Viscoelastic Wave (Under Development)

Viscoelastic Model

[Yang13]

Mathematical Model

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

(1)& \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}

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

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)\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

where

\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

and

(5)\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)

(6)\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)

(7)\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)

\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).