539.378:678.0 Mathematical model for finding finite deformations of rubber-like materials

Duishenaliev T. B. (NRU "MPEI"), Merkuryev I. V. (NRU "MPEI"), Duishembiev A. S. (Razzakov KSTU)


doi: 10.18698/2309-3684-2020-2-325

The article considers finite (geometrically nonlinear) elastic deformations of rubber-like materials and structures. Such deformations are described by a mathematical model developed on the basis of a non-classical approach to solving boundary-value problems of statics. Formulas are given for determining the final deformations of elastic rubber-like bodies based on the elements of spatial and material displacement gradients. Comparison of definitions for these two approaches is given. The validity of the above conclusions is confirmed by the example of one-dimensional, two-dimensional and three-dimensional transformations in the MathCad system. An example of determining the elements of the spatial displacement gradient is considered. It is known that a static boundary value problem has two formulations. The first one is put forward during its formulation and is used to derive the fundamental relations of the mechanics of a deformable body (Betty's theorem, general solution in the form of Somiliani formulas, etc.). The second is used in solving such problems. It is believed that the problems of both statements have the same solution. A non-classical solution of the boundary value problem of statics is proposed. It strictly corresponds to the generally accepted statement. The Chezaro method of representing the
displacement field using the deformation components is given. Further, this method is developed, it becomes possible to express the field of displacements also through the stress components. The problem of equilibrium of a rectangular plate of rubber-like material is solved. The expressions obtained determine the components of deformations, stresses, and displacements at any point of the plate. In all these expressions, only the coordinates of the finite region of the elastic body are present. There is no usual coordinate misunderstanding: in displacements and stresses are the same coordinates. This problem is also represented by the Navie equations. The uniqueness of its solution is proved.

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Дуйшеналиев Т.Б., Меркурьев И.В., Дуйшембиев А.С. Математическая модель для оценки конечных деформаций резиноподобных материалов. Математическое моделирование и численные методы, 2020, № 2, с. 3–25

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