Computational and experimental substantiation of the application of hybrid functions of the first kind in the mechanics of deformation and fracture

A mathematical definition of the concept of «hybrid function of the first kind» (GF1) is given, which provides a smooth controlled transition from one basic mathematical function to another and contains the characteristic features of these two functions. The transition is controlled both by the place of transition and by its speed. At the same time, the transition remains smooth and differentiable. The functions of GF1 are characterized by the coincidence of the arguments of both the basic functions and the argument of the control complex that is part of GF1. Vectors and tensors, as well as complex numbers and functions, can be used as basic functions. Based on GF1, it is possible to design chain GF1 or CGF1, which is essential for expanding the range of GF1 applications. GF1 itself can be used as basic functions. Hybrid functions have a surprisingly high potential for approximating experimental data. The article provides examples of using GF1 to approximate curves in relation to two-parameter crack resistance criteria. An algorithm for eliminating the singularity in calculations for crack resistance is described. Based on GF1, new probability distribution functions of statistical data are proposed. The possibility of qualitative analytical approximation by CGF1 of existing statistical data with subsequent differentiation of the obtained CGF1 to obtain the density of their distribution is shown. An attempt has been made to use GF1 to describe the phenomenon of bifurcation.

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