Numerical simulation of t-stresses and stress biaxiality factor for a centrally cracked specimen under mixed boundary conditions

The results of the numerical calculation of T-stresses and the stress biaxiality factor in a centrally cracked tensile plate obtained by the graph method are presented. The method of analysis of a deformable solid is based on the principles of graph theory used in mechanics for constructing discrete models that allow numerical modeling of the fields of displacements, deformations, and stresses of a solid. When analyzing the stress-strain state near the crack tip, the author uses the singular element of the graph model of an elastic medium. When calculating T-stresses, stress and displacement methods are used. Calculations are performed on a coarse grid, however, it allows one to get fairly accurate results. This is because the graph laws (Kirchhoff’s vertex and cyclic laws) provide conditions for equilibrium and compatibility of deformations for the element as a whole. In addition, a special procedure for determining the coefficients of approximating polynomials leads to the implementation of equilibrium equations for the volume of the element. The state of a sample with a crack is described by two dimensionless complexes, which makes it possible to evaluate the biaxiality of the sample. One of these complexes depends on the crack length, applied load, elastic modulus of the material and crack tip opening displacement. Another complex is related to the calculation of a non-singular term in the Williams decomposition. The computational experiment allowed establishing a relationship between these dimensionless complexes. It is important to note that these parameters can be easily determined from a full-scale experiment. As a result, using several full-scale measurements, an approximate estimate for T-stresses and the stress biaxiality factor was obtained.