On one method of solving the three-dimensional problem of physically non-linear deformation of transversal-isotropic multi-variable bodies

The method for solving the three-dimensional problem of elastic-plastic deformation of transversely isotropic multiply connected bodies by the finite element method (FEM) is presented in the article. The process of solving the problem consists of: determining the effective parameters of a transversely isotropic medium; construction of the finite element mesh of the body configuration, including the determination by the front method of the local minimum value of the width of the tape of non-zero coefficients of equation systems; constructing the coefficients of the stiffness matrix and the components of the node load vector of the equation of state of an individual finite element according to the theory of small elastic-plastic deformations for a transversely isotropic medium; the formation of a resolving symmetric-tape system of equations by summing the coefficients of the equations of state of all finite elements; solution of the system of symmetric-tape system of equations by means of the square root method; calculation of the elastic-plastic stress-strain state of the body by performing the iterative process of the method of elastic solutions AA Ilyushin. For each stage of solving the problem, effective computational algorithms have been developed that make it possible to reduce the number of computational operations by modifying existing methods of solving and taking into account the structure of the matrix coefficients. As an example, the solution of the problem of deformation of a transversally isotropic body in the form of a rectangle with a circular notch in the center is given.