ACCOUNTING OF PARTIAL SLIPPING IN CONTACT OF A STAMP WITH A CURVILINEAR ANISOTROPIC PLATE
DOI:
https://doi.org/10.32782/mathematical-modelling/2024-7-2-6Keywords:
asymptotic method, small parameter, elastic rod, dynamic loadAbstract
The purpose of this work is to study the stress-strain state of a finite elastic anisotropic plate in the form of a truncated curvilinear sector under the action of a rigid stamp, in the presence of areas of sliding and adhesion. The mathematical model of the given problem is described using equilibrium equations and Cauchy relations. The perturbation method is used for the solution, when a small parameter is set in the form of a ratio of physical characteristics of the material. The proposed transformations of coordinates and the required functions depending on a small parameter. The use of the indicated transformations allows you to decompose the original boundary value problems into two components that differ in their properties. The solution is in the form of a superposition of results of two types. Each of the stress-strain states contains a main function and an auxiliary one. Unknown functions are searched using series expansions with a small parameter. In each approximation, the main functions are derived from Laplace’s equations, the auxiliary functions are derived using integration. An analysis of the boundary conditions was carried out, which shows that they can almost always be formulated for the main functions. The stress distribution under the stamp and the size of the contact zone were obtained. The study showed that the anisotropic properties of the material significantly affect the mechanical condition of the plate. For example, changing the cut angle or distribution of material characteristics can lead to significant variations in stresses. This is especially important for practical applications, where the accuracy of the prediction of the stress-strain state is critical for ensuring the reliability of structures. In the future, the obtained results can be used to optimize the processes of forming parts from anisotropic materials, as well as to develop new materials that combine the required mechanical properties. Thus, this study opens new perspectives for further study of the influence of anisotropy on the mechanical characteristics of materials in various technological processes.
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