Remarks on isotropic extension of anisotropic constitutive functions via structural tensors

Abstract

The following is an elaboration on the linear non-local model of viscoelastic fluids proposed in a previous work (Int. J. Eng. Sci. 48 (2010), 1279-1288). As a recapitulation of that work, the basic theory is presented in terms of the temporal frequency and spatial wave number in the Laplace-Fourier domain. Taylor-series expansions in these variables provides a weakly non-local theory in spatio-temporal gradients that is more comprehensive than the "bi-velocity" model of Brenner. The linearized Chapman-Enskog kinetic theory is shown to provide a confirmation of the more general theory, from which one can reconstruct a fully non-local integral model. Following the work of Davis and Brenner (J. Acoust. Soc. Am. 132 (2012), 2963-2969), the general theory is employed to derive dispersion relations for acoustic, thermal and shear-wave propagation in compressible viscoelastic fluids. At Burnett order the Chapman-Enskog theory gives a cubic polynomial in wave number squared which reduces in the dissipative quasi-static limit to a quadratic like that given by the classical Navier-Stokes-Fourier model and the bi-velocity modification of that model. With minor modification, the present analysis applies to viscoelastic shear and dilatational wave propagation in solids with higher-gradient and Cosserat effects, where it may, for example, find application to the field of rotational seismology.

A structurally frame-indifferent model for anisotropic visco-hyperelastic materials

“…With the invariance requirements expressed by (4.17) and (4. 19), we are in the position of applying the representation theorem of isotropic functions to express δ and φ as polynomial functions of the scalar invariants of their arguments [35,36]. In particular, we consider the following integrity bases for φ…”

Untitled

“…As such, they describe Maxwell's viscoelasticity and Cataneo's heat conduction, which admit both mechanical shear waves and heat waves, reflecting a breakdown of purely diffusive, dissipative response on time scales τ . We recall that Ignaczak and Ostoja-Starzewski [2009] give a comprehensive treatment of the local theory of finite thermoelastic wave speeds, represented by terms O(k) in (19). By contrast, and as anticipated above, we expect dissipative response to arise in the small Deborah number limit De = τ 0 s 1.…”

“…Finally, we note that the present type of analysis can be extended to anisotropic media like those considered by [Suiker et al 2001] by appropriate symmetry restrictions and modification of the relations (6). One possibility is to employ the joint isotropic invariants of the wave vector k and a set of structure tensors to capture the anisotropy [Cowin 1985;Man and Goddard 2016].…”

Harmonic Factorization and Reconstruction of the Elasticity Tensor

“…The considered reconstruction problem is closely related to the so-called isotropic extension of anisotropic constitutive functions via structural tensors developed by Boehler [5] and Liu [34] independently (see also [6,36]). In the case of a linear constitutive law, an isotropic extension is just an equivariant reconstruction limited to a given symmetry class of the constitutive tensor.…”