and Computational Methods

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.

[1] Galerkin B.G. Sobranie sochinenij [Collected works]. Moscow, USSR Publ., 1952, 391 p.

[2] Green A.E., Adkins J.E. Bolshie uprugie deformacii v nelinejnoj mekhanike sploshnoj sredy [Large elastic deformations and non-linear continuum mechanics]. Мoscow, Mir Publ., 1965, 456 p.

[3] Duishenaliev T.B. Neklassicheskie resheniya mekhaniki deformiruemogo tela [Non-classical solutions of the mechanics of a deformable body]. Moscow, MPEI Publ., 2017, 400 p.

[4] Iliushin A.A. Mekhanika sploshnoj sredy [Continuum mechanics]. Moscow, Moscow University Publ., 1971, 248 p.

[5] Lurie A.I. Nelinejnaya teoriya uprugosti [Nonlinear theory of elasticity]. Мoscow, Nauka Publ., 1980, 512 p.

[6] Nowacki W. Teoriya uprugosti [Theory of elasticity]. Мoscow, Mir Publ., 1975, 256 p.

[7] Rabotnov Yu.N. Mekhanika deformiruemogo tverdogo tela [Mechanics of a deformable solid]. Мoscow, Nauka Publ., 1979, 744 p.

[8] Rudskoi A.I., Duishenaliev T.B. Prochnost i plastichnost materialov [Strength and ductility of materials]. Saint Petersburg, Polytechnic university Publ., 2016, 218 p.

[9] Trusdell C. Pervonachalnyj kurs racionalnoj mekhaniki sploshnyh sred [A first course in rational continuum mechanics]. Мoscow, Mir Publ., 1975, 592 p.

[10] Feodosev V.I. Soprotivlenie materialov [Strength of materials]. Мoscow, BMSTU Publ., 1999, 592 p.

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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