1. 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

2. 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 ValidationDamage ProgressionNumerical Homogenization Material Parameter Identification Polymer ModelExperimentsBackground/ Motivation July 23 rd 2009

3. Background & Motivation 3 Model ValidationDamage ProgressionNumerical Homogenization Material Parameter Identification Polymer ModelExperimentsBackground/ Motivation July 23 rd 2009

4. Typical Response of a Polymer 4 elastic hardening softening rehardening T=298K Compression Epon 862 Littel et al (2008) Model ValidationDamage ProgressionNumerical Homogenization Material Parameter Identification Polymer ModelExperimentsBackground/ Motivation July 23 rd 2009

5. Temperature & Rate sensitivity 5 Effect of Temperature (Epon 862) The behavior of polymers is temperature and strain-rate dependent Model ValidationDamage ProgressionNumerical Homogenization Material Parameter Identification Polymer ModelExperimentsBackground/ Motivation July 23 rd 2009 Tension 298K 323K 353K Littel et al (2008) Compression Littel et al (2008) Strain-rate effects (Epon 862)

6. Specification of plastic flow: Assume additive decompositionwhereand Pointwise tensor of elastic moduliJaumann rate of Cauchy stress Effective strain rate: (define direction of plastic flow) Flow rule: 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 Model ValidationDamage ProgressionNumerical Homogenization Material Parameter Identification Polymer ModelExperimentsBackground/ Motivation Modified Macromolecular Model (Chowdhury et al. CMAME 2008)

7. 7 Nota Bene: Original law (Boyce et al. 1988 )  Evolution of back stress:  Evolution of athermal shear strength s : Polymer Model July 2009 Model ValidationDamage ProgressionNumerical Homogenization Material Parameter Identification Polymer ModelExperimentsBackground/ Motivation

8. Material parameter identification 8 Material parameters :  Elastic constants :  Temperature sensitivity Strain-rate sensitivity Pressure sensitivity Small strain softening Large strain hardening, cyclic response Pre-peak hardening  Related to inelasticity :  E, s0s0 s1s1 s2s2 f h0h0 CRCR N A,   h3h3 Littell et al. (2008) Reverse flow stress Forward flow stress Model ValidationDamage ProgressionNumerical Homogenization Material Parameter Identification Polymer ModelExperimentsBackground/ Motivation July 18 th 2009

9. 9 1- Uniaxial tension, compression and torsion tests at fixed strain-rate : 2- Tensile data at various temperatures and strain-rates : 3- s 0 is determined from : 4- s 1 is determined from : (at lowest temperature at given strain-rate) 5- s 2 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 Model ValidationDamage ProgressionNumerical Homogenization Material Parameter Identification Polymer ModelExperimentsBackground/ Motivation July 18 th 2009

10. 10 Model validation Tension at T=323K 10 -1 /s 10 -3 /s 620/s Model ValidationDamage ProgressionNumerical Homogenization Material Parameter Identification Polymer ModelExperimentsBackground/ Motivation July 18 th 2009

11. 11 Model validation Tension at 10 -1 /s T=298K T=323K T=353K Model ValidationDamage ProgressionNumerical Homogenization Material Parameter Identification Polymer ModelExperimentsBackground/ Motivation July 18 th 2009

12. 12 Model validation Compression at T=298K 700/s 10 -1 /s 10 -3 /s 10 -5 /s 1600/s Model ValidationDamage ProgressionNumerical Homogenization Material Parameter Identification Polymer ModelExperimentsBackground/ Motivation July 18 th 2009

13. 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 conditions  Damage 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 x1x1 x2x2 a b Epon 862 C fiber interface Model ValidationDamage ProgressionNumerical Homogenization Material Parameter Identification Polymer ModelExperimentsBackground/ Motivation July 18 th 2009

14. 14 Geometries :  Height: b= 100  Cell aspect ratio: A c = 2  Fiber volume ratio: V w =0.1  Fiber aspect ratio: A w =variable Numerical Homogenization Model ValidationDamage ProgressionNumerical Homogenization Material Parameter Identification Polymer ModelExperimentsBackground/ Motivation July 18 th 2009

15. Numerical Homogenization 15 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-  method (Belytshko,1976) in an explicit FE code.  Constitutive updating is based on the rate tangent modulus method of Pierce et al (1984) Kirchhoff stress Green-Lagrange strain Surface traction Model ValidationDamage ProgressionNumerical Homogenization Material Parameter Identification Polymer ModelExperimentsBackground/ Motivation July 18 th 2009

16. 16 Calculations at E 22 =0.10:  Tension  Fiber : AS4 (sim. To T700) E t = 14 GPa  t =0.25 Geometries :  Height: b= 100  Cell aspect ratio: A c = 2  Fiber volume ratio: V w =0.2  Fiber aspect ratio: A w =1 (cyl.) Dramatic effect of fiber volume ratio on strengthening at all fiber aspect ratios Numerical Homogenization Model ValidationDamage ProgressionNumerical Homogenization Material Parameter Identification Polymer ModelExperimentsBackground/ Motivation July 18 th 2009

17. 17 Calculations at E 22 =0.10:  Compression  Fiber : AS4 (sim. To T700) E t = 14 GPa  t =0.25 Geometries :  Height: b= 100  Cell aspect ratio: A c = 2  Fiber volume ratio: V w =0.2  Fiber aspect ratio: A w =1 (cyl.) Plastic strains: Localization and maxima : same as in tension Hydrostatic stresses : Building-up in thin ligament between fiber and edge A w =6 : proximity of fiber to top surface where stresses are computed may explain strengthening? Numerical Homogenization Model ValidationDamage ProgressionNumerical Homogenization Material Parameter Identification Polymer ModelExperimentsBackground/ Motivation July 18 th 2009

18. 18 Damage Progression Model ValidationDamage ProgressionNumerical Homogenization Material Parameter Identification Polymer ModelExperimentsBackground/ Motivation July 18 th 2009 Objective: Develop an experimentally-valided matrix cracking model for use in mesoscale analyses

19. 19 Damage Progression Model ValidationDamage ProgressionNumerical Homogenization Material Parameter Identification Polymer ModelExperimentsBackground/ Motivation July 18 th 2009 Finding: Irrespective of the microscopic damage mechanisms, the fracture locus of the polymer matrix is pressure dependent and is temperature-dependent

20. 20 TENSION (PMMA) Benzerga et al. (JAE, 2009) DEBONDING : Asp et al., 1996 Damage Progression Model ValidationDamage ProgressionNumerical Homogenization Material Parameter Identification Polymer ModelExperimentsBackground/ Motivation July 18 th 2009

21. 21 COMPRESSION (PMMA) DEBONDING : Asp et al., 1996 Damage Progression Model ValidationDamage ProgressionNumerical Homogenization Material Parameter Identification Polymer ModelExperimentsBackground/ Motivation July 18 th 2009

22. Polymer Fracture Model 22 Sternstein et al, 1979 Gearing et Anand, 2004  Initiation: micro-void nucleation  Propagation: Drawing of new polymer from active zone Gearing et Anand, 2004  Breakdown: Chain scission and disentanglement Element Vanish Tech. of Tvergaard, 1981 Model ValidationDamage ProgressionNumerical Homogenization Material Parameter Identification Polymer ModelExperimentsBackground/ Motivation July 18 th 2009