Efficient method of solving the problems of steady-state oscillations of symmetric rectangular domains of general type

Abstract

When the wave processes in bounded elastic bodies are examined, we are faced with a significant complication of the structure of the wave field compared to the case of infinite bodies. This is due to the complex nature of the reflection of elastic waves from the boundaries of the body because the direction of the general flow of energy is changed. Even more complicated the structure of the wave field is, if there are inner boundaries between fields with different elastic properties. This entails the emergence of new wave effects associated with the dynamic stress concentration in the vicinity of the internal and external boundaries of the field. The nature of edge effects is changed too. They will depend not only from the size of the field but also from the geometric and elastic parameters defining the nature of heterogeneity. At the forefront are the questions of systematization of the results for the purpose of extradition of practical recommendations for optimal design of heterogeneous section details in particular conditions of its operation. Urgent enough is the question of the possibility of neglecting of structural heterogeneity and anisotropy of the section of the body in strengthening calculations and evaluation of possible errors. The mathematical basis for the study will be the expressions for particular solutions of equations of motion, constructed for infinite layers, which are sets of plane standing waves. When choosing the form of partial solutions, we must take into account not only the opportunity to satisfy the boundary conditions at the exterior boundary of the field, but also the mechanical properties at the interface of the sphere. This entails the complication of numerical-analytical algorithm of solving the problem