Well-posedness of the fractional Zener wave equation for heterogenous viscoelastic materials

Abstract

We explore the well-posedness of the fractional version of Zener's wave equation for viscoelastic solids, which is based on a constitutive law relating the stress tensor $\mathbf{\sigma }$ to the strain tensor $\mathbf{\epsilon }(\mathbf{u})$, with $\mathbf{u}$ being the displacement vector, defined by: $(1+\tau D_t^{\alpha })\mathbf{\sigma }=(1+\rho D_t^{\alpha })[2\mu \mathbf{\epsilon }(\mathbf{u})+\lambda \text{tr}(\mathbf{\epsilon }(\mathbf{u}))]$. Here $\mu ,\lambda \in {\mathrm{L}}^{\infty }(\Omega )$, $\mu$ is the shear modulus bounded below by a positive constant, and $\lambda \ge 0$ is first Lamé coefficient, $D_t^{\alpha }$, with $\alpha \in (0,1)$, is the Caputo time-derivative, $\tau >0$ is the characteristic relaxation time and $\rho \ge \tau$ is the characteristic retardation time. We show that, when coupled with the equation of motion $\varrho \ddot{\mathbf{u}}=\text{Div}\mathbf{\sigma }+\mathbf{F}$, considered in a bounded open Lipschitz domain $\Omega$ in ${\mathbb{R}}^3$ and over a time interval $(0,T]$, where $\varrho \in {\mathrm{L}}^{\infty }(\Omega )$ is the density of the material, bounded below by a positive constant, and $\mathbf{F}$ is a specified load vector, the resulting model is well-posed in the sense that the associated initial-boundary-value problem, with initial conditions $\mathbf{u}(0,\mathbf{x})=\mathbf{g}(\mathbf{x})$, $\dot{\mathbf{u}}(0,\mathbf{x})=\mathbf{h}(\mathbf{x})$, $\mathbf{\sigma }(0,\mathbf{x})=\mathbf{s}(\mathbf{x})$, for $\mathbf{x}\in \Omega$, and a homogeneous Dirichlet boundary condition, possesses a unique weak solution for any choice of $\mathbf{g}\in [{\mathrm{H}}_0^1(\Omega ){]}^3$, $\mathbf{h}\in [{\mathrm{L}}^2(\Omega ){]}^3$, and $\mathbf{S}={\mathbf{S}}^{\mathrm{T}}\in [{\mathrm{L}}^2(\Omega ){]}^{3\times 3}$, and any load vector $\mathbf{F}\in {\mathrm{L}}^2(0,T;[{\mathrm{L}}^2(\Omega ){]}^3)$, and that this unique weak solution depends continuously on the initial data and the load vector.