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 $\boldsymbol{\sigma}$ to the strain tensor $\boldsymbol\varepsilon(\bf u)$, with $\bf u$ being the displacement vector, defined by: $(1+\tau D_t^\alpha) {\boldsymbol{\sigma}}=(1+\rho D_t^\alpha)[2\mu {\boldsymbol\varepsilon}({\bf u})+\lambda\text{tr}(\boldsymbol\varepsilon(\bf u)) \bf ]$. Here $\mu,\lambda\in\mathrm{L}^\infty(\Omega)$, $\mu$ is the shear modulus bounded below by a positive constant, and $\lambda\geq 0$ is first Lam\'e coefficient, $D_t^\alpha$, with $\alpha \in (0,1)$, is the Caputo time-derivative, $\tau>0$ is the characteristic relaxation time and $\rho\geq\tau$ is the characteristic retardation time. We show that, when coupled with the equation of motion $\varrho \ddot{\bf u} = \text{Div}{\boldsymbol\sigma} + \bf F$, 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 $\bf F$ is a specified load vector, the resulting model is well-posed in the sense that the associated initial-boundary-value problem, with initial conditions ${\bf u}(0,\mathbf{x}) = {\bf g}(\mathbf{x})$, $\dot{\bf u}(0,\mathbf{x}) = \bf h(\mathbf{x})$, ${\boldsymbol\sigma}(0,\mathbf{x}) = {\bf s}(\mathbf{x})$, for $\mathbf{x} \in \Omega$, and a homogeneous Dirichlet boundary condition, possesses a unique weak solution for any choice of ${\bf g }\in [\mathrm{H}^1_0(\Omega)]^3$, ${\bf h}\in [\mathrm{L}^2(\Omega)]^3$, and ${\bf S} = {\bf S}^{\rm T} \in [\mathrm{L}^2(\Omega)]^{3 \times 3}$, and any load vector ${\bf F} \in\mathrm{L}^2(0,T;[\mathrm{L}^2(\Omega)]^3)$, and that this unique weak solution depends continuously on the initial data and the load vector.