Ramesh TalrejaAerospace Engineering

Texas A&M University, College Station, Texas

Contents

• The Engineering Motivation• Damage Scenarios• Multiple Scales of heterogeneities of damage entities• Hierarchical approach (“up-the-scales”)• Motivated (need-based) treatment of scales• Conclusion

Durability Analysisof Materials/Structures

Infrastructure• Buildings• Bridges• Roadways• Pipelines

Biomedical• Artificial Implants• Prosthetic Devices• Stents

Aerospace• Aircrafts• Launch Vehicles• Turbine Engines

Electronic Packages• Solder Joints• Substrate Materials• Wire Boards

Mechanical Loads

AqueousEnvironment

ThermalFluctuations

Component Durability AnalysisStress Analysis

Stress/Strain/Temp atCritical Sites

InitialDeformation

Models

ServiceLoading

Damage MechanismsMatrix cracking, delamination,

Viscoelasticity/aging, etc.

Damage MechanicsMicro/Meso/Macro Models

StiffnessDegradation

StrengthDegradation

Life Prediction

The Overall Approach

Criteria:

• Based on physical mechanisms

• Capable of structural analysis

• General (wide) applicability

Elements:

• Continuum thermodynamics with internalvariables, micromechanics, multiscale

Structure Substructure RVE Unit cell

CDM

Micromechanics

Multiscale Synergistic Damage Mechanics

Question:What is the best sequence of modeling: Right to Left, Or Left to Right, Or Combined (Synergistic)?

Damage classification

“Damage” in composites: Multiple cracking where shear-lag (at interfaces) is involved

• Pre-damage regime•Damage regime•Post-damage regime (Localization and fracture)

Pre-Damage RegimePre-Damage RegimeExample:

Unidirectional Composite in Transverse Tension

Debonding induces matrix cracking

Matrix cracking causes debonding

Length scales of microstructure:Fiber diameter, Inter-fiber spacing

σ

σ

Local Stress State resulting from transverse loading of fiber composites

Dilatational

Distortional

Depends on

• Fiber and matrix properties• Fiber distribution

σ

σ

Effect of Dilatational (hydrostatic tension) stress

Cavitation, presumably from free volume inpolymers

Unstable growth of cavitationat critical dilatational energy

When dilatational energyreaches a critical value,cavities burst opencausing debonding

Effect of Dilatational (hydrostatic tension) stress

Asp, Berglund, Talreja (1996)

Length scales of damage:Cavity diameter (before debonding)Fiber diameter (after debonding)

σ

σ

Distortional

Effect of Distortional stress

Matrix cracks form by Yielding, Void growth and Coalescence, crazing

Models:Rice, Tracey (1969)Boyce, Parks, Argon (1988)Gearing, Anand (2004)

Length scales of damage:Cavity diameter(before cracking)Inter-fiber spacing(after cracking)

σ

σ

Polymer Fracture Through CrazingPolymer Fracture Through Crazing

AB

D

E

C

ε

σhσ

Bulk Polymer

Active zone

Initiation

Defect Boundary

Bulk Polymer

Active zone

Initiation

Defect Boundary

Active zone

Bulk Polymer

Propagation

Initiation

Fibril Widening

Entanglement knots

Active zone

Bulk Polymer

Propagation

Initiation

Fibril Widening

Entanglement knots

Active zone

Bulk Polymer

Fibril Breakage

Active zone

Bulk Polymer

Fibril Breakage

Active zone

Bulk Polymer

Fibril Breakage

Fibril breakage Propagation

Active zone

Bulk Polymer

Fibril Breakage

Fibril breakage Propagation

Bulk PolymerBulk Polymer

CrazingCrazingFormation of fibrillar structure interspersed with micro cavity Formation of fibrillar structure interspersed with micro cavity due due to mechanical separation of polymer chains under tensile stressto mechanical separation of polymer chains under tensile stress

Craze LifeCraze Life

Craze InitiationCraze InitiationMicroMicro --void nucleationvoid nucleation

Craze WideningCraze WideningDrawing of new polymerDrawing of new polymer

from active zonefrom active zone

Craze BreakdownCraze BreakdownChain ScissionChain Scission

DisentanglementDisentanglement

ABCED

Damage RegimeDamage RegimeExample 1:

Unidirectional Ceramic Matrix Composite in Tension

Fibers

Fiber-bridgedmatrix crack

Increasing load

Damage RegimeDamage Regime

Example 2:Example 2:

Cross-Ply Polymer Matrix Composite in Fatigue

Delaminations

Transverse cracks

Axial splits

Cross Ply Composites and Woven Fabric Composites

P

P

0

0

L a y e r

_

1

0

L a y e r

_

2

0

L a y e r

S t r e s s F i e l d

Damage RegimeDamage Regime

Example 3:Example 3:

General laminate with off-axis ply cracking

Multiple matrix cracks, interfacial disbonds, delaminations, fiber breaks, microbuckled fibers, and more

Multiple orientations

Multiple scales of damage entities Multiple rates of evolution

Multiple effects on material response

Damage in Composites

What is the Lowest Damage Scale? • A Purist (Scientific) View: The first (basic) scale at which dissipative mechanism(s)

occur.

• A Pragmatist (Engineering) View:

The first significant scale (manifesting behavior of lower scales, if any) that governs the property of interest. Preferably, scale of observable entities.

The Multi-Scale Nature ofDamage in Composites

• Should be guided by the purpose (Model) -- To predict properties and performance, or -- To design properties for selected performance• Should account for the scale of inhomogeneities

(fibers, particles, plies, etc.)

-- Damage entities are often initiated by

inhomogeneities, and evolve under their influence

The Choice of Scales in an Engineering Approach

Damage MechanismsDamage Mechanisms

Unidirectional Ceramic Matrix Composite in Tension

Increasing Load Increasing Crack Density

Stress-Strain ResponseStress-Strain Response

Unidirectional Ceramic Matrix Composite in Tension

εL

σ

Stage I Stage II Stage III

εT

Stage II Damage MechanismStage II Damage Mechanism

εL

σ

Stage I Stage II Stage III

εTsliding

debonding

Fiber-bridged Matrix Cracking

Length Scales ofLength Scales ofStage II Damage Stage II Damage

MechanismMechanism

sliding

debonding

Damage Entity Length Scale: Crack lengthRVE Length Scale: Crack spacingMicrostructural Length Scale: Fiber diameter

Damage MechanismsDamage Mechanisms

Cross-Ply Polymer Matrix Composite in Fatigue

Multiple DamageModes:

Transverse Ply Cracks Axial Splits

Delaminations

Length Scales - Ply Cracking in Laminates

Damage Entity LengthScale: Ply thickness, tc

RVE Length Scale:Crack spacing, s

Microstructural LengthScale:Ply thickness, t0

Damage Entity LengthScale: Ply thickness, tc

RVE Length Scale:Crack spacing, s

Microstructural LengthScale:Ply thickness, t0

θ θ

Matrixcrack

Delamination

Ply Cracking with Delamination

Pn

a

Stationary microstructure

Evolvingmicrostructure

Homogenization of stationary microstructure

RVE

Homogenized continuumwith damage

Damage entity

A Continuum Characterization of Damage

Structure Substructure RVE Unit cell

CDM

Micromechanics

Multiscale Synergistic Damage Mechanics

Continuum Damage Mechanics - Elastic

Strain,Temperature,Temperature gradient,Damage,

ijε TiT,

ijD

Thεrmodynamic Statε,Stress ijσ ,Specific Helmholtz free energyy ,Specific entropyh

; ,Heat fluxiq

,Damage rateijD&

Response Functions

Isothermal Case

),( ijijDεψijij εψρσ ∂∂=

Truesdell’s Equipresence Principle

Clausius-Duhem Inequality

Material symmetry - irreducible integrity bases

ni

ai

S

P

X3

X2

X1

A Tensorial Representation of DamageA Tensorial Representation of Damage

RVE

nnii:: Unit normal to damage entity surfaceUnit normal to damage entity surface

aaii:: Represents pre-specified influence of Represents pre-specified influence of

damage entity on the surrounding mediumdamage entity on the surrounding medium

∫=S

jiij dSnad

∗ For “n ” distinct damage modes identified in the RVE a damage

mode tensor for each mode is given as

kkαα is the number of damage entities in the αα th mode, and VV is the

volume of the RVE

∗Only intralaminar-cracking mode is considered, i.e.

∑α

α=α

k kijij dVD )(1)(

€

D

1 1

≠ 0

ttcc: Thickness of the cracked ply

ttTT: Total laminate thickness

ss11: Spacing between cracks

κκ : Effect of constraint on the crack opening displacement imposed by the uncracked laminae

The damage tensor for intralaminar-cracking is given as follows:

D11 =κtc

2

s1tT

The Internal Variable of Damage

All terms are measurable, except κ

κ depends on “microstructure” and its length scales, and can be experimentally “identified” or calculated by analytical or computational micromechanics

D11 =κtc

2

s1tT

Structure Substructure RVE Unit cell

CDM

Micromechanics

Multiscale Synergistic Damage Mechanics

κ

Examples of SDM:

• Multiple cracking in UD CMC (Sørensen,Talreja, 1993)

• Multiple ply cracking in cross ply laminates (Varna, Akshantala, Talreja , 1999)

• Multiple transverse cracking with varying constraints (Varna, Akshantala, Talreja, 1999; Varna, Joffe, Talreja, 2001)

• Linear viscoelastic cross ply laminates with transverse cracks (Kumar, Talreja, 2003; Varna, Krasnikovs, Kumar, Talreja, 2004)

• Off-axis multiple cracking – one mode (Varna, Joffe, Akshantala, Talreja, 1999; Singh, Talreja, 2008)

• Off-axis multiple cracking – two modes (Singh, Talreja, 2009)

Review papers:• Talreja, R., Journal of Materials Science, 2006• Talreja, R. and Singh, C.V., In Multiscale Modelingand Simulation of Composite Materials and Structures, Y. Kwon, D.H. Allen and R. Talreja, Eds., Chapter 12, Springer, 2007.

ConclusionConclusion

Damage in composite materials is complex (multitude of size, shape, orientation) and not suited for “up-the scale” multi-scale approach

For application to complex shaped structures in service loading (time-varying multiaxial stress, temperature) continuum damage mechanics is the most suitable approach

Synergistic approach (CDM with “access” to judiciously selected micromechanics results) has been demonstrated for elastic and linear viscoelastic composites.

Damage evolution, not discussed here, is treated by micromechanics