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

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

3. PROBLEM: Knowing the mechanical response of a reinforced polymer under a given loading when the matrix is nonlinear viscoelastic SOLUTIONS: A Micromechanical Model for Non-Linear Viscoelastic Particle Reinforced Polymeric Composite Materials – Undamaged State 2.Building a theoretical model which accounts for : 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. OUTLINE 1. Homogenisation basics 3. Theoretical model 2. Constituents behaviour 4. Model theoretical predictions 5. Conclusion A Micromechanical Model for Non-Linear Viscoelastic Particle Reinforced Polymeric Composite Materials – Undamaged State

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

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

7. BASICS MODEL PREDICTION CONCLUSION BEHAVIOUR A Micromechanical Model for Non-Linear Viscoelastic Particle Reinforced Polymeric Composite Materials – Undamaged State 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 Equilibrium Behaviour Localisation

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

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

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

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

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

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

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

16. 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 BASICS MODEL PREDICTION CONCLUSION BEHAVIOUR A Micromechanical Model for Non-Linear Viscoelastic Particle Reinforced Polymeric Composite Materials – Undamaged State

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

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

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

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