Shear waves in a nonlinear relaxing media: A three-dimensional perspective

Nonlinear propagation of quasiplanar shear wave beams in soft elastic media with transverse isotropy

Model equations are developed for shear wave propagation in a soft elastic material that include effects of nonlinearity, diffraction, and transverse isotropy. A theory for plane wave propagation by Cormack [J. Acoust. Soc. Am. 150, 2566 (2021)] is extended to include leading order effects of wavefront curvature by assuming that the motion is quasiplanar, which is consistent with other paraxial model equations in nonlinear acoustics. The material is modeled using a general expansion of the strain energy density to fourth order in strain that comprises thirteen terms defining the elastic moduli. Equations of motion for the transverse displacement components are obtained using Hamilton's principle. The coupled equations of motion describe diffraction, anisotropy of the wave speeds, quadratic and cubic plane wave nonlinearity, and quadratic nonlinearity associated with wavefront curvature. Two illustrative special cases are investigated. Spatially varying shear vertical wave motion in the fiber direction excites a quadratic nonlinear interaction unique to transversely isotropic soft solids that results in axial second harmonic motion with longitudinal polarization. Shear horizontal wave motion in the fiber plane reveals effects of anisotropy on third harmonic generation, such as beam steering and dependence of harmonic generation efficiency on the propagation and fiber directions.

On the accuracy of one-way approximate models for nonlinear waves in soft solids

Simple strain-rate viscoelasticity models of isotropic soft solid are introduced. The constitutive equations account for finite strain, incompressibility, material frame-indifference, nonlinear elasticity, and viscous dissipation. A nonlinear viscous wave equation for the shear strain is obtained exactly and corresponding one-way Burgers-type equations are derived by making standard approximations. Analysis of the travelling wave solutions shows that these partial differential equations produce distinct solutions, and deviations are exacerbated when wave amplitudes are not arbitrarily small. In the elastic limit, the one-way approximate wave equation can be linked to simple wave theory and shock wave theory, thus, allowing direct error measurements.

Magneto-Viscoelastic Materials: Memory Functionals and Rate Equations

The properties of viscoelastic solids subject to a magnetic field are modelled within two thermodynamically consistent approaches that are typical of models with a non-instantaneous response. One is based on memory functionals: the reversible changes are described by the instantaneous response, while the dissipativity is expressed by the dependence on histories. The other approach involves objective rate equations. While memory functionals lead to the difficulty of determining thermodynamically consistent free energy functionals, rate equations result in a simpler scheme. The greater simplicity allows the discovery of, in particular, models of magneto-hyperelastic materials, magneto-hypoelastic materials, and various forms of magneto-viscoelastic behaviour. The novelty of the procedure is based on two features: a representation formula, originating from the entropy inequality, and the use of the entropy production as a constitutive function. Relations with other approaches in the literature are examined in detail.

Plane nonlinear shear wave propagation in transversely isotropic soft solids

Nonlinear wave equations are obtained for the two plane shear wave modes in a transversely isotropic soft solid. The material is modeled using a general expansion of the strain energy density up to fourth order in strain. Whereas, in an isotropic soft solid, leading order nonlinearity for plane wave propagation appears at cubic order in strain, elastic anisotropy in a transversely isotropic material introduces nonlinear effects at quadratic order, including interaction between the modes of a wave with two displacement components. Expressions for second harmonic generation in an elliptically polarized wave field illustrate the low efficiency of nonlinear interactions between the two displacement components, which results from the disparity between propagation speeds of the two shear wave modes. Coupled wave equations with up to cubic nonlinearity are presented and then approximated to describe linearly polarized waves by neglecting interaction between modes. Evolution equations are obtained for linearly polarized progressive waves, and explicit expressions are given in terms of elastic moduli and propagation direction for the coefficients of leading order nonlinearity. Expressions are presented for up to third harmonic generation from a time-harmonic source.