Abstract
We consider the statics case of the theory of linear thermoelasticity with microtemperatures materials. The representation formula of a general solution of the homogeneous system of differential equations obtained in this paper is expressed by means of four harmonic and three metaharmonic functions. These formulas are very convenient and useful in many particular problems for domains with concrete geometry. Here we demonstrare an application of these formulas to the III type boundary value problem for a half-space. Uniqueness theorems are proved. Solutions are obtained in quadratures.
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References
J. R. Barber, “The solution of elasticity problems for the half-space by the method of Green and Collins,” Appl. Sci. Res., 40, No. 2, 135–157 (1983).
M. Basheleishvili and L. Bitsadze, “Explicit solutions of the boundary-value problems of the theory of consolidation with double porosity for the half-plane,” Georgian Math. J., 19, No. 1, 41–48 (2012).
M. Basheleishvili and L. Bitsadze, “Explicit solutions of the BVPs of the theory of consolidation with double porosity for the half-space,” Bull. TICMI, 14, 9–15 (2010).
D. Burchuladze, M. Kharashvili, and K. Skhvitaridze, “An effective solution of the Dirichlet problem for a half-space with double porosity,” Georgian Int. J. Sci. Technol., 3, No. 3, 223–232 (2011).
P. S. Casas and R. Quintanilla, “Exponential stability in thermoelasticity with microtemperatures,” Int. J. Eng. Sci., 43, Nos. 1-2, 33–47 (2005).
E. Constantin and N. H. Pavel, “Green function of the Laplacian for the Neumann problem in ℝ n+ ,” Libertas Math., 30, 57–69 (2010).
L. Giorgashvili, K. Skhvitaridze, and M. Kharashvili, “Effective solution of the Neumann boundary value problem for a half-space with double porosity,” Georgian Int. J. Sci. Technol., 4, 143–154 (2012).
D. Ie,san, “Thermoelasticity of bodies with microstructure and microtemperatures,” Int. J. Solids Struct., 44, Nos. 25-26, 8648–8662 (2007).
D. Ie,san and R. Quintanilla, “On a theory of thermoelasticity with microtemperatures,” J. Thermal Stresses, 23, No. 3, 199–215 (2000).
R. Kumar and T. K. Chadha, “Plane problem in a micropolar thermoelastic half-space with stretch,” Indian J. Pure Appl. Math., 17, 827–842 (1986).
V. D. Kupradze, T. G. Gegelia, M. O. Basheleishvili, and T. V. Burchuladze, Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity [in Russian], Nauka, Moscow (1976).
H. H. Sherief and H. A. Saleh, “A half-space problem in the theory of generalized thermoelastic diffusion,” Int. J. Solids Struct., 42, No. 15, 4484–4493 (2005).
B. Singh and R. Kumar, Reflection of plane waves from the flat boundary of a micropolar generalized thermoelastic half-space,” Int. J. Eng. Sci., 36, Nos. 7-8, 865–890 (1998).
K. Skhvitaridze and M. Kharashvili, “Investigation of the Dirichlet and Neumann boundary-value problems for a half-space filled a viscous incompressible fluid,” in: Mechanics of the Continuous Environment Issues, Nova Sci. Publ., (2012), pp. 85–98.
M. Svanadze, “On the linear of thermoelasticity with microtemperatures,” Tec. Mech., 32, Nos. 2-5, 564–576 (2012).
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Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 94, Proceedings of the International Conference “Lie Groups, Differential Equations, and Geometry,” June 10–22, 2013, Batumi, Georgia, Part 1, 2014.
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Giorgashvili, L., Burchuladze, D. & Skhvitaridze, K. Explicit Solutions of Boundary-Value Problems of Thermoelasticity with Microtemperatures for a Half-Space. J Math Sci 216, 538–546 (2016). https://doi.org/10.1007/s10958-016-2911-1
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DOI: https://doi.org/10.1007/s10958-016-2911-1