Non-linear elasticity

Algorithm to solve the non-linear elasticity problem using a minimization technique.

Problem

Let $\Omega \subset \mathbb{R}^{3}$ denotes a hyperelastic material and $\partial \Omega= \Gamma_{0} \cup \Gamma_{1}$ its boundary, $\Gamma_{0}$ and $\Gamma_{1}$ denote respectively the disjoint parts of the boundary ($\Gamma_{0} \cap \Gamma_{1}= \emptyset$) where a null displacement and a surface traction $\mathbf{t}$ are applied.

The problem is to find the displacement field $\mathbf{u}$ of the body $\Omega$, which minimizes the total potential energy $\mathcal{E}$ given by:

$\displaystyle{ \mathcal{E}(\mathbf{v})= \int_{\Omega} \Psi \, dV - \int_{\Gamma_{1}} \mathbf{t}.\mathbf{v} \, dA }$

Therefore:

$\displaystyle{ \mathbf{u}= \underset{\mathbf{v} \in \mathcal{A}}{\text{argmin}}(\mathcal{E}(\mathbf{v})) }$

Where $\Psi$ is the strain energy function and $\mathcal{A}$ is the admissible displacements set defined by:

$\displaystyle{ \mathcal{A}=\left\lbrace \mathbf{v} \in \left( H^{1}(\Omega)\right)^{3} \; ; \; \mathbf{v}=0 \; \text{on} \; \Gamma_{0} \right\rbrace }$

Algorithms

Material parameters

File NeoHookean.idp

Elasticity law

Definition of the elasticity law in 2D.

File ElasticLaw2d.idp

Minimization algorithm

Result - Deformation and displacement field

Validation

See FRICTIONLESS CONTACT PROBLEM FOR HYPERELASTIC MATERIALS WITH INTERIOR POINT OPTIMIZER, H. Houssein, S. Garnotel, F. Hecht (upcoming article)

Authors

Author: Houssam Houssein