Convergence behavior of 3D finite elements for Neo‐Hookean material

The purpose of this paper is to examine quadratic convergence of finite element analysis for hyperelastic material at finite strains via Abaqus‐UMAT as well as classification of the rates of convergence for iterative solutions in regular cases.

Different formulations for stiffness – Hessian form of the free energy functionals – are systematically given for getting the rate‐independent analytical tangent and the numerical tangent as well as rate‐dependent tangents using the objective Jaumann rate of Kirchoff stress tensor as used in Abaqus. The convergence rates for available element types in Abaqus are computed and compared for simple but significant nonlinear elastic problems, such as using the 8‐node linear brick (B‐bar) element – also with hybrid pressure formulation and with incompatible modes – further the 20‐node quadratic brick element with corresponding modifications as well as the 6‐node linear triangular prism element and 4‐node linear tetrahedral element with modifications.

By using the Jaumann rate of Kirchoff stress tensor for both, rate dependent and rate independent problems, quadratic or nearly quadratic convergence is achieved for most of the used elements using Abaqus‐UMAT interface. But in case of using rate independent analytical tangent for rate independent problems, even convergence at all is not assured for all elements and the considered problems.

First time the convergence properties of 3D finite elements available in Abaqus sre systematically treated for elastic material at finite strain via Abaqus‐UMAT.