Anti-plane interfacial waves in a square lattice

Victor A. Eremeyev; Victor A. Eremeyev

doi:10.3934/nhm.2025004

Networks and Heterogeneous Media

2025, Volume 20, Issue 1: 52-64. doi: 10.3934/nhm.2025004

Previous Article Next Article

Research article Special Issues

Anti-plane interfacial waves in a square lattice

Victor A. Eremeyev ^,

Department of Civil and Environmental Engineering and Architecture (DICAAR), University of Cagliari, Via Marengo 2, Cagliari 09-123, CA, Italy

Received: 31 October 2024 Revised: 26 December 2024 Accepted: 06 January 2025 Published: 10 January 2025

Using the lattice dynamics approach, we discussed the propagation of interfacial waves localized near the interface in an infinite square lattice. The interface has been modeled as a single-particle layer of material particles with masses and elastic bonds different from those in the bulk. In this lattice structure there were anti-plane interface waves, i.e., waves that decayed exponentially with distance from the interface. Such waves could be useful for determining material properties in the vicinity of the interface. We obtained equations of motion and analyzed the corresponding dispersion relations for steady-state solutions. Here, the dispersion equation related the circular frequency to the wave number. In addition, we provided a comparison of the dispersion relations with those derived within the Gurtin-Murdoch surface elasticity. To do this, we have used the scaling law that links the continuum and discrete models. Unlike the continuum model, in the discrete model the wave number was limited by the first Brillouin zone, whereas in the continuum model it took a range from zero to infinity. The detailed parametric analysis was given for the discrete model. Finally, other models of interfaces in the case of a square lattice were discussed.
- surface elasticity,
- lattice dynamics,
- interfacial waves,
- square lattice,
- anti-plane motions
Citation: Victor A. Eremeyev. Anti-plane interfacial waves in a square lattice[J]. Networks and Heterogeneous Media, 2025, 20(1): 52-64. doi: 10.3934/nhm.2025004

Related Papers:

Abstract

Using the lattice dynamics approach, we discussed the propagation of interfacial waves localized near the interface in an infinite square lattice. The interface has been modeled as a single-particle layer of material particles with masses and elastic bonds different from those in the bulk. In this lattice structure there were anti-plane interface waves, i.e., waves that decayed exponentially with distance from the interface. Such waves could be useful for determining material properties in the vicinity of the interface. We obtained equations of motion and analyzed the corresponding dispersion relations for steady-state solutions. Here, the dispersion equation related the circular frequency to the wave number. In addition, we provided a comparison of the dispersion relations with those derived within the Gurtin-Murdoch surface elasticity. To do this, we have used the scaling law that links the continuum and discrete models. Unlike the continuum model, in the discrete model the wave number was limited by the first Brillouin zone, whereas in the continuum model it took a range from zero to infinity. The detailed parametric analysis was given for the discrete model. Finally, other models of interfaces in the case of a square lattice were discussed.

References

[1]	P. S. Laplace, Sur l'action capillaire, Supplément au livre X du Traité de mécanique céleste, Gauthier–Villars et fils, Paris, 4 (1805), 771–777.
[2]	P. S. Laplace, Supplément à la théorie de l'action capillaire, Traité de mécanique céleste, Gauthier–Villars et fils, Paris, 4 (1806), 909–945.
[3]	T. Young, An essay on the cohesion of fluids, Philos. Trans. R. Soc. London, 95 (1805), 65–87. https://doi.org/10.1098/rstl.1805.0005 doi: 10.1098/rstl.1805.0005
[4]	J. Willard Gibbs, The Collected Works, in Thermodynamics, Longmans, Green, New York, 1 (1928), 320.
[5]	H. L. Duan, J. Wang, B. L. Karihaloo, Theory of elasticity at the nanoscale, Adv. Appl. Mech., 42 (2008), 1–68. https://doi.org/10.1016/S0065-2156(08)00001-X doi: 10.1016/S0065-2156(08)00001-X
[6]	J. Wang, Z. Huang, H. Duan, S. Yu, X. Feng, G. Wang, et al., Surface stress effect in mechanics of nanostructured materials, Acta Mech. Solida Sin., 24 (2011), 52–82. https://doi.org/10.1016/S0894-9166(11)60009-8 doi: 10.1016/S0894-9166(11)60009-8
[7]	V. A. Eremeyev, On effective properties of materials at the nano- and microscales considering surface effects, Acta Mech., 227 (2016), 29–42. https://doi.org/10.1007/s00707-015-1427-y doi: 10.1007/s00707-015-1427-y
[8]	S. G. Mogilevskaya, A. Y. Zemlyanova, V. I. Kushch, Fiber-and particle-reinforced composite materials with the Gurtin–Murdoch and Steigmann–Ogden surface energy endowed interfaces, Appl. Mech. Rev., 73 (2021), 1–18. https://doi.org/10.1115/1.4051880 doi: 10.1115/1.4051880
[9]	M. E. Gurtin, A. I. Murdoch, A continuum theory of elastic material surfaces, Arch. Ration. Mech. Anal., 57 (1975), 291–323. https://doi.org/10.1007/BF00261375 doi: 10.1007/BF00261375
[10]	M. E. Gurtin, A. I. Murdoch, Surface stress in solids, Int. J. Solids Struct., 14 (1978), 431–440. https://doi.org/10.1016/0020-7683(78)90008-2 doi: 10.1016/0020-7683(78)90008-2
[11]	A. I. Murdoch, The propagation of surface waves in bodies with material boundaries, J. Mech. Phys. Solids, 24 (1976), 137–146. https://doi.org/10.1016/0022-5096(76)90023-5 doi: 10.1016/0022-5096(76)90023-5
[12]	A. I. Murdoch, The effect of interfacial stress on the propagation of stoneley waves, J. Sound Vib., 50 (1977), 1–11. https://doi.org/10.1016/0022-460X(77)90547-8 doi: 10.1016/0022-460X(77)90547-8
[13]	D. J. Steigmann, R. W. Ogden, Surface waves supported by thin-film/substrate interactions, IMA J. Appl. Math., 72 (2007), 730–747. https://doi.org/10.1093/imamat/hxm018 doi: 10.1093/imamat/hxm018
[14]	V. A. Eremeyev, G. Rosi, S. Naili, Surface/interfacial anti-plane waves in solids with surface energy, Mech. Res. Commun., 74 (2016), 8–13. https://doi.org/10.1016/j.mechrescom.2016.02.018 doi: 10.1016/j.mechrescom.2016.02.018
[15]	G. Mikhasev, B. Erbaş, V. A. Eremeyev, Anti-plane shear waves in an elastic strip rigidly attached to an elastic half-space, Int. J. Eng. Sci., 184 (2023), 103809. https://doi.org/10.1016/j.ijengsci.2022.103809 doi: 10.1016/j.ijengsci.2022.103809
[16]	S. Dhua, A. Maji, A. Nath, The influence of surface elasticity on shear wave propagation in a cylindrical layer structure with an imperfect interface, Eur. J. Mech. A. Solids, 106 (2024), 105318. https://doi.org/10.1016/j.euromechsol.2024.105318 doi: 10.1016/j.euromechsol.2024.105318
[17]	F. Jia, Z. Zhang, H. Zhang, X. Q. Feng, B. Gu, Shear horizontal wave dispersion in nanolayers with surface effects and determination of surface elastic constants, Thin Solid Films, 645 (2018), 134–138. https://doi.org/10.1016/j.tsf.2017.10.025 doi: 10.1016/j.tsf.2017.10.025
[18]	W. Wu, H. Zhang, F. Jia, X. Yang, H. Liu, W. Yuan, et al., Surface effects on frequency dispersion characteristics of Lamb waves in a nanoplate, Thin Solid Films, 697 (2020), 137831. https://doi.org/10.1016/j.tsf.2020.137831 doi: 10.1016/j.tsf.2020.137831
[19]	D. J. Steigmann, R. W. Ogden, Plane deformations of elastic solids with intrinsic boundary elasticity, Proc. R. Soc. Ser. A, 453 (1997), 853–877. https://doi.org/10.1098/rspa.1997.0047 doi: 10.1098/rspa.1997.0047
[20]	D. J. Steigmann, R. W. Ogden, Elastic surface–substrate interactions, Proc. R. Soc. Ser. A, 455 (1999), 437–474. https://doi.org/10.1098/rspa.1999.0320 doi: 10.1098/rspa.1999.0320
[21]	V. A. Eremeyev, Strongly anisotropic surface elasticity and antiplane surface waves, Phil. Trans. R. Soc. A, 378 (2020), 20190100. https://doi.org/10.1098/rsta.2019.0100 doi: 10.1098/rsta.2019.0100
[22]	C. Rodriguez, Elastic solids with strain-gradient elastic boundary surfaces, J. Elast., 156 (2024), 769–797. https://doi.org/10.1007/s10659-024-10073-w doi: 10.1007/s10659-024-10073-w
[23]	L. Brillouin, Wave Propagation in Periodic Structures: Electric Filters and Crystal Lattices, Mcgraw Hill Book Company, New York, 1953. https://doi.org/10.1038/158926a0
[24]	A. I. Murdoch, Some fundamental aspects of surface modelling, J. Elast., 80 (2005), 33–52. https://doi.org/10.1007/s10659-005-9024-2 doi: 10.1007/s10659-005-9024-2
[25]	V. A. Eremeyev, B. L. Sharma, Anti-plane surface waves in media with surface structure: Discrete vs. continuum model, Int. J. Eng. Sci., 143 (2019), 33–38. https://doi.org/10.1016/j.ijengsci.2019.06.007 doi: 10.1016/j.ijengsci.2019.06.007
[26]	B. L. Sharma, V. A. Eremeyev, Wave transmission across surface interfaces in lattice structures, Int. J. Eng. Sci., 145 (2019), 103173. https://doi.org/10.1016/j.ijengsci.2019.103173 doi: 10.1016/j.ijengsci.2019.103173
[27]	J. Achenbach, Wave Propagation in Elastic Solids, North Holland, Amsterdam, 1973.
[28]	G. B. Whitham, Linear and Nonlinear Waves, John Wiley & Sons, New York, 1999.
[29]	E. Barchiesi, M. Laudato, F. Di Cosmo, Wave dispersion in non-linear pantographic beams, Mech. Res. Commun., 94 (2018), 128–132. https://doi.org/10.1016/j.mechrescom.2018.11.002 doi: 10.1016/j.mechrescom.2018.11.002
[30]	N. Shekarchizadeh, M. Laudato, L. Manzari, B. E. Abali, I. Giorgio, A. M. Bersani, Parameter identification of a second-gradient model for the description of pantographic structures in dynamic regime, Z. Angew. Math. Phys., 72 (2021), 190. https://doi.org/10.1007/s00033-021-01620-9 doi: 10.1007/s00033-021-01620-9
[31]	A. Ciallella, I. Giorgio, S. R. Eugster, N. L. Rizzi, F. dell'Isola, Generalized beam model for the analysis of wave propagation with a symmetric pattern of deformation in planar pantographic sheets, Wave Motion, 113 (2022), 102986. https://doi.org/10.1016/j.wavemoti.2022.102986 doi: 10.1016/j.wavemoti.2022.102986
[32]	R. Fedele, L. Placidi, F. Fabbrocino, A review of inverse problems for generalized elastic media: Formulations, experiments, synthesis, Continuum Mech. Thermodyn., 36 (2024), 1413–1453. https://doi.org/10.1007/s00161-024-01314-3 doi: 10.1007/s00161-024-01314-3
[33]	L. Placidi, F. Di Girolamo, R. Fedele, Variational study of a Maxwell-Rayleigh-type finite length model for the preliminary design of a tensegrity chain with a tunable band gap, Mech. Res. Commun., 136 (2024), 104255. https://doi.org/10.1016/j.mechrescom.2024.104255 doi: 10.1016/j.mechrescom.2024.104255
[34]	B. L. Sharma, Diffraction of waves on square lattice by semi-infinite rigid constraint, Wave Motion, 59 (2015), 52–68. https://doi.org/10.1016/j.wavemoti.2015.07.008 doi: 10.1016/j.wavemoti.2015.07.008
[35]	B. L. Sharma, On linear waveguides of square and triangular lattice strips: An application of Chebyshev polynomials, Sādhanā, 42 (2017), 901–927. https://doi.org/10.1007/s12046-017-0646-4 doi: 10.1007/s12046-017-0646-4
[36]	G. I. Mikhasev, V. A. Eremeyev, Effects of interfacial sliding on anti-plane waves in an elastic plate imperfectly attached to an elastic half-space, Int. J. Eng. Sci., 205 (2024), 104158. https://doi.org/10.1016/j.ijengsci.2024.104158 doi: 10.1016/j.ijengsci.2024.104158
[37]	G. S. Mishuris, A. B. Movchan, D. Bigoni, Dynamics of a fault steadily propagating within a structural interface, Multiscale Model. Simul., 10 (2012), 936–953. https://doi.org/10.1137/110845732 doi: 10.1137/110845732
[38]	B. Lal Sharma, G. Mishuris, Scattering on a square lattice from a crack with a damage zone, Proc. R. Soc. A, 476 (2020), 20190686. https://doi.org/10.1098/rspa.2019.0686 doi: 10.1098/rspa.2019.0686
[39]	B. L. Sharma, Scattering of surface waves by inhomogeneities in crystalline structures, Proc. R. Soc. A, 480 (2024), 20230683. https://doi.org/10.1098/rspa.2023.0683 doi: 10.1098/rspa.2023.0683
[40]	M. Brun, A. B. Movchan, N. V. Movchan, Shear polarisation of elastic waves by a structured interface, Continuum Mech. Thermodyn., 22 (2010), 663–677. https://doi.org/10.1007/s00161-010-0143-z doi: 10.1007/s00161-010-0143-z
[41]	G. Rosi, L. Placidi, V. H. Nguyen, S. Naili, Wave propagation across a finite heterogeneous interphase modeled as an interface with material properties, Mech. Res. Commun., 84 (2017), 43–48. https://doi.org/10.1016/j.mechrescom.2017.06.004 doi: 10.1016/j.mechrescom.2017.06.004
[42]	A. Casalotti, F. D'Annibale, G. Rosi, Optimization of an architected composite with tailored graded properties, Z. Angew. Math. Phys., 75 (2024), 126. https://doi.org/10.1007/s00033-024-02255-2 doi: 10.1007/s00033-024-02255-2
[43]	F. dell'Isola, A. Madeo, L. Placidi, Linear plane wave propagation and normal transmission and reflection at discontinuity surfaces in second gradient 3D continua, ZAMM J. Appl. Math. Mech., 92 (2012), 52–71. https://doi.org/10.1002/zamm.201100022 doi: 10.1002/zamm.201100022
[44]	L. Placidi, G. Rosi, I. Giorgio, A. Madeo, Reflection and transmission of plane waves at surfaces carrying material properties and embedded in second-gradient materials, Math. Mech. Solids, 19 (2014), 555–578. https://doi.org/10.1177/1081286512474016 doi: 10.1177/1081286512474016
[45]	A. Berezovski, I. Giorgio, A. D. Corte, Interfaces in micromorphic materials: Wave transmission and reflection with numerical simulations, Math. Mech. Solids, 21 (2016), 37–51. https://doi.org/10.1177/1081286515572244 doi: 10.1177/1081286515572244
[46]	E. Barchiesi, L. Placidi, A review on models for the 3D statics and 2D dynamics of pantographic fabrics, Adv. Struct. Mater., (2017), 239–258. https://doi.org/10.1007/978-981-10-3797-9_14 doi: 10.1007/978-981-10-3797-9_14
[47]	E. Turco, E. Barchiesi, A. Ciallella, F. dell'isola, Nonlinear waves in pantographic beams induced by transverse impulses, Wave Motion, 115 (2022), 103064. https://doi.org/10.1016/j.wavemoti.2022.103064 doi: 10.1016/j.wavemoti.2022.103064
[48]	S. R. Eugster, Numerical analysis of nonlinear wave propagation in a pantographic sheet, Math. Mech. Complex Syst., 9 (2022), 293–310. https://doi.org/10.2140/memocs.2021.9.293 doi: 10.2140/memocs.2021.9.293
[49]	E. Turco, E. Barchiesi, F. dell'Isola, A numerical investigation on impulse-induced nonlinear longitudinal waves in pantographic beams, Math. Mech. Solids, 27 (2022), 22–48. https://doi.org/10.1177/10812865211010877 doi: 10.1177/10812865211010877
[50]	V. A. Eremeyev, G. Rosi, S. Naili, On effective surface elastic moduli for microstructured strongly anisotropic coatings, Int. J. Eng. Sci., 204 (2024), 104135. https://doi.org/10.1016/j.ijengsci.2024.104135 doi: 10.1016/j.ijengsci.2024.104135

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)