Abstract
In this paper, we address a problem of computing the integrals appearing in integral expressions of Laplace domain anisotropic elastic displacement fundamental solutions. The essence of the problem is that these integrals can become highly oscillatory for high values of frequency or large distance between source and observation points. The modified integral expressions for displacement fundamental solutions and their first derivative are given. We propose a procedure based on the quadrature rule developed by Evans and Webster for the evaluation of rapidly oscillatory integrals. For a triclinic anisotropic elastic material, we consider an illustrative numerical example which involves phase functions with stationary points.
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References
Alibert, J., Seppecher, P., & dell’Isola, F. (2003). Truss modular beams with deformation energy depending on higher displacement gradients. Mathematics and Mechanics of Solids, 8, 51–73. https://doi.org/10.1177/1081286503008001658.
Auffray, N., dell’Isola, F., Eremeyev, V., Madeo, A., & Rossi, G. (2013). Analytical continuum mechanics à la Hamilton-Piola: Least action principle for second gradient continua and capillary fluids. Mathematics and Mechanics of Solids, 20, 375–417. https://doi.org/10.1177/1081286513497616.
Barchiesi, E., Spagnuolo, M., & Placidi, L. (2018). Mechanical metamaterials: A state of the art. Mathematics and Mechanics of Solids, 24, 212–234. https://doi.org/10.1177/1081286517735695.
Barnett, D. (1972). The precise evaluation of derivatives of the anisotropic elastic Green’s functions. Physica Status Solidi (B), 49, 741–748. https://doi.org/10.1002/pssb.2220490238.
Bogaert, I. (2014). Iteration-free computation of Gauss-Legendre quadrature nodes and weights. SIAM Journal on Scientific Computing, 36, A1008–A1026. https://doi.org/10.1137/140954969.
Buroni, F., & Sáez, A. (2013). Unique and explicit formulas for green’s function in three-dimensional anisotropic linear elasticity. Journal of Applied Mechanics, 80, 051018. https://doi.org/10.1115/1.4023627.
Del Vescovo, D., & Giorgio, I. (2014). Dynamic problems for metamaterials: Review of existing models and ideas for further research. International Journal of Engineering Science, 80, 153–172. https://doi.org/10.1016/j.ijengsci.2014.02.022.
dell’Isola, F., Andreaus, U., & Placidi, L. (2015). At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: An underestimated and still topical contribution of Gabrio Piola. Mathematics and Mechanics of Solids, 20(8), 887–928. https://doi.org/10.1177/1081286513509811.
dell’Isola, F., Cuomo, M., Greco, L., & Della Corte, A. (2017). Bias extension test for pantographic sheets: Numerical simulations based on second gradient shear energies. Journal of Engineering Mathematics, 103(1), 127–157. https://doi.org/10.1007/s10665-016-9865-7.
dell’Isola, F., Della Corte, A., & Giorgio, I. (2016a). Higher-gradient continua: The legacy of Piola, Mindlin, Sedov and Toupin and some future research perspectives. Mathematics and Mechanics of Solids, 22(4), 852–872. https://doi.org/10.1177/1081286515616034.
dell’Isola, F., Della Corte, A., Greco, L., & Luongo, A. (2016b). Plane bias extension test for a continuum with two inextensible families of fibers: A variational treatment with Lagrange multipliers and a perturbation solution. International Journal of Solids and Structures, 81, 1–12. https://doi.org/10.1016/j.ijsolstr.2015.08.029.
dell’Isola, F., Giorgio, I., Pawlikowski, M., & Rizzi, N. (2016c). Large deformations of planar extensible beams and pantographic lattices: Heuristic homogenization, experimental and numerical examples of equilibrium. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 472, 20150790. https://doi.org/10.1098/rspa.2015.0790.
dell’Isola, F., Igumnov, L., Litvinchuk, S., Ipatov, A., Petrov, A., & Modin, I. (2019a). Surface waves in dissipative poroviscoelastic layered half space: Boundary element analyses. In H. Altenbach, A. Belyaev, V. Eremeyev, A. Krivtsov, & A. Porubov (Eds.), Advanced Structured Materials. Dynamical processes in generalized continua and structures (Vol. 103). Cham: Springer.
dell’Isola, F., Seppecher, P., Alibert, J., Lekszycki, T., Grygoruk, R., Pawlikowski, M., et al. (2019b). Pantographic metamaterials: An example of mathematically driven design and of its technological challenges. Continuum Mechanics and Thermodynamics, 31(4), 851–884. https://doi.org/10.1007/s00161-018-0689-8.
dell’Isola, F., Seppecher, P., & Madeo, A. (2012). How contact interactions may depend on the shape of Cauchy cuts in Nth gradient continua: Approach “à la D’Alembert.” Zeitschrift für Angewandte Mathematik und Physik, 63(6), 1119–1141. https://doi.org/10.1007/s00033-012-0197-9.
dell’Isola, F., Seppecher, P., Spagnuolo, M., Barchiesi, E., Hils, F., Lekszycki, T., et al. (2019c). Advances in pantographic structures: Design, manufacturing, models, experiments and image analyses. Continuum Mechanics and Thermodynamics, 31(4), 1231–1282. https://doi.org/10.1007/s00161-019-00806-x.
Dineva, P., Manolis, G., & Wuttke, F. (2019). Fundamental solutions in 3D elastodynamics for the BEM: A review. Engineering Analysis with Boundary Elements, 105, 47–69. https://doi.org/10.1016/j.enganabound.2019.04.003.
Dravinski, M., & Niu, Y. (2001). Three-dimensional time-harmonic Green’s functions for a triclinic full-space using a symbolic computation system. International Journal for Numerical Methods in Engineering, 53, 445–472. https://doi.org/10.1002/nme.292.
Dravinski, M., & Zheng, T. (2000). Numerical evaluation of three-dimensional time-harmonic Green’s functions for a nonisotropic full-space. Wave Motion, 32, 141–151. https://doi.org/10.1016/s0165-2125(00)00034-2.
Evans, G., & Webster, J. (1997). A high order, progressive method for the evaluation of irregular oscillatory integrals. Applied Numerical Mathematics, 23, 205–218. https://doi.org/10.1016/s0168-9274(96)00058-x.
Fooladi, S., & Kundu, T. (2019a). An improved technique for elastodynamic Green’s function computation for transversely isotropic solids. Journal of Nondestructive Evaluation, Diagnostics and Prognostics of Engineering Systems, 2, 021005. https://doi.org/10.1115/1.4043605.
Fooladi, S., & Kundu, T. (2019b). Distributed point source modeling of the scattering of elastic waves by a circular cavity in an anisotropic half-space. Ultrasonics, 94, 264–280. https://doi.org/10.1016/j.ultras.2018.09.002.
Fredholm, I. (1900). Sur les équations de l’équilibre d’un corps solide élastique. Acta Mathematica, 23, 1–42. https://doi.org/10.1007/bf02418668.
Iserles, A., Nørsett, S., & Olver, S. (2006) Highly oscillatory quadrature: The story so far. Numerical Mathematics and Advanced Applications, 97–118. https://doi.org/10.1007/978-3-540-34288-5_6.
Lee, V. (2009). Derivatives of the three-dimensional Green’s functions for anisotropic materials. International Journal of Solids and Structures, 46, 3471–3479. https://doi.org/10.1016/j.ijsolstr.2009.06.002.
Levin, D. (1982). Procedures for computing one- and two-dimensional integrals of functions with rapid irregular oscillations. Mathematics of Computation, 38, 531. https://doi.org/10.2307/2007287.
Lifshitz, I., & Rozenzweig, L. (1947). Construction of the Green tensor for the fundamental equation of elasticity theory in the case of unbounded elastically anisotropic medium. Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 17, 783–791.
Malén, K. (1971). A unified six-dimensional treatment of elastic green’s functions and dislocations. Physica Status Solidi (B), 44, 661–672. https://doi.org/10.1002/pssb.2220440224.
Mura, T., & Kinoshita, N. (1971). Green’s functions for anisotropic elasticity. Physica Status Solidi (B), 47, 607–618. https://doi.org/10.1002/pssb.2220470226.
Nakamura, G., & Tanuma, K. (1997). A formula for the fundamental solution of anisotropic elasticity. The Quarterly Journal of Mechanics and Applied Mathematics, 50, 179–194. https://doi.org/10.1093/qjmam/50.2.179.
Pan, E., & Chen, W. (2015). Static Green’s functions in anisotropic media. Cambridge: Cambridge University Press.
Phan, A., Gray, L., & Kaplan, T. (2004). On the residue calculus evaluation of the 3-D anisotropic elastic Green’s function. Communications in Numerical Methods in Engineering, 20, 335–341. https://doi.org/10.1002/cnm.675.
Phan, A., Gray, L., & Kaplan, T. (2005). Residue approach for evaluating the 3D anisotropic elastic Green’s function: Multiple roots. Engineering Analysis with Boundary Elements, 29, 570–576. https://doi.org/10.1016/j.enganabound.2004.12.012.
Placidi, L., Andreaus, U., & Giorgio, I. (2017). Identification of two-dimensional pantographic structure via a linear D4 orthotropic second gradient elastic model. Journal of Engineering Mathematics, 103, 1–21. https://doi.org/10.1007/s10665-016-9856-8.
Placidi, L., Barchiesi, E., Turco, E., & Rizzi, N. (2016). A review on 2D models for the description of pantographic fabrics. Zeitschrift für Angewandte Mathematik und Physik, 67, 121. https://doi.org/10.1007/s00033-016-0716-1.
Rahali, Y., Giorgio, I., Ganghoffer, J.-F., & dell’Isola, F. (2015). Homogenization à la Piola produces second gradient continuum models for linear pantographic lattices. International Journal of Engineering Science, 97, 148–172. https://doi.org/10.1016/j.ijengsci.2015.10.003.
Sáez, A., & Domı́nguez, J. (1999). BEM analysis of wave scattering in transversely isotropic solids. International Journal for Numerical Methods in Engineering, 44, 1283–1300. https://doi.org/10.1002/(sici)1097-0207(19990330)44:9<1283::aid-nme544>3.0.co;2-o.
Sáez, A., & Domı́nguez, J. (2000). Far field dynamic Green’s functions for BEM in transversely isotropic solids. Wave Motion, 32, 113–123. https://doi.org/10.1016/s0165-2125(00)00032-9.
Sales, M., & Gray, L. (1998). Evaluation of the anisotropic Green’s function and its derivatives. Computers & Structures, 69, 247–254. https://doi.org/10.1016/s0045-7949(97)00115-6.
Sciarra, G., dell’Isola, F., & Coussy, O. (2007). Second gradient poromechanics. International Journal of Solids and Structures, 44(20), 6607–6629. https://doi.org/10.1016/j.ijsolstr.2007.03.003.
Shiah, Y., Tan, C., & Lee, R. (2010). Internal point solutions for displacements and stresses in 3D anisotropic elastic solids using the boundary element method. Computer Modeling in Engineering & Sciences, 69, 167–179. https://doi.org/10.3970/cmes.2010.069.167.
Shiah, Y., Tan, C., & Wang, C. (2012). Efficient computation of the Green’s function and its derivatives for three-dimensional anisotropic elasticity in BEM analysis. Engineering Analysis with Boundary Elements, 36, 1746–1755. https://doi.org/10.1016/j.enganabound.2012.05.008.
Tan, C., Shiah, Y., & Wang, C. (2013). Boundary element elastic stress analysis of 3D generally anisotropic solids using fundamental solutions based on Fourier series. International Journal of Solids and Structures, 50, 2701–2711. https://doi.org/10.1016/j.ijsolstr.2013.04.026.
Ting, T. (1997). The three-dimensional elastostatic Green’s function for general anisotropic linear elastic solids. The Quarterly Journal of Mechanics and Applied Mathematics, 50, 407–426. https://doi.org/10.1093/qjmam/50.3.407.
Vavryčuk, V. (2007). Asymptotic Green’s function in homogeneous anisotropic viscoelastic media. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 463, 2689–2707. https://doi.org/10.1098/rspa.2007.1862.
Wang, C., & Achenbach, J. (1994). Elastodynamic fundamental solutions for anisotropic solids. Geophysical Journal International, 118, 384–392. https://doi.org/10.1111/j.1365-246x.1994.tb03970.x.
Wang, C., & Achenbach, J. (1995). Three-dimensional time-harmonic elastodynamic Green’s functions for anisotropic solids. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 449, 441–458. https://doi.org/10.1098/rspa.1995.0052.
Wilson, R., & Cruse, T. (1978). Efficient implementation of anisotropic three dimensional boundary-integral equation stress analysis. International Journal for Numerical Methods in Engineering, 12, 1383–1397. https://doi.org/10.1002/nme.1620120907.
Xie, L., Zhang, C., Sladek, J., & Sladek, V. (2016). Unified analytical expressions of the three-dimensional fundamental solutions and their derivatives for linear elastic anisotropic materials. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science, 472, 20150272. https://doi.org/10.1098/rspa.2015.0272.
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The work is financially supported by the Russian Science Foundation under grant No. 18-79-00082.
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Markov, I.P., Markina, M.V. (2021). Numerical Evaluation of Integrals in Laplace Domain Anisotropic Elastic Fundamental Solutions for High Frequencies. In: dell'Isola, F., Igumnov, L. (eds) Dynamics, Strength of Materials and Durability in Multiscale Mechanics. Advanced Structured Materials, vol 137. Springer, Cham. https://doi.org/10.1007/978-3-030-53755-5_11
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