The formulation and solution of the problem determining the least and largest values of the main characteristics of the particular class of two-link deformation

Abstract

The problem of determining the ultimate accumulated strain to fracture of a material for a particular class of the strain rate change according to the linear two-link trajectory is considered. All trajectories meet the following requirements: at the initial moment the strain rate is zero, reaches its maximum value at the break point and decreases during the second part of the trajectory until the material reaches its limit state. It is shown that for these trajectories the problem of determining the ultimate accumulated strain is reduced to a nonlinear programming problem with the target function and constraints that are nonlinear functions of three unknown parameters: the coordinates of the break point and the strain rate at the limit state. The case that is characterized by the simultaneous achievement of the limiting state of the material and zero strain rate is investigated in detail. It is shown that in two limited cases when the trajectory break tends to zero or to the time of fracture, the two-link trajectory degenerates into the single-link left and right limit trajectories. During defining for an analytical solution of the formulated nonlinear programming problem the theorem about the smallest and the largest values of the strain rate was formulated and proved. According to this theorem the maximum strain rate occurs for the left limit path and monotonously decreases with a gradual transition to the right limit path. The theorem is proved using elements of mathematical analysis and theory of functional series. Obtaining an analytical expression for the accumulated deformation together with the application of this theorem gave the possibility to find patterns of  the ultimate accumulated strain variation depending from the moment of reaching the break point. It is shown that the patterns of the ultimate accumulated strain variation, which are determined on the basis of the entire two-link trajectory, are similar to the patterns of change in the strain rate at the break point. It is emphasized that similar problem formulation and the obtained results is absent in publications related to the theory of summing damage.