Equations of anisotropic elastodynamics as a symmetric hyperbolic system: Deriving the time-dependent fundamental solution. (English) Zbl 1385.74002
Summary: The dynamic system of anisotropic elasticity from three second order partial differential equations is written in the form of the time-dependent first order symmetric hyperbolic system with respect to displacement velocity and stress components. A new method of deriving the time-dependent fundamental solution of the obtained system is suggested in this paper. This method consists of the following. The Fourier transform image of the fundamental solution with respect to a space variable is presented as a power series expansion relative to the Fourier parameters. Then explicit formulae for the coefficients of these power series are derived successively. Using these formulae the computer calculation of fundamental solution components (displacement velocity and stress components arising from pulse point forces) has been made for general anisotropic media (orthorhombic and monoclinic) and the simulation of elastic waves has been obtained. These computational examples confirm the robustness of the suggested method.
MSC:
| 74B05 | Classical linear elasticity |
| 74E10 | Anisotropy in solid mechanics |
| 74H10 | Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of dynamical problems in solid mechanics |
References:
| [1] | Ting, T. C.T., Anisotropic Elasticity: Theory and Applications (1996), Oxford University Press: Oxford University Press Oxford · Zbl 0871.73029 |
| [2] | Ting, T. C.T.; Barnett, D. M.; Wu, J. J., Modern Theory of Anisotropic Elasticity and Applications (1990), SIAM: SIAM Philadelphia · Zbl 0765.73002 |
| [3] | Dieulesaint, E.; Royer, D., Elastic Waves in Solids (1980), John Wiley and Sons: John Wiley and Sons Chichester |
| [4] | Federov, F. I., Theory of Elastic Waves in Crystals (1968), Plenum Press: Plenum Press New York |
| [5] | Poruchikov, V. B., Methods of the Classical Theory of Elastodynamics (1993), Springer Verlag: Springer Verlag Berlin |
| [6] | Donida, G.; Bernetti, R., Finite element approximation of dynamic problems for semi-infinite solids in elasticity, Computers and Structures, 41, 835-842 (1991) · Zbl 0850.73309 |
| [7] | Yakhno, V. G.; Akmaz, H. K., Anisotropic elastodynamics in a half space: an analytic method for polynomial data, Journal of Computational and Applied Mathematics, 204, 268-281 (2007) · Zbl 1112.74020 |
| [8] | Yakhno, V. G.; Akmaz, H. K., Initial value problem for the dynamic system of anisotropic elasticity, International Journal of Solids and Structures, 42, 855-876 (2005) · Zbl 1125.74306 |
| [9] | Stokes, G. G., On the dynamical theory of diffraction, Transactions of the Cambridge Philosophical Society, 9, 1-62 (1849) |
| [10] | Love, A. E.H., A Treatise on the Mathematical Theory of Elasticity (1944), Dover: Dover New York · Zbl 0063.03651 |
| [11] | Achenbach, J. D., Wave Propagation in Elastic Solids (1973), North-Holland: North-Holland Amsterdam · Zbl 0268.73005 |
| [12] | Kausel, E., Fundamental Solutions in Elastodynamics (2006), Cambridge University Press · Zbl 1225.74002 |
| [13] | Buchwald, V. T., Elastic waves in anisotropic media, Proceedings f the Royal Society of London, Series A, 253, 563-580 (1959) · Zbl 0092.42304 |
| [14] | Lighthill, M. J., Anisotropic wave motions, Philosophical Transactions of the Royal Society of London, Series A, 252, 397-470 (1960) · Zbl 0097.20806 |
| [15] | Burridge, R., The singularity on the plane lids of the wave surface of elastic media with cubic symmetry, The Quarterly Journal of Mechanics and Applied Mathematics, 20, 41-56 (1967) · Zbl 0189.26501 |
| [16] | Burridge, R., Lacunas in two-dimensional wave propagation, Mathematical Proceedings of the Cambridge Philosophical Society, 63, 819-845 (1967) · Zbl 0189.26502 |
| [17] | Burridge, R., Lamb’s problem for an anisotropic half-space, The Quarterly Journal of Mechanics and Applied Mathematics, 24, 81-98 (1971) · Zbl 0232.73022 |
| [18] | Kraut, E. A., Advances in the theory of anisotropic elastic wave propagation, Reviews of Geophysics, 1, 401-448 (1963) |
| [19] | Musgrave, M. J.P., Crystal Acoustics (1970), Holden-Day, Inc.: Holden-Day, Inc. San Francisco, CA · Zbl 0201.27601 |
| [20] | Willis, J. R., Self-similar problems in elastodynamics, Philosophical Transactions of the Royal Society of London, Series A. Mathematical, Physical and Engineering Sciences, 274, 435-491 (1973) · Zbl 0317.73062 |
| [21] | Payton, R. G., Elastic Wave Propagation in Transversely Isotropic Media (1983), Martinus Nijhoff: Martinus Nijhoff Dordrecht · Zbl 0574.73023 |
| [22] | Tsvankin, I. D.; Chesnokov, Y. M., Wave fields of point sources in arbitrarily anisotropic media, Izvestiya Earth Physics, 25, 528-540 (1989) |
| [23] | Wu, J. J.; Ting, T. C.T.; Barnett, D. M., Modern Theory of Anisotropic Elasticity and Applications (1990), SIAM: SIAM Philadelphia, PA · Zbl 0765.73002 |
| [24] | Payton, R. G., Wave propagation in a restricted transversely isotropic solid whose slowness surface contains conical points, The Quarterly Journal of Mechanics and Applied Mathematics, 45, 183-197 (1992) · Zbl 0775.73079 |
| [25] | Wang, C. Y.; Achenbach, J. D., A new look at 2D time domain elastodynamic Green’s functions for general anisotropic solids, Wave Motion, 16, 389-405 (1992) · Zbl 0764.73021 |
| [26] | Tewary, V. K.; Fortunko, C. M., Computationally efficient representation for propagation of elastic waves in anisotropic solids, Journal of the Acoustical Society of America, 91, 1888-1896 (1992) |
| [27] | Zhu, H., A method to evaluate three-dimensional time-harmonic elastodynamic Green’s functions in transversely isotropic media, Journal of Applied Mechanics, Transactions ASME, 59, 587-590 (1992) · Zbl 0760.73006 |
| [28] | Budreck, D. E., An eigenfunction expansion of elastic wave Green’s function for anisotropic media, The Quarterly Journal of Mechanics and Applied Mathematics, 46, 1-26 (1993) · Zbl 0780.73015 |
| [29] | Wang, C. Y.; Achenbach, J. D., Elastodynamic fundamental solutions for anisotropic solids, Geophysical Journal International, 118, 384-392 (1994) |
| [30] | Tewary, V. K., A computationally efficient representation for elastostatic and elastodynamic fundamental solutions for anisotropic solids, Physical Review B (1995), 51-22 |
| [31] | Wang, C. Y.; Achenbach, J. D., Three-dimensional time-harmonic elastodynamic Green’s functions for anisotropic solids, Proceedings f the Royal Society of London, Series A, 449, 441-458 (1995) · Zbl 0852.73011 |
| [32] | Yang, B.; Pan, E.; Tewary, V. K., Three-dimensional Green’s functions of steady-state motion in anisotropic half-spaces and bimaterial, Engineering Analysis with Boundary Elements, 28, 1069-1082 (2004) · Zbl 1130.74362 |
| [33] | Rangelov, T. V., Scattering from cracks in an elasto-anisotropic solid, Journal of Theoretical and Applied Mechanics, 33, 55-72 (2003) |
| [34] | Koçak, H.; Yıldırım, A., Numerical solution of 3D fundamental solution for the dynamic system of anisotropic elasticity, Physics Letters A, 373, 3145-3150 (2009) · Zbl 1233.74003 |
| [35] | Khojasteh, A.; Rahimian, M.; Pak, R. Y.S., Three-dimensional dynamic Green’s functions in transversely isotropic bi-materials, International Journal of Solids and Structures, 45, 4952-4972 (2008) · Zbl 1169.74443 |
| [36] | Wang, X.; Pan, E.; Feng, W. J., Time-dependent Green’s functions for an anisotropic bimaterial with viscous interface, Solid/European Journal of Mechanics-A/Solids, 26, 901-908 (2007) · Zbl 1117.74010 |
| [37] | Rangelov, T. V.; Manolis, G. D.; Dineva, P. S., Elastodynamic fundamental solutions for certain families of 2D inhomogeneous anisotropic domains: basic derivations, Solid/European Journal of Mechanics-A/Solids, 24, 820-836 (2005) · Zbl 1125.74344 |
| [38] | Manolis, G. D.; Shaw, R. P.; Pavlou, S., A first order system solution for the vector wave equation in a restricted class of heterogeneous media, Journal of Sound and Vibration, 209, 5, 723-752 (1998) · Zbl 1235.35186 |
| [39] | Lax, P. T., Hyperbolic Partial Differential Equations (2006), American Mathematical Society: American Mathematical Society Providence, Rhode Island · Zbl 1113.35002 |
| [40] | Courant, R.; Hilbert, D., Methods of Mathematical Physics (1962), Interscience: Interscience New York · Zbl 0729.35001 |
| [41] | Yakhno, V. G.; Yakhno, T. M.; Kasap, M., A novel approach for modelling and simulation of electromagnetic waves in anisotropic dielectrics, International Journal of Solids and Structures, 43, 6261-6276 (2006) · Zbl 1121.78322 |
| [42] | Aki, K.; Richard, P. G., Quantitative Seismology (2002), University Sciene Books |
| [43] | Dieulesaint, E.; Royer, D., Elastic Waves in Solids, vol. I (2000), Springer-Verlag: Springer-Verlag Berlin, Heidelberg · Zbl 0960.74002 |
| [44] | Batra, R. C.; Qian, L. F.; Chen, L. M., Natural frequencies of thick square plates made of orthotropic, trigonal, monoclinic, hexagonal and triclinic materials, Journal of Sound and Vibration, 270, 1074-1086 (2004) |
| [45] | Vladimirov, V. S., Generalized Functions in Mathematical Physics (1979), Mir Publishers: Mir Publishers Moscow · Zbl 0515.46034 |
| [46] | Hormander, L., Linear Partial Differential Operators (1963), Springer: Springer Berlin · Zbl 0171.06802 |
| [47] | Vladimirov, V. S., Equations of Mathematical Physics (1971), Marcel Dekker: Marcel Dekker New York · Zbl 0231.35002 |
| [48] | Reed, M.; Simon, B., Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness (1975), Academic Press: Academic Press New York · Zbl 0308.47002 |
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