Geometrically nonlinear analysis of hyperelastic solids by high-order tetrahedral finite elements

Abstract

The purpose of this study is to develop a computer code for accurate prediction of the mechanical behavior of hyperelastic solids under large deformation and finite strains. It is used, in the present work, the Lagrangian positional version of the Finite Element Method, in which the degrees of freedom are current positions instead of displacements. The main mechanical variables are the Green-Lagrange strain and the second Piola-Kirchhoff stress tensors. Isoparametric tetrahedral solid finite element of any approximation degree is employed with full integration in order to obtain accuracy. In order to be general, it is developed a numerical strategy to determine the shape functions coefficients of any order, following tetrahedral basis. The solid equilibrium is achieved at the current position by means of the Minimal Potential Energy Principle regarding positions and is solved by the Newton-Raphson iterative scheme. It is important to mention that the adopted constitutive laws are nonlinear, isotropic and homogeneous hyperelastic, normally used in mechanical analysis of elastomers, with the near compressibility assumption. It is applied the Flory decomposition, i.e., the multiplicative split of the deformation gradient into a volumetric and an isochoric part. A parallel processing strategy is used to increase the simulation speed and memory capacity, justifying the use of high order elements for solid analysis. Results confirm that the developed computer code is capable of predicting the static behavior of complex problems, such as Cook's membrane and partially loaded block. The shape functions generator code exhibits simplicity and, thus, it may be easily implemented in other finite elements. The increase in the order of approximation and the mesh refinement improve accuracy and, therefore, the proposed methodology together with parallel computing indicate a safe way for further developments in plasticity and damage mechanics for large strain and deformations.