A Model of Elasticity Taking into Account the Displacement Orientation in the Deformation

Aims/Objectives: In this work, we deal an elasticity model in 2D and 3D dimension for deformation under constraint by taking into account the deformation displacement orientation. This mathematical model can be used, for example, to describe the heart deformation taking into account the orientation of the fibers for estimating global and regional parameters of the left ventricular function.
Place and Duration of Study: Department of Mathematics and Computer Science, Faculty of Sciences An Chok, University HassanII-Casablanca and C. Jordan UMR CNRS 5208 Institut- INSA Lyon, from June 2012 to July 2015.
Methodology: In first, we start by studying a model of Poisson problem under constraint on a domain Ω ⊂ ℝn (n=2 or 3), considering a constraint on a part K of this domain. Secondly, we consider the proposed 2D and 3D elasticity model of deformation under constraint by taking into account the displacement orientation in the deformation. We treat only the case where the orientation of displacement is a given constant vector. The eigenvalues case is being finalized.

A major difficulty for those problems is to find a demonstration of existence and uniqueness of solution, which are given in both 2D and 3D dimension. A numerical approach of the solution using finite element method and its convergence is studied. Numerical simulations are performed with Free Fem software. Simulations results and comments are given in the end.
Results: Existence and uniqueness results are established in Sobolev spaces for the proposed models. Convergence of the FEM approached solution is given and numerical simulations are performed with success.

Conclusion: This work has been devoted to study a problem of elasticity under constraint in order to take account the orientation of structure displacement. Both analytical and numerical study of the proposed problem are realized. The numerical simulations give good results.