Non-linear vibration of hyperelastic axisymmetric solids by a mixed p-type method

Abstract

This paper presents finite amplitude transient vibration analysis of nearly incompressible hyperelastic axisymmetric solids by a mixed p-type method. In this method, displacement and pressure fields are separately defined using high degree polynomials and the solution is obtained with one or a few elements depending upon the nature of the problem. Geometry of the element is defined by polynomials of degrees much lower than that of displacement fields. The degrees of polynomials for pressure fields are lower than those used for displacement fields.Hyperelastic material is modelled by the Mooney–Rivlin material description. The total Lagrangian formulation is utilized to describe the deformations of axisymmetric solids subjected to pressure loads. Equations of motion are derived using the principle of virtual work and solved by the Newmark's method along with the Newton–Raphson iterative technique. The present formulation also includes the asymmetric tangent load matrix, resulting in linearization of deformation dependent load, which greatly reduces the number of equilibrium iterations to get the convergence of results at large strains.A convergence study of the results is presented with respect to the degrees of polynomials for displacement and pressure fields. The present method is verified by successfully comparing the results with those from finite element method using the commercial software ANSYS. The numerical simulations are conducted on circular plate, solid cylinder and spherical shells subjected to time dependent pressure loads and the highly non-linear behaviour of hyperelastic solids undergoing finite amplitude vibrations is studied. The method, presented herein, is very efficient, locking free, and accurate, which does not require a large number of elements or the total degrees of freedom as it is required in conventional finite element method for the convergence of results.