A phenomenological modelling of rocks based on the influence of damage initiation

Abstract

It is recognized that the stress-induced damage impacts the progressive failure behavior of rocks. A phenomenological model for the compressive failure of rocks is thus presented in this study. The model addresses the progressive growth of damage that leads to the strength weakening on a macroscopic scale. Considering dramatic difference between uniaxial compression and tension strengths for rocks, the admitted Mises–Schleiche Drucker–Prager strength criterion is adopted to characterize the damage initiation. On this basis, a two-parameter Weibull-type probability function is used to define the strength distribution of representative volume elements, followed by the use of damage variable for addressing the accumulated probability of failure. The proposed damage variable essentially characterizes both the critical stress level of damage initiation and progressive damage evolution law. Detailed comparisons have been carried out between the predictions and experimental observations, and issues related to the damage evolution are particularly addressed. In addition, the results further validate the proposed model considering damage initiation.

Appendix

In this section, a number of equations that denote intermediate variables were shown here for derivation requirement with respect to Eq. (23) with the boundary condition of ${\sigma _1}={\sigma _{{\text{1,p}}}}$ and ${\varepsilon _1}={\varepsilon _{{\text{1,p}}}}$. The following expression ensues

$$E(1 - {D_{\text{p}}}) - {\varepsilon _{{\text{1,p}}}}\frac{{\partial D}}{{\partial {\varepsilon _1}}}=0.$$

(33)

Rearranging Eq. (33) gives

$$\frac{{\partial D}}{{\partial {\varepsilon _1}}}=\frac{{1 - {D_{\text{p}}}}}{{{\varepsilon _{{\text{1,p}}}}}}.$$

(34)

In the equation, D_p is calculated using Eq. (26) at the situation of peak point, which reads

$${D_{\text{p}}}=1 - {{({\sigma _{\text{p}}} - 2\nu {\sigma _3})} \mathord{\left/ {\vphantom {{({\sigma _{\text{p}}} - 2\nu {\sigma _3})} {E{\varepsilon _{{\text{1,p}}}}}}} \right. \kern-0pt} {E{\varepsilon _{{\text{1,p}}}}}}.$$

(35)

By inspiring from Eq. (14), the partial differential of the damage variable with respect to the axial strain can be written as

$$\frac{{\partial D}}{{\partial {\varepsilon _1}}}=\frac{{m\exp [ - {{(F_{{\text{p}}}^{*}/{F_0})}^m}]{{(F_{{\text{p}}}^{*}/{F_0})}^m}}}{{F_{{\text{p}}}^{*}}}\frac{{\partial \left\langle {{F^*}} \right\rangle }}{{\partial {\varepsilon _1}}}.$$

(36)

In the equation, $F_{{\text{p}}}^{*}$ is calculated using Eq. (26) at the situation of peak point, which reads

$$F_{{\text{p}}}^{*}{\text{=}}\frac{{\sqrt 3 }}{3}\left[ {E{\varepsilon _{{\text{1,p}}}}+\frac{{(2\nu - 1){\sigma _3}}}{{1 - {D_{\text{p}}}}} - \sqrt {({\sigma _c} - {\sigma _t})f_{{\text{p}}}^{*}\left[ {E{\varepsilon _{{\text{1,p}}}}+\frac{{2(1+\nu ){\sigma _3}}}{{1 - {D_{\text{p}}}}}} \right]+{\sigma _c}{\sigma _t}} } \right],$$

(37)

with

$$f_{{\text{p}}}^{*}=1+\frac{{3{\alpha ^2}\left\langle {I_{{1,{\text{p}}}}^{*} - I_{{\text{T}}}^{{}}} \right\rangle }}{{{\sigma _{\text{c}}} - {\sigma _{\text{t}}}}},$$

(38)

$$I_{{1,{\text{p}}}}^{*}=E{\varepsilon _{{\text{1,p}}}}+[(2\nu - 1){\sigma _3}]/(1 - {D_{\text{p}}}).$$

(39)

These Eqs. (40) to (48) are shown to describe the solution of ${{\partial {F^*}} \mathord{\left/ {\vphantom {{\partial {F^*}} {\partial {\varepsilon _1}}}} \right. \kern-0pt} {\partial {\varepsilon _1}}}$ involved in Eq. (36):

$$\frac{{\partial \left\langle {{F^*}} \right\rangle }}{{\partial {\varepsilon _1}}}{\text{=}}{N_1}+{N_2}\frac{{\partial D}}{{\partial {\varepsilon _1}}},$$

(40)

with

$${N_1}={B_1} - \frac{{{B_3}{A_1}+{B_4}}}{{{B_6}}}$$

(41)

$${N_2}={B_2} - \frac{{{B_3}{A_2} - {B_5}}}{{{B_6}}}$$

(42)

$${A_1}=3E{\alpha ^2}(1 - {D_{\text{p}}})/({\sigma _{\text{c}}} - {\sigma _{\text{t}}})$$

(43)

$${A_2}=3E{\alpha ^2}{\varepsilon _{{\text{1,p}}}}/({\sigma _{\text{c}}} - {\sigma _{\text{t}}})$$

(44)

$${B_1}=\sqrt 3 /3E$$

(45)

$${B_2}=\frac{{\sqrt 3 (2\nu - 1){\sigma _{\text{3}}}}}{{3{{(1 - {D_{\text{p}}})}^2}}}$$

(46)

$${B_3}=\sqrt 3 ({\sigma _{\text{c}}} - {\sigma _{\text{t}}})\left[ {E{\varepsilon _{{\text{1,p}}}}+\frac{{2(1 - \nu ){\sigma _3}}}{{1 - {D_{\text{p}}}}}} \right]$$

(47)

$${B_4}=\sqrt 3 ({\sigma _{\text{c}}} - {\sigma _{\text{t}}})Ef_{{\text{p}}}^{*}$$

(48)

$${B_5}=\sqrt 3 ({\sigma _{\text{c}}} - {\sigma _{\text{t}}})\frac{{2(1+\nu ){\sigma _3}}}{{{{(1 - {D_{\text{p}}})}^2}}}f_{{\text{p}}}^{*}$$

(49)

$${B_6}=6\sqrt {({\sigma _{\text{c}}} - {\sigma _{\text{t}}})f_{{\text{p}}}^{*}\left[ {E{\varepsilon _{{\text{1,p}}}}+\frac{{2(1+\nu ){\sigma _3}}}{{1 - {D_{\text{p}}}}}} \right]+{\sigma _{\text{c}}}{\sigma _{\text{t}}}} .$$

(50)

Substituting Eqs. (34) and (40) into Eq. (36), Eqs. (28) and (29) in the main paper can be obtained.