Nonlinear shear constitutive model for peak shear-type joints based on improved Harris damage function

Abstract

The majority of jointed rock mass failures mainly occur along the joints in shear mode, which promotes a wide investigation on the proposal of a reasonable and reliable shear constitutive model of rock joints. In this paper, based on Improved Harris function and laboratory shear tests, a new constitutive model of saw-tooth joints was proposed. Firstly, a series of laboratory direct shear tests were carried out on saw-tooth joint specimens made of rock-like materials (cement mortar) to obtain the shear stress-displacement curves. Subsequently, the test results were divided into sliding failure type and peak shear type according to whether there is a significant stress drop between peak stress and residual stress. It is assumed that rock elements can be divided into undamaged parts and damaged parts during the shearing process. The stress-displacement relation of the undamaged part satisfies Hooke’s law, while the damaged part provides residual stress. Via the comparison with commonly used micro-element failure probability density functions, the Improved Harris distribution function was selected as the standard to characterize the strength of micro rock units. Finally, derived from the theory of damage statistical mechanics, a damage statistical constitutive model was proposed, which can reflect the deformation characteristics of rock joints. Compared with previous models and experimental data, the model proposed in this paper can represent the trend of peak shear curve variation with higher accuracy, the parameters are easy to be solved and have obvious physical significance, which verifies the advantages and applicability of this model.

Appendix A

In this appendix, we provide a detailed procedure for solving the parameters a and b.

By Eq. (13) to obtain the first derivative of the peak-strength point (u_p, τ_p) as follows:

$$ \frac{{{\text{d}}\tau }}{{{\text{d}}u}}\;{ = }\;{\frac{{\left[ {k_{\text{s}} { + }ab \times k_{\text{s}} \times \tau_{\text{r}} \left( {k_{s} u_{\text{p}} - {\tau_{y}} } \right)^{b - 1} } \right] \times \left[ {1 + a\left( {k_{s} {u_{\text{p}}} {- \tau_{y}} } \right)^{b} } \right] - \left[ {k_{\text{s}} {u_{\text{p}}} { + }a\tau_{\text{r}} \left( {k_{s} u_{\text{p}} - \tau_{y} } \right)^{b} } \right] \times ab \times k_{\text{s}} \left( {k_{\text{s}} {u_{\text{p}}} -{ \tau_{y}} } \right)^{b - 1} }}{{\left[ {1{\text{ + a}}\left( {k_{\text{s}} u_{\text{p}} - {\tau_{y}} } \right)^{b} } \right]^{2} }}}{ = }0 $$

(19)

The numerator of Eq. (19) is 0, that is:

$$ \begin{aligned} k_{\text{s}} &{ + }a \times k_{\text{s}} \times \left( {k_{\text{s}} u_{\text{p}} - \tau_{y} } \right)^{b} { + }ab \times k_{\text{s}} \tau_{\text{r}} \left( {k_{\text{s}} u_{\text{p}} - \tau_{y} } \right)^{b - 1} { + }a^{2} b \times k_{\text{s}} \tau_{\text{r}} \left( {k_{\text{s}} u_{\text{p}} - \tau_{y} } \right)^{2b - 1} \hfill \\ &- ab \times k_{\text{s}}^{2} \times u_{\text{p}} \left( {k_{\text{s}} u_{\text{p}} - \tau_{y} } \right)^{b - 1} - a^{2} b \times k_{\text{s}} \tau_{\text{r}} \left( {k_{\text{s}} u_{\text{p}} - \tau_{y} } \right)^{2b - 1}\,\, { = }\,\,0 \hfill \\ \end{aligned} $$

(20)

Then the $a^{2} b \times k_{\text{s}} {\tau_{\text{r}}} \left( {k_{\text{s}}{ u_{\text{p}}} - {\tau_{y} }} \right)^{2b - 1}$ term is eliminated from Eq. (20) to obtain:

$$ k_{\text{s}} { + }a \times k_{\text{s}} \times \left( {k_{\text{s}} u_{\text{p}} - \tau_{y} } \right)^{b} { + }\left( {\tau_{\text{r}} - k_{\text{s}} u_{\text{p}} } \right) \times ab \times k_{\text{s}} \left( {k_{\text{s}} u_{\text{p}} - \tau_{y} } \right)^{b - 1} { = }0 $$

(21)

According to Eq. (14), the following formula can be obtained:

$$ {k_{\text{s}}} {u_{\text{p}}} { + }a\left( {k_{\text{s}} {u_{\text{p}}} - \tau_{y} } \right)^{b} {\tau_{\text{r}}} { = }{\tau_{\text{p}}} { + }{\tau_{\text{p}}} \times a\left( {k_{\text{s}} {u_{\text{p}}} - \tau_{y} } \right)^{b} $$

(22)

Equation (22) can also be rewritten as:

$$ a\left( {{k_{\text{s}}} {u_{\text{p}}} - {\tau_{y}} } \right)^{b} { = }{\frac{{k_{\text{s}} u_{\text{p}} - \tau_{\text{p}} }}{{\tau_{\text{p}} - \tau_{\text{r}} }}} $$

(23)

Multiply both sides of Eq. (21) by ${k_{\text{s}}} {u_{\text{p}}} - \tau_{y}$ to get a new equation. In this new equation, replace the $a\left( {{k_{\text{s}}} {u_{\text{p}}} - \tau_{y} } \right)^{b}$ term with the ${\frac{{k_{\text{s}} {u_{\text{p}}} - {\tau_{\text{p}}} }}{{\tau_{\text{p}} - {\tau_{\text{r}}}} }}$ term (see Eq. (23)), and the following formula can be obtained:

$$ {k_{\text{s}}} \left( {k_{\text{s}} {u_{\text{p}}} - \tau_{y} } \right){ + }k_{\text{s}} \times \left( {k_{\text{s}} u_{\text{p}} - \tau_{y} } \right) \times \frac{{k_{\text{s}} u_{\text{p}} - \tau_{\text{p}} }}{{\tau_{\text{p}} - \tau_{\text{r}} }}{ + }b \times k_{\text{s}} \left( {\tau_{\text{r}} - k_{\text{s}} u_{\text{p}} } \right){\frac{{k_{\text{s}} u_{\text{p}} - \tau_{\text{p}} }}{{\tau_{\text{p}} - \tau_{\text{r}} }}}{ = }0 $$

(24)

The above expression can also be written:

$$ k_{\text{s}} \left( {k_{\text{s}} u_{\text{p}} - \tau_{y} } \right) \times \left[ {1{ + }{\frac{{k_{\text{s}} u_{\text{p}} - \tau_{\text{p}} }}{{\tau_{\text{p}} - \tau_{\text{r}} }}}} \right]{ + }b \times k_{\text{s}} \left( {\tau_{\text{r}} - k_{\text{s}} u_{\text{p}} } \right){\frac{{k_{\text{s}} u_{\text{p}} - \tau_{\text{p}} }}{{\tau_{\text{p}} - \tau_{\text{r}}} }}\;{ = }\;0 $$

(25)

Eq. (25) can be simplified to:

$$ \left( {k_{\text{s}} {u_{\text{p}}} - \tau_{y} } \right)\left( {k_{\text{s}} {u_{\text{p}}} - {\tau_{\text{r}}} } \right){ + }b\left( {\tau_{\text{r}} - {k_{\text{s}}} u_{\text{p}} } \right)\left( {k_{\text{s}} {u_{\text{p}}} - {\tau_{\text{p}}} } \right)\;{ = }\;0 $$

(26)

According to Eq. (26), the model expression of b can be solved as follows:

$$ b\;{ = }\;{\frac{{k_{\text{s}} u_{\text{p}} - \tau_{y} }}{{k_{\text{s}} u_{\text{p}} - \tau_{\text{p}} }}} $$

(27)

Substituting Eqs. (27) into (28), the model expression of parameter a can be expressed as

$$ a\;{ = }\;{\frac{{k_{\text{s}} u_{\text{p}} - \tau_{\text{p}} }}{{\left( {\tau_{\text{p}} - \tau_{\text{r}} } \right)\left( {k_{s} u_{\text{p}} - \tau_{\text{y}} } \right)^{b} }}} $$

(28)