Contents

1. A Constitutive Model Based on Meso and Micro Kinematics for Impregnated Woven Continuous Fibre Reinforced Composites P. Harrison M.J. Clifford A.C. Long C.D. Rudd University of Nottingham

2. Contents • Motivation • Introduction to textile composite models • Method • Results

3. Motivation: Simulate forming process Shear angle PAM-FORM ESI Software

4. Make force versus shear angle predictions using fibre diameter, fibre volume fraction, matrix viscosity and weave structure Reduce or eliminate the need for time-consuming and expensive characterisation experiments, e.g. picture frame and bias extension tests Specific aim

5. Continuum models based on Ideal Fibre Reinforced theory (Spencer, 1972) Non-continuum approach: homogenisation methods Background: modelling textile composites

6. Combine predictions of continuum and micromechanical models Incorporate observed meso-scale kinematics in model Proposed modelling approach

7. Stress tensor Extra stress tensor, depends on rate of deformation tensor, D, and fibre direction, a Reaction stresses due to fibre inextensibility and material incompressibility Background: Continuum theory, uniaxial model

8. Transverse viscosity Longitudinal viscosity Background: Continuum theory, uniaxial model continued The form of  can be derived from the general property that  must be form invariant with respect to rigid rotations and is linear with D, thus (b)

9. Biaxial models have been proposed to model textile composites here using general properties of tensors find Background: Continuum theory, biaxial model for textiles Unfortunately, the five model parameters can no longer be related to micro-mechanical mechanisms, thus not accessible to micromechanical modelling

10. Inter-tow region Tow + = Layer 2 Textile Layer 1 Current approach: Combine uniaxial theory with observed mesoscopic kinematics Need effective viscosities of tow and inter-tow regions

11. d Cross-section of tow showing fibres and kinematics

12. Calculating effective tow viscosity Consider affine deformation

13. After shear Before shear d d Unit cell go T go g Unit cell

14. Meso-scale kinematics: observation

15. Average strain profile across material Meso-scale strain profile Meso-mechanical observations

16. New rate of deformation tensor, D, required for use in uniaxial continuum theory Energy dissipation is produced between tow crossovers Effects of non-uniform meso-scale shear strain

17. Low friction High friction Crossover area Crossover kinematics

18. Element Y X Energy dissipation due to crossover shear

19. Glass/polypropylene 2 x 2 twill weave thermoplastic 180oC Comparison between picture frame test and model predictions: test conditions

20. Tow shear angle  = 90 – 73 = 17o Material shear angle = 90-42 = 48o Meso-scale observations

21. Carreau-Yassuda model Viscosity of polypropylene matrix

22. Experiment Uniaxial model Results of uniaxial model

23. Experiment Uniaxial model Results of uniaxial model Uniaxial model predicts similar form but wrong magnitude (viscosity predictions too low)

24. Experiment Crossover model Results of crossover model Crossover model predicts right magnitude but wrong form

25. Use magnitude predictions from crossover model Use curve shape prediction from uniaxial continuum model Combine predictions of two models

26. Combined model Experiment Result of combined model Combined model predicts right magnitude and similar form

27. Produced predictions of similar magnitude and form as experimental data without use of fitting parameters Need more comparisons between experiment and theory to investigate to generality of the results Conclusions

28. We would like to thanks the following organisations for their support: EPSRC, BAE SYSTEMS, BP Amoco, ESI Software, Ford Motor Company, QinetiQ, Saint-Gobain Vetrotex, MSC Software Ltd., and the Universities of Cambridge and Leeds Acknowledgements

29. Questions?