Effective Inelastic Response of Polymer Composites by Direct Numerical Simulations

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 862Littel 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/sε=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 sT 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 hs

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 hs s

Evolution of athermal shear

strength s :

Polymer Model

July 2009




8

2 12

13 ( )

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 Btr BN

B F F tr BN

L

Material parameter identification

8

• Material parameters :

Elastic constants : ,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 :

E, n

s0

pεs1

s2

f

h0

CR

N

A, 0ε

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

ss

1

2

( )( )

p p

d d

ss

,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=298KT=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

bEpon 862

C fiber

interface

02 1 1

220

220

1 ( , )

ln

a

T x b dxabEb




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 Ft

• 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 - dt

ij iij 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• t=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• t=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/So

(MPa

)

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

Effective Inelastic Response of Polymer Composites by Direct Numerical Simulations

Documents

strainrate effects epon

effective stress

list of material parameters

driving stress

tension 298k323k353klittel

compressionlittel et

kcompression epon

various scales li