Finite Element Formulation of Three-dimensional Nonlinear Elasticity Problem B. Soused´ ık 1) , P. Burda 2) 1) Department of Mathematics, Faculty of Civil Engineering, Czech Technical University in Prague, Th´ akurova 7, 166 29 Prague 6 sousedik@mat.fsv.cvut.cz 2) Department of Mathematics, Faculty of Mechanical Engineering, Czech Technical University in Prague, Karlovo n´ amˇ est´ ı 13, 121 35 Prague 2 burda@marian.fsik.cvut.cz Abstract The purpose of the present work is to give a brief description of the finite elasticity and of its approximation via finite element method. We formulate the problem for the case of compressible elasticity. Weak for- mulation allows to use any isotropic hyperelastic material model that satisfies polyconvexity assumptions. Discretization using FEM leads to systems of non-linear equations. Finally we also show the strategy of solving such systems of equations by modification of Newton’s method that can be used under some restrictions. 1 Introduction The main object of the finite three-dimensional elasticity is to predict changes in the geometry of solid bodies. The starting point of the classical theory of linear elasticity is the concept of small strains: the deformation of structures under working loads are not detectable by human eye. In contrast, many modern situations involve large deformations. The nonlinear behavior of polymers and synthetic rubbers are such examples. Applications in biome- chanics are even more critical because the most of vital organs such as eye, heart trachea or vocal apparatus fulfill their function only because of their large deformations. In this framework the concept of finite elasticity covers the simplest case where internal forces (stresses) depend only on the present deformation of the body and not on the history. In the paper we show the finite element approximation and strategy of solution of this nonlinear and ’visible’ stress-strain relationship.