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Ramesh TalrejaAerospace Engineering
Texas A&M University, College Station, Texas
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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
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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
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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
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The Overall Approach
Criteria:
• Based on physical mechanisms
• Capable of structural analysis
• General (wide) applicability
Elements:
• Continuum thermodynamics with internalvariables, micromechanics, multiscale
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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)?
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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)
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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
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Local Stress State resulting from transverse loading of fiber composites
Dilatational
Distortional
Depends on
• Fiber and matrix properties• Fiber distribution
σ
σ
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Effect of Dilatational (hydrostatic tension) stress
Cavitation, presumably from free volume inpolymers
Unstable growth of cavitationat critical dilatational energy
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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
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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
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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
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Damage RegimeDamage RegimeExample 1:
Unidirectional Ceramic Matrix Composite in Tension
Fibers
Fiber-bridgedmatrix crack
Increasing load
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Damage RegimeDamage Regime
Example 2:Example 2:
Cross-Ply Polymer Matrix Composite in Fatigue
Delaminations
Transverse cracks
Axial splits
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Cross Ply Composites and Woven Fabric Composites
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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
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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
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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
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• 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
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Damage MechanismsDamage Mechanisms
Unidirectional Ceramic Matrix Composite in Tension
Increasing Load Increasing Crack Density
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Stress-Strain ResponseStress-Strain Response
Unidirectional Ceramic Matrix Composite in Tension
εL
σ
Stage I Stage II Stage III
εT
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Stage II Damage MechanismStage II Damage Mechanism
εL
σ
Stage I Stage II Stage III
εTsliding
debonding
Fiber-bridged Matrix Cracking
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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
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Damage MechanismsDamage Mechanisms
Cross-Ply Polymer Matrix Composite in Fatigue
Multiple DamageModes:
Transverse Ply Cracks Axial Splits
Delaminations
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Length Scales - Ply Cracking in Laminates
Damage Entity LengthScale: Ply thickness, tc
RVE Length Scale:Crack spacing, s
Microstructural LengthScale:Ply thickness, t0
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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
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Pn
a
Stationary microstructure
Evolvingmicrostructure
Homogenization of stationary microstructure
RVE
Homogenized continuumwith damage
Damage entity
A Continuum Characterization of Damage
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Structure Substructure RVE Unit cell
CDM
Micromechanics
Multiscale Synergistic Damage Mechanics
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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
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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
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∗ 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
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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
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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
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Structure Substructure RVE Unit cell
CDM
Micromechanics
Multiscale Synergistic Damage Mechanics
κ
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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.
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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