Particle relabelling transformations in elastodynamics

Abstract

S U M M A R YThe motion of a self-gravitating hyperelastic body is described through a time-dependent mapping from a reference body into physical space, and its material properties are determined by a referential density and strain-energy function defined relative to the reference body. Points within the reference body do not have a direct physical meaning, but instead act as particle labels that could be assigned in different ways. We use Hamilton's principle to determine how the referential density and strain-energy functions transform when the particle labels are changed, and describe an associated 'particle relabelling symmetry'. We apply these results to linearized elastic wave propagation and discuss their implications for seismological inverse problems. In particular, we show that the effects of boundary topography on elastic wave propagation can be mapped exactly into volumetric heterogeneity while preserving the form of the equations of motion. Several numerical calculations are presented to illustrate our results.

Thermoelastic cloaking and the breaking of invisibility

“…The form invariance of finite thermoelasticity is demonstrated in Sect. 3, with the results generalising those obtained in [11] in the context of finite elasticity. In Sect.…”

“…A point x ∈ M acts as a label for a unique particle in the body, with this particle being also labelled by some y ∈ M. This one-to-one correspondence between points in the two reference bodies defines a mapping ξ : M → M which we assume to be an orientation-preserving diffeomorphism. In the terminology of Al-Attar & Crawford [11], the diffeomorphism ξ is a particle relabelling transformation, and by construction the two motions are related through…”

“…One approach to cloaking for linear elasticity was presented in [7] and developed in [8], though this required the introduction of non-standard constitutive relations that violate the principle of material frame indifference. Within the context of finite elasticity, form invariance of the governing equations has been discussed in [9], [10], [11] and [12], with the results of the latter two papers making clear that the form invariance does not, in this case, allow for invisibility cloaking. In brief, this is because the transformations under which the equations are form invariant simply relate two different referential descriptions of the same physical body.…”

Oceanic high-frequency global seismic wave propagation with realistic bathymetry

“…As AxiSEM3D requires an axisymmetric computational domain which is spherical at a global scale, these equations cannot then be solved directly. Leng et al (2019) describes the implementation of the particle relabelling transformation of Al-Attar & Crawford (2016) to handle such interface undulations.…”

Intrinsic non-uniqueness of the acoustic full waveform inverse problem

“…The particle relabeling method (Al-Attar & Crawford 2016) shows that the interior properties of an elastic or acoustic medium can be transformed in such a way that the surface waveforms are not changed. This is accomplished by the introduction of a smooth and invertible mapping of the body into itself that leaves the surface fixed.…”