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Keywords:
2D quasicrystals; inhomogeneous media; elastodynamic system
2D quasicrystals; inhomogeneous media; elastodynamic system
Summary:
Initial value problem for three dimensional (3D) elastodynamic system in two dimensional (2D) inhomogeneous quasicrystals is considered. An analytical method is studied for the solution of this problem. The system is written in terms of Fourier images of displacements with respect to lateral variables. The resulting problem is reduced to integral equations of the Volterra type. Finally, using Paley Wiener theorem it is shown that the solution of the initial value problem can be found by the inverse Fourier transform. A numerical example is considered for the comparison of the exact solution with the computed solution obtained by using the method.
References:
[1] Altunkaynak, M.: An analytical method for solving elastic system in inhomogeneous orthotropic media. Math. Methods Appl. Sci. 42 (2019), 2324-2333. DOI 10.1002/mma.5510 | MR 3936402 | Zbl 1420.35154
[2] Andersen, N. B.: Real Paley-Wiener theorems for the inverse Fourier transform on a Riemannian symmetric space. Pac. J. Math. 213 (2004), 1-13. DOI 10.2140/pjm.2004.213.1 | MR 2040247 | Zbl 1049.43004
[3] Chen, J., Liu, Z., Zou, Z.: Transient internal crack problem for a nonhomogeneous orthotropic strip (Mode I). Int. J. Eng. Sci. 40 (2002), 1761-1774. DOI 10.1016/S0020-7225(02)00038-1
[4] Chen, W. Q., Ma, Y. L., Ding, H. J.: On three-dimensional elastic problems of one-dimensional hexagonal quasicrystal bodies. Mech. Res. Commun. 31 (2004), 633-641. DOI 10.1016/j.mechrescom.2004.03.007 | MR 2092970 | Zbl 1098.74554
[5] Daros, C. H.: A fundamental solution for SH-waves in a class of inhomogeneous anisotropic media. Int. J. Eng. Sci. 46 (2008), 809-817. DOI 10.1016/j.ijengsci.2008.02.001 | MR 2427933 | Zbl 1213.74156
[6] Daros, C. H.: A time-harmonic fundamental solution for a class of inhomogeneous transversely isotropic media. Wave Motion 46 (2009), 269-279. DOI 10.1016/j.wavemoti.2009.02.001 | MR 2568578 | Zbl 1231.74172
[7] De, P., Pelcovits, R. A.: Linear elasticity theory of pentagonal quasicrystals. Phys. Rev. B 35 (1987), Article ID 8609, 12 pages. DOI 10.1103/PhysRevB.35.8609
[8] Ding, D.-H., Wang, R., Yang, W., Hu, C.: General expressions for the elastic displacement fields induced by dislocations in quasicrystals. J. Phys., Conden. Matt. 7 (1995), Article ID 5423, 14 pages. DOI 10.1088/0953-8984/7/28/003
[9] Ding, D.-H., Yang, W., Hu, C., Wang, R.: Generalized elasticity theory of quasicrystals. Phys. Rev. B 48 (1993), Article ID 7003, 8 pages. DOI 10.1103/PhysRevB.48.7003
[10] Ding, H., Chenbuo, Liangjian: General solutions for coupled equations for piezoelectric media. Int. J. Solids Struct. 33 (1996), 2283-2298. DOI 10.1016/0020-7683(95)00152-2 | Zbl 0899.73453
[11] Fan, T.: Mathematical Theory of Elasticity of Quasicrystals and Its Applications. Springer, Berlin (2011). DOI 10.1007/978-3-642-14643-5 | MR 2847904 | Zbl 1222.74003
[12] Fan, T.-Y., Mai, Y.-W.: Elasticity theory, fracture mechanics, and some relevant thermal properties of quasi-crystalline materials. Appl. Mech. Rev. 57 (2004), 325-343. DOI 10.1115/1.1763591
[13] Gao, Y.: Governing equations and general solutions of plane elasticity of cubic quasicrystals. Phys. Lett., A 373 (2009), 885-889. DOI 10.1016/j.physleta.2009.01.002 | Zbl 1236.74048
[14] Gao, Y., Xu, S.-P., Zhao, B.-S.: A theory of general solutions of 3D problems in 1D hexagonal quasicrystals. Phys. Scr. 77 (2008), Article ID 015601, 6 pages. DOI 10.1088/0031-8949/77/01/015601 | Zbl 1157.82395
[15] Gao, Y., Zhao, B.-S.: A general treatment of three-dimensional elasticity of quasicrystals by an operator method. Phys. Status Solid., B 243 (2006), 4007-4019. DOI 10.1002/pssb.200541400
[16] Gao, Y., Zhao, B.-S.: General solutions of three-dimensional problems for two-dimensional quasicrystals. Appl. Math. Modelling 33 (2009), 3382-3391. DOI 10.1016/j.apm.2008.11.001 | MR 2524125 | Zbl 1205.74023
[17] Holford, R. L.: Elementary source-type solutions of the reduced wave equation. J. Acoust. Soc. Am. 70 (1981), 1427-1436. DOI 10.1121/1.387099 | MR 0634323 | Zbl 0478.35030
[18] Hook, J. F.: Green's functions for axially symmetric elastic waves in unbounded inhomogeneous media having constant velocity gradients. J. Appl. Mech. 29 (1962), 293-298. DOI 10.1115/1.3640544 | MR 0138265 | Zbl 0107.41902
[19] Hu, C., Wang, R., Ding, D.-H.: Symmetry groups, physical property tensors, elasticity and dislocations in quasicrystals. Rep. Progr. Phys. 63 (2000), Article ID 63, 39 pages. DOI 10.1088/0034-4885/63/1/201 | MR 1732137
[20] Levine, D., Steinhardt, P. J.: Quasicrystals: A new class of ordered structures. Phys. Rev. Lett. 53 (1984), Article ID 2477, 4 pages. DOI 10.1103/PhysRevLett.53.2477 | MR 0831879
[21] Li, L.-H., Fan, T.-Y.: Final governing equation of plane elasticity of icosahedral quasicrystals and general solution based on stress potential function. Chin. Phys. Lett. 23 (2006), Article ID 2519, 3 pages. DOI 10.1088/0256-307X/23/9/047
[22] Liu, G. T., Fan, T. Y., Guo, R. P.: Governing equations and general solutions of plane elasticity of one-dimensional quasicrystals. Int. J. Solids Struct. 41 (2004), 3949-3959. DOI 10.1016/j.ijsolstr.2004.02.028 | MR 2215812 | Zbl 1079.74524
[23] Manolis, G. D., Shaw, R. P.: Fundamental solutions for variable density two-dimensional elastodynamic problems. Eng. Anal. Bound. Elem. 24 (2000), 739-750. DOI 10.1016/S0955-7997(00)00056-4 | Zbl 0971.74046
[24] Ovid'ko, I. A.: Plastic deformation and decay of dislocations in quasi-crystals. Mater. Sci. Eng., A 154 (1992), 29-33. DOI 10.1016/0921-5093(92)90359-9
[25] Peng, Y.-Z., Fan, T.-Y: Perturbation theory of 2D decagonal quasicrystals. Physica B 311 (2002), 326-330. DOI 10.1016/S0921-4526(01)00611-1
[26] Rangelov, T. V., Manolis, G. D., Dineva, P. S.: Elastodynamic fundamental solutions for certain families of 2D inhomogeneous anisotropic domains: Basic derivations. Eur. J. Mech., A, Solids 24 (2005), 820-836 \99999DOI99999 10.1016/j.euromechsol.2005.05.002 . DOI 10.1016/j.euromechsol.2005.05.002 | MR 2174322 | Zbl 1125.74344
[27] Rochal, S. B., Lebedyuk, I. V., Kozinkina, Y. A.: Linear continuous inhomogeneous strains in octagonal and decagonal quasicrystals. Phys. Rev., B 60 (1999), Article ID 865, 9 pages. DOI 10.1103/PhysRevB.60.865
[28] Shechtman, D., Blech, I., Gratias, D., Cahn, J. W.: Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53 (1984), Article ID 1951, 3 pages. DOI 10.1103/PhysRevLett.53.1951
[29] Slawinski, M. A.: Waves and Rays in Elastic Continua. World Scientific, Hackensack (2021). DOI 10.1142/11994 | Zbl 1459.74002
[30] Socolar, J. E. S., Lubensky, T. C., Steinhardt, P. J.: Phonons, phasons, and dislocations in quasicrystals. Phys. Rev., B 34 (1986), Article ID 3345, 16 pages. DOI 10.1103/PhysRevB.34.3345
[31] Wang, X.: The general solution of one-dimensional hexagonal quasicrystal. Mech. Res. Commun. 33 (2006), 576-580. DOI 10.1016/j.mechrescom.2005.02.022 | MR 2215812 | Zbl 1192.74069
[32] Watanabe, K.: Transient response of an inhomogeneous elastic solid to an impulsive SH-source: Variable SH-wave velocity. Bull. JSME 25 (1982), 315-320. DOI 10.1299/jsme1958.25.315
[33] Watanabe, K., Payton, R. G.: Green's function and its non-wave nature for SH-wave in inhomogeneous elastic solid. Int. J. Eng. Sci. 42 (2004), 2087-2106. DOI 10.1016/j.ijengsci.2004.08.001 | MR 2102742 | Zbl 1211.74123
[34] Yakhno, V.: A new method of solving equations of elasticity in inhomogeneous quasicrystals by means of symmetric hyperbolic systems. Math. Methods Appl. Sci. 44 (2021), 9487-9506. DOI 10.1002/mma.7373 | MR 4279862 | Zbl 1475.35338
[35] Yakhno, V. G., Sevimlican, A.: Solving an initial value problem in inhomogeneous electrically and magnetically anisotropic uniaxial media. Appl. Math. Comput. 215 (2010), 3839-3850. DOI 10.1016/j.amc.2009.11.027 | MR 2578849 | Zbl 1188.78003
[36] Yakhno, V., Sevimlican, A.: An analytic method for the initial value problem of the electric field system in vertical inhomogeneous anisotropic media. Appl. Math., Praha 56 (2011), 315-339. DOI 10.1007/s10492-011-0019-y | MR 2800581 | Zbl 1224.35392
[37] Yakhno, V. G., Yaslan, H. Ç: Three dimensional elastodynamics of 2D quasicrystals: The derivation of the time-dependent fundamental solution. Appl. Math. Modelling 35 (2011), 3092-3110. DOI 10.1016/j.apm.2010.12.019 | MR 2776264 | Zbl 1219.74010
[38] Yang, L. Z., Zhang, L. L., Gao, Y.: General solutions of plane problem in one-dimensional hexagonal quasicrystals. Appl. Mech. Mater. 275-277 (2013), 101-104. DOI 10.4028/www.scientific.net/AMM.275-277.101
[39] Zhang, L., Zhang, Y., Gao, Y.: General solutions of plane elasticity of one-dimensional orthorhombic quasicrystals with piezoelectric effect. Phys. Lett., A 378 (2014), 2768-2776. DOI 10.1016/j.physleta.2014.07.027 | MR 3250390 | Zbl 1298.74027
[40] Zhou, W.-M., Fan, T.-Y.: Axisymmetric elasticity problem of cubic quasicrystal. Chin. Phys. 9 (2000), Article ID 294, 10 pages. DOI 10.1088/1009-1963/9/4/009
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