1 Introduction

Time-dependent deformation properties of rocks are of fundamental importance for understanding the long-term behavior of rock mass in the underground mines, as well as predicting the long-term stability for rock engineering structures. Laboratory tests and in situ rheological observations are the major methods to study time-dependent deformation behaviors of rocks. Considerable creep tests (Scholz 1972; Heap et al. 2009, 2011) have been conducted, and three regimes (primary creep or transient creep, secondary creep or minimum creep, and tertiary creep or unstable creep) are usually observed during creep experiments. Many methods are used to model the time-dependent deformation of rocks. Constitutive laws based on laboratory experiments provide a relation between strain, stress, and strain rate (Voight 1989; Lockner 1998). Wang et al. (2016) simulated the time-dependent behavior of brittle rocks based on damage constitutive law at a mesoscale. Wang and Wong (2015) performed a series of triaxial compression creep tests of the saturated till to investigate the creep behavior at different effective stresses and strains. Zhao et al. (2017a, b) proposed a new viscoelastoplastic creep model to study the non-linear rheological mechanical properties of hard rocks under cyclic incremental loading and unloading. Zhao et al. (2017a, b) used an idealised microstructure mesh model (a random grains separated by idealised grain boundaries) to characterize crack growth at tertiary crack growth stage. Except for traditional experimental and creep model analysis, mathematical method of trust-region reflective least squares algorithm is used to obtain and optimize viscoplastic parameters at creep stage (Le and Fatahi 2016; Le et al. 2016). These models reproduce the creep behavior of different types of rocks under different loading conditions.

Recently, the numerical manifold method (NMM) (Shi 1992) has gained a wide attention and application in rock mechanics and rock engineering due to its efficient treatment to the problems involving continuous and discontinuous deformations in a unified way (Ning et al. 2011; An et al. 2014; Yang et al. 2016). He et al. (2013) put forward a customized contact algorithm in the 3-D NMM to study the jointed rock masses. Using time step-initial strain method, the creep theory is coupled with NMM to simulate the creep of rocks. High-order numerical manifold method is studied to improve the computational efficiency of the numerical manifold method, and solve the complicated geotechnical engineering problems (Liu and Chen 2012a, b).

Though great progress has been made in the application of NMM in rock engineering, there are still some difficult problems to be solved such as time-dependent deformation of rock. Thus, in the present paper, the ENM is incorporated into NMM to study the time-dependent deformation of stressed rock. The improved model is validated against experimental data and parameter sensitivity analysis is performed. Meanwhile, the influences of axial stress and confining pressure on time-dependent creep deformation are also investigated.

2 NMM-Based Nishihara Model

2.1 Numerical Manifold Method

Numerical manifold method is composed of block dynamics, cover system and numerical integration method. The block dynamics solve the mechanical behavior of block systems under loading. The cover system consists of a mathematical cover and a physical cover, and the simplex integration is an important method to calculate integral field, which transforms the area of polygon into many triangles and solve concave polygon problem that FEM cannot deal with, meanwhile, the analytic method is used to find the triangular integration of simplex. The mathematical and physical cover form the cover system of NMM, the former is composed of regular graph, and the latter consists of physical boundary. Compared with the differential manifold, the main function of NMM based on the finite covering technique is defined by the basis of covering and integration by parts. The function is almost discontinuous on the contact interface, however, the main function of differential manifold is highly differentiable, and it is independent on coverage.

In NMM, every physical cover set is a single cover displacement function. Main displacement function is composed of local displacement function, the main displacement function between the two sides of joint is discontinuous, which can adapt to wide variety from continuum to discontinuum medium. Cover functions on the physical cover Ui is ui(x)

$$ u\left( x \right) = \sum\limits_{i = 1}^{n} {w_{i} \left( x \right)u_{i} \left( x \right)} $$
(1)

where ui(x) can be constant, linear polynomial, high order polynomial and local series, ui(x) can connect with weight function wi(x). Main function u(x) is defined with wi(x) and ui(x). wi(x) means weighted average, which is the percent of each covering function ui(x) for all physical cover Ui containing x. As for each element, nodes have different weight functions with a summary of 1 in total.

$$ \sum\limits_{{(x,y) \in U_{j} }}^{{}} {w_{j} } \left( {x,y} \right) = 1 $$
(2)

local displacement function Ui of physical cover is defined ui(x,y) and vi(x,y),

$$ U_{i} = \left\{ {\begin{array}{*{20}c} {u_{i} \left( {x,y} \right)} \\ {v_{i} \left( {x,y} \right)} \\ \end{array} } \right\} = PD $$
(3)

where P is basic function of the cover function, D is displacement function. If the cover function is a first order function,

$$ U_{i} = \left\{ {\begin{array}{*{20}c} {u_{i} \left( {x,y} \right)} \\ {v_{i} \left( {x,y} \right)} \\ \end{array} } \right\} = \left( {\begin{array}{*{20}c} 1 & 0 & x & 0 & y & 0 \\ 0 & 1 & 0 & x & 0 & y \\ \end{array} } \right)\left\{ {\begin{array}{*{20}c} {d_{i1} } \\ {d_{i2} } \\ {d_{i3} } \\ {d_{i4} } \\ {d_{i5} } \\ {d_{i6} } \\ \end{array} } \right\} $$
(4)

and the whole displacement function U is calculated by wi(x,y) and Ui

$$ \begin{aligned} U = W_{i} {\kern 1pt} {\kern 1pt} U_{i} & = \sum\limits_{i = 1}^{n} {w_{i} \left( {x,y} \right)} \left\{ {\begin{array}{*{20}c} {u_{i} \left( {x,y} \right)} \\ {v_{i} \left( {x,y} \right)} \\ \end{array} } \right\} \\ & \quad {\kern 1pt} = \sum\limits_{i = 1}^{n} {\left[ {T{}_{i}\left( {x,y} \right)} \right]} \left\{ {D_{i} } \right\} \\ \end{aligned} $$
(5)

where T is displacement matrix. According to the minimum potential energy principle, total potential equation is

$$ KD = F $$
(6)

Simplex integration is the exact solution of general shape region in n-dimensional space, and the integration can be any n-dimensional polygon. One-dimensional simplex is line segment, two-dimensional simplex is directed triangle, and three-dimensional simplex is tetrahedron. Main integration difference between simplex and traditional is integration region. The simplex integration uses simplexes as integral region, positive and negative directions are designated as positive and negative volumes, respectively.

2.2 Formulation of Nishihara Creep Constitutive Model

The time-dependent deformation of rock is a common mechanical relationship between the magnitude of stress and strain and time in rock materials, which makes up of creep, stress relaxation and elastic aftereffect. The creep of rock is the most common phenomenon in rock engineering, so it is of great importance to study the creep deformation of rocks. Rheological component method is a common method that makes up of linear elastic spring, linearly viscous dashpot and slider. By combining these components, we can construct a model of linear viscoelasticity or viscoelastoplasticity. The models are all kinds of constitutive equations essentially, which can reflect rock deformation properties. The most representative creep models are the Generalized kelvin model, the Burgers model and the Nishihara model. The Generalized Kelvin model is a three-element model of a linear elastic spring and the other two-element (Kelvin model) in series. The Kelvin model making up of a spring and dash-pot in parallel, is oftern referred to as “solids”, since it reflects instantaneously as elastic materials and recover completely upon unloading, but this model can only simulate primary and secondary creep stages. The strain at secondary stage will not change and strain rate is zero. The Burgers model is a four-element model of a Maxwell and a Kelvin model. The difference between Generalized Kelvin and Burgers model is viscous creep strain at the secondary stage, the former holds constant and the latter increases with time. Traditional Nishihara model (Nishihara 1955) making up of Kelvin model and Bingham model in series, is a piecewise function and is unable to describe the accelerating stage of rock creep. The Bingham model just reflects the viscous deformation of the rock under a constant stress and strain rate is a certain definite value. However, the creep deformation of rock in actual condition changes against time. Only simulating the process of deformation, can we simulate the trend of rock failure. So, an extended Nishihara model is implemented in NMM to describe the accelerating creep properties of rock, in which a viscous component is replaced with the linear viscous components. In this way, ENM can be applied to analyze the problem of viscoelasto-ductile. In this part, ENM is considered to study the creep of rock based on NMM. The model of ENM is illustrated in Fig. 1.

Fig. 1.
figure 1

Extended Nishihara model (ENM)

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The differential piecewise creep constitutive equation is

$$ \varepsilon = \left\{ {\begin{array}{*{20}c} {\frac{{\sigma_{0} }}{{k_{1} }} + \frac{{\sigma_{0} }}{{\eta_{2} }}t + \frac{{\sigma_{0} }}{{k_{2} }}\left( {1 - e^{{ - \frac{{k_{2} }}{{\eta_{1} }}t}} } \right)\;,\quad {\kern 1pt} \sigma < \sigma_{s} {\kern 1pt} {\kern 1pt} } \\ {\frac{{\sigma_{0} }}{{k_{1} }} + \frac{{\sigma_{0} }}{{\eta_{2} }}t + \frac{{\sigma_{0} }}{{k_{2} }}\left( {1 - e^{{ - \frac{{k_{2} }}{{\eta_{1} }}t}} } \right) + \frac{{\left( {\sigma_{0} - \sigma_{s} } \right)t^{\text{n}} }}{{n!\eta_{ 3} }}\;,\quad \sigma \ge \sigma_{s} } \\ \end{array} } \right. $$
(7)

where k1 and k2 are spring stiffness, η1 and η2 are the viscosity of material, η3 is the viscosity of viscous component in the material, ε is total strain, n is time index (Yang et al. 2015), σs is the yield stress of plastic component. (σ0 − σs) × tn/n! × η3 in the equation denotes nonlinear component. Nonlinear means a relationship between stress and strain and strain rate in constitutive equation. Traditional creep model is linear model, which can simulate primary and secondary creep stage greatly, but it is unable to present creep failure process, so, nonlinear element can replace linear element to simulate creep nonlinear deformation. If σ < σs, ENM will reduce to Burgers model of a four-element model. If σ > σs, ENM will reflect rock deformation during accelerating creep stage, the dashpot η3 in Fig. 1 is the key component of deformation. In order to combine with NMM program, strain differential equation has to turn to the iteration by first order ordinary differential equation.

When σ < σs, the viscous strain iteration equation is shown in Eq. (8), which is the bridge of creep theory and NMM.

$$ \begin{aligned} & \varepsilon_{v} = \frac{{\sigma_{0} }}{{\eta_{2} }}t + \frac{{\sigma_{0} }}{{k_{2} }}\left( {1 - e^{{ - \frac{{k_{2} }}{{\eta_{1} }}t}} } \right)\quad \to e^{{ - \frac{{k_{2} }}{{\eta_{1} }}t}} = 1 - \frac{{k_{2} }}{{\sigma_{0} }}\left( {\varepsilon_{v} - \frac{{\sigma_{0} }}{{\eta_{2} }}t} \right) \\ & \varepsilon^{\prime}_{v} = \frac{{\sigma_{0} }}{{\eta_{2} }} + \frac{{\sigma_{0} }}{{\eta_{1} }}e^{{ - \frac{{k_{2} }}{{\eta_{1} }}t}} = \frac{{\sigma_{0} }}{{\eta_{2} }} + \frac{{\sigma_{0} }}{{\eta_{1} }} - \frac{{k_{2} }}{{\eta_{1} }}\varepsilon_{v} + \frac{{k_{2} \sigma_{0} }}{{\eta_{1} \eta_{2} }}t \\ & \varepsilon^{\prime}_{v} + \frac{{k_{2} }}{{\eta_{1} }}\varepsilon_{v} = \frac{{k_{2} \sigma_{0} }}{{\eta_{1} \eta_{2} }}t + \frac{{\sigma_{0} }}{{\eta_{2} }} + \frac{{\sigma_{0} }}{{\eta_{1} }} \\ \end{aligned} $$
(8)

According to the first order linear differential equation, the general solution of εv is

$$ \varepsilon_{v} = u(x)e^{{ - \frac{{k_{2} }}{{\eta_{1} }}t}} $$
(9)

Put Eq. (9) into Eq. (8),

$$ \begin{aligned} u^{\prime}(x) & = \frac{{k_{2} \sigma_{0} }}{{\eta_{1} \eta_{2} }}te^{{\frac{{k_{2} }}{{\eta_{1} }}t}} + \left( {\frac{{\sigma_{0} }}{{\eta_{2} }} + \frac{{\sigma_{0} }}{{\eta_{1} }}} \right)e^{{\frac{{k_{2} }}{{\eta_{1} }}t}} \\ u(x) & = \left( {\frac{{\sigma_{0} }}{{\eta_{2} }}t + \frac{{\sigma_{0} }}{{k_{2} }}} \right)e^{{\frac{{k_{2} }}{{\eta_{1} }}t}} \\ \end{aligned} $$
(10)
$$ \begin{aligned} & \varepsilon_{v} = u(x)e^{{ - \frac{{k_{2} }}{{\eta_{1} }}t}} { = }\frac{{\sigma_{0} }}{{\eta_{2} }}t_{1} + \frac{{\sigma_{0} }}{{k_{2} }}{\kern 1pt} {\kern 1pt} {\kern 1pt} \to {\kern 1pt} {\kern 1pt} {\kern 1pt} \varepsilon_{v} \left( {1 - e^{{ - \frac{{k_{2} }}{{\eta_{1} }}t}} } \right){ = }\left( {\frac{{\sigma_{0} }}{{\eta_{2} }}t_{1} + \frac{{\sigma_{0} }}{{k_{2} }}} \right)\left( {1 - e^{{ - \frac{{k_{2} }}{{\eta_{1} }}t}} } \right) \\ & \varepsilon_{v(t + \Delta t)} = \varepsilon_{v(t)} e^{{ - \frac{{k_{2} }}{{\eta_{1} }}\Delta t}} + \left( {\frac{{\sigma_{0} }}{{\eta_{2} }}t_{1} + \frac{{\sigma_{0} }}{{k_{2} }}} \right)A\left( {1 - e^{{ - \frac{{k_{2} }}{{\eta_{1} }}\Delta t}} } \right) \\ \end{aligned} $$
(11)

where t1 means total iterative time.

When σ > σs, ENM constitutive equation is the combination of Burgers and extended Bingham constitutive equation. For simplicity, we assume the time index n is 2.

$$ \begin{aligned} & \varepsilon_{v} = \frac{{\sigma_{0} }}{{\eta_{2} }}t + \frac{{\sigma_{0} }}{{k_{2} }}\left( {1 - e^{{ - \frac{{k_{2} }}{{\eta_{1} }}t}} } \right) + \frac{{\left( {\sigma_{0} - \sigma_{s} } \right)t^{2} }}{{2\eta_{ 3} }}\quad \to \;{\kern 1pt} e^{{ - \frac{{k_{2} }}{{\eta_{1} }}t}} = 1 - \frac{{k_{2} }}{{\sigma_{0} }}\left[ {\varepsilon_{v} - \frac{{\sigma_{0} }}{{\eta_{2} }}t - \frac{{\left( {\sigma_{0} - \sigma_{s} } \right)t^{2} }}{{2\eta_{ 3} }}} \right] \\ & \varepsilon^{\prime}_{v} = \frac{{\sigma_{0} }}{{\eta_{2} }} + \frac{{\sigma_{0} }}{{\eta_{1} }}\left[ {1 - \frac{{k_{2} }}{{\sigma_{0} }}\left( {\varepsilon_{v} - \frac{{\sigma_{0} }}{{\eta_{2} }}t - \frac{{\left( {\sigma_{0} - \sigma_{s} } \right)t^{2} }}{{2\eta_{ 3} }}} \right)} \right] \\ & \varepsilon^{\prime}_{v} + \frac{{k_{2} }}{{\eta_{1} }}\varepsilon_{v} = \frac{{\left( {\sigma_{0} - \sigma_{s} } \right)}}{{2\eta_{1} \eta_{2} }}k_{2} t^{2} + \left[ {\frac{{k_{2} \sigma_{0} }}{{\eta_{1} \eta_{2} }} + \frac{{\left( {\sigma_{0} - \sigma_{s} } \right)}}{{\eta_{ 3} }}} \right]t + \frac{{\sigma_{0} }}{{\eta_{2} }} + \frac{{\sigma_{0} }}{{\eta_{1} }} \\ \end{aligned} $$
(12)

Put Eq. (9) into Eq. (12),

$$ \begin{aligned} & u^{\prime}(x) = \left[ {\frac{{\left( {\sigma_{0} - \sigma_{s} } \right)}}{{2\eta_{1} \eta_{2} }}k_{2} t^{2} + \left[ {\frac{{k_{2} \sigma_{0} }}{{\eta_{1} \eta_{2} }} + \frac{{\left( {\sigma_{0} - \sigma_{s} } \right)}}{{\eta_{ 3} }}} \right]t + \frac{{\sigma_{0} }}{{\eta_{2} }} + \frac{{\sigma_{0} }}{{\eta_{1} }}} \right]e^{{\frac{{k_{2} }}{{\eta_{1} }}t}} \\ & u(x) = \left[ {\frac{{\left( {\sigma_{0} - \sigma_{s} } \right)}}{{2\eta_{ 3} }}t^{2} + \frac{{\sigma_{0} }}{{\eta_{2} }}t + \frac{{\sigma_{0} }}{{k_{2} }}} \right]e^{{\frac{{k_{2} }}{{\eta_{1} }}t}} \\ \end{aligned} $$
(13)
$$ \begin{aligned} & \varepsilon_{v} = \frac{{\left( {\sigma_{0} - \sigma_{s} } \right)}}{{2\eta_{ 3} }}t^{2} + \frac{{\sigma_{0} }}{{\eta_{2} }}t + \frac{{\sigma_{0} }}{{k_{2} }}{\kern 1pt} {\kern 1pt} \to {\kern 1pt} {\kern 1pt} {\kern 1pt} \varepsilon_{v} \left( {1 - e^{{ - \frac{{k_{2} }}{{\eta_{1} }}\Delta t}} } \right){ = }\left[ {\frac{{\left( {\sigma_{0} - \sigma_{s} } \right)}}{{2\eta_{ 3} }}t^{2} + \frac{{\sigma_{0} }}{{\eta_{2} }}t + \frac{{\sigma_{0} }}{{k_{2} }}} \right]\left( {1 - e^{{ - \frac{{k_{2} }}{{\eta_{1} }}\Delta t}} } \right) \\ & \varepsilon_{v(t + \Delta t)} = e^{{ - \frac{{k_{2} }}{{\eta_{1} }}\Delta t}} \varepsilon_{v(t)} + \left( {\frac{{\sigma_{0} }}{{k_{2} \left( {\sigma_{0} - \sigma_{s} } \right)}} + \frac{{\sigma_{0} }}{{\eta_{2} \left( {\sigma_{0} - \sigma_{s} } \right)}}t_{1} { + }\frac{{t_{1}^{n} }}{{n!{\kern 1pt} \eta_{ 3} }}} \right)A{\kern 1pt} {\kern 1pt} \left( {\sigma_{0} - \sigma_{s} } \right)\left( {1 - e^{{ - \frac{{k_{2} }}{{\eta_{1} }}\Delta t}} } \right) \\ \end{aligned} $$
(14)

So, the total viscous strain iteration equation is:

$$ \varepsilon_{v(t + \Delta t)} = \left\{ {\begin{array}{*{20}c} {e^{{ - \frac{{k_{2} }}{{\eta_{1} }}\Delta t}} {\kern 1pt} {\kern 1pt} \varepsilon_{v(t)} + \left( {\frac{1}{{k_{2} }} + \frac{{t_{1} }}{{\eta_{2} }}} \right)A{\kern 1pt} {\kern 1pt} \sigma_{0} \left( {1 - e^{{ - \frac{{k_{2} }}{{\eta_{1} }}\Delta t}} } \right)\;,\quad \sigma < \sigma_{s} {\kern 1pt} } \\ {e^{{ - \frac{{k_{2} }}{{\eta_{1} }}\Delta t}} {\kern 1pt} {\kern 1pt} \varepsilon_{v(t)} + \left( {\frac{{\sigma_{0} }}{{k_{2} \left( {\sigma_{0} - \sigma_{s} } \right)}} + \frac{{\sigma_{0} }}{{\eta_{2} \left( {\sigma_{0} - \sigma_{s} } \right)}}t_{1} { + }\frac{{t_{1}^{n} }}{{{\kern 1pt} n!\eta_{ 3} }}} \right)A{\kern 1pt} {\kern 1pt} \left( {\sigma_{0} - \sigma_{s} } \right)\left( {1 - e^{{ - \frac{{k_{2} }}{{\eta_{1} }}\Delta t}} } \right)\;,\quad \sigma \ge \sigma_{s} } \\ \end{array} } \right. $$
(15)

where Δt is the time step of creep simulation. A is constant matrix connected with Poisson’s ratio υ, A in the plane strain problem is

$$ A = \left[ {\begin{array}{*{20}c} 1 & {\frac{u}{u - 1}} & 0 \\ {\frac{u}{u - 1}} & 1 & 0 \\ 0 & 0 & {\frac{2}{1 - \mu }} \\ \end{array} } \right] $$
(16)

If σ < σs, unloading the stress at time t1, the elastic deformation of linear elastic spring recovers immediately, axial creep strain will take a long time to recover. According to the principle of stress superposition, when the time t > t1, the stress (−σ0) is coupled with σ0, the unloading equation is

$$ \varepsilon_{v(t + \Delta t)} = e^{{ - \frac{{k_{2} }}{{\eta_{1} }}\Delta t}} {\kern 1pt} {\kern 1pt} \varepsilon_{v(t)} {\kern 1pt} + A{\kern 1pt} {\kern 1pt} \sigma_{0} \frac{{t_{1} }}{{\eta_{2} }}\left( {1 - e^{{ - \frac{{k_{2} }}{{\eta_{1} }}\Delta t}} } \right) $$
(17)

Viscous strain will generate the additional viscous load F′ according to the strain energy equation,

$$ \begin{aligned} \left\{ {D_{e} } \right\}^{T} \left[ {F^{\prime} } \right] & = \iint\limits_{\varOmega } {\frac{1}{2}\left[ \varepsilon \right]}^{T} \left[ \sigma \right]dxdy \\ & \quad = \iint\limits_{\varOmega } {\frac{1}{2}}\left\{ {D_{e} } \right\}^{T} \left[ {B_{e} } \right]^{T} \left[ E \right]{\kern 1pt} {\kern 1pt} \varepsilon^{\prime} {\kern 1pt} {\kern 1pt} dxdy \\ \to \quad \left[ {F^{\prime} } \right] & = \iint\limits_{\varOmega } {\left[ {B_{e} } \right]^{T} \left[ E \right]{\kern 1pt} {\kern 1pt} \varepsilon^{\prime} {\kern 1pt} {\kern 1pt} dxdy} \\ \end{aligned} $$
(18)

where Be is strain matrix, E is elastic stiffness matrix, Ω is integral area. By putting the viscous load into total potential energy equation (Eq. 6), we will get De, and then we will get displacement U by Eq. (4).

2.3 Implementation of ENM into NMM

Time-dependent strain can be divided into a series of incremental values in each time step, and parameters are regarded as constants in each incremental time step, that is to say, a series of linear problems are used to approximate the solution of problem, this approach is named as time step-viscous strain method. For time-dependent creep problem, we add viscous strain to each time step and solve creep problems finally, then we get the relationship between time and both stress and strain, as well as the deformation of rock.

We take two steps to incorporate time-dependent creep into NMM. Firstly, the program will calculate the initial elastic deformation of rock specimen under constant stress, the system will gradually balance according to block dynamics, and calculate elastic strain. The time of elastic strain is negligible comparing with creep time, so it can be regarded as instantaneous. Secondly, according to viscous strain iteration equation (Eq. 15) and strain energy equation (Eq. 18), new additional viscous load will break the balance of system, the new viscous displacement is generated until the system balances again. The stress acting on the specimen will induce a deformation, which increase to a specified limit (i.e. the strain rate approaches to a constant) with time.

The improved NMM program is used to simulate the time-dependent deformation of rock specimen under constant stress loading and unloading condition. In order to obtain the input parameters of rock samples, the uniaxial compressive experiment was conducted numerically. The sandstone specimen of 50 mm in diameter and 100 mm in length is divided into 236 elements by physical cover system of NMM. The physical and mechanical parameters are listed in Table 1 (Liu et al. 2012). The specimen is under the single-stage axial stress load of 40 MPa, the bottom of the specimen is fixed, and the both sides of the specimen are free. Volumetric creep is the sum of axial strain and lateral strain of specimen under triaxial compression condition. The volumetric strain experiences a non-linear change process. Shear creep is the study of the creep deformation of specimens under the action of constant shear stress in direct shear test. The creep in this paper is mainly the axial creep under constant axial stress of uniaxial compressive condition. Figure 2 shows the simulation model under uniaxial compression test. The measuring point A (25,100) at the middle upper end of specimen is set to record the deformation of specimen under loading. Normal direction is same as cartesian coordinate system.

Table 1. Physico-mechanical parameters of numerical model
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Fig. 2.
figure 2

Numerical model in uniaxial compression

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Traditional Nishihara model cannot capture accelerating creep stage, so, the linear Newton component in Fig. 1 is replaced by a new nonlinear viscoelastoplastic component. According to the viscous strain iteration equation (Eq. 15), we applied 40 MPa axial stress and 36 MPa yield stress. We get the creep strain curve of ENM in Fig. 3, which shows the changing process of creep strain, strain rate and strain acceleration against time, in which the strain will go through the primary, secondary and tertiary creep stages. At primary creep, with the decrease of strain rate and strain acceleration, the axial strain increased slowly, at secondary creep, strain increases with time ae a constant strain value, and at tertiary creep, the strain increases faster with the increasing of strain rate and strain acceleration.

Fig. 3.
figure 3

Axial strain, strain rate and strain acceleration curve

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Plasticity is the property that the deformation of an object after being unloaded cannot be fully recovered, this property has been considered in this paper. According to the unloading equation (Eq. 17) and the creep program of NMM, we can get the unloading curve of the specimen in Fig. 4. Elastic deformation OA was produced by spring k1, which was generated immediately when stress applied to the specimen. Viscous strain AB was transient creep stage, the longer the time, the larger the strain, and the slower the strain rate until approaching to zero. Then the deformation would go into a minimum strain rate stage. If unloading the stress at time t1, elastic strain would recover immediately at CD, viscous strain would slowly return to zero over time, DE was strain recovery stage, the strain generated in minimum creep would not recover.

Fig. 4.
figure 4

Stress unloading curve of extended Nishihara model under uniaxial compression

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Many parameters in ENM control the curve, different parameters have different effects on experimental results. Parameter sensitivity analysis is performed to study the influence of parameters on calculation result. The parameters of elastic modulus, viscosity coefficient, Poisson’s ratio and time index are shown in Table 2. The values of parameters are −40, −20, 0, 20, 40% of initial values, respectively. Axial stress is 40 MPa, yield stress is 36 MPa, the effect of the parameters on experimental creep results are shown in Fig. 5.

Table 2. Parametric sensitivity analysis scheme
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Fig. 5.
figure 5

Strain-time curves with different parameters

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In Fig. 5, the elastic strain of rock specimen increases with the decreasing of spring stiffness k1. It is indicated that k1 is negative with the axial strain, and has little influence to strain rate and strain acceleration, while, which has more effect on elastic strain. Spring stiffness k2 is a complicated parameter, accelerating strain rate increases with the increase of k2 and the creep strain also increases, it has no influence on elastic, decelerating and minimum creep strain, but it has a significant impact on tertiary creep stage. Both viscosity η1 and η2 have the same effect on creep strain, the elastic strain doesn’t change, the creep strain will decrease at the same time but not at the same amplitude with the increasing of viscosity, the influence of η1 on strain is more remarkable than that of η2. As for viscosity coefficient η3, we can find that there are more remarkable effect on accelerating creep stage, the smaller the η3, the larger of tertiary creep strain and the strain rate. Time index n is a factor controlling the strain acceleration, the effect of time index on tertiary strain is positive if n is greater than 1. The effect of Poisson’s ratio υ is just like η3, the smaller Poisson’s ratio υ, the greater the tertiary creep strain.

3 Numerical Simulation of Creep Deformation

3.1 Model Set-up

The improved NMM program is validated against the experimental results of sandstone tested by a servo-controlled rock testing system. The geometrical dimension of red sandstone specimen is 50 mm in diameter and 100 mm in length. Figure 6 is the complete stress-strain curve of red sandstone in uniaxial compression test (Zhao et al. 2013). The uniaxial compressive strength, elastic modulus and Poisson’s ratio of the sandstone specimen are 58.37 MPa, 9.37 GPa and 0.14, respectively. The numerical model of specimen is the same as the model in Fig. 2. The multi-step axial stresses of 28.03, 33.63, 39.24, 44.84 and 50.45 MPa are applied on the top of the specimen. For triaxial compressive creep tests, confining pressure is applied to the present value and hold constant. After axial stress loading, the confining pressure will be used to test the characteristics of rock. Creep test are performed to study the influence of different confining pressures on the creep behavior of the sandstone specimen.

Fig. 6.
figure 6

Stress-strain curve in uniaxial compression

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3.2 Modelling of Creep Under Uniaxial Stress

Multi-step creep experimental curve is shown in Fig. 7. The load at each stress-step is maintained constant for several hours until the specimen fails. At the first 2.65 h, the stress remains at 28.03 MPa, instantaneous displacement is 0.20 mm, creep displacement is 0.013 mm. When the stress increases to 33.63 MPa, creep deformation in rock specimen increases gradually, instantaneous axial displacement is 0.0022 mm, creep displacement is 0.011 mm, this period lasts 3.26 h. When the stress increases to 39.24 MPa, instantaneous displacement is 0.024 mm, creep displacement is 0.012 mm and creep time will continue 3.68 h. The next stage will last 4.62 h, stress is 44.84 MPa, instantaneous displacement is 0.028 mm, creep displacement is 0.014 mm. Under the stress level of 50.45 MPa, the creep curve at every stage makes up of the primary creep and secondary creep phase. When the stress is up to 50.45 MPa, accelerating stage, can be found and the specimen finally fails. Based on numerical model and ENM, numerical simulations will be performed on red sandstone at laboratory test. Axial stresses of 28.03, 33.63, 39.24, 44.84 and 50.45 MPa are applied on model specimen. The influence of the model parameters on the axial strain is shown in Fig. 5, some adjustments to the parameters (Zhao et al. 2013) are made to simulate the creep of rock. The input mechanical parameters for the numerical model based on the least squares method are listed in Table 3.

Fig. 7.
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Multi-step creep test curve

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Table 3. Physico-mechanical parameters of numerical model
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The simulation results are shown in Fig. 8. When σ < 50.45 MPa, the simulated curve is in good agreement with the test curve at each stage. With the increase of the stress, creep deformation becomes more and more obvious and the numerical creep curve agrees well with experimental curve. When σ > 50.45 MPa, the specimen finally fails, and creep curve exhibits typical three creep stage, and axial strain increases sharply at tertiary creep stage. Figure 9 is the effect of axial stress on minimum axial strain rate. We can clearly see that axial stress has a great influence on the minimum axial creep strain rate, the small increase of axial stress leads to a great increase of the minimum axial creep strain rate, a very small stress perturbation can have a great effect on the minimum creep strain rate. Figure 10 is creep trimodal strain curve under the axial stress of 50.45 MPa, the specimen undergoes three stages of deformation, i.e., primary creep, secondary creep, and tertiary creep, and finally fails. Figure 11 is the strain rate curve under axial stress of 50.45 MPa, it is a classic bathtub curve. We can find that primary creep phase is characterized by an initially high strain rate that decreases with time to reach a quasilinear secondary phase that is often interpreted as minimum creep. After an extended period of time, a tertiary creep phase is reached, characterized by the accelerating strain. This eventually results in macroscopic failure of the samples by propagation of a shear fault. The displacement of the specimen under axial stress of 44.84 MPa is balanced in Fig. 12. The displacement of the specimen decreases from the top to the bottom of the specimen. The maximum displacement is 0.32 mm, it is similar to the experimental strain curve of the rock sample under axial stress of 44.84 MPa.

Fig. 8.
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Comparisons between numerical and experimental curves for multi-step creep

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Fig. 9.
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Effect of axial stress on minimum axial strain rate

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Fig. 10.
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Creep curve under 50.45 MPa

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Fig. 11.
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Creep strain rate curve under 50.45 MPa

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Fig. 12.
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Stress cloud image of rock specimen under uniaxial compression

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3.3 Modelling of Creep Under Confining Pressures

To study the influence of confining pressure on rock creep, we use stress-stepping method to investigate the influence of confining pressure at 5, 10 and 25 MPa. The axial stresses of 28.03, 33.63, 39.24 and 44.84 MPa are applied on the specimen. Axial creep strain curves under different confining pressures are plotted in Fig. 13. It can be seen that the variation of creep strain in different stress stages and confining pressures. The red circle line stands for the confining pressure of 5 MPa, the blue up triangle line represents confining pressure of 10 MPa, the pink down triangle line represents confining pressure of 25 MPa, and the black square line stands for controlled test with the confining pressure of 0 MPa. As the stress increases, axial creep strain increases gradually. The greater the stress, the grater the strain. We can see from Fig. 19 that as the applied axial stress increases, axial creep strain increase gradually, the greater the applied axial stress, the greater the axial creep strain. While the strain is decreased with the increasing confining pressure, the greater the confining pressure, the smaller the axial strain value. Comparing with strain values in four stress levels under different constant confining pressures, we can find that with the increase of confining pressure, the strain has a certain reduction, but the confining pressure does not change the creep deformation trend, the greater the confining pressure, the greater the strain reduction. Figure 14 is the effect of confining pressure on minimum axial strain rate. We can see clearly from Fig. 14 that confining pressure has a weakening effect on minimum axial strain rate under the same axial stress condition, which inhibits the increase of axial strain. On the other hand, the steady strain rate increases with the increase of axial stress under the same confining pressure condition, the greater the axial stress, the higher the steady strain rate.

Fig. 13.
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Influence of confining pressure on axial creep strain behavior of sandstone specimen at 5, 10 and 25 MPa

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Fig. 14.
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Effect of confining pressure on minimum axial strain rate

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4 Discussions

We have present an ENM for creep to replicate time-dependent deformation under loading conditions and different confining pressures. In order to reflect the accelerating creep stage, the linear Newton component was replaced by a new viscoelastoplastic component, to achieve the tertiary creep. Traditional creep differential equation cannot couple with numerical manifold method, and the creep time is much longer than the time step of numerical manifold method. To solve this problem, there are two methods (time step-viscous initial stress and linear non-homogeneous differential equation) to deal with them. The former is used to divide a series of incremental values in each time step, and the latter is used to change creep differential equation to iteration equation, which can couple with NMM. After the construction of constitutive model, we performed parametric sensitivity analysis to study the correctness of model and characteristics of creep. By this way, we find that spring stiffness k1 plays a major role in elastic deformation, k2 has a great influence on tertiary creep stage, with the increase of k2, the strain gradually decreases. Both viscosity η1 and η2 affect the creep strain of first and secondary creep stages, which is the same as k2, but the influence of η1 was greater than that of η2. As viscoelastoplastic component, η3 mainly affects the tertiary creep stage, with the increase of η3, the strain and strain rate gradually decreases. Time index n is a factor controlling the accelerating strain rate, if the iteration time was larger than 1 (based on current unit), the larger the factor n, the larger the accelerating rate. Poisson’s ratio has little effect on primary and secondary creep stage, with the increase of υ, the tertiary creep strain gradually decreases. After the sensitivity analysis of parameter, we perform a comparative test between laboratory tests and numerical simulation under axial pressure and confining pressure by the improved NMM. According to the results, we can see that the simulation results are in good agreement with the test, the ENM can greatly simulate the trend of creep deformation. The present creep model allows us to investigate the creep deformation with different axial stresses and confining pressures, and we will get the full strain deformation curve to study the creep behavior of rock specimen.

5 Conclusions

In the present paper, the Nishihara creep constitutive equations are incorporated into numerical manifold method by rheological differential equations on MATLAB platform to study the creep properties of rocks, in which the linear Newton component is replaced by a viscoelastoplastic component (in Fig. 1). By this way, the tertiary creep is achieved in the program. In the improved model, the viscous strain is devided into a series of increment time-step values due to time inconsistency between the creep theory and NMM, and the relative parameters are regarded as constants in each time-step increment, this method can solve the time problem between creep and NMM. Parameter sensitivity analysis is performed and the improved NMM is validated against experimental data. Parameter sensitivity analysis reveals the influence of different parameters on the creep deformation. Meanwhile, the influences of axial stress and confining pressure on time-dependent creep deformation are also investigated. The fact that numerical simulations agree well with experimental results shows that the improved NMM based Nishihara model is suitable for modelling of time-dependent creep deformation of rocks.