The effective solution of two-dimensional integro-differential equations and their applications in the theory of viscoelasticity. (English) Zbl 1335.45004
Summary: The effective solutions for integro-differential equations related to problems of interaction of an elastic thin finite inclusion with a plate, when the inclusion and plate materials possess the creep property are constructed. If the geometric parameter of the inclusion is measured along its length according to the parabolic and linear law we have managed to investigate the obtained boundary value problems of the theory of analytic functions and to get exact solutions and establish behavior of unknown contact stresses at the ends of an elastic inclusion.
MSC:
| 45J05 | Integro-ordinary differential equations |
| 35R09 | Integro-partial differential equations |
| 74B05 | Classical linear elasticity |
| 74D05 | Linear constitutive equations for materials with memory |
| 74K20 | Plates |
Keywords:
contact problem; integro-differential equation; viscoelasticity; Carleman type boundary value problem; Fourier’s integral transformation; Volterra’s integral equationReferences:
| [1] | V. M.Aleksandrov and S. M.Mkhitaryan, Contact Problems for Bodies with thin Coverings and Layers (in Russian), (“Nauka”, Moscow, 1983). |
| [2] | Yu. N.Rabotnov, Elements of Continuum Mechanics of Materials with Memory (in Russian), (“Nauka”, Moscow, 1977). |
| [3] | N. Kh.Arutyunyan, V. B.Kolmanovskii, The Creep Theory for Nonhomogeneous Bodies (in Russian), (“Nauka”, Moscow, 1983). |
| [4] | N. Kh.Arutyunyan and A. V.Manzhirov, Contact Problems of the Creep Theory (in Russian), (Publ. house Acad. Sci. Armenian SSR, Erevan, 1990). |
| [5] | V. N.Akopyan, On the transference of load from an elastic nonhomogeneous semi‐infinite plate to a half‐plane possessing the creep property, Proc. Nat. Acad. Sci. Armenii46-50, (1999). |
| [6] | N. N.Shavlakadze, A contact problem of the interaction of a semi‐finite inclusion with a plane, Georgian Math. J.6(5), 489-500 (1999). · Zbl 0960.74052 |
| [7] | N. N.Shavlakadze, The solution of system of integral differential equation and its application in the theory of elasticity, ZAMM. Z. Angew. Math. Mech.91(12), 979-992 (2011). · Zbl 1298.74080 |
| [8] | B.Gross, Mathematical Structure of the Theories of Viscoelasticity (Paris. Hermann.1953). · Zbl 0052.20901 |
| [9] | R. M.Christensen, Theory of Viscoelasticity (New York, Academic, 1971). |
| [10] | A. C.Pipkin, Lecture on Viscoelasticity Theory (New York, Springer‐Verlag, 1971). |
| [11] | M.Fabrizio and A.Morro, Mathematical Problems in Linear Viscoelasticity (SIAM, Philadelphia, 1992). · Zbl 0753.73003 |
| [12] | H.Altenbach and J. J.Skrzypek, Creep and Damade in Materials and Structures, CISM Courses and Lectures. No. 399 (Springer, WienNew York, 1999). |
| [13] | H.Altenbach, Creep analysis of thin‐walled structures, ZAMM. Z. Angew. Math. Mech.82(8), 507-533 (2002). · Zbl 1040.74504 |
| [14] | H.Altenbach, Y.Gorash, and K.Naumenko, Steady‐state creep of a pressurized thick cylinder in both the linear and power law ranges, Acta Mechanica195(1‐4), 263-274 (2008). · Zbl 1136.74036 |
| [15] | R.Lakes, Viscoelastic Materials (Cambridge Univ. Press, 2009). |
| [16] | N. I.Muschelishvili, Singular Integral Equations. Boundary Problems of Function Theory and Their Application to Mathematical Physics (in Russian). 2nd. (Fizmatgiz, Moscow, 1962) English translation: Wolters‐Noordhoff Publishing, Groningen, 1967. · Zbl 0103.07502 |
| [17] | F. D.Gakhov and Ju. I.Cherski, Equations of Convolution Type (in Russian), (“Nauka”, Moscow, 1978). · Zbl 0458.45002 |
| [18] | R. D.Bantsuri, A contact problem for a wedge with elastic bracing (in Russian), Dokl. Akad. Nauk. SSSSR211, 797-800 (1973). |
| [19] | R. D.Bantsuri, A certain boundary value problem in analytic function theory (in Russian), Soobshch. Akad. Nauk Grzin. SSR73, 549-552 (1974). · Zbl 0309.30035 |
| [20] | R. D.Bantsuri, A contact problem for an anisotropic wedge with an elastic stringer (in Russian), Dokl. Akad. Nauk. SSSR222, 568-571 (1975). |
| [21] | M.Abramowitz (ed.) and I.Stegun (ed.) (eds.), Reference Book on Special Function (in Russian), (“Nauka”, Moscow, 1979). |
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