Abstract
Boundary integral formulation of 3D problems on the normal incidence of time-harmonic longitudinal elastic wave on a one-periodic array of coplanar rigid disk-shaped inclusions is given. A boundary integral equation for the jump of normal stresses across the reference inclusion surfaces is deduced by applying the periodicity conditions. The dynamic interaction in a periodic chain of wave scatterers is described by Green’s function written in terms of the exponentially convergent Fourier integrals. Regular analog of the equation is obtained involving the singularity subtraction and map** techniques, which allows its numerical solution for a wide range of wave numbers. The dynamic translation of the inclusions as rigid units and the dynamic stress intensity factor in the inclusion vicinities are computed and analyzed depending on the wave number, the periodicity length of the array, and the mass of constituent inclusions.
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References
Bruno OP, Delourme B (2014) Rapidly convergent two-dimensional quasi-periodic Green function throughout the spectrum - including Wood anomalies. J Comput Phys 262:262–290
Golub MV, Doroshenko OV (2019a) Boundary integral equation method for simulation scattering of elastic waves obliquely incident to a doubly periodic array of interface delaminations. J Comput Phys 376:675–693
Golub MV, Doroshenko OV (2019b) Wave propagation through an interface between dissimilar media with a doubly periodic array of arbitrary shaped planar delaminations. Math Mech Solids 24:483–498
Gradshteyn IS, Ryzhik IM (2000) Tables of integrals, series and products. Academic Press, New York
Guz AN (2022) Problem 7. Brittle fracture of materials with cracks under action of dynamic loads (with allowance for contact interaction of the crack edges). Adv Struct Mater 159:335–349
Isakari H, Niino K, Yoshikawa H, Nishimura N (2012) Calderon’s preconditioning for periodic fast multipole method for elastodynamics in 3D. Int J Numer Methods Eng 90:484–505
Itou S (2000) Three-dimensional dynamic stress intensity factors around two parallel square cracks in an infinite elastic medium subjected to a time-harmonic stress wave. Acta Mech 143:79–90
Kadic M, Milton GW, Van Hecke M (2019) 3D metamaterials. Nat Rev Phys 1:198–210
Kassir MK, Sih GC (1968) Some three-dimensional inclusion problems in elasticity. Int J Solids Struct 4:225–241
Kit HS, Mykhas’skiv VV, Khay OM (2002) Analysis of the steady oscillations of a plane absolutely rigid inclusion in a three-dimensional elastic body by the boundary element method. J Appl Math Mech 66:817–824
Li FL, Wang YS, Zhang Ch, Yu GL (2013) Boundary element method for band gap calculation of two-dimensional solid phononic crystals. Eng Anal Bound Elem 37:225–235
Linton CM (2010) Lattice sums for the Helmholtz equation. SIAM Rev 52:630–674
Linton CM, McIver P (2001) Handbook of mathematical techniques for wave/structure interactions. Chapman & Hall/CRC, London
Mahmood MS, Laghrouche O, Trevelyan J, El Kacimi A (2017) Implementation and computational aspects of a 3D elastic wave modelling by PUFEM. Appl Math Model 49:568–586
Martin PA (2006) Multiple scattering: interaction of time-harmonic waves with N obstacles. Cambridge University Press, Cambridge
Maslov K, Kinra VK, Henderson BK (2000) Elastodynamic response of a coplanar periodic layer of elastic spherical inclusions. Mech Mater 32:785–795
Menshykova MV, Menshykov OV, Mikucka VA, Guz IA (2012) Interface cracks with initial opening under harmonic loading. Compos Sci Technol 72:1057–1063
Mykhailova II, Menshykov OV, Guz IA (2011) Cracks’ closure in 3-D fracture dynamics: the effect of relative location of two coplanar cracks. Arch Appl Mech 81:1215–1230
Mykhas’kiv VV, Khay OM, Zhang Ch, Boström A (2010) Effective dynamic properties of 3D composite materials containing rigid penny-shaped inclusions. Waves Random Complex Media 20:491–510
Mykhas’kiv V, Kunets Ya, Matus V, Khay O (2018) Elastic wave dispersion and attenuation caused by multiple types of disc-shaped inclusions. Int J Struct Integr 9:219–232
Mykhas’kiv VV, Zhbadynskyi IYa, Zhang Ch (2014) Dynamic stresses due to time-harmonic elastic wave incidence on doubly periodic array of penny-shaped cracks. J Math Sci 203:114–122
Mykhas’kiv V, Zhbadynskyi I, Zhang Ch (2010) Elastodynamic analysis of multiple crack problem in 3-D bi-materials by a BEM. Int J Numer Methods Biomed Eng 26:1934–1946
Mykhas’kiv VV, Zhbadynskyi IYa, Zhang Ch (2019) On propagation of time-harmonic elastic waves through a double-periodic array of penny-shaped cracks. Eur J Mech Solids 73:306–317
Remizov MYu, Sumbatyan MA (2017) Three-dimensional one-mode penetration of elastic waves through a doubly periodic array of cracks. Math Mech Solids 23:636–650
Wang YF, Wang YS, Su XX (2011) Large bandgaps of two-dimensional phononic crystals with cross-like holes. J Appl Phys 110:113520
Zhang X, Liu Z, Liu Y, Wu F (2003) Elastic wave band gaps for three-dimensional phononic crystals with two structural units. Phys Lett A 313:455–460
Zhao J, Li Y, Liu WK (2015) Predicting band structure of 3D mechanical metamaterials with complex geometry via XFEM. Comput Mech 55:659–672
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Mykhas’kiv, V., Zhbadynskyi, I. (2023). Boundary Integral Equation Method for 3D Elastodynamic Problems with Chain-Arranged Rigid Disk-Shaped Inclusions. In: Guz, A.N., Altenbach, H., Bogdanov, V., Nazarenko, V.M. (eds) Advances in Mechanics. Advanced Structured Materials, vol 191. Springer, Cham. https://doi.org/10.1007/978-3-031-37313-8_22
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