 non-con

450 1.24x105 100,000 550 1.05x10

5 6.68 25,000 5,318 con

Table 6.9 Marin’s prediction for two level-stress-tests.

1 N1 n1 2 N2 n2 = nf

Experimental

n2 = nf

Manson

Comments

(Conservative /

Non-

conservative)

High- 650 1.0x104 8,000 580 1.85x10

4 2,500 2,038 con

Low 600 1.45x104 6,000 500 1.15x10

5 92,000 39,121 very con

Tests 600 9x103 6,000 500 1.6x10

4 62,000 6,985 very con

Low- 580 1.85x104 2,500 650 1.0x10

4 14,000 8,994 con

High 550 1.05x105 60,000 580 1.85x10

4 8,500 11,440 non-con

Tests 550 1.05x105 20,000 580 1.85x10

4 33,500 16,410 con

450 1.24x105 100,000 550 1.05x10

5 25,000 21,695 con

Table 6.10 Manson’s prediction for two level-stress-tests.


70

1 N1 n1 2 N2 n2 = nf

Experimental

n2 = nf

Modified

Henry

Comments

(Conservative /

Non-

conservative)

High- 650 1.0x104 8,000 580 1.85x10

4 2,500 7,000 non-con

Low 600 1.45x104 6,000 500 1.15x10

5 92,000 45,000 con

Tests 600 9x103 6,000 500 1.6x10

4 62,000 12,300 very con

Low- 580 1.85x104 2,500 650 1.0x10

4 14,000 5,500 con

High 550 1.05x105 60,000 580 1.85x10

4 8,500 10,400 non-con

Tests 550 1.05x105 20,000 580 1.85x10

4 33,500 11,700 con

450 1.24x105 100,000 550 1.05x10

5 25,000 10,800 con

Table 6.11 Modified Henry’s prediction for two level-stress-tests.


71

1 N1 n1 2 N2 D1 D2 E1 n2 = nf

Experimental

n2 = nf

Henry

Comments (Conservative /

Non-conservative)

High- 650 1.0x104 8,000 580 1.85x10

4 .694 .306 85.4 2,500 6,304 non-con

Low 600 1.45x104 6,000 500 1.15x10

5 .273 .727 203.4 92,000 94,053 non-con

Tests 600 9x103 6,000 500 1.6x10

4 .516 .484 75 62,000 83,907 non-con

Low- 580 1.85x104 2,500 650 1.0x10

4 .074 .926 259 14,000 9,543 con

High 550 1.05x105 60,000 580 1.85x10

4 .395 .605 169.2 8,500 12,652 non-con

Tests 550 1.05x105 20,000 580 1.85x10

4 .103 .897 251 33,500 17,372 con

450 1.24x105 100,000 550 1.05x10

5 .611 .389 108.7 25,000 46,446 non-con

Table 6.12 Henry’s prediction for two level-stress-tests.

1 N1 n1 2 N2 C D2 Se1 n2 = nf

Experimental

n2 = nf

Gatts

Comments (Conservative /

Non-conservative)

High- 650 1.0x104 8,000 580 1.85x10

4 .338 .306 233.3 2,500 7,852 non-con

Low 600 1.45x104 6,000 500 1.15x10

5 .338 .727 252 92,000 96,984 non-con

Tests 600 9x103 6,000 500 1.6x10

4 .218 .484 148.4 62,000 91,466 non-con

Low- 580 1.85x104 2,500 650 1.0x10

4 .338 .926 270.9 14,000 9,553 con

High 550 1.05x105 60,000 580 1.85x10

4 .338 .605 233.2 8,500 13,304 non-con

Tests 550 1.05x105 20,000 580 1.85x10

4 .338 .897 266 33,500 17,425 con

450 1.24x105 100,000 550 1.05x10

5 .338 .389 189.9 25,000 51,771 non-con

Table 6.13 Gatts’s prediction for two level-stress-tests.


72

1 N1 n1 2 N2 1 1 u1 2 D1 e2 n2 = nf

Experimental

n2 = nf

Unified Theory

Comments

(Conservative / Non-

conservative)

High- 650 1.0x104 8,000 580 1.85x10

4 .8 2.32 2.95 2.07 .708 .820 2,500 3,330 non-con

good

Low 600 1.45x104 6,000 500 1.15x10

5 .41 2.14 2.95 1.78 .277 .464 92,000 61,640 con

Tests 600 9x103 6,000 500 1.6x10

4 .66 2.14 4.59 1.78 .508 .599 62,000 64,070 non-con

good

Low- 580 1.85x104 2,500 650 1.0x10

4 .14 2.07 2.95 2.32 .079 .124 14,000 8,760 con

High 550 1.05x105 60,000 580 1.85x10

4 .57 1.96 2.95 2.07 .398 .554 8,500 8,251 con

good

Tests 550 1.05x105 20,000 580 1.85x10

4 .19 1.96 2.95 2.07 .105 .181 33,500 15,152 con

450 1.24x105 100,000 550 1.05x10

5 .81 1.61 2.95 1.96 .618 .764 25,000 24,780 con

good

Table 6.14 The Unified Theory prediction for two level-stress-tests.

Analysis of Data Cumulative Damage Analysis – Kieran J. Claffey

73

Chapter 7

7.1 Determination of S-N Curve

An S-N curve for cast vitalliumtm was constructed from data provided by the previous

researcher, Michael Moloney [29] and the author. The experimental curve was

insufficient to do constant life calculations because it existed only between 104 and

106 cycles.

The first part of the curve was plotted on the basis of Juvinall & Marsheks’ estimate

[1], which is that the material loses one tenth of its fatigue strength between 1 cycle

and 103 cycles. This can be done as it yields a good correlation with the experimental

results obtained at 104 cycles.

It was evident from exploratory tests that the S-N curves for machined vitalliumtm and

cast vitallium tm were different. Refer to figure 7.1. The curve for machined

vitalliumtm is to the right and above the cast S-N curve, indicating that it has a greater

fatigue resistance than cast vitalliumtm.


74

0

100

200

300

400

500

600

700

800

900

1.00E+00 1.00E+01 1.00E+02 1.00E+03 1.00E+04 1.00E+05 1.00E+06 1.00E+07

Str

es

s A

mp

litu

de (

Mp

a)

Number of cycles to Failure

Figure 7.1 S-N Curves for Machined and Cast Vitallium

Juvinall and Marshek

Machined

Cast


75

7.2 Mean Stress Effects on Machined Vitalliumtm

It is important to quantify the effect of maximum allowable mean stress. This is done

by using the mean stress correction equations of Goodman, Gerber and Soderberg.

The Soderberg equation yields conservative allowable mean stresses, Gerber yields

realistic allowable mean stresses and Goodman yields a result somewhere in-between.

The Goodman criterion was chosen for this analysis.

Goodman a = N (1 - ( m / ts) Equation 7.1

Where a Stress Amplitude

N Nominal Stress

m Mean Stress

ts Ultimate Tensile Strength

The effects of applying a mean stress can be seen in figure 7.2. The maximum

allowable mean stress of 280 MPa yields an S-N curve much lower than the original

S-N curve. This curve was constructed by using the Goodman criteria of mean stress.

0

200

400

600

800

1000

1.00E+02 1.00E+05 1.00E+08

Log N (no. of cycles to failure)

Str

ess A

mp

litu

de

Mean Stress = 0

Mean Stress = 100

Mean Stress = 200

Mean Stress = 280

Figure 7.2 The reduction of the fatigue strength of machined Vitalliumtm

due mean stress effects,

according to Goodman’s mean stress criteria.


76

7.3 Constant Life Diagram for Machined Vitaliumtm

Engineers use constant life diagrams to predict the combination of stress amplitude

and mean stress that a material can take at a specified lifespan. These diagrams are

useful because they can be used to find the maximum allowable cyclic stresses that

can be applied to a material for their required life expectancy.

A Haigh constant life diagram was constructed for machined vitalliumtm (figure 7.3)

with the use of it’s S-N curve [1].

Figure 7.3 Haigh constant life diagram for machined Vitalliumtm

.

This diagram shows that a maximum of 280 MPa of mean stress can be applied for

infinite life. This is substantially less than the estimated endurance limit of 377 MPa

for machined vitalliumtm, when zero mean stress is applied.

0

827

0

827

0

827

0

827

00

0

Ultimate Tensile

Strength531-Yield Strength0

100

200

300

400

500

600

700

800

0 200 400 600 800 1000

Mean Stress (MPa)

Str

es

s A

mp

litu

de

(M

Pa

)

1.00E+03

1.00E+04

1.00E+05

3.00E+05

1.00E+06

1.00E+08

Yield Stress


77

7.4 Constant Life Diagram with Mean Stress Correction for Machined

Vitalliumtm

Another constant life diagram was constructed (figure 7.4) using the S-N curve for a

mean stress of 280 MPa.

Haigh Constant Life Diagram

0

200

400

600

0 200 400 600 800 1000

Mean Stress (MPa)

Str

es

s A

mp

litu

de

(M

Pa

)

1.00E+03

1.00E+04

1.00E+05

3.00E+05

1.00E+06

1.00E+08

Yield Stress

Figure 7.4 Constant life diagram for machined Vitalliumtm

with mean stress correction.

This constant life diagram shows that a maximum mean stress of 402 MPa can be

applied, for infinite life conditions so long as the stress amplitude does not exceed

127 MPa. Having mean stress correction in a constant life diagram has the effect of

increasing the materials chances of yielding before actual failure. This reduces the

possibility of catastrophic failure due to mean stress.

This chart can be used if the application of a mean stress is unavoidable. This constant

life diagram still is not complete, as stress concentration and fatigue notch factors

have not been considered.


78

7.5 Fatigue Strength Diagram for Cast VitalliumTM

A Haigh constant life diagram was also constructed for non-machined cast vitalliumtm

with the use of its own S-N curve.

The constant life diagram was then converted into a fatigue strength diagram [5], by

considering the relationship between the maximum stress [Smax], minimum stress [S-

min] and the mean stress [Sm].

Figure 7.5 Relationship between mean stress, maximum stress and minimum stress in a typical fatigue

situation.

It can be seen that Sm = ½ (Smax + Smin) Equations 7.2

Sa = ½ (Smax - Smin) = Smax - Sm

When there is zero mean stress Smax and Smin both have a magnitude of Sa. .Refer to

figure 7.5 above. Points A and B were plotted giving one point on each curve. If the

mean stress were equal to the ultimate tensile stress ‘Su’ (827 MPa), the specimen can

withstand no further load, hence the corresponding stress amplitude ‘Sa’ is zero.

Then Smax and Smin are the same and equal to Su. Thus both curves pass through the

point C. The curves AC and BC were assumed to be straight [17]. The line OC in

figure 7.5 slopes upwards at 45° because C was plotted with both coordinates equal to

Su. The vertical distance to an arbitrary point on the line OC is the mean stress Sm


79

while the additional vertical distance to the line AC is the stress amplitude Sa. This

allows the stress amplitude (also known as alternating stress) to be determined, once

the maximum and mean stresses are known. This information was superimposed on

the constant life diagram to produce the fatigue strength diagram for cast vitalliumtm,

shown in figure 7.6.


80

Figure 7.6 Fatigue strength diagram constructed for heat treated, cast vitalliumtm

.


81

7.6 Cumulative Damage

Figure 7.7 Comparison of cumulative damage theories with experimental data obtained for machined vitalliumtm

.


82

There are several damage theories available. The author has investigated each theory

to see which cumulative damage theory is best suited to machined vitalliumtm. This

was done by comparing the experimental and theoretical results obtained from the

previous author. It can be seen from figure 7.7 that Subramanyan’s theory is the

closest to the experimental curve, especially in the region of S1/S2 > 1, where high-

low testing occurred.

A plot of Damage vs. Cycle Ratio for each two level step test was configured (figure

7.8).

Figure 7.8 Relationship between damage fraction and cycle ratio for vitalliumtm

according to Henry’s

damage theory.

Damage was calculated using Henry’s damage equation.

According to Henry’s theory, damage can be quantified as a reduction in the

endurance limit after the initial loading has occurred. He postulated that the new

endurance limit was a function of the applied stress, the cycle ratio and the original

fatigue limit.

D

n

N

S

S S

n

N

fo

nom fo

1 1

Equation 7.3

Relationship between Damage Fraction and Cycle Ratio for

Vitallium

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Cycle Ratio

Dam

ag

e F

racti

on

High-low Sequence

Low-high Sequence


83

Where D = Damage

n/N = Cycle ratio

Sfo = Original endurance limit

Snom = Nominal stress

The plot shows that the low-high tests yield a straight line. This is almost identical to

Miner’s linear damage rule, which states that the damage accumulated is a linear

function of cycle ratio. This explains why Miner’s theory yielded relatively accurate

results. The plot also shows that the high-low tests yield a polynomial curve below

the straight line. This curve is similar to the curve that Subramanyan’s damage theory

would provide.

The plot suggests; for cast vitalliumtm, that Miner’s damage theory should be applied

to low-high stressing situations and that Subramanyan’s damage theory should be

applied to high-low stressing situations. More tests need to be conducted at higher

values of cycle ratio to confirm that this graph is a true representation of the

cumulative damage behaviour of cast vitalliumtm.

7.7 Notched Vitalliumtm

The S-N curve for notched vitalliumtm is to the left and below that for plain

vitalliumtm. Refer to figure 7.9. This indicates that vitalliumtm is a notch sensitive

material.

The endurance limit of the notched specimens is below 200 MPa and is estimated to

be 180 MPa. Using this information and Peterson’s stress concentration design charts

[13], the notch sensitivity ‘q’ of each notch geometry was determined.


84

Figure 7.9 S-N curves for plain/unnotched and notched vitalliumtm


85

Figure 7.10 Peterson’s stress concentration chart used to determine the stress concentration factors of

both notch geometries [13].

The results were as follows.

At notch radius = 1.5 mm q = 0.585

At notch radius = 2.5 mm q = 0.855

Specimens with the smaller notch radii were more notch sensitive than those with the

larger radii. This was to be expected. However, the specimens with the smaller radii

seem to have a better resistance to fatigue than do the specimens with the larger notch

radii (figure 7.9). This suggests that vitalliumtm is notch sensitive because of the

stress concentration factor, not the fatigue notch factor.

7.8 HIPed Vitalliumtm

It can be seen from figure 7.11 that the specimens that were hot isostatically pressed

had increased fatigue resistance compared with specimens that were not HIPed. The

benefits of HIPing vitalliumtm are discussed further in Chapter 9.5.


86

Figure 7.11 S-N curves for plain/unnotched and HIPed vitalliumtm

0

100

200

300

400

500

600

700

800

900

1.00E+00 1.00E+01 1.00E+02 1.00E+03 1.00E+04 1.00E+05 1.00E+06 1.00E+07

Str

ess

Am

pli

tud

e (M

Pa)

Number of Cycles to Failure

S-N Curves for Plain and HIPed Vitallim

Plain Vitallium

HIPed Vitallium

Inspection Cumulative Damage Analysis – Kieran J. Claffey

87

Chapter 8

8.1 Microscopic Examination

Microscopic analysis entails a visual examination under a microscope. Samples of

failed specimens were bonded in bakelite plastic, ground and polished. They were

then etched electrolytically in a nitric acid and methanol solution. It was difficult to

get a good view of the microstructure. It is suspected that more current is necessary

to etch the surface better. However, some reasonably good magnified photographs

were taken of the fatigued surface.

8.2 Procedure for Microstructure Analysis

1. Small pieces of vitalliumtm were cut away from the fractured surface of two

broken test specimens.

2. They were bonded in a bakelite matrix under high temperature and pressure.

3. The bakelite specimens were ground flat using 600 grit silicon carbide paper.

4. The specimens were polished using a six micron diamond based polishing

lubricant. White spirits was sprayed on the rotating polishing wheel to allow

waste particles to disperse. This lasted for approximately ten minutes.

5. This was repeated using a one micron polishing agent for twenty five minutes.

6. The specimens were etched electrolytically for eight seconds with five per cent

Nital etching agent.

7. Once etched, they were examined under the electron microscope and

photographed.


88

8.3 Results of Microstructural Inspection

Figure 8.1 shows tiny cracks which propagated around the crystal boundaries at the

fatigued surface. This follows the theory of crack propagation which states that

cracks will propagate in the direction of least resistance, i.e. along crystal boundaries.

Figure 8.2 shows a magnified view of the tiny cracks from figure 8.1.

Figure 8.1 Tiny cracks propagate around the crystal boundaries at the fatigued surface of vitalliumtm

.

Figure 8.2 A magnified view of the tiny cracks in figure 8.1.


89

Casting voids are shown in figures 8.3a and 8.3b. A cluster of defects can be seen in

figure 8.3a while a single defect next to a crystal boundary can be seen in figure 8.3b.

a b

Figure 8.3 8.3a Cluster of casting voids. 8.3b Single void close to a crystal boundary line.

The photograph in figure 8.4 shows polishing scratches across a specimen’s surface.

The specimens were re-polished until there were no more visible scratches.

Figure 8.4 Polishing scratches across a specimen’s surface.


90

8.4 Macroscopic Inspection

Macroscopic inspection entailed a visual examination of the fracture surface of

specimens. It was noted that all specimens failed close to the minimum cross-

sectional area, as expected. The specimens failed in a brittle fashion. This is evident

due to the flaking of the material at the fracture line and by the clean break across the

specimen. Refer to figure 8.5.

Figure 8.5 Photograph of fractured specimen, showing a flaked fracture line. The scale at the bottom is

in millimetres.

Figure 8.6 Fatigue surface characterised by fibrous and crystalline areas.


91

This brittle fracture is unexpected because vitalliumtm is a ductile material. This

could suggest that, cast vitalliumtm may work harden under completely reversed

bending conditions. Work hardening can make a metal less ductile and more brittle.

Completely reversed bending is an extreme case of maximum bending stress which

tests the material to its limits. Work hardening under completely reversed bending

could explain why ductile vitalliumtm fatigued in a brittle manner. Little or no

macroscopic distortion occurred during the fatigue process. This was to be expected

as it is a distinguishing characteristic of fatigue failure of ductile materials [1].

The picture of a fractured surface of vitalliumtm can be seen in figure 8.6. It is

apparent that it has failed by fatigue because of the two types of surface areas, the

fibrous and the crystalline. Cracks propagated from the specimen surface at the

smooth fibrous areas and catastrophic failure occurred at the bumpy crystalline areas.

Discussion Cumulative Damage Analysis – Kieran J. Claffey

92

Chapter 9

Discussion

The test program had to be modified because of the late arrival of specimens. Certain

high cycle S-N curve tests, low stress cumulative damage two-level-stress tests and

cycle ratio hardness tests were thus eliminated from the program. These tests were

chosen because of test time considerations and lower priority. The various aspects of

the entire fatigue study and cumulative damage analysis are discussed below.

9.1 Machining Effects on Fatigue Resistance

The machined endurance limit, estimated to be 377 MPa, is much higher than the cast

endurance limit which lays in the range 250 to 300 MPa. It is apparent from figure

7.1 that the machined specimens have an overall fatigue resistance superior to that of

the plain cast specimens. The heavy machining may have introduced compressive

residual stresses on the specimen’s surface and made the specimens more resistant to

cyclic loads [4]. In other words, crack nucleation occurred at a slower rate in the

machined specimens than that in the cast specimens. Since all points on both S-N

curves, machined and cast, are consistent with this theory, it may be concluded that

heavy machining has a positive influence on the fatigue resistance of vitalliumtm.

9.2 Notch Analysis

It has been suggested to introduce a notch into the stem of bonded and press fitted hip

prosthesis, so as to reduce the effects of bone resorption. The theory is that the bone

in the femur will grow into the gap and halt bone loss due to stress shielding. The

notch would have to be of a gradual slope so as to reduce stress concentration which

might cause bone microfracture.


93

The S-N curves for both notch geometry’s were lower than that for unnotched

vitalliumtm. Therefore, it can be said that vitalliumtm is a notch sensitive material.

The notch sensitivity increased with a decrease in notch radius. This was due to the

stress concentration, not the fatigue notch factor. It can be seen from figure 7.9 that

the notch with the smaller radius seems to have a better resistance to fatigue than the

larger notch. Fatigue resistance suffers with increased notches. There are not enough

points on the notched S-N curve to make any definite conclusions quantifying the

fatigue resistance of different size notches.

The endurance limit of notched vitalliumtm is a value below 200 MPa. It is estimated

to be approximately 180 MPa. More high cycle testing is required to obtain the true

value. The threshold notch sensitivity is a value in the range 0 to 0.585. A sharp

notch analysis with notch radii less than 1.5 mm is required to determine this value.

9.3 Damage Assessment

In order to apply a particular damage theory to find remaining life, it is quite helpful

to know whether the material is sensitive to microcrack formation, to dislocation

migration, or to both. It is hypothesised that brittle materials are more susceptible to

microcracks and that ductile materials are susceptible to dislocation migration.

Vitalliumtm is a ductile material and so should be more sensitive to dislocation

migration than microcrack formation. The author suggests, if at all possible, using

density or hardness as a damage indicator property, as opposed to endurance limit,

because these properties best describe the relationship between damage and cycle

ratio for ductile materials (Refer to literature survey, Chapter 3.9.2).


94

9.4 Cumulative Damage Theories

The cumulative damage tests were reproducibility tests to validate the work of

Michael Moloney, the previous author [29]. His tests were based on the S-N curve

for machined vitalliumtm. The present authors’ tests were based on the S-N curve for

cast vitalliumtm. Therefore, the same two-level-stress tests yielded different numerical

results. Moloney’s cumulative damage results were accurate to within 20% for most

theories.

The extremity of prediction accuracy for cast vitalliumtm was enormous. The most

accurate prediction (error of 0.76%) was for a low-high test using Subramanyan’s

theory 9 (table 6.5). The least accurate prediction (error of 1224%) was for a high-

low test using Corten-Dolan’s theory (Table 6.8). The author decided not to use

percentage difference, but instead orders of magnitude. All of the predictions of

remaining life were of the same order of magnitude. These predictions are acceptable

when dealing with the random nature of fatigue damage. Some theories were more

accurate than others.

9.4.1 Marco Starkey’s Theory

Marco Starkey’s theory represents a better approximation of cumulative damage than

Miner’s linear damage rule because it takes the non-linear relationship of damage into

consideration. However, it is completely dependent on established damage curves at

each stress level for each different material. These damage curves are simply not

available for vitalliumtm, so the author could not apply Marco Starkey’s cumulative

damage theory. The previous author approximated these curves from the

experimentally determined S-N curve for vitalliumtm. This was considered to be

inaccurate because it was an approximation of an already approximate method of

damage evaluation (as are all cumulative damage theories). To apply this theory it

would be necessary to experimentally determine each damage curve for each stress

level for vitalliumtm. This would require a method of damage evaluation for different


95

cycle ratios. Vitalliumtm is a ductile material. Therefore hardness or density would be

the best damage indicators if this experimental work were to be conducted.

Hardness testing would be the easier of the two methods of damage evaluation to

implement at the University of Limerick because the testing equipment is readily

available. This work was not conducted because the test specimens came too late and

too few. This hardness testing was sidelined for tests that had a higher priority.

9.4.2 Miner’s, Subramanyan’s and Ben-Amoz’s Theories

Miner’s theory yields conservative estimates for both types of test which are high-low

tests and low-high tests, (table 6-4). Subramanyan’s theory, is less conservative but

more accurate, (table 6-5). These two theories are combined to produce Ben-Amoz’s

theory.

Ben-Amoz’s theory takes advantage of the two very useful theories of Miner and

Subramanyan. The cumulative damage scatter is taken into consideration and so

provides a conservative tool for failure prediction. The theory works best when crack

initiation information is available. This was not the case for vitalliumtm. However, it

was observed during testing that once a crack started to propagate, it failed

catastrophically very soon afterwards. This suggested that the vast majority of cycles

to failure are in the crack initiation phase. Therefore, it was considered quite

satisfactory to apply the simpler form of Ben-Amoz’s theory to vitalliumtm (Equations

4.10 and 4.1).

From the seven tests, two predictions were correct, four conservative and one non-

conservative. This is excellent by any standard. This accuracy can be further

increased by applying Ben-Amoz’s full theory using crack initiation information. The

author recommends the use of this theory for the prediction of the remaining life of

vitalliumtm.


96

9.4.3 Corten-Dolan’s and Marin’s Theories

Corten and Dolan’s theory is suited to conditions where there is several varied stress

levels contained in periodic blocks. It does not apply well to basic two-level-stress

tests, where the stress is only varied once. This is a distinct disadvantage when

carrying out two-level-stress tests. It is advantageous in modeling fatigue

accumulation, once the loading pattern is known in actual components subject to

repeated cycles of varied loads, such as those in prosthetic knee implants.

Corten-Dolan’s theory yields poor results when the constant ‘d’ is equal to 6.3 (i.e.

average between steel and aluminium) Refer to table 6.7. The overall accuracy is

greatly increased by using the average value of ‘d’ equal to 9.82 for vitalliumtm. (table

6.8). More tests need to be conducted to validate this value. It is a poor theory for

two-level stress testing because it does not take the initial number of stress cycles into

account. It’s application to vitalliumtm cannot be dismissed yet because it is a theory

designed for variable stress amplitude loading, not two-level-stress tests.

Marin’s theory yields random predictions and should not be applied to vitalliumtm.

Refer to table 6.9

9.4.4 Manson’s Theory

The main problem with this theory is the resolution of where the point ‘P’ lies on an

S-N curve for vitalliumtm. To determine ‘NP’ a high stress program would have to be

conducted for various cycle ratios at the ultimate tensile strength of the material. The

Avery-Denison fatigue machine was unable to apply the ultimate tensile strength of

812 MPa required for vitalliumtm.

The theory proposed by Manson yields very conservative predictions for high-

low tests, (table 6.10). It was the only theory which predicted failure at 2,500 cycles

for the first high-low test. Even though it is not accurate it can be considered as a safe


97

theory. However, this is true only when the value of “Np” is known for vitalliumtm.

The previous author used a value of Np equal to 1900 cycles. It is unknown whether

this is correct or not. More fatigue test are required to determine the value of “Np”.

9.4.5 Henry’s, Henry’s Modified and Gatts’s Theories

Henry’s cumulative damage theory utilises the new endurance limit ‘E’ as the damage

indicator. This may not be such a good idea for vitalliumtm because it has not yet

been shown experimentally that damage is a function of the new endurance limit. It is

unknown if the basic assumption of Henry’s theory applies to vitalliumtm.

That is, if DE E

E1

0 1

0

Equation 9.1

A possible test procedure to determine this would be to load eight test specimens at

the same stress, say 500MPa, for the same cycle ratio (n/N D) of say 0.5, then to

apply different stress levels to the eight specimens to determine an S-N curve and a

new endurance limit ‘E1’. If the left hand side of equation 9.1 approximately equals

the right hand side of the equation then it could be said that Henry’s theory applies to

vitalliumtm.

Henry’s theory generally predicts non-conservative values, yet it is relatively

accurate. However, non-conservative estimates are dangerous. It is a theory that is

based on steels and possibly should not be applied to vitalliumtm (table 6.12).

Henry’s modified theory makes use of an S-N curve for each new damage level. It

probably should not be applied to vitalliumtm for two reasons. The first reason is that

the theory is approximate in its nature because the S-N curve is assumed to be linear.

This is rarely ever the case as most S-N curves are ogee-shaped. The second reason is


98

that the modified theory suggests that the slope of the S-N curve should start at 106

cycles. The use of this theory would produce false S-N curves for each damage level

because the S-N curve for vitalliumtm is not asymptotic to the endurance limit at 106

cycles. If the theory were to be modified to produce an S-N curve at say 5x106 cycles

instead of 106 cycles, it would yield a better approximation.

Henry’s modified theory yields poor predictions for remaining life. Refer to table

6.11. This is because it assumes a linear S-N curve. This assumption cannot be made

for vitalliumtm because of the definite curved nature of its S-N curve.

Gatts’s and Henry’s cumulative damage theories are similar in that they both use the

reduction in endurance limit as their damage indicator. Gatts’s theory cannot be

readily applied to vitalliumtm for the same reason that Henry’s theory which is that

damage may not be a function of the new endurance limit for vitalliumtm.

The same points for Henry’s theory are valid for Gatts’ theory. The values of “C”,

0.338 and 0.218, for unnotched and notched vitalliumtm are questionable. However,

these values give better results than do the constants for the steel equivalent. Gatts’

theory generally yields non-conservative predictions and should therefore not be

applied to vitalliumtm.

9.4.6 Unified Theory

The unified theory presented very good overall accuracy for both types of tests. This

suggests that making use of non-dimensionalised fatigue data is useful to determine

the equivalent cycle ratio “Be2” caused by residual stressing. From seven tests, there

were five conservative predictions and two non-conservative predictions. The non-

conservative predictions however were quite accurate. The author believes that this

theory should be applied to vitalliumtm when accurate life predictions are required.


99

To conclude, Ben-Amoz’s cumulative damage theory is the best theory to apply to

vitalliumtm because it is conservatively accurate and simple to use. When absolute

accuracy is required the more complex unified theory should be used to predict

remaining life.

9.5 HIPed Specimens

It was expected that HIPing would increase the fatigue life of the cast specimens.

Overall, this was observed except in one case. At a stress of 650 MPa the HIPed

specimen failed earlier than the as-cast plain specimen, at the same stress. As soon as

a tiny crack was observed on the surface a loud banging noise and catastrophic failure

occurred. There was little or no crack propagation phase at this high stress. The

author suggests this may be the reason why.

A crack will propagate until it reaches a stress absorber, such as a casting void. This

increases the crack propagation time because the crack has to overcome the void

before it can continue. HIPing vitalliumtm had the effect of densifying the structure.

At the high stress of 650 MPa the crack energy was extremely high. This energy

could not have been absorbed in HIPed vitalliumtm because casting voids were

eliminated. Instead this high energy was used to cut straight through the material

without any resistance and caused premature catastrophic failure. HIPing vitalliumtm

may reduce fatigue life at stresses above 580 MPa. More high stress tests are required

to make this conclusive.

However, for lower stresses, the opposite case seemed to exist. The energy required

to cause crack initiation was much higher for HIPed specimens than for plain

specimens. This was due to the lack of internal voids to act as stress raisers. This

meant that the crack initiation phase was increased and therefore the overall fatigue

life.


100

It was observed for HIPed specimens, at stresses below 580 MPa, that once a crack

started to propagate it failed much slower than specimens that were not HIPed. The

fracture line was straight and well defined for the HIPed specimens, whereas the

crack line was more random for the plain specimens. From this it was deduced that

HIPing had the effect of increasing the crack propagation phase and the overall

ductility of vitalliumtm. The HIPed vitalliumtm was more resilient to bending and

therefore, more ductile.

HIPing, on average increased the fatigue life of specimens by 92%. This can be

observed in figure 7.11 ;( take note that this is not completely obvious from the figure

because the abscissa of the graph has a logarithmic scale).

Despite the increase in fatigue life, it is still unsure whether HIPing actually increases

the endurance limit. Intuitively, the author believes this to be so. However, this is

opinion and not fact. It is estimated that the endurance limit exists in the range 280

MPa to 300 MPa, but it may be somewhat lower than this. A more conservative

estimate is 250 MPa to 300 MPa. More high cycle testing is required to make a final

determination on the endurance limit.

9.6 Mean Stress Effects

A Haigh constant life diagram was constructed for machined vitalliumtm (figure 7.3)

with the use of its S-N curve (Appendix IV). This diagram shows that a maximum of

280 MPa of mean stress can be applied for infinite life. A fatigue strength diagram

was constructed fore cast vitalliumtm (figure 7.6). It gives the failure envelopes for

different life requirements and can be used as an approximate design chart fore cast

vitalliumtm.

9.7 Inspection

The magnified photographs in figures 8.1 and 8.2 suggest that cracks propagate along

the crystal boundaries of vitalliumtm. This is in the direction of least resistance.


101

The macroscopic inspection indicates that vitalliumtm fails in a brittle fashion (figure

8.5). This may be caused by work hardening of the specimen surface under cyclic

loading. The specimens’ surfaces were characterised by fibrous areas caused during

progressive crack propagation and crystalline areas which occurred during

catastrophic failure. Catastrophic failure did not occur in all cases. Therefore, the

surfaces of these specimens were not examined and failure was deemed to occur

when a deep crack was observed around the entire perimeter of the specimen.

9.8 The Fatigue Testing Machine

The stresses required, to conduct proper fatigue tests on vitalliumtm, seems to have

damaged the machine. Before further tests can be conducted the machine must be

repaired. The following repairs are required

1. The second adjuster needs to be re-bored and re-threaded to size M10.

2. The front damping plate needs to be replaced by a new one.

Note: The front damping plate supports the torsion bar housing.

3. The worn area on the torsion bar arm needs to be permanently repaired.

Although not totally necessary, it would be desirable to replace the dashpot switching

mechanism, to make it more reliable for high cycle tests.

Conclusions Cumulative Damage Analysis – Kieran J. Claffey

102

Chapter 10

Conclusions

1. Ben-Amoz’s cumulative damage theory works well for vitalliumtm.

2. The Unified cumulative damage theory yields accurate life prediction data for

vitalliumtm.

3. Vitalliumtm is a notch sensitive material.

4. HIPing increases the fatigue resistance and ductility of cast vitalliumtm.

5. Heavily machined vitalliumtm has a better fatigue resistance than cast vitalliumtm,

which has not been machined.

6. Vitalliumtm work hardens and fails in a brittle manner when subjected to

completely reversed bending cyclic loading.

7. Cracks propagate along the crystal boundaries of vitalliumtm.

8. The endurance limit of cast vitalliumtm lays within the range of 250 MPa to 300

MPa.

9. The endurance limit of HIPed vitalliumtm is estimated to lay within the range of

250 MPa to 300 MPa (and is expected to be greater than endurance limit of cast

vitalliumtm upon completion of further high cycle tests).

10. The endurance limit of the notched specimens is below 200 MPa and is estimated

to be 180 MPa.

11. The endurance limit of machined vitalliumtm is approximately 377 MPa.

12. The maximum allowable mean stress for infinite life that can be applied to

machined vitalliumtm under completely reversed bending conditions is 280 MPa.

13. The average value of ‘d’ for vitalliumtm, in Corten-Dolan’s cumulative theory was

found to be 9.82.

14. The threshold notch sensitivity of vitalliumtm is a value in the range 0 to 0.585.

Recommendations Cumulative Damage Analysis – Kieran J. Claffey

103

Chapter 11

Recommendations

1. Repair the fatigue testing machine as outlined in the discussion.

2. Conduct more tests on HIPed specimens to quantify better their increased fatigue

resistance.

3. The microstructure of HIPed vitalliumtm could be examined to check if any

casting defects exist.

4. Conduct more high cycle tests to properly determine the true endurance limit of

cast, notched and HIPed vitalliumtm.

5. The threshold notch sensitivity of vitalliumtm can be found by conducting another

notch analysis, with notch radii less than 1.5mm.

6. Conduct experiments to find the crack initiation information required to apply

Ben-Amoz’s full cumulative damage theory.

7. Conduct a test programme to determine whether a reduction in endurance limit is

a valid damage indicator for vitalliumtm.

8. Test the hardness of vitalliumtm for different cycle ratios of fatigue to discover if

hardness can be used as a damage indicator.

9. A variable stress amplitude fatigue test programme could be conducted to see if

Corten- Dolan’s cumulative damage theory applies to vitalliumtm.

10. It may be useful to conduct strain controlled fatigue tests. This type of test would

show the influence of randomised high strains that may occur in cast vitalliumtm.


104

References

1. JUVINALL, R.C., and MARSHEK, K.M., ‘Fundamentals of Machine Component

Design’, John Wiley & Sons, USA. 1991, p. 257-293.

2. HIGGINS, R.A., Materials for the Engineering Technician, Second Edition, p. 30-35, 71.

3. BARNBY, J.T, Fatigue M&B Technical Library, pp.13-31.

4. HEARN, E.J., ‘Mechanics of Materials’, Second Edition, Pergamon Press, 1992, p.842-

858.

5. BUCH, A., ‘Fatigue Strength Calculation’, Trans Tech SA, 1988, p.1-227.

6. AGOGINO, A.M., ‘Notch Effects, Stress State and Ductility’, Transactions of the

ASME, October 1978, V100, p.349-350.

7. SMITH, R.A., ‘Fatigue Crack Growth’, pp.117-129

8. WEI, R.P., 'Fracture Mechanics Approach to Fatigue Analysis in Design', Journal of

Engineering Materials and Technology, April 1978, V. 100, p. 113 - 120.

9. HARRISON, J.D., ' Damage Tolerant Design', Fatigue Crack Growth', (SMITH, R.A.).

pp. 117 - 129.


105

10. KITAGAWA, H., NAKASONE, Y., MIYASHITA, S., ‘Measurement of Fatigue

Damage by Randomly Distributed Small Crack Data’, Fatigue Mechanisms: Advances in

Quantitative Measurement of Physical Damage, ASTM STP 811, LANKFORD, J.,

DAVIDSON, D.L., MORRIS, W.L., WEI, R.P., Eds., ASTM, 1983, pp233-263.

11. McNAMARA, B.P., VICECONTI, M., CRISTOFOLINI, L., TONI, A., TAYLOR, D.,

‘Experimental and numerical pre-clinical evaluation relating to total hip Arthroplasty’,

Archives of Orthopaedic and Traumatic Surgery.

12. PLUVINAGE, G.C., RAGUET, M.N., 'Physical and Mechanical Measurements of

Damage in Low - Cycle Fatigue: Applications for Two-Level Tests, ' Fatigue

Mechanisms: Advances in Quantitative Measurement of Physical Damage, ASTM STP

811, LANKFORD, J., DAVIDSON, P.L. MORRIS, W.L., WEI, R.P., ASTM, 1983, PP.

139-150.

13. PETERSON, R.E., ‘Stress Concentration Factors’, John Wiley & Sons, 1953.

14. JAMES, P.J., ‘Isostatic Pressing Technology’, Applied Science Publishers, p.169-203,

221-238.

15. BS 3518: Part 1: 1993 ‘Fatigue Test Standards’.

16. ASTM, Handbook of Fatigue Testing, p. 13 - 14, 90-97.

17. MINER, M.A., ‘Cumulative Damage in Fatigue’, Journal of Applied Mechanics, 1945,

V12, p.159-164.

18. JOHNSON, J.B., OBERG, T, T., 'Airplane Propeller Blade Life', Metals and Alloys,

1938, V8, pp. 259 - 262.


106

19. BENHAM, P.P., CRAWFORD, R.J., 'Mechanics of Engineering Materials', p.559.

20. SUBRAMANYAN, S., ‘A Cumulative Damage Rule based on the Knee-point of the S-N

curve’, Transactions of the ASME, V98, October 1976, p.316-321.

21. BEN-AMOZ, M., ‘A Cumulative Damage Theory for Fatigue Life Prediction’,

Engineering Fracture Mechanics, 1990, V37, No. 2, pp. 341-347.

22. CORTEN, H.T., DOLAN, T.J., ‘Cumulative Fatigue Damage’, Proceedings of

International conference on Fatigue of Metals, ASME and IME, 1956.

23. COLLINS, J.A., ‘Failure of Materials in Mechanical Design’, John Wiley & Sons, USA,

1981, p. 255-274.

24. MARIN, J., ‘Mechanical Behaviour of Engineering Materials’, Prentice Hall 1962.

25. MANSON, S.S., FRECHE, J.C., ‘Application of a Double Linear Damage Rule to

Cumulative Fatigue’ Fatigue Crack Propagation, STP- 415, ASTM, Philadelphia, 1967,

p.384.

26. HENRY, D.L., ‘A Theory of Fatigue Damage Accumulation in Steel’, Transactions of

the ASME, 1955, V77, p. 913- 918.

27. GATTS, R.R., ‘Application of a Cumulative Damage Concept to Fatigue’, Transactions

of the ASME, 1961, p.531.

28. DUBUC, J., THANG, B.Q., BAZERGUI, A., BIRON, A., ‘Unified Theory of

Cumulative Damage in Metal Fatigue’, Canadian Welding Research Council (WRC),

Bulletin 162, p.1-19.


107

29. MOLONEY, M., ‘A Cumulative Damage Analysis of the Surgical Alloy, Vitalliumtm’,

Final year project report, 1996, p.1-145. (Not Published).

30. ASTM, ‘Handbook of Fatigue Testing’, p 90-97.

Appendix Cumulative Damage Analysis – Kieran J. Claffey

APPENDIX


A.1 Derivation of Equation 5.1

From bending theory

But the second moment of area I = BD3/12 for a rectangular beam

And the distance from the neutral axis to the surface is, y = D/2

Thus


A.2



A.3


A.4


A.5


A.6 Sample Calculations

All sample calculations shown in appendix are for the first high-low, two-level-stress test in

the cumulative damage analysis.

The details of which are

S1 = 650 MPa

N1 = 10,000

n1 = 8,000

S2 = 580 MPa

N2 = 18,500

n2 experimental = 2,500

Miner’s Life Prediction Theory

n Nn

N2 2

1

1

1 18 500 18 000

10 0003 700

,

,

,, cycles

Subramanyan’s Life Prediction Theory

580 280

650 280081

18 500 18 000

10 0003 0622

0 81

.

,,

,,

.

n

n2 = 3,062 cycles

Corten-Dolan’s Life Prediction Theory

The constant ‘d’ is taken as 6.3.

nN

S

S

2

2

1

6 3 6 3

10 000

580

650

20 500

. .

,, cycles


Marin’s Life Prediction Theory

q y x

y

x

n Nn

N

S

S

q

9 82

18 500

10 000

650

580

4 42

1 18 500 18 000

10 000

650

5806 1222 2

1

1

1

2

4 42

.

log,

,

log

.

,,

,,

.

n2 = 6,122 cycles

Manson’s Life Prediction Theory

A value of Np = 1,900 was assumed.

n2

18 5001 900

10 0001 90018 500 1

8 000

10 0002 038

,

,

,,

log,

,

log,

, cycles


Henry’s Life Prediction Theory

D

D D

E MPa

n

1

2 1

1

2

8 000

10 000

1280

650 2801

8 000

10 000

0 694

1 0 306

650 18 000

10 000650 280

2801

8 000

10 000

85 4

0 306 306 85 4

580 85 418 500

1306 85 4

580 85 4

6 304

,

,,

,

.

.

,

,,

,

.

. . .

.,

. .

.

,

n2 = 6,304 cycles

Gatt’s Life Prediction Theory

For most steels the original fatigue limit ‘Seo’ is about one half the ultimate tensile strength, giving a value

of ‘C’ equal to 0.5.

Therefore C = 280/827 = 0.338 for vitalliumtm

.

The new endurance limit is given by

S Sn

N

C

SS

SS

nN

e

eo

eo

11

1 11

1

1 1

1.


S MPa

n

e

650 1 18 000

10 000

1 0 338

650280

650280 1

18 000

10 000

2333

0 306306 2333

580 233318 500

1306 2333

580 2333

7 8522

,,

.

,

,

.

.. .

.,

. .

.

,

n2 = 7,852 cycles

Unified Life Prediction Theory

D

n N

e

e

1 8

2

8

8

2 2 2

0 8

0 8

1 0 8 2 322 32

2 95

132

0 708

708 2 072 07

2 95

107

1

0 708 12 07

2 95

107

0 82

1 18 500 1 0 82 3 330

.

.

. ..

.

.

.

. ..

.

.

..

.

.

.

, . ,

n2 = 3,330 cycles

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