Copyright 2005 ABAQUS, Inc.
CompTest2006compTest2006
Copyright 2006 ABAQUS, Inc.
compTest2006
Progressive Damage Modeling In Fiber-Reinforced Materials
Ireneusz Lapczyk, Juan Hurtado
Progressive Damage Modeling in Fiber-Reinforced Materials 2
Copyright 2006 ABAQUS, Inc.
Outline
Overview of Damage Model for Fiber-Reinforced Materials
Example
Progressive Damage Modeling in Fiber-Reinforced Materials 3
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Damage Model for Fiber-Reinforced Materials: Overview
The damage in the material is anisotropic
Four different failure modes are taken into account: fiber tension, fiber compression, matrix tension, matrix compression
The behavior of the undamaged material is linearly elastic
The model must be used with elements with a plane stress formulation (plane stress, shell, continuum shell, and membrane elements)
Post-localization response is regularized (Crack Band Model)
The model can be used in conjunction with a viscous regularization scheme to improve the convergence rate in the softening regime
Progressive Damage Modeling in Fiber-Reinforced Materials 4
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Damage Initiation (Hashin’s criteria)
10,ˆˆ 2
122
11 ≤≤⎟⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛= ασασ
whereSX
f LTI
MODE I: fiber tension
MODE II: fiber compression
MODE III: matrix tension
MODE IV: matrix compression
211ˆ⎟⎠⎞
⎜⎝⎛= CII X
f σ
212
222 ˆˆ
⎟⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛= LTIII SY
fσσ
21222
2222 ˆˆ
122
ˆ⎟⎠⎞
⎜⎝⎛+
⎥⎥⎦
⎤
⎢⎢⎣
⎡−⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟
⎠⎞
⎜⎝⎛= LCT
C
TIV SYSY
Sf
σσσ
strengthsmaterial,,,,, −CTCTTL YYXXSS
(1)
(2)
(3)
(4)
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Damaged Material Response (Matzenmiller, Lubliner, Taylor)
operatordamage−M
Mσσ =ˆ
⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−
−
−
=
s
m
f
d
d
d
1100
01
10
001
1
M
(1)
Effective Stress
(2)
variablesdamage,, −smf ddd
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Damaged Material Response, cont.
( )( )( )( )mcmtfcfts ddddd −−−−−= 11111
( )
( )
( ) ⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−
−−
−−
=
Gd
EdE
EEd
s
m
f
1100
01
1
01
1
21
12
2
21
1
ν
ν
H
moduli undamaged,, 21 −sGEE
Damaged Elasticity Matrix:
( ) ( )( )( )( ) ( )
( ) ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−−−
−−−=
GdDEdEdd
EddEd
Ds
mmf
mff
10001110111
12212
1211
νν
C
( )( ) 0111where 2112 >−−−= ννmf ddD
(1)
(2)
Damaged Compliance Matrix:
ratiossPoisson', 2112 −νν
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Damaged Material Response, cont.
( )smcmtfcft dGEddEddEG
−+−
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−−
+−
+⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−−
+−
=1112
1112
1 212
1
2211122
222
22
2
211
211
1
σσσνσσσσ
0mod
1
≥= ∑=
esnumber
i
ii dYD
forcesmicthermodynaarewherei
i dGY
∂∂=
(1)
(2)
Gibbs Free Energy:
Energy Dissipation:
( )2
aseveryfordefinedoperator,bracketMacauleyiswhereαα
αα+
=ℜ∈
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Damage Evolution
The modeling approach is a generalization of that used to model cohesive elements, which is based on the work of C. Davila and P. Camanho
The evolution law is based on the energy dissipated during the process Linear material softening is assumed
( )( )0
0
eqf
eqeq
eqeqf
eqdδδδδδδ
−−
=
Figure 1.
(1)
equivalent displacement
equivalentstress
A
B
C0
Gc
δ eqo δ eq
f
Progressive Damage Modeling in Fiber-Reinforced Materials 9
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Equivalent Displacements and Stresses
11δδ −=fceq fc
eq
cteq δ
δσσ 1111 −−
=
212
222 δδδ +=mt
eq
212
211 αδδδ +=ft
eq fteq
fteq δ
δασδσσ 12121111 +
=
mteq
mteq δ
δσδσσ 12122222 +
=
212
222 δδδ +−=mc
eq mceq
mceq δ
δσδσσ 12122222 +−−
=
Fiber Tensile Mode
Fiber Compressive Mode
Matrix Tensile Mode
Matrix Compressive Mode
– is the characteristic lengthcLand
where ijcij L εδ =
Progressive Damage Modeling in Fiber-Reinforced Materials 10
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Damage Evolution: Procedure
Ifeqeq σδ ,
0
2
eq
Iff
eq
Gσ
δ =
Evaluate the initiation criterion,
Compute equivalent displacement and stress,
Compute equivalent displacement and stress at the onset of damage
Store
Compute equivalent displacement at full damage
Update the damage variable
I
eq
I
eq
ff eqeq
σσ
δδ == 00 and
( )( )⎟⎟⎠
⎞
⎜⎜
⎝
⎛
−−
= 0
0
, I ,maxd
eqf
eq
eqeqf
eqOLDI
eq
dδδδδδδ
00 and eqeq σδ
Progressive Damage Modeling in Fiber-Reinforced Materials 11
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Mesh Dependence
Example: uniaxial tension
Figure 1. Bar subjected to axial load Figure 2. Localization of deformation
Figure 3. Force-displacement for different discretizations
the energy dissipated is specified per unit volumethe results are mesh dependentdeformation localizes into one layer of elements
Progressive Damage Modeling in Fiber-Reinforced Materials 12
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Mesh Dependency, cont.
Figure 1.
The energy dissipated is regularized using Crack Band Model (Bazant and Oh)The energy dissipated, Gf, is expressed per unit area instead of per unit volumeCharacteristic length, Lc, is introduced, computed as
Lc = √AI, where AI is the area at an integration point
The post-localization stress-displacement response is computed correctly
Progressive Damage Modeling in Fiber-Reinforced Materials 13
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Viscous Regularization
( )νν
η ftftft
ft ddd −= 1 ( )νν
η fcfcfc
fc ddd −= 1
ννννmcmtfcft dddd ,,, - are used to compute damaged stiffness matrix
ηft, ηfc, ηmt, ηmc - are viscosities
( )νν
η mtmtmt
mt ddd −= 1 ( )νν
η mcmcmc
mc ddd −= 1
(1a-b)
(2a-b)
Generalization of the Duvaut-Lions regularization model
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Viscous Regularization, cont.
Updating “Regularized” Damage Variables
Jacobian
modedamageadenotesIwhere,000 t
vI
I
IttI
Itt
vI d
td
ttd
Δ++
Δ+Δ=
Δ+Δ+ ηη
η
ttd
d I
I
IvI Δ+
Δ∂
∂∂∂
+=∂∂ ∑ ηε
εCCεσ d
d
(1)
(2)
tt
dd
II
vI
Δ+Δ=
∂∂
η
Progressive Damage Modeling in Fiber-Reinforced Materials 15
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Viscous Regularization, cont.
energyfree−ψ
( )vIddd CC =
( ) ( )00
00
2 tttttt
DE ψψ +−Δ+
=ΔΔ+
Δ+ εεCεC dd
Viscous Energy Dissipation
Damage Energy
( ) ( ) ( ) ( )ε
εCεCε
εCεC 0d
0ddd
Δ+
−Δ+
=Δ Δ+Δ+
220000 tttttt
VE
( )Id0d
0d CC =
(1)
(2)
Progressive Damage Modeling in Fiber-Reinforced Materials 16
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Output
Initiation Criteria VariablesHSNFTCRT – tensile fiber Hashin’s criterionHSNFCCRT – compressive fiber Hashin’s criterionHSNMTCRT – tensile matrix Hashin’s criterionHSNMCCRT – compressive matrix Hashin’s criterion
Damage VariablesDAMAGEFT – tensile fiber damageDAMAGEFC – compressive fiber damageDAMAGEMT – tensile matrix damageDAMAGEMC – compressive matrix damageDAMAGESHR - shear damage
StatusSTATUS – element status (1 – present, 0 – removed)
EnergiesDamage energy (ALLDMD,DMENER,ELDMD,EDMDDEN)Viscous regularization (ALLCD, CENER, ELCD, ECDDEN)
Progressive Damage Modeling in Fiber-Reinforced Materials 17
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Example: Failure of Blunt Notched Fiber Metal Laminates
y
xz
1/8 part model
50 mm
300 mm1.406 mm
d = 4.8 mm
Plate Geometry
Progressive Damage Modeling in Fiber-Reinforced Materials 18
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Example: Failure of Blunt Notched Fiber Metal Laminates
Through-thickness view of the laminate
Progressive Damage Modeling in Fiber-Reinforced Materials 19
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Example: Results
1
2
3
Figure 1. Fiber damage pattern
Figure 2. Load-displacement curve
Progressive Damage Modeling in Fiber-Reinforced Materials 20
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Example: Results
Figure 1. Matrix tension damage pattern Figure 2. Matrix compression damage pattern
Progressive Damage Modeling in Fiber-Reinforced Materials 21
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Example: Energy dissipation
Figure 2.Figure 1.