Numerical Simulation of T-Stresses and Stress Biaxiality Factor for a Centrally Cracked Specimen under Mixed Boundary Conditions

摘要

Abstract The T-stresses and the stress biaxiality factor in a centrally cracked tensile plate are numerically calculated using the graph method. The analysis method is based on the principles of graph theory used in mechanics to construct discrete models for numerical computation of displacement, deformation, and stress fields in solids. The stress-strain state near the crack tip is analyzed using a singular element proposed by the author for the graph model of an elastic medium. The T-stresses are calculated by applying stress and displacement methods. The calculations are performed on a coarse grid; nevertheless, fairly accurate results are obtained. 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 equilibrium equations satisfied over the volume of the element. The state of the cracked specimen is described by two dimensionless complexes, which make it possible to evaluate the biaxiality of the specimen. One complex depends on the crack length, the applied load, the elastic modulus of the material, and the crack tip opening displacement and can easily be determined from a full-scale experiment. The other complex is related to the calculation of a nonsingular term in the Williams series expansion. The relationship between these dimensionless complexes is established in a numerical experiment. As a result, the T-stresses and the stress biaxiality factor are approximately estimated obtained using several full-scale measurements.