An inhomogeneous problem for an elastic half-strip: an exact solution. (English) Zbl 07589904
Summary: We construct exact solutions of two inhomogeneous boundary value problems in the theory of elasticity for a half-strip with free long sides in the form of series in Papkovich-Fadle eigenfunctions: (a) the half-strip end is free and (b) the half-strip end is firmly clamped. Initially, we construct a solution of the inhomogeneous problem for an infinite strip. Subsequently, the corresponding solutions for a half-strip are added to this solution, whereby the boundary conditions at the end are satisfied. The Papkovich orthogonality relation is used to solve the inhomogeneous problem in a strip.
MSC:
| 74-XX | Mechanics of deformable solids |
Keywords:
half-strip; inhomogeneous boundary value problem; Papkovich-Fadle eigenfunctions; Papkovich orthogonality relation; exact solutionReferences:
| [1] | Papkovich, PF. On one form of solution of the plane problem of the theory of elasticity for a rectangular strip. Dokl Akad Nauk SSSR 1940; 27: 335-339 (in Russian). (German abstract in Jb Fortschr Math 1940; 66: 453; English abstract in Math Rev 1941; 2: 332) |
| [2] | Grinberg, GA . On a method applied by P.F. Papkovich for the solution of the plane problem of the theory of elasticity for a rectangular region and of problem of bending of a rectangular thin plate with two clamped edges, and on some of its generalizations. Prikl Mat Mekh 1953; 17: 211-228 (in Russian). (English abstract in Math Rev 1953; 14: 1143) |
| [3] | Gregory, RD. A note on bi-orthogonality relations for elastic cylinders of general cross section. J Elast 1983; 13: 351-355. · Zbl 0522.73052 · doi:10.1007/BF00043002 |
| [4] | Nuller, BM . On the generalized orthogonality relation of P.A. Schiff. J Appl Math Mech 1969; 33(2): 364-372. · Zbl 0194.25801 · doi:10.1016/0021-8928(69)90049-5 |
| [5] | Patra, B. On a generalised orthogonality relation and its use in the problem of elasticity of a truncated cylindrical wedge. J Indian Inst Sci 1981; 63(B): 25-33. · Zbl 0468.73024 |
| [6] | Prakash, BG. Generalised orthogonality relation for rectangular strips in elastodynamics. Mech Res Comm 1978; 5(5): 251-255. · Zbl 0397.73023 · doi:10.1016/0093-6413(78)90019-8 |
| [7] | Kovalenko, MD, Shulyakovskaya, TD. Expansions in Fadle-Papkovich functions in a strip. Theory Foundations. Mech Solids 2011; 46(5): 721-738. · doi:10.3103/S0025654411050074 |
| [8] | Steketee, JA. On Volterra’s dislocations in a semi-infinite elastic medium. Can J Phys 1958; 36(2): 192-205. · Zbl 0079.44605 · doi:10.1139/p58-024 |
| [9] | Kasahara, K. Earthquake mechanics. Cambridge: Cambridge University Press, 1981. |
| [10] | Landau, LD, Lifshitz, EM. Theory of Elasticity. Course of Theoretical Physics, vol. 7. Oxford: Butterworth-Heinemann, 1986. |
| [11] | Slepyan, LI. Mechanics of cracks. Leningrad: Sudostroenie, 1990. (in Russian) |
| [12] | Kovalenko, MD, Menshova, IV, Shulyakovskaya, TD. Expansions in Fadle-Papkovich functions: examples of solutions in a half-strip. Mech Solids 2013; 48(5): 584-602. · doi:10.3103/S0025654413050154 |
| [13] | Vlasov, VV. Method of initial functions in problems of the theory of elasticity and structural mechanics. Moscow: Stroiizdat, 1975. (in Russian) |
| [14] | Prudnikov, AP, Brychkov Yu, A, Marichev, OI. Integrals and Series, vol. 1. Elementary Functions. New York: Gordon and Breach Science Publishers, 1986. · Zbl 0733.00004 |
| [15] | Matrosov, AV, Kovalenko, MD, Menshova, IV, Kerzhaev, AP. Method of initial functions and integral Fourier transform in some problems of the theory of elasticity. Z Angew Math Phys 2020; 71: 24. · Zbl 1431.74018 · doi:10.1007/s00033-019-1247-3 |
| [16] | Ketch, V, Teodorescu, P. Introduction to the theory of generalized functions with applications in engineering. New York: Wiley, 1978. |
| [17] | Timoshenko, SP, Goodier, JN. Theory of elasticity. New-York: McGraw-Hill, 1951. · Zbl 0045.26402 |
| [18] | Kovalenko, MD, Menshova, IV, Kerzhaev, AP, Yu, G. Mixed boundary value problems in the theory of elasticity in an infinite strip. Acta Mech 2018; 229(11): 4339-4356. · Zbl 1430.74019 · doi:10.1007/s00707-018-2244-x |
| [19] | Kovalenko, MD, Menshova, IV, Kerzhaev, AP. On the exact solutions of the biharmonic problem of the theory of elasticity in a half-strip. Z Angew Math Phys 2018; 69: 121. · Zbl 1401.74036 · doi:10.1007/s00033-018-1013-y |
| [20] | Kovalenko, MD, Abrukov, DA, Menshova, IV, Kerzhaev, AP, Yu, G. Exact solutions of boundary value problems in the theory of plate bending in a half-strip: Basics of the theory. Z Angew Math Phys 2019; 70: 98. · Zbl 1415.74038 · doi:10.1007/s00033-019-1139-6 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
