д-р физ.-мат. наук, заведующий кафедрой теоретической и прикладной механики Донского государственного технического университета, профессор кафедры математического моделирования Южного федерального университета
аспирант кафедры теоретической и прикладной механики Донского государственного технического университета
Problems arise when considering new types of materials, led the development of several new measuring techniques, as well as the adaptation of existing methods. One such approach is the combination of evolutionary algorithms with the method of solving direct problems, such as the finite element method (FEM), and others. One of the ways to describe the mechanical behavior of composite materials is the transition to the effective elastic properties. At the same time, depending on the microstructure of the composite material with advantageous properties can have a certain symmetry and described as part of a transversely isotropic, orthotropic, etc. material. In this paper, we consider several methods for determining the elastic constants of anisotropic material. In particular, described as experimental and analytical as well as experimental and numerical methods. We consider the natural oscillations of the plates with a variety of securing conditions, and determine a set of resonance frequencies, which is additional information for solving inverse problems. These natural frequencies are analytical or numerical solution using the finite element method. The inverse coefficient problem of determining the elastic constants is achieved by minimizing the residual functional system of nonlinear algebraic equations. As an example, we consider orthotropic material from which the cut three rectangular plate in a plane orthogonal to the crystallographic axes. Each plate is pivotally secured and conducted an experiment to determine the first three natural frequencies. Experimental measurement of resonance frequencies of vibration of its own in this paper is replaced by their numerical calculation in the complexes FlexPDE and ANSYS. The described methods allow us to determine the elastic constants of anisotropic material, in the case of experimental and numerical method, with an error equal to the measurement error, or less than 1% for the experimental and numerical method.
В работе рассмотрено несколько методов определения упругих констант анизотропного материала. Описаны как экспериментально-аналитические, так и экспериментально-численные методы. Рассматриваются собственные колебания пластин с различными условиями закрепления и определяется набор резонансных частот, который служит дополнительной информацией для решения обратной коэффициентной задачи. Эти собственные частоты находятся аналитически или в результате численного решения с помощью МКЭ. Обратная коэффициентная задача определения упругих постоянных решается на основе минимизации функционала невязки системы нелинейных алгебраических уравнений.