# Non-stationary problem of elasticity for a quarter-plane

## DOI:

https://doi.org/10.17721/1812-5409.2021/3.2## Keywords:

quarter-plane, non-stationary loading, integral transform, singular integral equation, Green’s matrix-function## Abstract

The plane problem for an elastic quarter-plane under the non-stationary loading is solved in the article. The method for solving was proposed in the previous authors’ papers, but it was used for the stationary case of the problem there. The initial problem is reduced to the one-dimensional problem by using the Laplace and Fourier integral transforms. The one-dimensional problem in transform space is written in vector form. Its solution is constructed as the superposition of the general solution for the homogeneous equation and the partial solution for the inhomogeneous equation. The general solution for the homogeneous vector equation is derived using the matrix differential calculations. The partial solution is found through Green’s matrix-function. The derived expressions for displacements and stresses are inverted by using of mutual inversion of Laplace-Fourier transforms. The solving of the initial problem is reduced to the solving of the singular integral equation regarding the displacement function at the one of the boundary of the quarter-plane. The time discretization is used, and the singular integral equation is solved using the orthogonal polynomials method at the fixed time moments. Based on numerical research some important mechanical characteristics depending on the time and loading types were derived.

** Pages of the article in the issue**: 28 - 33

** Language of the article**: English

## References

UFLYAND, Ya. S. (1967) Integralnie preobrazovaniya v zadachah teorii uprugosti. Leningrad: Nauka.

HANSON, M.T. and KEER L.M. (1989) Stress analysis and contact problems for an elastic quarter-plane. Q.J. Mech. Appl. Math., 42(3). p. 363-383.

YASINKYY, TOKOVYY, Yu. And IEROKHOVA, O. (2016) Optimization of two-dimensional nonstationary thermal stresses and displacements in a half-space through the use of internal heat sources. Journal of Thermal Stresses, 39(9). p. 1084-1097.

MYKHAS’KIV, V. and STANKEVYCH, V. (2019) Elastodynamic problem for a layered composite with penny-shaped crack under harmonic torsion. ZAMM.

MENSHYKOV, O. et al. (2016) Cracked materials under dynamic loading: effects of crack’s closure. The 12th World Congress on Computational Mechanics and The 6th Asia-Pacific Congress on Computational Mechanics.

TARLAKOVSKII, D.V. and FEDONTENKOV G.V. (2016) Non-stationary problems for elastic half-plane with moving point of changing boundary conditions. PNRPU Mechanics Bulletin, 3. р. 188-206. DOI: 10.15593/perm.mech/2016.3.13

Fu, Y., KAPLUNOV and J., PRIKAZCHIKOV, D. (2020) Reduced model for the surface dynamics of a generally anisotropic elastic half-space. Proceedings of the Royal Society A, 476(2234)

KUBENKO, V.D. (2015) Stress state of an elastic half-plane under nonstationary loading. International Applied Mechanics, 51(2). p. 121-130.

XIN, Zh. and XU, W.-Q. Boundary conditions and boundary layers for a class of linear relaxation systems in a quarter plane. Part of the The IMA Volumes in Mathematics and its Applications book series, 135.

SHMEGERA, S.V. The initial boundary-value mixed problems for elastic half-plane with the conditions of contact friction.

DOETSCH, G. (1974) Introduction to the Theory and Application of the Laplace Transformation, New York: Springer-Verlag.

BERBERAN-SANTOS, M.N. (2005) Analytical inversion of the Laplace transform without contour integration: application to luminescence decay laws and other relaxation functions. Journal of Mathematical Chemistry, 38(2). p. 165-173.

SLEPYAN, L.I. (1972) Nestazionarnie uprugie volni, Leningrad: Sudostroenie.

VESTYAK, V.A. et al. (2014) Elastic half place under the action of nonstationary surface kinematic perturbations. Journal of Mathematical Sciecens, 203(2). p. 202-214.

SHAFIEI, A.R. (2003) Wave propagation in an elastic quarter plane. International journal of engineering science (English), 14(3). p. 119-134.

HEJAZI, A.A. et al. (2014) Dislocation technique to obtain the dynamic stress intensity factors for multiple cracks in a half-plane under impact load. Arch. Appl. Mech, 84. p. 95-107.

VAYSFEL’D, N.D. and ZHURAVLOVA, Z.Yu. (2015) On one new approach to the solving of an elasticity mixed plane problem for the semi-strip. Acta Mechanica. 226 (12). p. 4159-4172.

POPOV, G.Ya., ABDIMANOV S.A. and EFIMOV V.V. (1999) Funkcii I matrici Grina odnomernih kraevih zadach. Almatu: Racah.

SLEPYAN, L.I. and YAKOVLEV, Yu.S. (1980) Integralnie preobrazovaniya v nestazionarnih zadachah mehaniki, Leningrad: Sudostroenie.

POPOV, G. Ya. (1982) Koncentraciya Uprugih Napryazheniy Vozle Shtampov, Razrezov, Tonkih Vklyucheniy i Podkrepliniy. Moskva: Nauka.

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*Bulletin of Taras Shevchenko National University of Kyiv. Physical and Mathematical Sciences*, (3), 28–33. https://doi.org/10.17721/1812-5409.2021/3.2

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