Numerical modeling of nonlinear deformation processes for shells of medium thickness

Abstract

When modeling a nonlinear isotropic eight-node finite element, the main kinematic and physical relationships are determined. In particular, isoparametric approximations of the geometry and an unknown displacement increment vector, covariant and contravariant components of basis vectors, metric tensors, strain tensors (Cauchy - Green and Almansi) and true Cauchy stresses in the initial and current configuration are introduced. Next, a variational equation is introduced in the stress rates in the actual configuration without taking into account body forces and the Seth material is considered, where the Almansi strain tensor is used as the finite strain tensor. Linearization of this variational equation, discretization of the obtained relations (stiffness matrix, matrix of geometric stiffness) is carried out. The resulting expressions are written as a system of linear algebraic equations. Several test cases are considered. The problem of bending a strip into a ring is presented. This problem is solved analytically, based on kinematic and physical relationships. Examples of nonlinear deformation of cylindrical and spherical shells are also shown. The method proposed in this paper for constructing a three-dimensional finite element of the nonlinear theory of elasticity, using the Seth material, makes it possible to obtain a special finite element, with which it is quite realistic to calculate the stress state of shells of medium thickness using a single-layer approximation in thickness. The obtained results of test cases demonstrate the operability of the proposed technique.