The Dynamic Mechanical Properties of a Hard Rock Under True Triaxial Damage-Controlled Dynamic Cyclic Loading with Different Loading Rates: A Case Study

Highlights

• A dynamic constitutive model for rock materials suited to dynamic cyclic loading was established.

• The numerical tests on a hard rock under true triaxial dynamic cyclic loading with different loading rates were conducted.

• The dynamic deformation and mechanical properties of a hard rock were studied.

Appendices

Appendix

Numerical Implementation of the Dynamic Constitutive Model

Based on the idea of elastic prediction-plasticity correction, through self-programming Finite Element Method (FEM), the flow of the numerical implementation of the dynamic constitutive model is introduced in detail (as shown in Fig. 

Fig. 15

Flow chart of numerical implementation of the dynamic constitutive model

15).

1. Setting the initial internal variables:

$$ s_{n + 1}^{k} = s_{n} ,\alpha_{n + 1}^{k} = \alpha_{n} ,R_{n + 1}^{k} = R_{n} ,Q_{n + 1}^{k} = Q_{n} ,\kappa_{n + 1}^{k} = \kappa_{n} ,\dot{\varepsilon }_{n + 1}^{k} = \dot{\varepsilon }_{n} ,\left( {D^{el} } \right)_{n + 1}^{k} = \left( {D^{el} } \right)_{n} $$

(11)

where n is the time step is and k is iterative step. $D^{el}$ is the elastic matrix.

2. Elastic prediction: the stress can be calculated as followed.

$$ \sigma_{n + 1}^{k + 1} = \sigma_{n + 1}^{k} + \left( {D^{el} } \right)_{n + 1}^{k} d\varepsilon_{n + 1}^{k + 1} $$

(12)

$$ \overline{\sigma }_{n + 1}^{k + 1} = \sigma_{n + 1}^{k} { - }R_{n + 1}^{k} \alpha_{n + 1}^{k} + s_{n + 1}^{k} \left( {R_{n + 1}^{k} - 1} \right) $$

(13)

where $d\varepsilon$ is the strain increment.

3. Yield judgments: if $\left\| {\frac{{d\varepsilon_{n + 1}^{k + 1} }}{dt}} \right\| < \dot{\varepsilon }_{n + 1}^{k}$, then $\dot{\varepsilon }_{n + 1}^{k + 1} = \dot{\varepsilon }_{n + 1}^{k}$; otherwise, $\dot{\varepsilon }_{n + 1}^{k + 1} = \left\| {\frac{{d\varepsilon_{n + 1}^{k} }}{dt}} \right\|$

$$ {\text{If}}\,\,\,f\left( {\overline{\sigma }_{n + 1}^{k + 1} ,\dot{\varepsilon }_{n + 1}^{k + 1} ,\kappa_{n + 1}^{k} } \right) - R_{n + 1}^{k} Q_{n + 1}^{k} \le 0 $$

(14)

then the stress is the calculated stress and go to (iii) in Step 4.; otherwise, the corresponding plasticity correction should be done. Where dt is the time increment.

4. Plasticity correction:

i. Solving the plasticity factor:

$$ d\lambda_{n + 1}^{k + 1} = \frac{{f\left( {\overline{\sigma }_{n + 1}^{k + 1} ,\dot{\varepsilon }_{n + 1}^{k + 1} ,\kappa_{n + 1}^{k} } \right) - R_{n + 1}^{k} Q\left( {\dot{\varepsilon }_{n + 1}^{k + 1} ,\kappa_{n + 1}^{k} } \right)}}{{\left\| {\frac{{\partial f\left( {\overline{\sigma }_{n + 1}^{k + 1} ,\dot{\varepsilon }_{n + 1}^{k + 1} ,\kappa_{n + 1}^{k} } \right)}}{{\partial \overline{\sigma }_{n + 1}^{k + 1} }}} \right\|\left( {M_{n + 1}^{k} + N_{n + 1}^{k} \cdot \left( {D^{el} } \right)_{n + 1}^{k} \cdot N_{n + 1}^{k} } \right)}} $$

(15)

$$ N_{n + 1}^{k} = \frac{{\frac{{\partial f\left( {\overline{\sigma }_{n + 1}^{k + 1} ,\dot{\varepsilon }_{n + 1}^{k + 1} ,\kappa_{n + 1}^{k} } \right)}}{{\partial \overline{\sigma }_{n + 1}^{k + 1} }}}}{{\left\| {\frac{{\partial f\left( {\overline{\sigma }_{n + 1}^{k + 1} ,\dot{\varepsilon }_{n + 1}^{k + 1} ,\kappa_{n + 1}^{k} } \right)}}{{\partial \overline{\sigma }_{n + 1}^{k + 1} }}} \right\|}} $$

(16)

$$ M_{n + 1}^{k} = N_{n + 1}^{k} \cdot \left\{ \begin{gathered} \frac{{\left( {{\raise0.7ex\hbox{${\partial Q_{n + 1}^{k} }$} \!\mathord{\left/ {\vphantom {{\partial Q_{n + 1}^{k} } {\partial \kappa_{n + 1}^{k} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\partial \kappa_{n + 1}^{k} }$}}} \right)}}{{Q_{n + 1}^{k} }}L_{n + 1}^{k} \frac{{\overline{\sigma }_{n + 1}^{k + 1} }}{{R_{n + 1}^{k} }} + {\raise0.7ex\hbox{${\partial \alpha_{n + 1}^{k} }$} \!\mathord{\left/ {\vphantom {{\partial \alpha_{n + 1}^{k} } {\partial \kappa_{n + 1}^{k} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\partial \kappa_{n + 1}^{k} }$}} - u\ln R_{n + 1}^{k} \frac{{\tilde{\sigma }_{n + 1}^{k} }}{{R_{n + 1}^{k} }} + C\left( {1 - R_{n + 1}^{k} } \right)\left[ {\frac{{\overline{\sigma }_{n + 1}^{k + 1} }}{{R_{n + 1}^{k} }} - \frac{{\hat{s}_{n + 1}^{k} }}{\chi }} \right] - \hfill \\ \frac{{1 - R_{n + 1}^{k} }}{{\chi Q_{n + 1}^{k} }}\frac{{\partial f\left( {s_{n + 1}^{k} ,\dot{\varepsilon }_{n + 1}^{k + 1} ,\kappa_{n + 1}^{k} } \right)}}{{\partial \kappa_{n + 1}^{k} }}W_{n + 1}^{k} \hat{s}_{n + 1}^{k} - \frac{1}{{R_{n + 1}^{k} Q_{n + 1}^{k} }}\frac{{\partial f\left( {\overline{\sigma }_{n + 1}^{k + 1} ,\dot{\varepsilon }_{n + 1}^{k + 1} ,\kappa_{n + 1}^{k} } \right)}}{{\partial \kappa_{n + 1}^{k} }}L_{n + 1}^{k} \overline{\sigma }_{n + 1}^{k + 1} \hfill \\ \end{gathered} \right\} $$

(17)

$$ L_{n + 1}^{k} = G\sqrt {\frac{2}{3}\left( {K \cdot \frac{{\partial f\left( {\overline{\sigma }_{n + 1}^{k + 1} ,\dot{\varepsilon }_{n + 1}^{k + 1} ,\kappa_{n + 1}^{k} } \right)}}{{\partial \overline{\sigma }_{n + 1}^{k + 1} }}} \right)\left( {K \cdot \frac{{\partial f\left( {\overline{\sigma }_{n + 1}^{k + 1} ,\dot{\varepsilon }_{n + 1}^{k + 1} ,\kappa_{n + 1}^{k} } \right)}}{{\partial \overline{\sigma }_{n + 1}^{k + 1} }}} \right)} $$

(18)

$$ W_{n + 1}^{k} = G\sqrt {\frac{2}{3}\left( {K \cdot \frac{{\partial f\left( {s_{n + 1}^{k} ,\dot{\varepsilon }_{n + 1}^{k + 1} ,\kappa_{n + 1}^{k} } \right)}}{{\partial s_{n + 1}^{k} }}} \right)\left( {K \cdot \frac{{\partial f\left( {s_{n + 1}^{k} ,\dot{\varepsilon }_{n + 1}^{k + 1} ,\kappa_{n + 1}^{k} } \right)}}{{\partial s_{n + 1}^{k} }}} \right)} $$

(19)

$$ \hat{s}_{n + 1}^{k} = s_{n + 1}^{k} - \alpha_{n + 1}^{k} $$

(20)

$$ \tilde{\sigma }_{n + 1}^{k} = \sigma_{n + 1}^{k} - s_{n + 1}^{k} $$

(21)

where $\chi$ is the maximum value of the ratio of the size of the similarity-centre surface to that of the normal-yield surface. u and C are material parameters.

ii. Updating the internal variables

$$ {\text{d}}\varepsilon_{n + 1}^{{p\left( {k + 1} \right)}} = {\text{d}}\lambda_{n + 1}^{k + 1} N_{n + 1}^{k} $$

(22)

$$ \kappa_{n + 1}^{k + 1} = \kappa_{n + 1}^{k} + G\sqrt {\frac{2}{3}\left( {d\varepsilon_{n + 1}^{p(k + 1)} - \frac{1}{3}tr\left( {d\varepsilon_{n + 1}^{p(k + 1)} } \right)} \right)\left( {d\varepsilon_{n + 1}^{p(k + 1)} - \frac{1}{3}tr\left( {d\varepsilon_{n + 1}^{p(k + 1)} } \right)} \right)} $$

(23)

$$ \left( {D^{el} } \right)_{n + 1}^{k + 1} = \left\{ {\begin{array}{*{20}c} {\left( {D^{el} } \right)_{0} \left( {a_{0} \left( {\lg \left( {\frac{{\dot{\varepsilon }_{n + 1}^{k + 1} }}{{\dot{\varepsilon }_{s} }}} \right)} \right)^{b} + 1} \right) \, \kappa = 0 \, } \\ {\left( {D^{el} } \right)_{0} \left( {a_{0} \left( {\lg \left( {\frac{{\dot{\varepsilon }_{n + 1}^{k + 1} }}{{\dot{\varepsilon }_{s} }}} \right)} \right)^{b} + 1} \right)\left( {1 - \left( {1 - \frac{{E_{E} }}{{E_{s} }}} \right)\frac{{\kappa_{n + 1}^{k + 1} }}{{\kappa_{E} }}} \right){ 0} < \kappa_{n + 1}^{k + 1} \le \kappa_{E} \, } \\ {\left( {D^{el} } \right)_{0} \left( {a_{0} \left( {\lg \left( {\frac{{\dot{\varepsilon }_{n + 1}^{k + 1} }}{{\dot{\varepsilon }_{s} }}} \right)} \right)^{b} + 1} \right)\left( {\frac{{E_{E} }}{{E_{s} }}} \right) \, \kappa_{n + 1}^{k + 1} > \kappa_{E} } \\ \end{array} } \right. $$

(24)

$$ \sigma_{n + 1}^{k + 1} = \sigma_{n + 1}^{k} + \left( {D^{el} } \right)_{n + 1}^{k} \left( {d\varepsilon_{n + 1}^{k + 1} { - }d\varepsilon_{n + 1}^{{p\left( {k + 1} \right)}} } \right) $$

(25)

$$ c\left( {\dot{\varepsilon }_{n + 1}^{k + 1} ,\kappa_{n + 1}^{k + 1} } \right) = \left\{ {\begin{array}{*{20}c} {c_{0} \left( {d\lg \left( {\frac{{\dot{\varepsilon }_{n + 1}^{k + 1} }}{{\dot{\varepsilon }_{s} }}} \right) + 1} \right) \, \left( {\kappa_{n + 1}^{k + 1} \le \kappa_{{c_{0} }} } \right)} \\ \begin{gathered} c_{0} \left( {d\lg \left( {\frac{{\dot{\varepsilon }_{n + 1}^{k + 1} }}{{\dot{\varepsilon }_{s} }}} \right) + 1} \right)\left( {\frac{{c_{r} }}{{c_{0} }} \, + \, \frac{{\kappa_{{c_{1} }} - \kappa_{n + 1}^{k + 1} }}{{\kappa_{{c_{1} }} - \kappa_{{c_{0} }} }}\left( {1 - \frac{{c_{r} }}{{c_{0} }}} \right)} \right) \, \left( {\kappa_{{c_{0} }} < \kappa_{n + 1}^{k + 1} < \kappa_{{c_{1} }} } \right) \, \hfill \\ c_{r} \left( {d\lg \left( {\frac{{\dot{\varepsilon }_{n + 1}^{k + 1} }}{{\dot{\varepsilon }_{s} }}} \right) + 1} \right) \, \left( {\kappa_{n + 1}^{k + 1} > \kappa_{{c_{1} }} } \right) \, \hfill \\ \end{gathered} \\ \end{array} } \right. $$

(26)

$$ \phi_{n + 1}^{k + 1} = \left\{ {\begin{array}{*{20}c} \begin{gathered} \phi_{0} \, \left( {\kappa_{n + 1}^{k + 1} \le \kappa_{{\phi_{0} }} } \right) \hfill \\ \phi_{0} + \frac{{\kappa_{{\phi_{0} }} - \kappa_{n + 1}^{k + 1} }}{{\kappa_{{\phi_{0} }} - \kappa_{{\phi_{1} }} }}\left( {\phi_{r} - \phi_{0} } \right) \, \left( {\kappa_{{\phi_{0} }} < \kappa_{n + 1}^{k + 1} < \kappa_{{\phi_{1} }} } \right) \hfill \\ \end{gathered} \\ {\phi_{r} \, \left( {\kappa_{n + 1}^{k + 1} > \kappa_{{\phi_{1} }} } \right) \, } \\ \end{array} } \right. $$

(27)

$$ Q_{n + 1}^{k + 1} = \frac{{6c\left( {\dot{\varepsilon }_{n + 1}^{k + 1} ,\kappa_{n + 1}^{k + 1} } \right)\cos \phi_{n + 1}^{k + 1} }}{{\sqrt 3 \left( {3 - \sin \phi_{n + 1}^{k + 1} } \right)}} $$

(28)

$$ \alpha_{n + 1}^{k + 1} = \alpha_{n + 1}^{k} + a\left( {rQ_{n + 1}^{k + 1} N_{n + 1}^{k} - \sqrt{\frac{2}{3}} \alpha_{n + 1}^{k} } \right)\left\| {d\varepsilon_{n + 1}^{p(k + 1)} } \right\| $$

(29)

$$ s_{n + 1}^{k + 1} = s_{n + 1}^{k} + \left( {C\left[ {\frac{{\overline{\sigma }_{n + 1}^{k + 1} }}{{R_{n + 1}^{k} }} - \frac{{\hat{s}_{n + 1}^{k} }}{\chi }} \right] + {\raise0.7ex\hbox{${\partial \alpha_{n + 1}^{k + 1} }$} \!\mathord{\left/ {\vphantom {{\partial \alpha_{n + 1}^{k + 1} } {\partial \kappa_{n + 1}^{k + 1} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\partial \kappa_{n + 1}^{k + 1} }$}} + \frac{{\left( {{\raise0.7ex\hbox{${\partial Q_{n + 1}^{k + 1} }$} \!\mathord{\left/ {\vphantom {{\partial Q_{n + 1}^{k + 1} } {\partial \kappa_{n + 1}^{k + 1} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\partial \kappa_{n + 1}^{k + 1} }$}}} \right)}}{{Q_{n + 1}^{k + 1} }}\hat{s}_{n + 1}^{k} } \right)\left\| {d\varepsilon_{n + 1}^{{p\left( {k + 1} \right)}} } \right\| $$

(30)

where a and r are material parameters. $\left( {D^{el} } \right)_{0}$ is the initial elastic matrix.

iii. Solving similarity ratio R:

$$ R_{n + 1}^{k + 1} = \frac{{ - B_{n + 1}^{k + 1} + \sqrt {B_{n + 1}^{{\mathop {k + 1}\nolimits^{2} }} - 4A_{n + 1}^{k + 1} z_{n + 1}^{k + 1} } }}{{2A_{n + 1}^{k + 1} }} $$

(31)

$$ A_{n + 1}^{k + 1} = \frac{1}{2}\left\| {\left( {s_{n + 1}^{k + 1} { - }\alpha_{n + 1}^{k + 1} } \right)^{\prime } } \right\|^{2} - 9\left( {\beta_{n + 1}^{k + 1} } \right)^{2} \left( {s_{n + 1}^{k + 1} { - }\alpha_{n + 1}^{k + 1} } \right)_{m}^{2} - \left( {Q_{n + 1}^{k + 1} } \right)^{2} + 6\beta_{n + 1}^{k + 1} Q_{n + 1}^{k + 1} \left( {s_{n + 1}^{k + 1} { - }\alpha_{n + 1}^{k + 1} } \right)_{m} $$

(32)

$$ B_{n + 1}^{k + 1} = \left( {\left( {s_{n + 1}^{k + 1} - \alpha_{n + 1}^{k + 1} } \right)^{\prime } \cdot \left( {\sigma_{n + 1}^{k + 1} { - }s_{n + 1}^{k + 1} } \right)^{\prime } } \right) - 18\left( {\beta_{n + 1}^{k + 1} } \right)^{2} \left( {s_{n + 1}^{k + 1} - \alpha_{n + 1}^{k + 1} } \right)_{m} \left( {\sigma_{n + 1}^{k + 1} { - }s_{n + 1}^{k + 1} } \right)_{m} + 6\beta_{n + 1}^{k + 1} Q_{n + 1}^{k + 1} \left( {\sigma_{n + 1}^{k + 1} - s_{n + 1}^{k + 1} } \right)_{m} $$

(33)

$$ z_{n + 1}^{k + 1} = \frac{1}{2}\left\| {\left( {\sigma_{n + 1}^{k + 1} - s_{n + 1}^{k + 1} } \right)^{\prime } } \right\|^{2} - 9\left( {\beta_{n + 1}^{k + 1} } \right)^{2} \left( {\sigma_{n + 1}^{k + 1} - s_{n + 1}^{k + 1} } \right)_{m}^{2} $$

(34)

where $\left( {} \right)_{m}$ represents the mean value, and $\left( {} \right)^{\prime }$ represents the deviatoric stress.

Determining balance:

$$ {\text{If}}\,\,\,\left\| {g\left( {\sigma_{n + 1}^{k + 1} } \right)_{{\text{int}}} - g\left( {\sigma_{n + 1}^{k + 1} } \right)_{ext} } \right\| < ToL $$

(35)

the process proceeds to the next step [Step (1)], until the end. Otherwise, jump to Step (2) to continue the iteration. Where $g\left( {\sigma_{n + 1}^{k + 1} } \right)_{{\text{int}}}$, $g\left( {\sigma_{n + 1}^{k + 1} } \right)_{ext}$ are the internal force and external force of analysis object, respectively, and $ToL$ is a permissible minimum (set to 10^–5).