Viscoelastic Wave (Under Development)¶

[Yang13]

Mathematical Model¶

For isothermal viscoelastic material, the model equations consist conservation of mass and momentum as follows,

(1)\begin{align}\begin{aligned}\newcommand{\dpd}[3][]{\mathinner{ \dfrac{\partial{^{#1}}#2}{\partial{#3^{#1}}} }}\\\begin{split}& \dpd{v_i}{t} - \frac{1}{\rho} \sum_{j=1}^3\dpd{\sigma_{ji}}{x_j} = 0 \\ & \dpd{\sigma_{ij}}{t} - \delta_{ij} \left( G^{\psi}_e + \sum^L_{l=1} G^{\psi}_l \right) \sum_{k=1}^3 \dpd{v_k}{x_k} + \left( G^{\mu}_e + \sum^L_{l=1}G^{\mu}_l \right) \left( 2 \delta_{ij} \sum_{k=1}^3 \dpd{v_k}{x_k} - \dpd{v_i}{x_j} - \dpd{v_j}{x_i} \right) = \sum^L_{l=1}\gamma^l_{ij} \\ & \dpd{\gamma^l_{ij}}{t} + \delta_{ij} \frac{G^{\psi}_l - G^{\mu}_l}{\tau_{\sigma l}} \sum_{k=1}^3 \dpd{v_k}{x_k} + \frac{G^{\mu}_l}{\tau_{\sigma l}} \left( \dpd{v_i}{x_j} + \dpd{v_j}{x_i} \right) = -\frac{1}{\tau_{\sigma l}}\gamma^l_{ij}\end{split}\end{aligned}\end{align}

where $$v_i$$ are the Cartesian component of the velocity, $$\rho$$ the density, $$\sigma_{ij}$$ the stress tensor, $$\gamma_{ij}$$ the internal variables, and $$\delta_{ij}$$ the Kronecker delta. Subscripts $$i, j, k = 1, 2, 3$$ are for the Cartesian tensors. $$G^{\psi}_l, G^{\mu}_e, G^{\mu}_l$$, and $$\tau_{\sigma l}$$ are the constants of the standard linear solid (SLS) model with $$l = 1, 2, \ldots, L$$. $$L$$ is the number of the employed SLS model components.

Equation (1) can be further organized to a vector form:

(2)\begin{align}\begin{aligned}\newcommand{\bvec}[1]{\mathbf{#1}}\\\dpd{\bvec{u}}{t} + \sum_{k=1}^3 \dpd{\bvec{f}^{(k)}}{x_k} = \bvec{s}\end{aligned}\end{align}

where $$\bvec{u}$$ is the solution variable, $$\bvec{f}^{(1)}$$, $$\bvec{f}^{(2)}$$, and $$\bvec{f}^{(3)}$$ flux functions, and $$\bvec{s}$$ the source term.

Jacobian Matrices¶

By applying the chain rule to Eq. (2), we can derive the Jacobian matrices:

(3)$\dpd{\bvec{u}}{t} + \sum_{k=1}^3 \mathrm{A}^{(k)} \dpd{\bvec{u}}{x_k} = \bvec{s}$

where $$\mathrm{A}^{(1)}$$, $$\mathrm{A}^{(2)}$$, and $$\mathrm{A}^{(3)}$$ are $$(9+6L)\times(9+6L)$$ are the Jacobian matrices:

(4)\begin{align}\begin{aligned}\newcommand{\defeq}{\buildrel{\text{def}}\over{=}}\\\begin{split}\mathrm{A}^{(i)} \defeq \dpd{\bvec{f}^{(i)}}{\bvec{u}} = \left( \begin{array}{c|c|c} \mathrm{0}_{3\times3} & \mathrm{C}^{(i)} & \mathrm{0}_{3\times(6L)} \\ \hline \mathrm{B}^{(i)} & \mathrm{0}_{(6+6L)\times6} & \mathrm{0}_{(6+6L)\times(6L)} \end{array} \right), \quad i = 1, 2, 3\end{split}\end{aligned}\end{align}

where

$\begin{split}\mathrm{B}^{(i)} \defeq \left( \begin{array}{ccc} \left[ 2(G^{\mu}_e + \sum^L_{l=1} G^{\mu}_l) - (G^{\psi}_e + \sum^L_{l=1} G^{\psi}_l) \right] \mathrm{M}^{(i)} - (G^{\psi}_e+\sum^L_{l=1}G^{\psi}_l) \mathrm{K}^{(i)} \\ \frac{G^{\phi}_1 - G^{\mu}_1}{\tau_{\sigma 1}} \mathrm{M}^{(i)} + \frac{G^{\phi}_1}{\tau_{\sigma 1}} \mathrm{N}^{(i)} + \frac{G^{\mu}_1}{\tau_{\sigma 1}} \mathrm{K}^{(i)} \\ \vdots \\ \frac{G^{\phi}_L - G^{\mu}_L}{\tau_{\sigma 1}} \mathrm{M}^{(i)} + \frac{G^{\phi}_L}{\tau_{\sigma 1}} \mathrm{N}^{(i)} + \frac{G^{\mu}_L}{\tau_{\sigma 1}} \mathrm{K}^{(i)} \end{array} \right), \, \mathrm{C}^{(i)} \defeq -\frac{1}{\rho} {\mathrm{K}^{(i)}}^t, \quad i = 1, 2, 3\end{split}$

and

(5)$\begin{split}\mathrm{M}^{(1)} \defeq \left( \begin{array}{ccc} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right), \, \mathrm{M}^{(2)} \defeq \left( \begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right), \, \mathrm{M}^{(3)} \defeq \left( \begin{array}{ccc} 0 & 0 & 1 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right)\end{split}$
(6)$\begin{split}\mathrm{N}^{(1)} \defeq \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right), \, \mathrm{N}^{(2)} \defeq \left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right), \, \mathrm{N}^{(3)} \defeq \left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right)\end{split}$
(7)$\begin{split}\mathrm{K}^{(1)} \defeq \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{array} \right), \, \mathrm{K}^{(2)} \defeq \left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{array} \right), \, \mathrm{K}^{(3)} \defeq \left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right)\end{split}$

$$\mathrm{B}^{(1)}$$, $$\mathrm{B}^{(2)}$$, and $$\mathrm{B}^{(3)}$$ are $$(6+6L)\times3$$ matrices. $$\mathrm{C}^{(1)}$$, $$\mathrm{C}^{(2)}$$, and $$\mathrm{C}^{(3)}$$ are $$3\times6$$ matrices.

Hyperbolicity¶

The left hand side of the model equation Eq. (3) can be proved as a hyperbolic system. The method of proof is similar to the Hydro-Acoustics (Under Development). The list of the eigenvalues is provided:

(8)$\lambda_{1,2,3,4,5,6\cdots} = \pm\sqrt{ar(k^2_1+k^2_2+k^2_3)}, \pm\sqrt{br(k^2_1+k^2_2+k^2_3)}, \pm\sqrt{br(k^2_1+k^2_2+k^2_3)}, 0,\cdots,$

where $$r = \frac{1}{\rho}, a = G^{\psi}_e+\sum^L_{l=1}G^{\psi}_l$$, and $$b = G^{\mu}_e+\sum^L_{l=1}G^{\mu}_l$$. The $$k_1, k_2$$, and $$k_3$$ are the components of a direction vector, as used in Hydro-Acoustics (Under Development).
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