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).