Dynamic problems of the theory of elasticity for layers and semilayers with cavities. (English) Zbl 1397.74082
Summary: We present a solution methodology for dynamic problems of the theory of elasticity based on the fundamental (F)-solutions approach for layers and semilayers containing cavities. Under the proposed solution framework boundary-value problems for three-dimensional cylindrical bodies are reduced to well-studied systems of one-dimensional singular integral equations. With the aid of the integral Fourier transform in time, we study the problem of impulse loading at the sides of cavities. We also demonstrate how the combination of the proposed methodology with the approach of reflections can be used for the solution of analogous problems for semi-infinite layers.
MSC:
| 74H10 | Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of dynamical problems in solid mechanics |
| 74B99 | Elastic materials |
| 74E05 | Inhomogeneity in solid mechanics |
| 42A38 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
References:
| [1] | Achenbach J.D.: Wave Propagation in elastic Solids. North-Holland, Amsterdam (1973) · Zbl 0268.73005 |
| [2] | Achenbach J.D.: Transient shear waves in two joint elastic quarter spaces. Int. Appl. Mech. 36, 491–496 (1969) · Zbl 0185.53303 |
| [3] | Achenbach J.D.: Shear waves in an elastic wedge. Int. J. solids Struct. 6, 379–388 (1970) · Zbl 0187.23704 · doi:10.1016/0020-7683(70)90090-9 |
| [4] | Bardzokas, D.I., Zobnin, A.I.: Mathematical modelling of Physical processes in composite materials of periodical structure. Moscow Edit. URSS (2003) · Zbl 1054.80006 |
| [5] | Bardzokas, D.I., Zobnin, A.I., Senik, N.A., Filshtisnky, M.L.: Mathematical modelling in the problems of the mechanics of coupled fields. Moscow Edit. URSS (2006) |
| [6] | Bardzokas D.I., Filshtinsky M.L., Filshtinsky L.A.: Mathematical Methods in Electro-magneto-elasticity. Springer, Berlin (2007) · Zbl 1147.74001 |
| [7] | Bardzokas, D.I., Kudryartsev, B.A., Senik, N.A.: Wave propagation in electromagnetoelastic media. Moscow Edit. URSS (2003) |
| [8] | Brock L.M.: Two basic problems of plane crack extension: a unified treatment. Int. J. Eng. Sci. 15, 527–536 (1977) · Zbl 0364.73082 · doi:10.1016/0020-7225(77)90049-0 |
| [9] | Brock L.M.: Sliding and indentation by a rigid half-wedge with friction and displacement coupling effects. Int. J. Eng. Sci. 19, 33–40 (1981) · Zbl 0444.73096 · doi:10.1016/0020-7225(81)90047-1 |
| [10] | Gaginard L.: Reflection and Refraction of Progressive Seismic Waves in Layered Waves. McGraw-Hill, NY (1957) |
| [11] | Freund L.B.: Dynamic Fracture Mechanics. Cambridge University Press, London (1989) · Zbl 0712.73072 |
| [12] | Freund L.B., Achenbach J.D.: Diffraction of a plane pulse by a closed crack at the interface of elastic solids. Z. Angew. Math. Mech. 48, 173–185 (2006) · Zbl 0181.53402 · doi:10.1002/zamm.19680480304 |
| [13] | Freund L.B.: The initial wave emitted by a suddenly extending crack in an elastic solid. J. Appl. Mech. 39, 601–602 (1972) · doi:10.1115/1.3422728 |
| [14] | Freund L.B.: Crack propagation in an elastic solid subjected to general loading. III. Stress wave loading. J. Mech. Phys. Solids 21, 47–61 (1973) · Zbl 0265.73080 · doi:10.1016/0022-5096(73)90029-X |
| [15] | Freund L.B.: The response of an elastic solid to non-uniformly moving surface loads. J. Appl. Mech. 40, 699–704 (1973) · Zbl 0276.73019 · doi:10.1115/1.3423076 |
| [16] | Georgiadis H.G., Barber J.R.: Steady-state transonic motion of a line load over an elastic half-space. The corrected Cole/Huth solution. J. Appl. Mech. Trans. ASME 60, 772–774 (1993) · Zbl 0805.73020 · doi:10.1115/1.2900872 |
| [17] | Georgiadis H.G., Barber J.R.: On the super-Rayleigh/subseismic elastodynamic indentation problem. J. Elast. 31, 141–161 (1993) · Zbl 0782.73068 · doi:10.1007/BF00044967 |
| [18] | Georgiadis H.G., Lykotrafitis G.: A method based on the Radon transform for three-dimensional elastodynamic problems of moving loads. J. Elast. 65, 87–129 (2002) · Zbl 1205.74064 · doi:10.1023/A:1016135605598 |
| [19] | Gurtin M.E., Stenberg E.: A note on uniqueness in classical elastodynamics. Quart. Appl. Math. 19, 169–171 (1961) · Zbl 0100.37703 |
| [20] | Gurtin M.E., Toupin R.A.: A uniqueness theorem for the displacement boundary value problem of linear elastodynamics. Quart. Appl. Math. 23, 79–81 (1965) · Zbl 0133.17702 |
| [21] | Graff K.F.: Wave Motion in Elastic Solids. Ohio State University Press, USA (1975) · Zbl 0314.73022 |
| [22] | Hudson J.A.: The Excitation and Propagation of Elastic Waves. Cambridge University Press, London (1980) · Zbl 0435.35002 |
| [23] | Harris J.G.: Linear Elastic Waves. Cambridge University Press, London (2004) |
| [24] | DeHoop A.T.: A modification of Canginard’s method for solving the seismic pulse problem. Appl. Sci. Res. B 8, 349–356 (1960) · Zbl 0100.44208 · doi:10.1007/BF02920068 |
| [25] | Ewing W.M., Jardetzky W.S., Press F.: Elastic Waves in Layered Media. McGraw-Hill, NY (1957) · Zbl 0083.23705 |
| [26] | Eringen A.C., Suhubi E.S.: Elastodynamics. Finite Motions, vol. 1. Academic Press, NY (1975) · Zbl 0344.73036 |
| [27] | Eringen A.C., Suhubi E.S.: Elastodynamics. Linear Theory, vol. 2. Academic Press, NY (1975) · Zbl 0344.73036 |
| [28] | Kolsky H.: Stress waves in solids. Oxford University Press (Clarendon), London (1953) · Zbl 0052.42502 |
| [29] | Jupradge V.D.: Dynamical problems in elasticity. In: Sneddon, I.N., Hill, R. (eds) Progress in Solid Mechanics, vol. III, North-Holland Publications, Amsterdam (1973) |
| [30] | Miklowitz J.: The Theory of Elastic Waves and Wave Guides. North-Holland, NY (1978) · Zbl 0565.73025 |
| [31] | Pao Y.H., Mow C.C.: Diffraction of Elastic Waves and Dynamic Stress Concentrations. Crane, Russak and Co., NY (1973) |
| [32] | Poruchikov V.B.: Methods of the Classical Theory of Elastodynamics. Springer, Berlin (1993) |
| [33] | Lamb H.: On the propagation of tremors over the surface of separation of two solids. Proc. Roy. Soc., London Ser. A 203, 1–42 (1904) · JFM 34.0859.02 |
| [34] | Royer D., Dieulesaint E.: Elastic waves in solids. I. Free and guided propagation. Springer, Berlin (1996) · Zbl 0960.74002 |
| [35] | Stoneley R.: Elastic waves at the surface of separation of two solids. Proc. Roy. Soc. London Ser. A 106, 416–428 (1924) · JFM 51.0644.01 · doi:10.1098/rspa.1924.0079 |
| [36] | Lurie A.I.: On the theory of thick plates. Appl. Math. Mech. 6, 151–167 (1942) |
| [37] | Prokopov V.K.: Application of the symbolic method to the derivation of the equations of the theory of plates. J. Appl. Math. Mech. 29, 1064–1083 (1965) · Zbl 0208.53305 · doi:10.1016/0021-8928(65)90127-9 |
| [38] | Aksentyan O.K., Vorovich I.I.: Stress state of a thin plate. Appl. Math. Mech. 27, 1057–1074 (1963) · Zbl 0146.22002 |
| [39] | Vorovich I.I., Malkina O.S.: The stress state of thick layers. Appl. Math. Mech. 31, 230–241 (1967) · Zbl 0153.56102 |
| [40] | Vorovich I.I.: Certain stress concentration problems. Appl. Math. Mech. 2, 45–53 (1968) |
| [41] | Kosmodamianskii A.S.: Three-dimensional problems of the theory of elasticity for multi-connected plates. Appl. Mech. 19, 3–21 (1983) |
| [42] | Altuhov E.V.: The stress state of thick plates under homogeneous boundary conditions of mixed type on the faces. J. Math. Sci. 76, 2339–2342 (1995) · doi:10.1007/BF02362894 |
| [43] | Altuhov E.V., Misovshii V.V., Panchenko Y.V.: Three-dimensional problems of steady vibrations of isotropic plates. J. Math. Sci. 86, 3095–3098 (1997) · doi:10.1007/BF02355703 |
| [44] | Altuhov E.V., Goltsev A.S., Hiznyak V.K.: Stress state of isotropic layers with crack. Int. Appl. Mech. 33, 39–46 (1997) · doi:10.1007/BF02700885 |
| [45] | Bodnya Y.N., Shaldyrvan V.A.: Elastic deformation of thick plates with sliding restraint of the end faces. J. Math. Sci. 57, 2835–2840 (1991) · Zbl 0729.73819 · doi:10.1007/BF01100894 |
| [46] | Karapetian E., Kachanov M.: Green’s functions for the isotropic or transversely isotropic space containing a circular crack. Acta Mech. 126, 169–187 (1998) · Zbl 0897.73050 · doi:10.1007/BF01172806 |
| [47] | Bulanov G.S.: Multisided tension of thick partial homogeneous plates. J. Theor. Appl. Mech. 8, 19–23 (1977) |
| [48] | Zhirov V.E.: Electro-elastic equilibrium of a piezoceramic plate. Appl. Math. Mech. 41, 1114–1121 (1977) |
| [49] | Fridman L.I.: Dynamic problem of the theory of elasticity for bodies of canonical form. Int. Appl. Mech. 23, 1195–1201 (1987) · Zbl 0668.73049 |
| [50] | Ulitko A.F.: The method of vector eigenfunctions in three-dimensional problems of elasticity theory. Int. Appl. Mech. 3, 1–7 (1967) |
| [51] | Grinchenko V.T., Ulitko A.F.: Exact solution of the Kirsch problem. Int. Appl. Mech. 6, 455–461 (1970) |
| [52] | Kupradze V.D., Gegelia T.G., Bagheleishvili H.O., Burchuladze T.V.: Three-dimensional problems of the mathematical theory of elasticity and thermo-elasticity. North-Holland, NY (1979) |
| [53] | Kit, G.S., Hai, M.V.: The method of potential in three-dimensional problems of thermo-elasticity with cracks. Naukova Dumka, Kiev (In Russian) (1989) |
| [54] | Parton V.Z., Perlin V.I.: Integral Equations in Elasticity. MIR, Moscow (1982) · Zbl 0497.73002 |
| [55] | Voreshko P.P.: Effective construction of integral equations of potential theory for basic boundary problems of the theory of elasticity. Strength Mater. 5, 83–90 (1996) |
| [56] | Tranter C.J.: Integral Transforms in Mathematical Physics. Wiley, NY (1956) · Zbl 0074.31901 |
| [57] | Stenberg E.: Three-dimensional stress concentrations in the theory of elasticity. Appl. Mech. Rev. 1, 1–4 (1958) |
| [58] | Landau L.D., Lifshitz E.M.: Theory of Elasticity. Pergamon, NY (1986) · Zbl 0178.28704 |
| [59] | Alexandrov, A.Y., Soloviev, Y.I.: Three-dimensional problems of theory of elasticity. Nauka, Moscow (In Russian) (1978) |
| [60] | Shapiro G.S.: On the stress distribution in an unconstrained layer. Appl. Math. Mech. 8, 167–168 (1944) · Zbl 0063.06768 |
| [61] | Wang W., Shi M.X.: Thick plate theory based on general solutions of elasticity. Acta Mech. 123, 27–36 (1997) · Zbl 0902.73048 · doi:10.1007/BF01178398 |
| [62] | Filshtinskii L.A.: Periodic solutions of the theory of elasticity and electro-elasticity for cylinders in R3. J. Theor. Appl. Mech. 21, 13–20 (1990) |
| [63] | Filshtinskii L.A., Kovalev Y.D.: Bending of a layer with through tunnel cuts and free-sliding end faces. Mater. Sci. 36, 570–574 (2000) · doi:10.1023/A:1011322423868 |
| [64] | Grigolyuk E.I., Filshtinskii L.A.: Bending of an elastic plate weakened by a doubly periodic system of circular holes. Mech. Comp. Mater. 4, 1–3 (1968) |
| [65] | Grigoluk E.I., Kovalev Y.D., Filshtinskii L.A.: Mixed oblique-symmetric problem of the theory of elasticity for weakened from tunnel cracks layers. Rep. Univ. North-Caucasus 3, 46–47 (2000) (in Russian) |
| [66] | Filshtinskii L.A., Kovalev Yu.D.: Mixed antisymmetric problem of an elastic layer weakened by through cavities. Mater. Sci. 37, 833–837 (2001) · doi:10.1023/A:1015069130863 |
| [67] | Filshtinskii L.A., Oleinik V.M.: Boundary value problems of electroelasticity for layers weakened from tunnel cracks. Int. Appl. Mech. 27, 21–26 (1991) |
| [68] | Filshtinskii L.A., Oleinik V.M.: Three-dimensional problems of electroelasticity for layers weakened from tunnel cracks. Phys. Chem. Mech. Mater. 27, 66–72 (1991) |
| [69] | Filshtinskii L.A.: Tension of a layer weakened from tunnel cracks. Appl. Math. Mech. 59, 827–835 (1995) · Zbl 0900.73315 · doi:10.1016/0021-8928(95)00114-X |
| [70] | Filshtinskii L.A.: Fundamental solutions for the equations of electroelasticity for piezoceramic layers in R3. Mech. Comp. Mater. 31, 377–388 (2001) |
| [71] | Filshtinskii L.A.: Periodic in time fundamental solutions for the equations of thermoelasticity for anisotropic layers in R3. Math. Methods Phys. Mech. 46, 147–154 (2003) |
| [72] | Filshtinskii L.A., Kovalev Yu.D., Ventsel E.S.: Solution of the elastic boundary value problems for a layer with tunnel stress raisers. Int. J. Solids Struct. 39, 6385–6402 (2002) · Zbl 1032.74632 · doi:10.1016/S0020-7683(02)00490-0 |
| [73] | Filshtinksii L.A., Sirenko Y.V.: Homogeneous solutions of the Neumann problem for a composite layer in R3. Curr. problems Continuum Mech. 1, 200–2005 (2005) |
| [74] | Filshtinskii L.A., Shramko L.V., Shramko Y.V.: Fundamental solutions for piezoceramic layers in R3-symmetric case, mixed boundary conditions. Curr. Problems Continuum Mech. 2, 199–203 (2003) |
| [75] | Filshtinskii L.A.: Fundamental solutions of electroelasticity equations for a piezoceramic layer in R3. Mech. Comp. Mater. 37, 234–237 (2001) · doi:10.1023/A:1010694518294 |
| [76] | Nazarenko A.M., Filshtinskii L.A.: Interaction of elastic waves with a curvilinear crack in a half-plane. J. Math. Sci. 57, 2887–2891 (1991) · Zbl 0729.73830 · doi:10.1007/BF01100905 |
| [77] | Tikhonov A.N., Samarskii A.A.: Equations of Mathematical Physics. Dover Publications, NY (1990) · Zbl 0044.09302 |
| [78] | Theocaris P.S.: Numerical solution of singular integral equations: methods. J. Eng. Mech. Div. ASCE 107, 733–752 (1981) |
| [79] | Theocaris P.S.: Numerical solution of singular integral equations: applications. J. Eng. Mech. Div. ASCE 107, 753–771 (1981) |
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.
