Nonlinear finite element analysis of incompressible hyperelastic materials using symmetric stiffness matrix

Abstract

The objective of this research is to present a symmetric stiffness matrix for incompressible hyperelastic materials in the case of plane strain, axisymmetric and three-dimensional problems solved by the finite element method. The constraint of incompressibility is incorporated into the finite element method through the use of a Lagrange multiplier. An incremental form of equilibrium equations and incompressible conditions with a symmetric stiffness matrix is obtained using the total Lagrangian formulation. Various isoparametric finite elements: 4-node element with a constant Lagrange multiplier and 8-node element with a linear Lagrange multiplier field (3 constraints) in a two-dimensional case, and 8-node element with a constant Lagrange multiplier and 20-node element with a linear Lagrange multiplier field (4 constraints) in a three-dimensional case, are considered. To demonstrate the applicability of these elements, numerical analyses of linear and nonlinear problems are carried out and numerical results are compared with analytical solutions. The Mooney-Rivlin form of the strain energy function is considered. It is shown that 8-node quadrilateral elements give more rapidly convergent solutions than 4-node quadrilateral elements, the integration points in the 8-node elements using the reduced integration are the best sampling points for stresses, while element averages of stresses in 4-node elements are in agreement with exact solutions, and in linear analyses of incompressible materials 8-node elastic elements with the Poisson's ratio approximating to 0.5 using the reduced integration give reasonable solutions.