A thermomechanical anisotropic model for shock loading of elastic-plastic and elastic-viscoplastic materials with application to jointed rock

Abstract

A large deformation thermomechanical model is developed for shock loading of a material that can exhibit elastic and inelastic anisotropy. Use is made of evolution equations for a triad of microstructural vectors $\mathbf{m}_{\mathrm{i}}\,(\hbox {i}=1,2,3)$ which model elastic deformations and directions of anisotropy. Specific constitutive equations are presented for a material with orthotropic elastic response. The rate of inelasticity depends on an orthotropic yield function that can be used to model weak fault planes with failure in shear and which exhibits a smooth transition to isotropic response at high compression. Moreover, a robust, strongly objective numerical algorithm is proposed for both rate-independent and rate-dependent response. The predictions of the continuum model are examined by comparison with exact steady-state solutions. Also, the constitutive equations are used to obtain a simplified continuum model of jointed rock which is compared with high fidelity numerical solutions that model a persistent system of joints explicitly in the rock medium.

Appendices

Appendix 1: Constitutive coefficients for the linearized purely mechanical theory of an orthotropic material

1.1 An orthotropic material with seven material constants

For the purely mechanical theory, the Helmholtz free energy ${\uppsi }_{\mathrm{d}}$ is taken in the form (4.7) and the Helmholtz free energy ${\uppsi }_{\mathrm{e}}$ is taken in the form (4.18). Since $\hbox {K}_{\mathrm{ijkl}}$ satisfies the restrictions (4.8), the constants $\{\hbox {K}_{\mathrm{1122}}$, $\hbox {K}_{\mathrm{1133}}$, $\hbox {K}_{\mathrm{2233}}\}$ are given by (7.3) so this orthotropic material is characterized by the reference density ${\uprho }_{\mathrm{0}}$ and seven independent constants

$$\begin{aligned} \{\hbox {C}_{\mathrm{0}} , \hbox {K}_{\mathrm{1111}} , \hbox {K}_{\mathrm{2222}},^{\mathrm{}} \hbox {K}_{\mathrm{3333}} , \hbox {K}_{\mathrm{1212}} , \hbox {K}_{\mathrm{1313}}, \hbox {K}_{\mathrm{2323}}\} . \end{aligned}$$

(11.1)

Moreover, using (3.14), (4.10) and (4.18) yields an expression for the stress in the form

$$\begin{aligned} \mathbf{T} = -\, {\uprho }_{\mathrm{0}}\hbox {C}_{0}^{2} \left( \frac{1}{\hbox {J}} - 1\right) \mathbf{I} + \hbox {J}^{\mathrm{-1}} \hbox {K}_{\mathrm{ijkl}} \hbox {e}_{\mathrm{kl}}^{\prime } \left( \mathbf{m}_{\mathrm{i}}\otimes \mathbf{m}_{\mathrm{j}}\right) . \end{aligned}$$

(11.2)

For the purely mechanical linearized theory the strain energy function $\Sigma$ per unit mass and the stress components $\hbox {T}_{\mathrm{ij}}$ can be expressed in the forms

$$\begin{aligned} {\uprho }_{\mathrm{0}}\Sigma = \frac{1}{2} \tilde{\hbox {K}}_{\mathrm{ijkl}} \hbox {e}_{\mathrm{ij}} \hbox {e}_{\mathrm{kl}} , \hbox {T}_{\mathrm{ij}} = \tilde{\hbox {K}}_{\mathrm{ijkl}} \hbox {e}_{\mathrm{kl}} , \end{aligned}$$

(11.3)

where $\tilde{\hbox {K}}_{\mathrm{ijkl}}$ has the same symmetries as $\hbox {K}_{\mathrm{ijkl}}$ associated with the first set of Eq. in (4.8) but $\{\tilde{\hbox {K}}_{\mathrm{nnkl}}, \tilde{\hbox {K}}_{\mathrm{ijnn}}\}$ do not vanish. For small elastic strains $\hbox {e}_{\mathrm{ij}}$

$$\begin{aligned} \hbox {J} \approx 1 + \hbox {e}_{\mathrm{mm}} , \mathbf{m}_{\mathrm{i}} \approx \mathbf{e}_{\mathrm{i}} , \end{aligned}$$

(11.4)

so ${\uppsi }_{\mathrm{e}}$ in (4.18) can be approximated by

$$\begin{aligned} {\uprho }_{\mathrm{0}}{\uppsi }_{\mathrm{e}} = \frac{1}{2} {\uprho }_{\mathrm{0}}\hbox {C}_{0}^{2} \hbox {e}_{\mathrm{mm}} \hbox {e}_{\mathrm{nn}} , \end{aligned}$$

(11.5)

Then, with the help of (4.7) the linearized strain energy is given by

$$\begin{aligned}&\Sigma = {\uppsi }_{\mathrm{e}}+{\uppsi }_{\mathrm{d}} , {\uprho }_{\mathrm{0}}{\uppsi }_{\mathrm{e}} = \frac{1}{2} {\uprho }_{\mathrm{0}}\hbox {C}_{0}^{2} \hbox {e}_{\mathrm{mm}} \hbox {e}_{\mathrm{nn}} ,\nonumber \\&{\uprho }_{\mathrm{0}}{\uppsi }_{\mathrm{d}}= \frac{1}{2} \hbox {K}_{\mathrm{ijkl}}\hbox {e}_{\mathrm{ij}}^{\prime }\hbox {e}_{\mathrm{kl}}^{\prime }, \end{aligned}$$

(11.6)

where $\hbox {e}_{\mathrm{kl}}^{\prime }$ is the deviatoric part of the linearized strain $\hbox {e}_{\mathrm{ij}}$. Consequently, for the linearized version of the orthotropic model discussed in Sect. 7 it follows that

$$\begin{aligned} \tilde{\hbox {K}}_{\mathrm{1111}}= & {} {\uprho }_{\mathrm{0}}\hbox {C}_{0}^{2} + \hbox {K}_{\mathrm{1111}} , \tilde{\hbox {K}}_{\mathrm{2222}} = {\uprho }_{\mathrm{0}}\hbox {C}_{0}^{2} + \hbox {K}_{\mathrm{2222}} ,\nonumber \\ \tilde{\hbox {K}}_{\mathrm{3333}}= & {} {\uprho }_{\mathrm{0}}\hbox {C}_{0}^{2} + \hbox {K}_{\mathrm{3333}} ,\nonumber \\ \tilde{\hbox {K}}_{\mathrm{1122}}= & {} {\uprho }_{\mathrm{0}}\hbox {C}_{0}^{2} + \hbox {K}_{\mathrm{1122}} , \tilde{\hbox {K}}_{\mathrm{1133}} = {\uprho }_{\mathrm{0}}\hbox {C}_{0}^{2} + \hbox {K}_{\mathrm{1133}} ,\nonumber \\ \tilde{\hbox {K}}_{\mathrm{2233}}= & {} {\uprho }_{\mathrm{0}}\hbox {C}_{0}^{2} + \hbox {K}_{\mathrm{2233}} ,\nonumber \\ \tilde{\hbox {K}}_{\mathrm{1212}}= & {} \hbox {K}_{\mathrm{1212}} , \tilde{\hbox {K}}_{\mathrm{1313}} = \hbox {K}_{\mathrm{1313}} , \tilde{\hbox {K}}_{\mathrm{2323}} = \hbox {K}_{\mathrm{2323}} . \end{aligned}$$

(11.7)

Moreover, the standard expressions for strain as a function of stress for an orthotropic material can be written in the forms [12]

$$\begin{aligned} \hbox {e}_{\mathrm{11}}= & {} \frac{\hbox {T}_{11}}{\hbox {E}_{1}} - \frac{{\upnu }_{21}\hbox {T}_{22}}{\hbox {E}_{2}} - \frac{{\upnu }_{31}\hbox {T}_{33}}{\hbox {E}_{3}} ,\nonumber \\ \hbox {e}_{\mathrm{22}}= & {} - \frac{{\upnu }_{12}\hbox {T}_{11}}{\hbox {E}_{1}} + \frac{\hbox {T}_{22}}{\hbox {E}_{2}}- \frac{{\upnu }_{32}\hbox {T}_{33}}{\hbox {E}_{3}} ,\nonumber \\ \hbox {e}_{\mathrm{33}}= & {} - \frac{{\upnu }_{13}\hbox {T}_{11}}{\hbox {E}_{1}} - \frac{{\upnu }_{23}\hbox {T}_{22}}{\hbox {E}_{2}} + \frac{\hbox {T}}{\hbox {E}_{3}},\nonumber \\ \hbox {e}_{\mathrm{12}}= & {} \frac{\hbox {T}_{12}}{2\hbox {G}_{12}} , \hbox {e}_{\mathrm{13}} = \frac{\hbox {T}_{13}}{2\hbox {G}_{13}} , \hbox {e}_{\mathrm{23}} = \frac{\hbox {T}_{23}}{2\hbox {G}_{23}}, \end{aligned}$$

(11.8)

where $\{\hbox {E}_{\mathrm{1}}$, $\hbox {E}_{\mathrm{2}}$, $\hbox {E}_{\mathrm{3}}\}$ are Young’s moduli, $\{{\upnu }_{\mathrm{12}}$, ${\upnu }_{\mathrm{21}}$, ${\upnu }_{\mathrm{13}}$, ${\upnu }_{\mathrm{31}}$, ${\upnu }_{\mathrm{23}}$, ${\upnu }_{\mathrm{32}}\}$ are Poisson’s ratios and $\{\hbox {G}_{\mathrm{12}}$, $\hbox {G}_{\mathrm{13}}$, $\hbox {G}_{\mathrm{23}}\}$ are shear moduli. Also, the symmetry of the constitutive equation requires

$$\begin{aligned} \frac{{\upnu }_{\mathrm{21}}}{\hbox {E}_{2}} = \frac{{\upnu }_{\mathrm{12}}}{\hbox {E}_{1}} , \frac{{\upnu }_{\mathrm{31}}}{\hbox {E}_{3}} = \frac{{\upnu }_{\mathrm{13}}}{\hbox {E}_{1}} , \frac{{\upnu }_{\mathrm{32}}}{\hbox {E}_{3}} = \frac{{\upnu }_{\mathrm{23}}}{\hbox {E}_{2}}. \end{aligned}$$

(11.9)

It can be shown that the expressions (7.3), (7.5)-(7.7) cause (11.7) to be consistent with (11.8) when the values of the Poisson ratios are given by

$$\begin{aligned} {\upnu }_{\mathrm{12}}= & {} \frac{-1+9\left( 1-2{\upbeta }_{3}\right) \sqrt{{\upalpha }_{1}{\upalpha }_{2}}}{1+9{\upalpha }_{1}},\quad {\upnu }_{\mathrm{21}} = \left( \frac{1+9{\upalpha }_{1}}{1+9{\upalpha }_{2}}\right) {\upnu }_{\mathrm{12}} ,\nonumber \\ {\upnu }_{\mathrm{13}}= & {} \frac{-1+9{\upalpha }_{1}-9\left( 1-2{\upbeta }_{3}\right) \sqrt{{\upalpha }_{1}{\upalpha }_{2}}}{1+9{\upalpha }_{1}},\nonumber \\ {\upnu }_{\mathrm{31}}= & {} \left( \frac{1+9{\upalpha }_{1}}{1+9\left( {\upalpha }_{1}+{\upalpha }_{2}\right) -18\left( 1-2{\upbeta }_{3}\right) \sqrt{{\upalpha }_{1}{\upalpha }_{2}}}\right) {\upnu }_{\mathrm{13}} ,\nonumber \\ {\upnu }_{\mathrm{23}}= & {} \frac{-1+9{\upalpha }_{2}-9\left( 1-2{\upbeta }_{3}\right) \sqrt{{\upalpha }_{1}{\upalpha }_{2}}}{1+9{\upalpha }_{2}},\nonumber \\ {\upnu }_{\mathrm{32}}= & {} \left( \frac{1+9{\upalpha }_{2}}{1+9\left( {\upalpha }_{1}+{\upalpha }_{2}\right) -18\left( 1-2{\upbeta }_{3}\right) \sqrt{{\upalpha }_{1}{\upalpha }_{2}}}\right) {\upnu }_{\mathrm{23}}.\nonumber \\ \end{aligned}$$

(11.10)

Moreover, using the fact that $\hbox {e}_{33}^{\prime }= - (\hbox {e}_{11}^{\prime }+\hbox {e}_{22}^{\prime })$, the strain energy function ${\uprho }_{\mathrm{0}}{\uppsi }_{\mathrm{d}}$ can be expressed in the simplified form

$$\begin{aligned} {\uprho }_{\mathrm{0}}{\uppsi }_{\mathrm{d}}= & {} \frac{1}{2} \hbox {K}_{\mathrm{ijkl}}\hbox {e}_{\mathrm{ij}}^{\prime }\hbox {e}_{\mathrm{kl}}^{\prime }=\frac{1}{2} [\hbox {K}_{\mathrm{11}}\hbox {e}_{\mathrm{11}}^{\prime }\hbox {e}_{\mathrm{11}}^{\prime }+ 2\hbox {K}_{\mathrm{12}}\hbox {e}_{\mathrm{11}}^{\prime }\hbox {e}_{\mathrm{22}}^{\prime }\nonumber \\&+\, \hbox {K}_{\mathrm{22}}\hbox {e}_{\mathrm{22}}^{\prime }\hbox {e}_{\mathrm{22}}^{\prime }+ 4\hbox {G}_{\mathrm{12}}\hbox {e}_{\mathrm{12}}^{\prime }\hbox {e}_{\mathrm{12}}^{\prime }\nonumber \\&+\, 4\hbox {G}_{\mathrm{13}}\hbox {e}_{\mathrm{13}}^{\prime }\hbox {e}_{\mathrm{13}}^{\prime } + 4\hbox {G}_{\mathrm{23}}\hbox {e}_{\mathrm{23}}^{\prime }\hbox {e}_{\mathrm{23}}^{\prime }], \end{aligned}$$

(11.11)

with

$$\begin{aligned} \hbox {K}_{\mathrm{11}}= & {} \frac{{\uprho }_{0}\hbox {C}_{0}^{2}}{4{\upalpha }_{1} {\upbeta }_{3}(1-{\upbeta }_{3})}, \hbox {K}_{\mathrm{22}} =\frac{{\uprho }_{0}\hbox {C}_{0}^{2}}{4{\upalpha }_{2} {\upbeta }_{3}(1-{\upbeta }_{3})},\nonumber \\ \hbox {K}_{\mathrm{12}}= & {} \frac{{\uprho }_{0}\hbox {C}_{0}^{2}(1-2{\upbeta }_{3})\sqrt{{\upalpha }_{1}{\upalpha }_{2}}}{4{\upalpha }_{1}{\upbeta }_{3}(1-{\upbeta }_{3})}. \end{aligned}$$

(11.12)

It then follows that ${\uprho }_{\mathrm{0}}\Sigma$ in (11.6) will be a positive definite function of the strain $\hbox {e}_{\mathrm{ij}}$ when

$$\begin{aligned}&{\uprho }_{\mathrm{0}}\hbox {C}_{0}^{2} > 0 , \hbox {K}_{\mathrm{11}} > 0 , \hbox {K}_{\mathrm{22}} > 0 ,\nonumber \\&\hbox {K}_{\mathrm{11}}\hbox {K}_{\mathrm{22}}- \hbox {K}_{\mathrm{12}}\hbox {K}_{\mathrm{12}} = \frac{{\uprho }_{\mathrm{0}}^{2}\hbox {C}_{0}^{4}}{4{\upalpha }_{1}{\upalpha }_{2}{\upbeta }_{3}(1-{\upbeta }_{3})} > 0 ,\nonumber \\&\hbox {G}_{\mathrm{12}} > 0 , \hbox {G}_{\mathrm{13}} > 0 , \hbox {G}_{\mathrm{23}} > 0 , \end{aligned}$$

(11.13)

which are satisfied by the restrictions (7.8).

In summary, the orthotropic material is characterized by the seven material constants

$$\begin{aligned} \{ \hbox {C}_{\mathrm{0}} , {\upalpha }_{\mathrm{1}} , {\upalpha }_{\mathrm{2}}, {\upbeta }_{\mathrm{3}} , \hbox {G}_{\mathrm{12}} , \hbox {G}_{\mathrm{13}} , \hbox {G}_{\mathrm{23}} \} . \end{aligned}$$

which satisfy the restrictions (7.8). Then, the non-trivial values of $\hbox {K}_{\mathrm{ijkl}}$ are determined by (7.3), (7.6) and (7.7), and the values of $\{\hbox {E}_{\mathrm{1}}$, $\hbox {E}_{\mathrm{2}}$, $\hbox {E}_{\mathrm{3}}$, ${\upnu }_{\mathrm{12}}$, ${\upnu }_{\mathrm{21}}$, ${\upnu }_{\mathrm{13}}$, ${\upnu }_{\mathrm{31}}$, ${\upnu }_{\mathrm{23}}$, ${\upnu }_{\mathrm{32}}\}$ are determined by (7.5) and (11.10). Moreover, it can be seen from (7.5) and (7.8) that $\{\hbox {E}_{\mathrm{1}}$, $\hbox {E}_{\mathrm{2}}$, $\hbox {E}_{\mathrm{3}}\}$ have the ranges

$$\begin{aligned} 0 < \hbox {E}_{\mathrm{1}} < 9{\uprho }_{\mathrm{0}}\hbox {C}_{0}^{2} , 0 < \hbox {E}_{\mathrm{2}} < 9{\uprho }_{\mathrm{0}}\hbox {C}_{0}^{2} , 0 < \hbox {E}_{\mathrm{3}} < 9{\uprho }_{\mathrm{0}}\hbox {C}_{0}^{2}.\nonumber \\ \end{aligned}$$

(11.14)

1.2 An orthotropic material with five material constants

For the special case when

$$\begin{aligned} {\upalpha }_{\mathrm{1}} = {\upalpha }_{\mathrm{2}} = {\upalpha }, {\upbeta }_{\mathrm{3}} = \frac{1}{4}, \end{aligned}$$

(11.15)

it follows that the Young’s moduli are equal to E and the Poisson’s ratios are equal to ${\upnu }$

$$\begin{aligned} \hbox {E}_{\mathrm{1}}= & {} \hbox {E}_{\mathrm{2}} = \hbox {E}_{\mathrm{3}} = \hbox {E} , \nonumber \\ {\upnu }_{\mathrm{12}}= & {} {\upnu }_{\mathrm{21}}= {\upnu }_{\mathrm{13}} = {\upnu }_{\mathrm{31}} = {\upnu }_{\mathrm{23}} = {\upnu }_{\mathrm{32}} = {\upnu }, \end{aligned}$$

(11.16)

with

$$\begin{aligned} \hbox {E}= & {} 3{\uprho }_{\mathrm{0}}\hbox {C}_{0}^{2}(1-2{\upnu }) , {\upalpha }= \frac{2(1+{\upnu })}{9(1-2{\upnu })} ,\nonumber \\ {\upnu }= & {} \frac{9{\upalpha }-2}{2(9{\upalpha }+1)} , - 1 < {\upnu }< \frac{1}{2}, \end{aligned}$$

(11.17)

which yields an orthotropic material characterized by five material constants

$$\begin{aligned} \{\hbox {C}_{\mathrm{0}}, {\upnu }, \hbox {G}_{\mathrm{12}}, \hbox {G}_{\mathrm{13}}, \hbox {G}_{\mathrm{23}}\} . \end{aligned}$$

(11.18)

1.3 An orthotropic material with three material constants

The special case of (11.18) with all shear moduli being equal

$$\begin{aligned} \hbox {G}_{\mathrm{12}} = \hbox {G}_{\mathrm{13}} = \hbox {G}_{\mathrm{23}} = \hbox {G}_{\mathrm{d}} , \end{aligned}$$

(11.19)

yields an orthotropic material characterized by three material constants

$$\begin{aligned} \{\hbox {C}_{\mathrm{0}}, {\upnu }, \hbox {G}_{\mathrm{d}}\} . \end{aligned}$$

(11.20)

1.4 An isotropic material with two material constants

The special case of (11.20) with Poisson’s ratio specified by

$$\begin{aligned} {\upnu }=\frac{3{\uprho }_{0}\hbox {C}_{0}^{2}-2\hbox {G}_{\mathrm{d}}}{2(3{\uprho }_{0}\hbox {C}_{0}^{2}+\hbox {G}_{\mathrm{d}})}, \end{aligned}$$

(11.21)

yields an isotropic material.

Appendix 2: Approximate elastic wave speeds

In order to develop approximate expressions for relevant wave speeds it is assumed that the pressure $\hbox {p}_{\mathrm{e}}$ can be approximated by the Hugoniot pressure $\hbox {p}_{\mathrm{H}}$ in (4.14) so with the help of (3.14) and (4.10) the stress T is approximated by

$$\begin{aligned} \mathbf{T} = - \,\hbox {p}_{\mathrm{H}}(\hbox {J}) \mathbf{I} + \mathbf{T}_{\mathrm{d}} , \mathbf{T}_{\mathrm{d}} = \hbox {J}^{\mathrm{-1}} \hbox {K}_{\mathrm{ijkl}}\hbox {e}_{\mathrm{kl}}^{\prime } (\mathbf{m}_{\mathrm{i}}\otimes \mathbf{m}_{\mathrm{j}}) . \end{aligned}$$

(11.22)

For elastic response, the rate of inelasticity $\mathbf{L}_{\mathrm{p}}$ is neglected in (1.8) and use is made of (3.1), (3.5), (4.8) and (4.14) to deduce that

$$\begin{aligned} \dot{\mathbf{T}}= & {} \hbox {K}_{\mathrm{H}} (\mathbf{D} {\cdot } \mathbf{I}) \mathbf{I} + \hbox {J}^{\mathrm{-1}} \hbox {K}_{\mathrm{ijkl}}(\mathbf{D} {\cdot } \mathbf{m}_{\mathrm{k}}\otimes \mathbf{m}_{\mathrm{l}}) (\mathbf{m}_{\mathrm{i}}\otimes \mathbf{m}_{\mathrm{j}})\nonumber \\&+ \mathbf{LT}_{\mathrm{d}} + \mathbf{T}_{\mathrm{d}}{} \mathbf{L}^{\mathrm{T}} - (\mathbf{D} {\cdot } \mathbf{I}) \mathbf{T}_{\mathrm{d}} , \end{aligned}$$

(11.23)

where the Hugoniot bulk modulus $\hbox {K}_{\mathrm{H}}$ is given by

$$\begin{aligned} \hbox {K}_{\mathrm{H}}= & {} - \hbox {J}\frac{\hbox {dp}_{\mathrm{H}}}{\hbox {dJ}}, \hbox {K}_{\mathrm{H}} = \frac{\hbox {J}{\uprho }_{0}\hbox {U}_{\mathrm{s}}^{2}}{1+\frac{{\upgamma }_{0}}{2}(\hbox {J}-1)} \quad \hbox {for} \quad \hbox {J} > 1 ,\nonumber \\ \hbox {K}_{\mathrm{H}}= & {} - \hbox {J} \frac{\hbox {dp}_{\mathrm{H}}}{\hbox {dJ}}\nonumber \\= & {} \hbox {J}{\uprho }_{\mathrm{0}}\hbox {U}_{\mathrm{s}}^{2} \bigg [1+2(\hbox {S}_{\mathrm{1}}+\hbox {S}_{\mathrm{2}}+\hbox {S}_{\mathrm{3}})(1-\hbox {J})\left( \frac{\hbox {U}_{\mathrm{s}}}{\hbox {C}_{0}}\right) \bigg ] \nonumber \\&\quad \hbox {for} \quad \hbox {J} \le 1. \end{aligned}$$

(11.24)

Next, consider small deformations from a hydrostatic state of stress with

$$\begin{aligned} \mathbf{T}_{\mathrm{d}} = 0, \end{aligned}$$

(11.25)

and let $\hbox {D}_{\mathrm{ij}}$ be the components of the rate of deformation tensor D relative to rectangular Cartesian base vectors $\mathbf{e}_{\mathrm{i}}$ which are parallel to $\mathbf{m}_{\mathrm{i}}$, such that

$$\begin{aligned} \hbox {D}_{\mathrm{ij}} = \mathbf{D} {\cdot } (\mathbf{e}_{\mathrm{i}}\otimes \mathbf{e}_{\mathrm{j}}) , \mathbf{m}_{\mathrm{i}} = \hbox {J}^{\mathrm{1/3}} \mathbf{e}_{\mathrm{i}} , \end{aligned}$$

(11.26)

with the components $\hbox {T}_{\mathrm{ij}}$ of the stress T given by (9.7). Then, for the orthotropic material characterized by (7.3), (7.6) and (7.7) it follows that for uniaxial strains in the $\mathbf{e}_{\mathrm{i}}$ directions

$$\begin{aligned} \dot{\hbox {T}}_{\mathrm{11}}= & {} (\hbox {K}_{\mathrm{H}}+\hbox {J}^{\mathrm{1/3}}\hbox {K}_{\mathrm{1111}}) \hbox {D}_{\mathrm{11}} , \dot{\hbox {T}}_{\mathrm{22}} = (\hbox {K}_{\mathrm{H}}+\hbox {J}^{\mathrm{1/3}}\hbox {K}_{\mathrm{2222}}) \hbox {D}_{\mathrm{22}} ,\nonumber \\ \dot{\hbox {T}}_{\mathrm{33}}= & {} (\hbox {K}_{\mathrm{H}}+\hbox {J}^{\mathrm{1/3}}\hbox {K}_{\mathrm{3333}}) \hbox {D}_{\mathrm{33}} , \end{aligned}$$

(11.27)

and for simple shears

$$\begin{aligned} \dot{\hbox {T}}_{\mathrm{12}}= & {} 2\hbox {J}^{\mathrm{1/3}}\hbox {G}_{\mathrm{12}} \hbox {D}_{\mathrm{12}} , \dot{\hbox {T}}_{\mathrm{13}} = 2\hbox {J}^{\mathrm{1/3}}\hbox {G}_{\mathrm{13}} \hbox {D}_{\mathrm{13}} , \nonumber \\ \dot{{\hbox {T}}}_{\mathrm{23}}= & {} 2\hbox {J}^{\mathrm{1/3}}\hbox {G}_{\mathrm{23}} \hbox {D}_{\mathrm{23}} . \end{aligned}$$

(11.28)

Thus, the uniaxial strain wave speeds for waves in the $\mathbf{e}_{\mathrm{i}}$ directions are approximated by

$$\begin{aligned} \hbox {C}_{1}^{2}= & {} \frac{\hbox {K}_{\mathrm{H}}+\hbox {J}^{1/3}\hbox {K}_{1111}}{{\uprho }}, \hbox {C}_{2}^{2}=\frac{\hbox {K}_{\mathrm{H}}+\hbox {J}^{1/3}\hbox {K}_{2222}}{{\uprho }},\nonumber \\ \hbox {C}_{3}^{2}= & {} \frac{\hbox {K}_{\mathrm{H}}+\hbox {J}^{1/3}\hbox {K}_{3333}}{{\uprho }}, \end{aligned}$$

(11.29)

and the shear wave speeds are approximated by

$$\begin{aligned} \hbox {C}_{4}^{2}=\frac{\hbox {J}^{1/3}\hbox {G}_{12}}{{\uprho }}, \hbox {C}_{5}^{2}=\frac{\hbox {J}^{1/3}\hbox {G}_{13}}{{\uprho }}, \hbox {C}_{6}^{2}=\frac{\hbox {J}^{1/3}\hbox {G}_{23}}{{\uprho }} . \end{aligned}$$

(11.30)

These values can be used to limit the time step in explicit wave propagation codes.