Computational and Experimental Substantiation of Application of Hybrid Functions of the First Kind in Deformation and Fracture Mechanics

Abstract—A mathematical definition of the concept of “first-kind hybrid function” (HF1) is provided. This function ensures a smooth controlled transition from one basic mathematical function to another and incorporates the characteristic features of these two functions. The transition is managed both at the point of transition and in terms of its speed, while remaining smooth and differentiable. It is characteristic of HF1 that the arguments coincide for both the basic functions and the control complex included in HF1. Vectors and tensors, as well as complex numbers and functions, are used as basic functions. Based on HF1, it is possible to construct chain hybrid functions (CHF1), which is significant for expanding the range of applications of HF1. HF1 itself can be used as basic functions. Hybrid functions have great potential for approximating experimental data. The work presents examples of using HF1 for curve approximation concerning two-parameter fracture toughness criteria. An algorithm for eliminating singularities in fracture toughness calculations is described. New probability distribution functions for statistical data are proposed based on HF1. The possibility of qualitative analytical approximation of existing statistical data using CHF1 is demonstrated, followed by differentiating the obtained CHF1 to derive the density of their distribution. An attempt is made to apply HF1 to describe the phenomenon of bifurcation.