Third- and fourth-order constants of incompressible soft solids and the acousto-elastic effect

Abstract

Acousto-elasticity is concerned with the propagation of small-amplitude waves in deformed solids. Results previously established for the incremental elastodynamics of exact non-linear elasticity are useful for the determination of third-and fourth-order elastic constants, especially in the case of incompressible isotropic soft solids, where the expressions are particularly simple. Specifically, it is simply a matter of expanding the expression for ρv 2 , where ρ is the mass density and v the wave speed, in terms of the elongation e of a block subject to a uniaxial tension. The analysis shows that in the resulting expression: ρv 2 = a + be + ce 2 , say, a depends linearly on µ; b on µ and A; and c on µ, A, and D, the respective second-, third, and fourth-order constants of incompressible elasticity, for bulk shear waves and for surface waves.

Characterizing the nonlinear interaction of S- and P-waves in a rock sample

“…The choice of a shear wave pump in this paper aims to consider a realistic pump strain field in a subsurface experiment, which includes shear components. As for experiments, we found no theoretical studies on the effect of shear strain on nonlinear elasticity; to rectify this a fourth order elasticity model is presented in the Appendix, inspired by a series of papers by Destrade et al [34][35][36] The nonlinear characterization technique is presented in Sec. II, by discussing the experimental set-up, signal acquisition procedure, and strain estimation method.…”

A finite element model to study the effect of tissue anisotropy onex vivoarterial shear wave elastography measurements

“…Analytically describing and solving the case of the orthotropic material subjected to stress is challenging due to the splitting of the effective elastic constants in the push-forward elasticity tensor. Another approach in the theory of acoustoelasticity could be using third-order elastic constants (Duquennoy et al, 2008;Destrade et al, 2010b), but such constants were not available for our tissue and are difficult to measure.…”

“…Analytical studies of SW propagation in stressed bulk linear elastic media (Gennisson et al, 2008;Destrade et al, 2010b;Abiza et al, 2012) or general anisotropic/transverse isotropic bulk media (Destrade et al, 2010a;Cheviakov & Ganghoffer, 2015;Shams et al, 2011) were previously performed. These studies did not focus on the SW propagation within restricting boundaries of the plate and considering the orthotropic material.…”

Untitled

“…In more recent years they have been recognized as highly anisotropic due to the presence of collagen fibers [Holzapfel et al 2000]. Determination of the acoustoelastic coefficients in incompressible solids and the limiting values of the coefficients of nonlinearity for elastic wave propagation, among other studies, has very recently attracted a lot of attention since these analyses give an opportunity to capture the mechanical properties of these materials (see, for instance, Destrade et al [Destrade et al 2010b] and references therein). For other applications dealing with linearized dynamics we refer to [Bigoni et al 2007;2008] and the references therein.…”

“…waves propagating in incompressible solids subject to homogeneous deformations [Destrade et al 2010b] (linearized waves). It should be noted that if the analysis of a material only includes the small deformation regime then it is enough to consider (1) up to fourth order in strains.…”