Progressive Damage Modeling In Fiber-Reinforced Materialscomptest/proc/files/presentations/... ·...

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Progressive Damage Modeling In Fiber-Reinforced Materials

Ireneusz Lapczyk, Juan Hurtado

Progressive Damage Modeling in Fiber-Reinforced Materials 2


Outline

Overview of Damage Model for Fiber-Reinforced Materials

Example



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



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)



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



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



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∈



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



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 =



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



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



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



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



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



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)



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)



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



Example: Failure of Blunt Notched Fiber Metal Laminates

Through-thickness view of the laminate



Example: Results

1

2

3

Figure 1. Fiber damage pattern

Figure 2. Load-displacement curve



Example: Results

Figure 1. Matrix tension damage pattern Figure 2. Matrix compression damage pattern



Example: Energy dissipation

Figure 2.Figure 1.



Example: Results

Blunt Notch Strength (MPa) for Different Viscosities

446454.2461.3466.6ηf=0.00025ηf=0.0005ηf=0.001

Experimental Results (De Vries, 2001)

Numerical Results (SC8R, 0º angle)

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