1. 16/12/2002 1 Texture alignment in simple shear Hans Mühlhaus,Frederic Dufour and Louis Moresi

2. 16/12/2002 2 Outline I) Introduction II) Numerical methods III) Rheological models IV) Applications V) Conclusions and perspectives

3. 16/12/2002 3 Problem To model in large transformations a large range of materials at equilibrium.

4. 16/12/2002 4 Idea of the Particle in Cell method (PIC) Eulerian finite element mesh Lagrangian particles used as integration points Time variables are stored on particles Updated Lagrangian formulation

5. 16/12/2002 5 Setting of the plastic viscosity Experimental flow time of 5,2 s

6. 16/12/2002 6 Numerical results on mortar

7. 16/12/2002 7 Integration scheme Unlike a standard FEM scheme where the integration points are worked out in advance for the master element and weights computed accordingly, the PIC case, the particle positions are imposed by the deformation only the weights are unknown. Balance between time consumption, accuracy and uniqueness of the solution (negative weight) Keep in mind that it is still an approximation of an iterative process Balance between time consumption, accuracy and uniqueness of the solution (negative weight) Keep in mind that it is still an approximation of an iterative process Constant terms Linear terms Quadratic temrs, etc… FEM PIC

8. 16/12/2002 8 Forced convection A reference solution is calculated over a 36800 node mesh, with Gauss integration. Parametric study over a 2300 node mesh with particles regularly distributed and weighted initially.

9. 16/12/2002 9 Solution de référence 4 particules Solution de référence Reference solution 4 particles 16 particles How many particles?  Initially 4×4 particles

10. 16/12/2002 10 Termes linéaires Termes constants What condition on the weight?  Conditions to the linear terms Constant terms

11. 16/12/2002 11 Particle separation Particle = Integration point  Concentrated representative volume

12. 16/12/2002 12 Particle fusion

13. 16/12/2002 13 Maxwell viscoelasticity Deviatoric part Isotropic part Integration scheme Deviatoric relaxation time And volumic Law Jaumann derivative

14. 16/12/2002 14 Viscoelastic oscillations Shear Constitutive relationship Using Second order PDE Solution

15. 16/12/2002 15 Stability/accuracy of the scheme  = 1,0s h 0 = 1,0m  t = 0,01s V h(t) Compression : 0  t  9s  V=0,1m/s Relaxation : A t=9s  V=0 m/s Stability  Accurace 

16. 16/12/2002 16 Cosserat theory Stress approach Kinematic approach

17. 16/12/2002 17 Cosserat rheology in 2D Bending viscosityWhere d is the internal length of the material

18. 16/12/2002 18 Flow of a Cosserat continuum d/a=0 d/a=1/3 d/a=2/3 d/a=5/3 d/a=10/3

19. 16/12/2002 19 Finite Anisotropy Director evolution n : the director of the anisotropy W, W n : spin and director spin D, D ’: stretching and its deviatoric part

20. 16/12/2002 20 Anisotropy (kinematic) Evolution of the director Evolution of the thickness of the layer

21. 16/12/2002 21  Couple stresses Elastic: Viscous: X1X1 t h F

22. 16/12/2002 22 Virtual Power Requires continuously differentiable shape functions: inconvenient ! Penalty formulation:

23. 16/12/2002 23 P : penalty term Anisotropy (C 0 reconstruction) Principle of virtual power

24. 16/12/2002 24 Convergence of Penalty Method

25. 16/12/2002 25 Anisotropic rheology Constitutive relationship for the deviatoric stress with

26. 16/12/2002 26 Flambement d’une couche anisotrope Isotropic background medium with viscosity : 0,001 Pa.s Anisotropic layer of normal viscosity 1 Pa.s And tangenial viscosity 0,001 Pa.s Initial perturbation of the director’s orientation Change of major mode for Mühlhaus et al, 2002

27. 16/12/2002 27 Simple Shear and Convection Problems Constitutive equations: Temperature dependent viscosities: Stress and Thermal equilibrium: Non-dimensionalisation:

28. 16/12/2002 28 Heat Equation cont. C is the heat capacity; 800-1000 W/(kg K) k is the thermal conductivity ; 2.3-3.5 W/(m K) Reference viscosity     Pa Density Activation Energy Q= 180-550 KJ/mol Universal Gas Constant R=0.00831J/(mol K)

29. 16/12/2002 29 Shear Histories simple shear and shear alignment with shear heating and temperature dependent viscosity

30. 16/12/2002 30 Shear-Heating : Director Field and Temperature Contours

31. 16/12/2002 31 Shear Alignment with Shear Heating and Temperature Dependent Viscosity Alignment=0 if n is parallel to v and = 1 if n is orthogonal (steady state!) to v. Initial configuration Final (aligned) configuration

32. 16/12/2002 32 Conclusions + Benchmark for the Particle in Cell method 3 16 particles initially for a good integration 3 Constraints on the weight to the linear terms 3 Appetite of 0,8 + Developing/implementing new rheologies 3 Cosserat continuum 3 Viscoelasticity 3 Anisotropic model (classical or in a Cosserat context) 3 Bingham’s law for mortar Benchmarks were successfull in the context of comparison with theory of with other numerical methods.

33. 16/12/2002 33 Conclusions + Applications 3 Performant tool to study geological folding 3 Promising first steps on the study of fresh concrete flow + Drawbacks of the method 3 Traction boundary conditions 3 Diffusion of the interface by separation of the particles 3 Uncertainty on the quality of the numerical integration 3 Expertise needed