Geometric Optics for Surface Waves in Nonlinear Elasticity

Abstract

This work is devoted to the analysis of high frequency solutions to the equations of nonlinear elasticity in a half-space. We consider surface waves (or more precisely, Rayleigh waves) arising in the general class of isotropic hyperelastic models, which includes in particular the Saint Venant-Kirchhoff system. Work has been done by a number of authors since the 1980s on the formulation and well-posedness of a nonlinear evolution equation whose (exact) solution gives the leading term of an approximate Rayleigh wave solution to the underlying elasticity equations. This evolution equation, which we refer to as "the amplitude equation", is an integrodifferential equation of nonlocal Burgers type. We begin by reviewing and providing some extensions of the theory of the amplitude equation. The remainder of the paper is devoted to a rigorous proof in 2D that exact, highly oscillatory, Rayleigh wave solutions u ε to the nonlinear elasticity equations exist on a fixed time interval independent of the wavelength ε, and that the approximate Rayleigh wave solution provided by the analysis of the amplitude equation is indeed close in a precise sense to u ε on a time interval independent of ε. The paper focuses mainly on the case of Rayleigh waves that are pulses, which have profiles with continuous Fourier spectrum, but our method applies equally well to the case of wavetrains, whose Fourier spectrum is discrete.

Geometric Optics for Rayleigh Wavetrains in d-Dimensional

“…The task of rigorously justifying these approximate solutions, that is, showing that they are close in a precise sense to true exact solutions of the traction problem was begun in [CW16], which provided a justification in the case of pulses in two space dimensions; see Remark 0.2. In this paper we treat the case of wavetrains in any space dimension d ≥ 2 by entirely different methods.…”

“…Although Kreiss symmetrizers are now available in a number of situations where strict hyperbolicity fails, including some cases of variable multiplicity [Mét00,MZ05], the linearized SVK problem in higher dimensions lies beyond the reach of current Kreiss-symmetrizer technology. The method of [CW16] can also be used to justify approximate wavetrain solutions ε 2 U 2 in 2D, but the lack of Kreiss symmetrizers prevents us from using that method for either pulses or wavetrains in space dimensions ≥ 3. 0.1.…”

“…The problem of constructing oscillatory, approximate solutions to the traction problem in nonlinear elasticity (0.1) has been considered by a number of authors including [Lar83,Hun06,BGC12,BGC16,CW16]. These papers construct 2-scale, WKB-type, approximate solutions consisting of a leading term and, in the case of [CW16], a first corrector as well. This paper provides a "rigorous justification" (explained below) of such approximate solutions for wavetrains in any number of space dimensions d ≥ 2.…”

“…In the paper [CW16] approximate, leading-term pulse solutions ε 2 U 2 were justified in the case of two space dimensions by a different method that relied on precise microlocal energy estimates for certain "singular systems" associated to (0.4). These estimates were proved using Kreiss symmetrizers [Kre70] for the linearized SVK problem whose construction relied on the fact that in 2D, the linearized problem is strictly hyperbolic.…”

“…Let m ≥ m 1 > d 2 + 2. Then sufficiently smooth solutions of the Cauchy problem In isotropic elastodynamics, the kernel b that appears in (11.6) satisfies the bound (12.30) as can be seen immediately by inspection of the basic kernels written down in formulas (2.56)-(2.58) of [CW16] (or formula (3.20) of [Hun06]).…”