Martin Lévesque *, Katell Derrien*, Didier Baptiste* Michael D. Gilchrist**

1. A Micromechanical Model for Non-Linear Viscoelastic Particle Reinforced Polymeric Composite Materials – Undamaged State Micromechanical approach Martin Lévesque*, Katell Derrien*, Didier Baptiste* Michael D. Gilchrist** *Laboratoire LM3, ENSAM Paris **Department of Mechanical Engineering, NUI Dublin

2. 1. Phenomenological behaviour law identification under specific loading. Loading changes (strain rate effects ?) Reinforcement change (shape, nature, volume faction, etc.) ? A Micromechanical Model for Non-Linear Viscoelastic Particle Reinforced Polymeric Composite Materials – Undamaged State PROBLEM: Knowing the mechanical response of a reinforced polymer under a given loading when the matrix is nonlinear viscoelastic SOLUTIONS: What happens if : Time consuming, expensive

3. 2. Building a theoretical model which accounts for : A Micromechanical Model for Non-Linear Viscoelastic Particle Reinforced Polymeric Composite Materials – Undamaged State PROBLEM: Knowing the mechanical response of a reinforced polymer under a given loading when the matrix is nonlinear viscoelastic SOLUTIONS: Different loading cases Different reinforcement situations (nature, shape, volume fraction, etc.) Based on : Knowledge of each reinforcement behaviour Knowledge of the microstructure HOMOGENISATION APPROACH

4. A Micromechanical Model for Non-Linear Viscoelastic Particle Reinforced Polymeric Composite Materials – Undamaged State OUTLINE 1. Homogenisation basics 2. Constituents behaviour 3. Theoretical model 4. Model theoretical predictions 5. Conclusion

5. A Micromechanical Model for Non-Linear Viscoelastic Particle Reinforced Polymeric Composite Materials – Undamaged State BASICS Behaviour law identification of each constituent BEHAVIOUR MODEL Description of the microstructure PREDICTION CONCLUSION 3 BASIC STEPS: 1. Description Elastic, plastic, viscoelastic, nonlinear, etc. Shape of the reinforcements Orientation of the reinforcements Volume fraction of the reinforcements Position of the reinforcements

6. A Micromechanical Model for Non-Linear Viscoelastic Particle Reinforced Polymeric Composite Materials – Undamaged State BASICS Matrix (phase 0) BEHAVIOUR Solution of a structure problem MODEL PREDICTION Functions of the description and the homogenisation scheme (hypothesis on which the solution is based) CONCLUSION Reinforcement (phase 1) 3 BASIC STEPS: 2. Localisation: What are the stresses and strains in each of the constituent phases when a macroscopic loading is applied ?

7. A Micromechanical Model for Non-Linear Viscoelastic Particle Reinforced Polymeric Composite Materials – Undamaged State BASICS Equilibrium Localisation BEHAVIOUR MODEL PREDICTION Behaviour CONCLUSION 3 BASIC STEPS: 3. Homogenisation: How to relate the loading in each phase to the macroscopic loading ? Transition from the micro scale to the macro scale

8. A Micromechanical Model for Non-Linear Viscoelastic Particle Reinforced Polymeric Composite Materials – Undamaged State BASICS BEHAVIOUR MODEL PREDICTION CONCLUSION Material studied Glass bead reinforced polypropylene (10% to 30 %) Beads assumed to be randomly distributed Glass assumed to be linear elastic Polypropylene assumed to obey a Schapery type behaviour law

9. A Micromechanical Model for Non-Linear Viscoelastic Particle Reinforced Polymeric Composite Materials – Undamaged State BASICS Elastic part Viscoelastic part BEHAVIOUR MODEL PREDICTION CONCLUSION Linear viscoelastic creep compliance Time-shift factor Generalised stress Schapery type behaviour law:

10. A Micromechanical Model for Non-Linear Viscoelastic Particle Reinforced Polymeric Composite Materials – Undamaged State BASICS BEHAVIOUR MODEL PREDICTION + CONCLUSION Nonlinear viscoelastic ? MODEL ELABORATION: Homogenisation schemes well established for linear elastic materials What happens if : One constituent is linear viscoelastic ? Nonlinear (elastic, plastic) ?

11. A Micromechanical Model for Non-Linear Viscoelastic Particle Reinforced Polymeric Composite Materials – Undamaged State BASICS Linear viscoelastic behaviour law BEHAVIOUR Laplace – Carson transform MODEL PREDICTION CONCLUSION Analogy with linear elastic behaviour Homogenisation for linear viscoelastic materials: Homogenisation carried out in the Laplace – Carson space Time domain solution obtained by inverse transform of the symbolic solution

12. A Micromechanical Model for Non-Linear Viscoelastic Particle Reinforced Polymeric Composite Materials – Undamaged State Homogenisation for nonlinear materials: Affine formulation BASICS BEHAVIOUR Tangent modulus calculated for a given loading  MODEL PREDICTION CONCLUSION Thermoelastic problem 0  Linearisation Non-zero initial loading

13. A Micromechanical Model for Non-Linear Viscoelastic Particle Reinforced Polymeric Composite Materials – Undamaged State 1. Linearisation of the nonlinear viscoelastic material into a linear viscoelastic material BASICS 2. Solve the homogenisation problem with the Laplace – Carson transforms BEHAVIOUR MODEL PREDICTION Tangency in tn for the same load history CONCLUSION Stress free deformation history Homogenisation for nonlinear viscoelastic materials: Suggested linearisation :

14. A Micromechanical Model for Non-Linear Viscoelastic Particle Reinforced Polymeric Composite Materials – Undamaged State BASICS Tensile loading at constant STRESS rate BEHAVIOUR MODEL PREDICTION CONCLUSION Homogenisation for nonlinear viscoelastic materials: Linearisation example

15. A Micromechanical Model for Non-Linear Viscoelastic Particle Reinforced Polymeric Composite Materials – Undamaged State Laplace – Carson transform Matrix behaviour Linearisation BASICS Laplace – Carson transform Glass behaviour BEHAVIOUR Reinforcements shape and orientation Localisation as per the Mori – Tanaka scheme in the Laplace space MODEL Reinforcements volume fraction PREDICTION CONCLUSION Homogenisation Inverse Laplace Carson Transform Homogenisation for nonlinear viscoelastic materials: Macroscopic response of the whole composite

16. A Micromechanical Model for Non-Linear Viscoelastic Particle Reinforced Polymeric Composite Materials – Undamaged State BASICS BEHAVIOUR MODEL PREDICTION CONCLUSION Simulated response to a uniaxial stress load applied at a stress rate of 5 MPa / sec for various volume fractions of glass beads reinforced polypropylene

17. A Micromechanical Model for Non-Linear Viscoelastic Particle Reinforced Polymeric Composite Materials – Undamaged State BASICS BEHAVIOUR MODEL PREDICTION CONCLUSION Effect on the strain at 25 MPa of the stress rate for simulated uniaxial loadings at a constant stress rate for various volume fractions of reinforcements

18. A Micromechanical Model for Non-Linear Viscoelastic Particle Reinforced Polymeric Composite Materials – Undamaged State BASICS Matrix behaviour BEHAVIOUR Reinforcements behaviour MODEL Reinforcements shape and orientation PREDICTION CONCLUSION Global composite behaviour Reinforcements volume fraction CONCLUSION Theoretical model taking into account the nonlinear viscoelasticity

19. A Micromechanical Model for Non-Linear Viscoelastic Particle Reinforced Polymeric Composite Materials – Undamaged State BASICS 11 Creep - recovery tests where 11 and 22 are sufficient to identify the full 3D behaviour law BEHAVIOUR MODEL   PREDICTION 1 CONCLUSION 2 t t Identification of a Schapery type behaviour law:

20. A Micromechanical Model for Non-Linear Viscoelastic Particle Reinforced Polymeric Composite Materials – Undamaged State BASICS the behaviour is linear viscoelastic If BEHAVIOUR Evaluated for a low level of stress MODEL  PREDICTION Evaluated by nonlinear regression CONCLUSION Evaluated with the transverse data t Identification of a Schapery type behaviour law:

21. A Micromechanical Model for Non-Linear Viscoelastic Particle Reinforced Polymeric Composite Materials – Undamaged State BASICS BEHAVIOUR Prony series MODEL PREDICTION CONCLUSION Identification of a Schapery type behaviour law: Some results: