A new model of shear creep and its experimental verification

Abstract

Creep exerts a significant role in rock engineering safety. In engineering practice, it is helpful to develop a mathematical model representing rock creep behaviors. We used a shear rheometer to carry out shear creep tests of cubic mudstone specimens. The experimental results indicate that the mudstone creep can be divided into steady and accelerated stages when the applied shear stress is greater than the long-term shear strength. To capture the creep characteristics of the mudstone samples, in this paper, we propose a variable-parameter fractional derivative element based on fractional theory. A viscosity coefficient is considered to be a variable. We develop a mathematical model describing rock shear creep. The new model shows a visco-elastic-plastic property, which can effectively characterize two stages of rock creep process. The shear creep results show that the new model of shear creep is more accurate than that of the Nishihara model and the constant-parameter fractional creep model. Sensitivity analysis demonstrates that derivative orders have the greatest effect on shear creep.

Appendices

Appendix A

Before deriving the equations, we give necessary mathematical definitions.

• a. Fractional integral definition

  Let $\beta \in R^{ +}$. If $f(x) \in L^{1}(R^{ +} )$, then the $\beta$th-order Riemann–Liouville (R-L) integral is defined as:

  $$ \mathrm{I}^{\beta } f(x) = \frac{1}{\Gamma (\beta )}\int _{0}^{x} \frac{f(s)}{(x - s)^{1 - \beta }} ds. $$

  (A.1)

• b. Gamma function and its properties

  The gamma function is defined as:

  $$ \Gamma (p) = \int _{0}^{\infty } e^{ - x}x^{p - 1}dx,\qquad p>0. $$

  (A.2)

  The properties of the gamma function:

  $$\begin{aligned} &\Gamma (1) = 1, \end{aligned}$$

  (A.3)
  $$\begin{aligned} &\Gamma (p + 1) = p\Gamma (p). \end{aligned}$$

  (A.4)

• c. Beta function

  The beta function $\beta (m, n)$ is defined as:

  $$ \beta (m,n) = \int _{0}^{1} x^{m - 1}(1 - x)^{n - 1}dx = \frac{\Gamma (m)\Gamma (n)}{\Gamma (m + n)}. $$

  (A.5)

1.1 A.1 Derivation of Eq. (7)

Equations (6a)–(6b) can be transformed into

$$ \frac{d^{\beta } \varepsilon }{dt^{\beta }} = \frac{\tau _{0}}{At^{B}}. $$

(A.6)

Solving Eq. (A.6), based on the definition a, b, and c above, we have:

$$\begin{aligned} \varepsilon (t) =& I^{\beta } ( \frac{\tau _{0}}{A t^{B}} )\rightarrow \frac{\tau _{0}}{\Gamma (\beta )} \int _{0}^{t} \frac{1}{A s^{B} (t-s)^{1-\beta }} ds \rightarrow \frac{\tau _{0}}{\Gamma (\beta )} \int _{0}^{1} \frac{t}{A t^{B} s^{B} (t-t\theta )^{1-\beta }} d\theta \\ \rightarrow & \frac{\tau _{0} t^{\beta -B}}{A\Gamma (\beta )} \int _{0}^{1} s^{-B} (1-\theta )^{\beta -1} d\theta \rightarrow \frac{\tau _{0} t^{\beta -B}}{A\Gamma (\beta )} \cdot \frac{\Gamma \left ( \beta \right ) \Gamma \left ( 1-B \right )}{\Gamma \left ( \beta +1-B \right )} \\ \rightarrow & \frac{\tau _{0}}{A} \cdot \frac{\Gamma \left ( 1-B \right ) t^{\beta -B}}{\Gamma \left ( \beta +1-B \right )}. \end{aligned}$$

(A.7)

1.2 A.2 Proof of two special forms of new components

The variable-parameter elastic element can be expressed as:

$$ \varepsilon (t) = \frac{\tau _{0}}{At^{B}}. $$

(A.8)

The variable-parameter viscous element can be expressed as:

$$ \varepsilon (t) = \frac{\tau _{0}}{A(1 - B)}t^{1 - B}. $$

(A.9)

This can be proved by letting $\beta=0$, 1 in Equation (7).

Appendix B: Three-dimensional form of the new model

A three-dimensional constitutive relationship is expressed as (Jiang et al. 2013; Liao et al. 2017):

$$ \left\{ \textstyle\begin{array}{l} e_{ij} = \frac{1}{2G} S_{ij},\\ \varepsilon _{kk} = \frac{1}{3K} \sigma _{kk}, \end{array}\displaystyle \right. $$

(B.1)

where $e_{ij}$ and $S_{ij}$ are the deviatoric strain and stress tensors, respectivelym and $\sigma _{kk}$ and $\varepsilon _{kk}$ are the first stress tensor invariant and first strain tensor invariant, respectively. The total strain is

$$ \varepsilon _{ij} = e_{ij} + \varepsilon _{kk}. $$

(B.2)

Substituting Eqs. (B.1) and (B.2) into Eq. (10) and noting that $\sigma _{kk}$ and $\varepsilon _{kk}$ only affect the instantaneous creep, we can obtain Eq. (11). Substituting the Mises yield condition and the DP yield criterion into Eq. (11), we can derive Eqs. (12) and (13).