On contact problems with a deformable punch and variable rheology
DOI:
https://doi.org/10.21638/spbu01.2023.401Abstract
The paper presents for the first time one of the methods for studying and solving contact problems with a deformed stamp for those cases when there is a need to change the rheology of the stamp material. It is based on a new universal modeling method previously published by the authors, which is used in boundary value problems for systems of partial differential equations. With its help, solutions of complex vector boundary value problems for systems of differential equations can be decomposed into solutions of scalar boundary value problems for individual differential equations. Among them, the Helmholtz equations are the simplest. Solutions to scalar boundary value problems are represented as fractals, self-similar mathematical objects first introduced by the American mathematician B. Mandelbrot. The role of fractals is performed by packed block elements. The transition from systems of differential equations in partial derivatives to individual equations is carried out using the transformation of academician B. G. Galerkin or representation by potentials. It is known that the solutions of dynamic contact problems with a deformable stamp of complex rheology are cumbersome, and their study is always difficult. The problem is complicated by the presence of discrete resonant frequencies in such problems, which were once discovered by Academician I. I. Vorovich.
Keywords:
block element, factorization, integral equations, external forms, cracks of a new type
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Литература
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2. Zhou S., Gao X. L. Solutions of half-space and half-plane contact problems based on surface elasticity. Z. Angew. Math. Phys. 64, 145-166 (2013). https://doi.org/10.1007/s00033-012-0205-0
3. Almqvist A. An LCP solution of the linear elastic contact mechanics problem. Available at: https://www.mathworks.com/matlabcentral/fileexchange/43216-an-lcp-solution-of-the-linearelastic-contact-mechanics-problem (2013). http://doi.org/10.13140/RG.2.1.3960.7200
4. Cocou M. A class of dynamic contact problems with Coulomb friction in viscoelasticity. Nonlinear Analysis. Real World Applications 22, 508-519 (2015). https://doi.org/10.1016/j.nonrwa.2014.08.012
5. Ciavarella M. The generalized Cattaneo partial slip plane contact problem. I. Theory. Int. J. Solids Struct 35 (18), 2349-2362 (1998). https://doi.org/10.1016/S0020-7683(97)00154-6
6. Ciavarella M. The generalized Cattaneo partial slip plane contact problem. II. Examples. Int. J. Solids Struct 35 (18), 2363-2378 (1998). https://doi.org/10.1016/S0020-7683(97)00155-8
7. Guler M.A., Erdogan F. The frictional sliding contact problems of rigid parabolic and cylindrical stamps on graded coatings. Int. J. Mech. Sci 49 (2), 161-182 (2007). https://doi.org/10.1016/j.ijmecsci.2006.08.006
8. Ke L.-L.,Wang Y.-S. Two-dimensional sliding frictional contact of functionally graded materials. Eur. J. Mech. A/Solids 26 (1), 171-188 (2007). https://doi.org/10.1016/j.euromechsol.2006.05.007
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Published
2023-12-23
How to Cite
Babeshko, V. A., Evdokimova, O. V., Babeshko, O. M., Zaretskaya, M. V., & Evdokimov, V. S. (2023). On contact problems with a deformable punch and variable rheology. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 10(4), 588–599. https://doi.org/10.21638/spbu01.2023.401
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On the anniversary of A. K. Belyaev
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Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.
