Effective Inelastic Response of Polymer Composites by Direct Numerical Simulations A. Amine Benzerga Aerospace Engineering, Texas A&M University With:

Effective Inelastic Response of Polymer Composites by Direct Numerical Simulations

A. Amine Benzerga

Aerospace Engineering, Texas A&M University

With: R. Talreja, K. Chowdhury, X. Poulain, A. DeCastro and B. Burgess

Background & Motivation

2

Example: Composite blade containment casing for jet engines

Wide range of temperatures (service conditions)

Wide range of strain-rates (design for impact applications)

Ideal for implementing a multiscale modeling strategy:

(i) the material is heterogeneous at various scales;

(ii) the physical processes of damage occur at various scales

Li et al. (JAE, July 2009)

Goal: Develop a strategy aimed at predicting durability of structural components

Basic ingredient: Reliable physics-based inelastic constitutive models

Model Validation Damage ProgressionNumerical Homogenization

Material Parameter Identification

Polymer ModelExperimentsBackground/Motivation

July 23rd 2009

Background & Motivation

3




July 23rd 2009

Typical Response of a Polymer

4

elastic

hardening

softening

rehardening

T=298K

Compression

510 / s

Epon 862

Littel et al (2008)




July 23rd 2009

Temperature & Rate sensitivity

5

Effect of Temperature (Epon 862)

The behavior of polymers is temperature and strain-rate dependent




July 23rd 2009

Tension -3ε=10 /s

298K

323K

353K

Littel et al (2008)

CompressionLittel et al (2008)

- 5ε=10 /s - 3ε=10 /s

- 1ε=10 /s

ε=700/s

ε=1600/s

Strain-rate effects (Epon 862)

Specification of plastic flow:

e pD D DAssume additive decomposition

5/6

0 exp 1kk e

kk

A s

T s

2

:3

p pD D

3

2 de

p

3

:2e d d d b

where and

1 :eD L pD p

Pointwise tensor of elastic moduli Jaumann rate of Cauchy stress

Effective strain rate:

(define direction of plastic flow)

Flow rule:

3

2p

de

D

Effective stress: Deviatoric part of driving stress:

Back stress tensor

Strain rate effects

Material parameters

Describe pressure sensitivity

Internal variable

6

Polymer model

July 2009




Modified Macromolecular Model (Chowdhury et al. CMAME 2008)

7

1-ss

ss h

s

Nota Bene: Original law(Boyce et al. 1988 )

p

3 ch 8 ch

:1

b R DR R R

Evolution of back stress:

1 21 2

( ) 1 ( ) 1

s ss h h

s s

Evolution of athermal shear

strength s :

Polymer Model

July 2009




8

2 12

1

3 ( )1

. , ( ), , 3 1 csch

i j klijkl R ik jl jl ikc c cch

c c

T c cc c c

c c

B BR C N g B g B

tr BN

B F F tr BN

L

Material parameter identification

8

• Material parameters :

Elastic constants : ,E

e

Temperaturesensitivity

Strain-ratesensitivity

Pressuresensitivity

Small strainsoftening

Large strain hardening,

cyclic response

Pre-peak hardening

1 0 , 00 2 3,, , , ,, ,,, , , pR s s hA C Nm s f h Related to inelasticity :

s

E, n

s0

pε

s1

s2

f

h0

CR

N

A, 0ε

s

e

h3

Littell et al. (2008)

Reverse flow stress

Forward flow stress




July 18th 2009

9

1- Uniaxial tension, compression and torsion tests at fixed strain-rate :

2- Tensile data at various temperatures and strain-rates :

3- s0 is determined from :

4- s1 is determined from : (at lowest temperature at given strain-rate)

5- s2 is determined from : (at lowest temperature at given strain-rate)

6- Large strain compressive response and/or unloading response at fixed strain-rate and temperature :

7- Specific shape of stress-strain curve around peak :

Material parameter identification

0,A

( )( )

2(1 )

E TT

0 0.077

(1 )

s

log

( )ref

ref

ET T

E T

1

0

( )

( )p p

y y

s

s

1

2

( )

( )p p

d d

s

s

,RC N

0 , ,h f




July 18th 2009

10

Model validation

Tension at T=323K

10-1/s

10-3/s

620/s




July 18th 2009

11

Model validation

Tension at 10-1/s

T=298K

T=323K

T=353K




July 18th 2009

12

Model validation

Compression at T=298K

700/s

10-1/s

10-3/s

10-5/s

1600/s




July 18th 2009

Numerical Homogenization

13

• Principles of Numerical Simulations :

Unit cell composed of Epon 862 matrix (not optimized set), interface of fixed thickness and carbon fiber

Plane strain conditionsDamage not included

• Objectives :

Investigate evolution of mechanical fields (strains, stresses) in unit-cells

Relate micro/macroscopic behaviors Input for understanding of

onset/propagation of fracture

x1

x2

a

b

Epon 862

C fiber

interface

0

2 1 122

0

220

1( , )

ln

a

T x b dxa

bE

b




July 18th 2009

14

• Geometries :

Height: b= 100Cell aspect ratio: Ac= 2Fiber volume ratio: Vw =0.1Fiber aspect ratio: Aw=variable

Numerical HomogenizationModel Validation Damage ProgressionNumerical

HomogenizationMaterial Parameter

IdentificationPolymer ModelExperimentsBackground/

Motivation

July 18th 2009

Numerical Homogenization

15

2

2

UM F

t

• Numerical implementation :

Convective representation of finite deformations (Needleman, 1989)

Dynamic principle of Virtual Work:

FEM : Linear displacement triangular elts arranged in quadrilaterals of 4 crossed triangles.

Equations of Motions :

They are integrated numerically by Newmark-B method (Belytshko,1976) in an explicit FE code.

Constitutive updating is based on the rate tangent modulus method of Pierce et al (1984)

2 i

2

ud dS - d

tij i

ij i

V S V

V T u V

Kirchhoff stress

Green-Lagrange strain

Surface traction




July 18th 2009

16

• Calculations at E22=0.10:

TensionFiber : AS4 (sim. To T700)• Et= 14 GPa• ut=0.25

• Geometries :

Height: b= 100Cell aspect ratio: Ac= 2Fiber volume ratio: Vw =0.2 Fiber aspect ratio: Aw=1 (cyl.)

• Dramatic effect of fiber volume ratio on strengthening at all fiber aspect ratios




Motivation

July 18th 2009

17

• Calculations at E22=0.10:

CompressionFiber : AS4 (sim. To T700)• Et= 14 GPa• ut=0.25

• Geometries :

Height: b= 100Cell aspect ratio: Ac= 2Fiber volume ratio: Vw =0.2 Fiber aspect ratio: Aw=1 (cyl.)

• Plastic strains: Localization and maxima : same as in tension

• Hydrostatic stresses : Building-up in thin ligament between fiber and

edge Aw=6 : proximity of fiber to top surface where

stresses are computed may explain strengthening?




Motivation

July 18th 2009

18

Damage ProgressionModel Validation Damage ProgressionNumerical



Motivation

July 18th 2009

Objective: Develop an experimentally-valided matrix cracking model for use in mesoscale analyses

19




Motivation

July 18th 2009

Finding: Irrespective of the microscopic damage mechanisms, the fracture locus of the polymer matrix is pressure dependent and is temperature-dependent

-5 0 5 10 15 20 25 30 35

-20

0

20

40

60

80

100

120

StrainRate_10e-1eng

StrainRate_10e-3eng

StrainRate_10e-5eng

Maximum local Strain (%)

F/S

o (

MP

a)

0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.70

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

f(x) = NaN x + NaN

Room Temperature; strain rate 10e-3/s

Notched Bars

Linear (Notched Bars)

Smooth Bars

Stress Triaxiality Ratio

20

TENSION (PMMA)

Benzerga et al. (JAE, 2009)

DEBONDING : 0crit kv v kU U

Asp et al., 1996




Motivation

July 18th 2009

21

COMPRESSION (PMMA) DEBONDING : 0crit kv v kU U

Asp et al., 1996

No debonding :

0kk




Motivation

July 18th 2009

Polymer Fracture Model

22

2 1

, 0

,

c k kI k k

c kk k

k

T

c TT c T

Sternstein et al, 1979Gearing et Anand, 2004

Initiation:micro-void nucleation

fC

Propagation:Drawing of new polymer from active zone

1/

02(1 ( ) )

m

p cr II I

crc

D e es

Gearing et Anand, 2004

1f

Breakdown:Chain scission and disentanglement

c

Element Vanish Tech. of Tvergaard, 1981




July 18th 2009

Documents

Effective Inelastic Response of Polymer Composites by Direct Numerical Simulations A. Amine Benzerga Aerospace Engineering, Texas A&M University With: