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

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

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

16/12/ 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

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

16/12/ Numerical results on mortar

16/12/ 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

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

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

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

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

16/12/ Particle fusion

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

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

16/12/ 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 

16/12/ Cosserat theory Stress approach Kinematic approach

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

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

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

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

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

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

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

16/12/ Convergence of Penalty Method

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

16/12/ 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

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

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

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

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

16/12/ 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

16/12/ 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.

16/12/ 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