fatigue,)damage)and)failure)of) composite)materials...
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Fatigue, Damage and Failure of Composite Materials:
Mechanisms, Fatigue Life Diagrams and Life Prediction
Ramesh TalrejaDepartment of Aerospace Engineering
Department of Materials Science and EngineeringTexas A&M University, College Station, Texas, USA
UTMIS Autumn Course, Gothenburg, Sweden, 15-‐16 October 2019
Lecture 5 : DAMAGE MECHANICS
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Contents
•Multiscale approach to damage and failure•Macro damage mechanics•Micro damage mechanics• Synergistic damage mechanics• Defect damage mechanics• Virtual testing, computational micromechanics
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Multi-‐scale analysis -‐ Methodology
5 55𝜎"
Loading direction
Near the 0° ply at the free edge((y, z)=(W, h))
Central plane of the free edge((y, z)=(W, 0))
Internal section((y, z)=(0, 0))
Cracking positions
W
1/8model
h
(y, z)=(0, 0)
L
Lxx¥e!0°ply
90°ply
Resin
xy
z
Macro
Micro
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Macro damage mechanics (also called continuum damage mechanics, CDM)
Damage
Initiation
Homogenization
Undamaged continuum
Continuum with damage
Homogenized continuum
uS
tSt
uuS
tSt
uuS
tSt
u
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Early concepts of damage as an “internal state”, Kachanov (1958)
dd
m
Atf s
fæ ö
= - ç ÷è ø
Kachanov considered creep rupture of metals. He assumed the internal structure was degrading in time (becoming discontinuous) and introduced an internal variable calleddiscontinuity, f, f = 1 at t = 0, whose rate of change was assumed as a power law:
Robotnov later (1969) defined damage as w = 1 - f as a damage parameter representingnet area reduction. Thus, 0 < w < 1 was born as an internal state variable.
Today, “damage” denoted by D has become an abused concept, and D is used arbitrarilyas needed, mostly driven by convenience.
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Damage as an internal variable for composite materials (Talreja, 1985)
P
na
Stationary microstructure
Evolving
microstructure
Homogenization of stationary microstructure
RVE
Homogenized continuum
with damage
Damage entity
Note:1. Internal damage in compositematerials is in the form of distributed, orientedmicrocracks.2. All variables, stress, strain anddamage must be described usinga representative volume element(RVE).
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Damage characterization for composites
RVE Damage entity
o
x1
x2
x3
Pn
a
S
Damage Entity Tensor
Damage Mode Tensor
dij i jSd a n S= ò
Note: Instead of a tensora vector can be used(see Talreja, Proc. R. Soc,1985)
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2-‐Sep-‐19
Damage Mode Tensors
kk+1
kt kl
RVE of volume Vwhere kα is the number of damage entities in the αth mode
na
S
a: crack opening displacementb: crack sliding displacement
Assume b = 0
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Damage Tensor Components(One Damage Mode)
(one damage mode)
κ (kappa): Constraint parameter
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Computational Materials Engineering lab, Univ Toronto. Canada
Three Damage Modes:Cracking in θ, -‐θ, and 90° Plies
Singh and Talreja, 2013
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Multiscale Synergistic Damage Mechanics (SDM)
Constraint Parameter κθ
Damage Constants forreference laminate ai
Continuum Damage Mechanics
Multiscale Modeling
Overall Structural Analysis
ComputationalMicromechanics
Experiments/ Numerical
FE Analysis
Macro-‐level
Micro-‐level
Meso-‐ level StiffnessChanges
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Results: quasi-‐isotropic laminate
Longitudinal Modulus Poisson’s Ratio
Refs. Singh & Talreja, Mech Mat (2009) ; Singh & Talreja, Int J Solids Struct (2008)
Steps1. Fit the damage model with experimental data for crossply laminateèGives us phenomenological constants ai2. Compute constraint parameters by calculating CODs from FEM 3. Employ the model for quasi-‐isotropic laminate.
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Computational structural analysis
PRELIMINARY STRESS ANALYSIS
INPUTGeometry, laminate configuration, material
properties and loading conditions
Create geometric model Meshing Apply loading &
service conditions Stress analysis
DAMAGE ANALYSIS
Identify regions where damage might have
developed
Predict damage initiation & evolution
Evaluate stiffness properties of
damaged regions
UPDATED STRESS ANALYSIS
OUTPUTFailure characteristics, stress-‐strain response, deformation behavior, life and durability
Update stiffness of damaged regions Perform stress analysis of whole structure again
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Analytical/TheoryProvide a mathematical representation of physical
mechanisms;; predict for similar cases
ExperimentsProvide understanding of real material behavior;;
Calibrate, verify/validate models
ComputationalCombined with accurate
modeling provide predictions for complex configurations;;
Virtual testing;; Multiscale analysis
SYNERGISTIC DAMAGE MECHANICS
(SDM)Analytical/TheoryProvide a mathematical representation of physical
mechanisms;; predict for similar cases
ExperimentsProvide understanding of real material behavior;;
Calibrate, verify/validate models
ComputationalCombined with accurate
modeling provide predictions for complex configurations;;
Virtual testing;; Multiscale analysis
SYNERGISTIC DAMAGE MECHANICS
(SDM)
SDM methodology
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The “Big Picture”
Process'modeling,Simulation,
Tooling,'assembly,…
1.#Manufacturing
Real'initial'and'current'material'state'(RIMS'+'RCMS): Microstructure,'
Defects,'RVE
2.#Material
5.#Performance
Integrity,'Durability,Damage'tolerance
Cost/PerformanceTrade:offs#
Specification
PhysicalModeling
Multiphysics excitation(Mechanical,'thermal,'electromagnetic'etc.)
4.#Loading
Length'scale,'Shape,'Boundary'conditions
3.#Geometry • Define Material State (RIMS):
-‐ Fiber misalignment-‐ Fiber waviness-‐ Ply waviness-‐ Matrix voids• Construct RVE• Apply ply level
boundary conditions to RVE surfaces
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Defect Damage Mechanics
• Fiber Defects
Ø Misalignment, waviness
Ø Breakage
• Matrix Defects
Ø Incomplete curing
Ø Voids
• Interface Defects
Ø Fiber/matrix disbonds
Ø Delamination
• Fiber volume fraction
• Fiber Distribution
Ø Length
Ø Orientation
Idealized models:
Heterogeneities, no defects
Real composites: Defects
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Cost-‐performance trade-‐off
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Manufacturing Defects: NonuniformFiber Distribution and Matrix Voids
Debond link-‐up to transverse cracking
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Simulation of manufacturing induced fiber distribution nonuniformityApproach 1: Quantification of fiber mobility (radial and angular)
Dry fiber bundle
Resininfusion
Sudhir and Talreja, ASC, 2017
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Initial uniformpattern
Intermediate stepsto “shake” fibers
Final nonuniformpattern
Elnekhaily and Talreja, CST, 2018
Simulation of manufacturing induced fiber distribution nonuniformityApproach 2: Quantification of degree of nonuniformity
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RVE realizations for different degrees of nonuniformity 100% Nonuniformity 60% Nonuniformity 30% Nonuniformity
(a) 40% fiber volume fraction
(b) 30% fiber volume fraction
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Minimum RVE sizeApproach 1: Pair distribution function G(r)
NOR=5
NOR: Number of Rings around the central fiber
𝑑𝑟 = ±0.25𝑟𝑑𝜃 = ±15°
NOR=9
As NOR ≥ 9 the avg G function stabilizes è the min size of RVEis NOR=9
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Minimum RVE sizeApproach 2: Stabilization of nearest neighbor statistics
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 100 200 300 400 500 600
Freq
uenc
y M
ean
Val
ue
Number of Fibers-RVE size
Stable statistical content
24 x 24 fibers RVE20 x 20 fibersInner window
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Stress analysis for Approach 1: Embedded cell method
R R
0 0.2 0.4 0.6 0.8 1 1.2Normalized Traction
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Thickness of the RVE
Traction at the BoundaryL=2.05RL=2.10RL=2.20RL=2.50RL=3.00RL=4.0RL=8.0RL=12RL=24RL=32RL=40R
Δ𝑇 = −82,Uniform boundary displacement
Size of the embedding composite determinedby Hill’s criterion: Uniform boundary traction
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Failure analysis:Multi level energy hierarchy approach
Assumptions (Statements of truths):
• The FIRST failure event occurs at the LOWEST critical energy level• Local stress states are generally TRIAXIAL• “Strength” criteria (max tension, max shear, “effective” stress, etc.)
are not rational unless derived from energy criteria• Progression of failure can be by a sequence of failure events or by
incremental advance of already occurred failure (e.g. crack growth)
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Debonding induces matrix cracking
Matrix cracking causes debondingOR
Failure under transverse tension
Fiber/Matrix debondingand matrix cracking
σ
σ
Gamstedt et al (1999)
How is damage initiated under transverse tension?
Wood & Bradley (1997)
σ
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Dilatational
Distortional
σ
σ
Failure under transverse tension
Dilatational Energy Density
Distortional Energy Density
Total Stain Energy Density
Criticality:Brittle cavitationSubsequent failure:Fiber/matrix debonding
Criticality:YieldingSubsequent failure:Shear bands, cavitation,cracking
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Asp, Berglund, Talreja (1996)
poker-chip test
Failure analysis: Brittle cavitation
Uv=GHIJKL
(σ1+ σ2 + σ3)2 = Uv,crit ≈0.2MPa for epoxiesMuch lower than energy for yielding
σ
σ
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0.000.501.001.502.002.503.003.504.004.505.00
0 0.2 0.4 0.6 0.8 1Max
imum
Str
ain
Ene
rgy
Den
sitie
s M
Pa
Strain %
Uv Ud
Example: 50% fiber volume fraction and 100 % degree of nonuniformity
Dilatational strain energy density Criterion 0.2MPa
0.00
1.00
2.00
3.00
0 0.2 0.4 0.6 0.8 1
Prin
cipa
lStr
ess R
atio
s
Strain %
σ1/σ2σ1/σ3σ2/σ3σmax/σmean
Brittle cavitation under transverse tension
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Point of damage initiation (cavitation) by dilatational energy density criteria
Location of Dilatation Induced Brittle Cavitation
30IMECE 2017
𝑑𝑟 = ±0.25𝑟𝑑𝜃 = ±15°
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Crack Formation by Debond Coalescence
𝑑𝑟 = ±0.25𝑟𝑑𝜃 = ±15°
Increasing load
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Applied strain to transverse cracking
Clusteredfibers
Dispersedfibers
Note: These resultscannot be obtainedby homogenizingcomposites,or by consideringuniform fiberdistributions
Sudhir and Talreja, ASC, 2018
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Effect of fiber volume fraction and degree of nonuniformity
0
0.1
0.2
0.3
0.4
0.5
0.6
0 20 40 60 80 100
Aver
age
Cav
itatio
n M
echa
nica
l Str
ain
%
Percentage Degree of Nonuniformity
60VF50VF40VF
Transverse strain to initiation of debonding
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Debond initiation depends on fiber stiffness
0
0.1
0.2
0.3
0.4
0 0.01 0.02 0.03 0.04 0.05 0.06
Aver
age
Cav
itatio
n M
echa
nica
l St
rain
%
Em/Ef
Carbon/epoxy
Glass/epoxy
50% fiber volume fraction100% degree of nonuniformity
Elnekhaily & Talreja, CST, 2017
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Summarizing remarks
• Manufacturing defects cannot be fully eliminated without making composite structures prohibitively expensive• Stress and failure analysis of early failure events must include defects• Defect severity depends on the failure mode, e.g., for transverse crack formation, it is the degree of nonuniformity of fiber distribution (degree of fiber clusters)
Other examples:• Fiber misalignment for axial compression failure• Voids for matrix crack initiation• Fiber surface defects for axial tension failure