A Caputo variable-order fractional damage creep model for sandstone considering effect of relaxation time

Abstract

Establishing a fractional creep model with few parameters and explicit physical interpretation is of significant meaning for predicting rheological deformation of rock. In this study, based on the Caputo variable-order fractional derivative, a Caputo variable-order fractional creep model is proposed, whose physical interpretation is clearly stated by setting a varying-order function related to relaxation time. The significance of relaxation time is firstly highlighted to reveal the evolution mechanism of viscoelasticity of creep and relaxation response by constructing equivalence between rheological responses of constant-order fractional Maxwell model and that of time-varying viscosity Maxwell model. Meanwhile, considering the importance of relaxation time in rheology, a modified damage factor is also presented and introduced in proposed model. Next, for verifying the applicability of proposed damage creep model, a series of uniaxial creep experiments were conducted on sandstone under step by step loading, the creep data predicted by proposed damage creep model are well agreement with experimental creep data. And then, a comparative study with constant-order fractional damage creep model was performed to present the advantages of proposed Caputo variable-order fractional damage creep model, which gives further references for application of Caputo variable-order fractional derivative in rheological model. Finally, the variations and influence of elastic modulus and relaxation time on creep response based on proposed Caputo variable-order fractional damage creep model are discussed and expounded deeply.

Appendix

The discretization of Caputo VOF1 derivative is expressed as follows [26]:

$$ \begin{aligned} {}_{{{\text{VOF}}1}}^{C} D_{0}^{\alpha \left( t \right)} f\left( t \right) & = \mathop \int \limits_{0}^{t} \frac{1}{{{\Gamma }\left( {1 - \alpha \left( \tau \right)} \right)}}\frac{{f^{\prime}\left( \tau \right)}}{{\left( {t - \tau } \right)^{\alpha \left( \tau \right)} }}{\text{d}}\tau \\ & \approx \mathop \sum \limits_{i = 0}^{k} \mathop \int \limits_{i\tau }^{{\left( {i + 1} \right)\tau }} \frac{\partial f\left( \mu \right)}{{\partial \mu }}\frac{1}{{{\Gamma }\left( {1 - \alpha \left( {t_{i + 1} } \right)} \right)}}\frac{{{\text{d}}\mu }}{{\left( {t_{k + 1} - \mu } \right)^{{\alpha \left( {t_{i + 1} } \right)}} }} \\ & = \mathop \sum \limits_{i = 0}^{k} \frac{{f\left( {t_{i + 1} } \right) - f\left( {t_{i} } \right)}}{\tau }\mathop \int \limits_{i\tau }^{{\left( {i + 1} \right)\tau }} \frac{1}{{{\Gamma }\left( {1 - \alpha \left( {t_{i + 1} } \right)} \right)}}\frac{{{\text{d}}\mu }}{{\left( {t_{k + 1} - \mu } \right)^{{\alpha \left( {t_{i + 1} } \right)}} }} \\ & = \mathop \sum \limits_{i = 0}^{k} \frac{{f\left( {t_{i + 1} } \right) - f\left( {t_{i} } \right)}}{\tau }\mathop \int \limits_{{\left( {k - i} \right)\tau }}^{{\left( {k - i + 1} \right)\tau }} \frac{1}{{{\Gamma }\left( {1 - \alpha \left( {t_{k - i + 1} } \right)} \right)}}\frac{{{\text{d}}\varphi }}{{\varphi^{{\alpha \left( {t_{k - i + 1} } \right)}} }} \\ & = \mathop \sum \limits_{i = 0}^{k} \frac{{f\left( {t_{k - i + 1} } \right) - f\left( {t_{k - i} } \right)}}{\tau }\mathop \int \limits_{\left( i \right)\tau }^{{\left( {i + 1} \right)\tau }} \frac{1}{{{\Gamma }\left( {1 - \alpha \left( {t_{i + 1} } \right)} \right)}}\frac{{{\text{d}}\varphi }}{{\varphi^{{\alpha \left( {t_{i + 1} } \right)}} }} \\ & = \mathop \sum \limits_{i = 0}^{k} \frac{{f\left( {t_{k - i + 1} } \right) - f\left( {t_{k - i} } \right)}}{\tau }\frac{1}{{{\Gamma }\left( {1 - \alpha \left( {t_{i + 1} } \right)} \right)}}\mathop \int \limits_{\left( i \right)\tau }^{{\left( {i + 1} \right)\tau }} \frac{{{\text{d}}\varphi }}{{\varphi^{{\alpha \left( {t_{i + 1} } \right)}} }} \\ & = \mathop \sum \limits_{i = 0}^{k} \left[ {f\left( {t_{k - i + 1} } \right) - f\left( {t_{k - i} } \right)} \right]\frac{{\tau^{{ - \alpha \left( {t_{i + 1} } \right)}} }}{{{\Gamma }\left( {2 - \alpha \left( {t_{i + 1} } \right)} \right)}}\left[ {\left( {i + 1} \right)^{{1 - \alpha \left( {t_{i + 1} } \right)}} - i^{{1 - \alpha \left( {t_{i + 1} } \right)}} } \right] \\ \end{aligned} $$

(19)

where $\left( {k + 1} \right)\tau = t$ and $\tau$ is the time step.

The discretization of Caputo VOF2 derivative is expressed as follows [1]:

$$ \begin{aligned} {}_{{{\text{VOF}}2}}^{C} D_{0}^{\alpha \left( t \right)} f\left( t \right) & = \frac{1}{{{\Gamma }\left( {1 - \alpha \left( t \right)} \right)}}\mathop \int \limits_{0}^{t} \frac{{f^{\prime}\left( \tau \right)}}{{\left( {t - \tau } \right)^{\alpha \left( t \right)} }}{\text{d}}\tau \\ & \approx \frac{1}{{{\Gamma }\left( {1 - \alpha_{n} } \right)}}\mathop \sum \limits_{i = 0}^{n - 1} \mathop \int \limits_{i\Delta t}^{{\left( {i + 1} \right)\Delta t}} \frac{{f^{\prime}\left( \tau \right){\text{d}}\tau }}{{\left( {t - \tau } \right)^{{\alpha_{n} }} }} \\ & = \frac{1}{{{\Gamma }\left( {1 - \alpha_{n} } \right)}}\mathop \sum \limits_{i = 0}^{n - 1} \mathop \int \limits_{i\Delta t}^{{\left( {i + 1} \right)\Delta t}} \frac{{f_{i + 1} - f_{i} }}{\Delta t}\frac{{{\text{d}}\tau }}{{\left( {t - \tau } \right)^{{\alpha_{n} }} }} \\ & = \frac{1}{{{\Gamma }\left( {1 - \alpha_{n} } \right)}}\mathop \sum \limits_{i = 0}^{n - 1} \frac{{f_{i + 1} - f_{i} }}{\Delta t}\mathop \int \limits_{i\Delta t}^{{\left( {i + 1} \right)\Delta t}} \frac{{{\text{d}}\tau }}{{\left( {t - \tau } \right)^{{\alpha_{n} }} }} \\ & = \frac{1}{{{\Gamma }\left( {2 - \alpha_{n} } \right)\left( {\Delta t} \right)^{{\alpha_{n} }} }}\mathop \sum \limits_{i = 0}^{n - 1} \left( {f_{i + 1} - f_{i} } \right)\left[ {\left( {n - i} \right)^{{1 - \alpha_{n} }} - \left( {n - i - 1} \right)^{{1 - \alpha_{n} }} } \right] + \frac{{f_{n} - f_{n - 1} }}{{{\Gamma }\left( {2 - \alpha_{n} } \right)\left( {\Delta t} \right)^{{\alpha_{n} }} }} \\ \end{aligned} $$

(20)

where $\left( {k + 1} \right)\tau = t$ is the time step and $t = n\Delta t$

The R–L fractional integral with the order $\alpha$ is defined by

$$ D^{ - \alpha } f\left( t \right) = \frac{1}{\Gamma \left( \alpha \right)}\mathop \int \limits_{0}^{t} \left( {t - s} \right)^{\alpha - 1} f\left( s \right){\text{d}}s $$

(21)

And the R–L fractional derivative is expressed as follows:

$$ D^{\alpha } f\left( t \right) = \frac{{{\text{d}}^{n} \left[ {D^{{ - \left( {n - \alpha } \right)}} f\left( t \right)} \right]}}{{{\text{d}}t^{n} }} $$

(22)

where $D^{ - \alpha }$ and $D^{\alpha }$ are R–L fractional integral and derivative. And $n$ is positive integers.