A stress recovery procedure for solving geometrically non-linear problems in the mechanics of a deformable solid by the finite element method

A stress recovery procedure is presented for non-linear and linearized problems, based on the determination of the forces at the mesh points using a stiffness matrix obtained by the finite element method for the Lagrange variational equation written in the initial configuration using an asymmetric Piola–Kirchhoff stress tensor. Vectors of the forces reduced to the mesh points are constructed using the displacements at the mesh points found by solving this equation and for the known stiffness matrices of the elements. On the other hand, these forces at the mesh points are defined in terms of unknown forces distributed over the surface of an element and given shape functions. As a result, a system of Fredholm integral equations of the first kind is obtained, the solution of which gives these distributed forces. The values of the Piola–Kirchhoff stress tensor of the first kind at the mesh points are determined using the values found for the distributed forces on the surfaces of the finite element mesh (including at the mesh points) using the Cauchy relations for the initial configuration. The linearized representation of this tensor enables all the derivatives of the increment in the strain vector with respect to the coordinates to be found without invoking the operation of differentiation. The particular features of the use of the stress recovery procedure are demonstrated for a plane problem in the non-linear theory of elasticity.