Nonlinear Incompressible Shear Wave Models in Hyperelasticity and Viscoelasticity Frameworks, with Applications to Love Waves

Abstract

General equations describing shear displacements in incompressible hyperelastic materials, holding for an arbitrary form of strain energy density function, are presented and applied to the description of nonlinear Love-type waves propagating on an interface between materials with different mechanical properties. The model is valid for a broad class of hyper-viscoelastic materials. For a cubic Yeoh model, shear wave equations contain cubic and quintic differential polynomial terms, including viscoelasticity contributions in terms of dispersion terms that include mixed derivatives $u_{xxt}$ of the material displacement. Full (2+1)-dimensional numerical simulations of waves propagating in the bulk of a two-layered solid are undertaken and analyzed with respect to the source position and mechanical properties of the layers. Interfacial nonlinear Love waves and free upper surface shear waves are tracked; it is demonstrated that in the fully nonlinear case, the variable wave speed of interface and surface waves generally satisfies the linear Love wave existence condition $c_1 < \abs{v} < c_2$, while tending to the larger material wave speed $c_1$ or $c_2$ for large times.

Figures (28)

• Figure 2.1: An isotropic layer overlying an isotropic elastic half-space. The $y$ direction is perpendicular to this space spanned by $x,z$. The positive $z$ direction points downwards, $L$ is the depth of the isotropic layer, waves travelling through the isotropic layer (medium $1$) have speed $c_1^2$, and waves travelling through the isotropic elastic half-space (medium $2$) have speed $c_2^2$.
• Figure 2.2: Left: plots of $f_0(z; v)$, $f_1(z; v)$, $f_2(z; v)$, and $f_3(z; v)$ for $z\in[0,3L]$ and $v=(c_1+c_2)/2$. Right: plot of $k_n(v)$ over $v\in[c_1,c_2]$ for $n=0,1,2,3$. Recall that $k_n(v)$ strictly increases with $n$. Other parameters are $L=1$, $c_1 = 1$, $c_2 = 2$
• Figure 2.3: The real part of fundamental solutions $f_n(z;v)e^{ik_n(v)(x-vt)}$ to equation \ref{['eqn:linearlove']} for various $v$ and $n$, with $c_1=1$, $c_2 = 2$, and $L = 1$.
• Figure 3.1: A general illustration of the deformation $\mathbf{x}(\mathbf{X},t)$ acting on the reference configuration after some time $t$.
• Figure 6.1: Plots of a numerical solution $u(x,z,t)$ of equation \ref{['eqn:linearlove']} with $c_1<c_2$ after a Gaussian explosion centered in the lower half-space (i.e. $z_0<0$ in equation \ref{['eqn:gaussian_IC']}). In this and any similar surface plots in this section, $x$ changes over the horizontal axis and $z$ changes over the vertical axis. The parameters are $c_1 = 0.1$, $c_2 = 0.2$, $z_0=-0.1$, $r=0.05$, $A=1$, $L=0.15$, $W=3.6$, $H=1.8$, and $h = 0.005$.