General problems of elastic wave propagation in structured anisotropic materials

Anisotropy of crystals, which implies large number of independent material parameters, significantly diversifies acoustic phenomena but also complicates direct derivation of their characteristics and actually precludes finding exact closed-form solutions.  At the same time, anisotropy is also prone to impede numerical calculations. This is all the more so in the case of materials with discretely or continuously varying properties. That is why any kind of analytical insight into the general properties of elastic waves in anisotropic and/or structured materials remains indispensable even though it is usually confined to approximate or averaged evaluations, asymptotical trends, upper and lower bounds, analysis of existence of wave solutions, etc. It is also a means for developing tractable models for evaluating wave propagation in the new generation of composite materials.

Our research record,  which has been and is being conducted in the above context, includes the following topics and milestones:

• asymptotics of the dispersion spectra of guided waves travelling in anisotropic plates made of elastic or piezoelectric, uniform or functionally graded materials under free or loaded  boundary conditions;
• general formalism for wave propagation in cylindrically and spherically anisotropic radially inhomogeneous elastic media;
• homogenization into the Willis-type model and the effective speed of sound in 1D-, 2D-  and 3D-periodic elastic and piezoelectric continua (phononic crystals);
• methods of monodromy-matrix and resolvent for analysing and calculating dispersion spectra of surface waves in 1D- and 2D-periodic substrates and waveguides;
• tunable wave properties in piezoelectric crystals with periodically embedded electrodes controlled via external network of capacitors;
• existence and possible number of surface and interfacial wave solutions in elastic and piezoelectric superlattices of arbitrary anisotropy.

Related publications:

• A.L. Shuvalov, O. Poncelet, “On the backward Lamb waves near thickness resonances in anisotropic plates”, Int. J. Solids Struct. 45 (11-12), 3430-3448 (2008)
• C. Aristégui, A.L. Shuvalov, O. Poncelet, M. Caleap, “Trapping of shear acoustic waves by a near-surface distribution of cavities”, J. Acoust. Soc. Am. 125(2), 628-631 (2009)
• A.N. Norris, A.L. Shuvalov, “Wave impedance matrices for cylindrically anisotropic radially inhomogeneous elastic solids”, Q. J. Mech. Appl. Math. 63(4), 401-435 (2010)
• A.L. Shuvalov, A.A. Kutsenko, A.N. Norris, O. Poncelet, “Effective Willis constitutive equations for periodically stratified anisotropic elastic media”, Proc. R. Soc. A 467(2130), 1749-1769 (2011)
• A.N. Norris, A.L. Shuvalov, A.A. Kutsenko, “Analytical formulation of three-dimensional dynamic homogenization for periodic elastic systems”, Proc. R. Soc. A  468(2142), 1629-1651 (2012)
• M.E. Korotyaeva, A.A. Kutsenko, A.L. Shuvalov, O.Poncelet. “Resolvent method for calculation dispersion spectra of the shear waves in the phononic plates and waveguides”, J. Comput. Acoust. 22(3), 1450008 (2014)
• A.N. Darinskii, A.L. Shuvalov, O.Poncelet, A.A. Kutsenko. “Bulk longitudinal wave reflection/transmission in periodic piezoelectric structures with metallized interfaces”, Ultrasonics 63, 118-125 (2015)
• A.A. Kutsenko, A.J. Nagy, X. Su, A.L. Shuvalov, A.N. Norris. “Wave propagation and homogenization in 2d and 3d lattices: A semi-analytical approach”, Q. J. Mech. Appl. Math. 70(2), 131-151 (2017)
• A.A. Kutsenko, A.L. Shuvalov, O.Poncelet. “Dispersion spectrum of acoustoelectric waves in 1D piezoelectric crystal coupled with 2D infinite network of capacitors”, J. Appl. Phys. 123, 044902 (2018)
• A. N. Darinskii, A. L. Shuvalov. “Existence of surface acoustic waves in one-dimensional piezoelectric phononic crystals of general anisotropy”, Phys. Rev. B 99, 174305 (2019).