Paper detail

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

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 $\boldsymbolσ$ to the strain tensor $\boldsymbol\varepsilon(\bf u)$, with $\bf u$ being the displacement vector, defined by: $(1+τD_t^α) {\boldsymbolσ}=(1+ρD_t^α)[2μ{\boldsymbol\varepsilon}({\bf u})+λ\text{tr}(\boldsymbol\varepsilon(\bf u)) \bf ]$. Here $μ,λ\in\mathrm{L}^\infty(Ω)$, $μ$ is the shear modulus bounded below by a positive constant, and $λ\geq 0$ is first Lamé coefficient, $D_t^α$, with $α\in (0,1)$, is the Caputo time-derivative, $τ>0$ is the characteristic relaxation time and $ρ\geqτ$ is the characteristic retardation time. We show that, when coupled with the equation of motion $\varrho \ddot{\bf u} = \text{Div}{\boldsymbolσ} + \bf F$, considered in a bounded open Lipschitz domain $Ω$ in $\mathbb{R}^3$ and over a time interval $(0,T]$, where $\varrho\in \mathrm{L}^\infty(Ω)$ 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 ${\bf u}(0,\mathbf{x}) = {\bf g}(\mathbf{x})$, $\dot{\bf u}(0,\mathbf{x}) = \bf h(\mathbf{x})$, ${\boldsymbolσ}(0,\mathbf{x}) = {\bf s}(\mathbf{x})$, for $\mathbf{x} \in Ω$, and a homogeneous Dirichlet boundary condition, possesses a unique weak solution for any choice of ${\bf g }\in [\mathrm{H}^1_0(Ω)]^3$, ${\bf h}\in [\mathrm{L}^2(Ω)]^3$, and ${\bf S} = {\bf S}^{\rm T} \in [\mathrm{L}^2(Ω)]^{3 \times 3}$, and any load vector ${\bf F} \in\mathrm{L}^2(0,T;[\mathrm{L}^2(Ω)]^3)$, and that this unique weak solution depends continuously on the initial data and the load vector.