Approximators for rapid computation of the stress state parameters in non-standard samples with cracks

An actual trend in the field of experimental solid mechanics is the extension of the test specimen range. This article considers a force-controlled loading test of a hollow "Brazilian disk" with two inclined cracks. Tests of this type of specimens provide important information on brittle and quasi-brittle fracture under mixed loading conditions (Modes I and II). For practical application of these specimens, it is necessary to know the stress state parameters in the vicinity of the crack tip, such as KI, KII and T-stress. Unfortunately, due to the complex geometry of the specimens, there are no analytical expressions for evaluating these parameters, which are generally computed via finite element modeling with post-processing of the results. However, due to a significant algorithmic complexity of this procedure, the application of new types of specimens remains limited. To simplify the computational experiments for new types of specimens, we propose an approach, which is based on the approximation of the dependence of the desired stress state parameters on the problem parameters, namely, the dimensions of the disk, the length of the cracks, and their inclination angle with respect to the loading axis. The approximation of the desired parameters is found by solving a linear problem of the least root-mean-square deviation. To obtain an accurate approximation, it is necessary to use higher degree polynomials, although an excessive number of monomials leads to a rapid increase in the number of coefficients in the approximator and, as a result, to a rapid deterioration of the problem conditioning, which eventually reduces the accuracy and stability of the approximation. To avoid overparameterization, we consider three methods for constructing bases with a specified number of monomials.. The accuracy of the constructed approximators is evaluated by making comparison with the data obtained from numerical modeling and verified experimentally. As numerical experiments have shown, the approximation error is about 1% for each of the desired stress state parameters. The obtained approximators are available as a MATLAB script, open for free download at GitHub.