Розв’язок статичної плоскої задачі теорії пружності для неоднорідних ізотропних тіл
Journal Title: Математичне моделювання - Year 2018, Vol 1, Issue 2
Abstract
THE SOLUTION OF THE STATIC FLAT TASK OF ELASTICITY THEORY FOR HETEROGENEOUS ISOTROPIC SHAPES Kalinin E.I., Polyashenko S.O. Abstract It is known that operators that determine the constituent laws for visco-elastic materials contain parameters that are very sensitive to changes in temperature. In the case of an inhomogeneous thermal field, these parameters depend on spatial coordinates. The influence of this inductive heterogeneity on the distribution of stresses caused by external forces is much longer and longer than the effect of stresses caused by the most thermal field. Neglecting this effect, even in the simplest cases, leads to physically inappropriate solutions. A series of works is devoted to the study of the plane problem of inhomogeneous elastic bodies. Some of these works derive from the simplifying hypothesis of modifying one modulus of elasticity, assuming that the Poisson constant is constant. Another considers the case of bodies consisting of the union of disjunctive homogeneous elastic regions. Were proposed formulas for the complex mapping of stresses and displacements that are valid for elastic and visco-elastic bodies with continuous uniformity of the general form in a plane and axisymmetric case. In this paper, the method of solving the static plane problem of the theory of elasticity for non-homogeneous bodies is described by successive approximations based on the use of reflections of the Kolosov type, Muschelishvili and conformal transformations. The aim of the work is to construct a quasistatic problem of visco-elasticity theory in the presence of a stationary thermal field to an elastic-static problem for a non-uniform body using Laplace transform methods. It is established that in the most general case, the quasistatic problem of visco-elasticity theory, in the presence of a stationary thermal field, can be formally reduced by means of Laplace's transformation into an elastic-static problem for a nonhomogeneous body, and the latter can be solved also by the method indicated in the work. It is noted that obtaining a solution of a visco-elastic problem requires the implementation of a reverse transformation, which is associated with rather large computational difficulties, and obtaining a solution for areas with heterogeneity of general appearance requires the definition of a solution that corresponds to the same areas in a homogeneous environment. References [1] Freudenthal А., H. Geiringer “The mathematical theories of the inelastic continuum” Encyclopedia of Physics.Springer, Vol. 6. pp. 15–22., 2005 (references) [2] Nоwinsкi J., Turski S. “Studium nad stanami naprzenia w cialach sprzystych niejendorodnych” Arch. Mech. Stos. Vol. 5. No 3. pp. 54–63. 2000 (references) [3] Teodorescu P., Predeleanu M. “Über das ebene Problem nichthomogener elastischer Körper” Acta tech. Acad. Sei. Hung. Vol. 27. No 3. pp. 95–103. 1995 (references) [4] Sherman D.I. On the problem of plane strain in non-homogeneous media. Nonhomogeneity in Elasticity and Plasticity, London: Pergamon Press, 1995, 354 p. [5] Milieu M. “Asupra ieprezentärii vectorului asociat cuasistatici dinamic ai echilibrului mediilor continue neomogene, cu proprietati reologice cuasiliniare” Comun. Acad. R.P.R. Vol. 12. No 8. pp. 13–24. 1996 (references) [6] Misicu M., Teodosiu C. “Asupra problemei axial-simetrice si a problemei plane a teoriei elasticitatii pentru corpuri izotrope neomogene” Com. Acad. R.P.R. Vol. 12. No 8. pp. 354-362. 1992 (references) [7] Mushelishvili N.I. Nekotorie osnovnie zadachi matematicheskoj teorii uprugosti [Some basic problems in the mathematical theory of elasticity]. Moskva, 1954. 709 p. [8] M. Guгtin, E. Sternberg “On the linear theory of viscoelasticity” Arch. Rational Mech. Annal. Vol. 11. No 4. pp. 156-162. 1962 (references)
Authors and Affiliations
Є. І. Калінін, С. О. Поляшенко
Keywords
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- EP ID EP444622
- DOI 10.31319/2519-8106.2(39)2018.154228
- Views 83
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