1 Introduction

In recent years, more and more high-fill embankments and dams in mountainous areas have been planned and constructed [13,14,15, 22, 29]. Rockfills have been extensively used for the construction of these high-fill embankments and dams. In general, rockfills exhibit many complex mechanical properties, i.e., nonlinear stress–strain behavior, stress-level-dependent behavior, and other properties, which have been investigated extensively in past studies [2, 5, 31, 32]. Furthermore, these embankments and dams have been observed to deform with time [8, 19, 28], which may cause serviceability problems and even failures. To ensure the long-term safety of such high-fill structures, a study on the creep characteristics of rockfills is desirable. Finite element analysis is usually adopted for realistic deformation prediction of these engineering problems, which requires an accurate constitutive model that considers the time-dependent properties of rockfills [4, 7, 24,25,26, 30, 35]. Discrete element method has also been used in recent years to study the micro-mechanism of the creep behavior of rockfills [21, 33, 38].

Laboratory tests have been conducted to study the time-dependent behavior of rockfills [3, 6, 10, 11]. Anhdan et al. [3] investigated the viscous behavior of gravelly soils subjected to drained triaxial compression and found that particle crushing produces more creep deformation at higher stresses. Enomoto et al. [10] conducted a large-scale triaxial compression test on an air-dried well-graded Shinanogawa riverbed gravel, which consists of round gravels and relatively angular sand particles. Creep deformation was revealed to increase with increasing sustained deviator stress levels, which is defined as \(D = (\sigma_{1} - \sigma_{3} )/(\sigma_{1} - \sigma_{3} )_{f}\), where \((\sigma_{1} - \sigma_{3} )_{f}\) is the peak deviator stress. Chen and Zhang [6] developed a new large triaxial creep apparatus, and tertiary rheological behavior was observed when redstone granular soils were subjected to a high deviator stress level (\(D = 0.8\)). Enomoto et al. [11] evaluated the creep failure behavior of a natural gravelly soil by performing drained sustained tests at the prescribed stress states after monotonic loading at different loading rates. These test results indicate that the creep behaviors of rockfills are influenced by the stress state (deviator stress level and confining pressure) and loading rate.

Based on experimental observations, typical strain–time curves for rockfills are presented in Fig. 1, where the creep deformation can be generally categorized into three stages: (1) the attenuated creep stage with a decreasing strain rate (segment AB); (2) the steady creep stage with a constant strain rate (segment BC); and (3) the accelerated creep stage with an increasing strain rate (segment CE).

Fig. 1
figure 1

Schematic view of the strain–time curve for rockfills

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Developing appropriate constitutive models to simulate the creep behavior of rockfills based on physical tests is important. In general, the creep models of geomaterials can be classified into three categories: empirical models, rheological models, and general stress–strain–time models [20]. Zhou et al. [37] proposed an empirical model based on experimental results, in which the creep strain of rockfills is expressed as a function of elapsed time, confining pressure, and deviator stress level. Then, a numerical simulation procedure was proposed such that this empirical model can be used with the general finite element method to predict the long-term deformation of high rockfill dams. The differential approach of the rheological model (i.e., Maxwell model, Kelvin model, Burgers model, etc.) is composed of elastic springs, plastic sliders, and viscous dashpots. These types of models have been extensively used in geotechnical engineering due to their simple creep constitutive equations and the clear physical meanings of the model parameters. However, most rheological models have been developed for rock and fine-grained soils (e.g., sand, clay, etc.). Few rheological models can generate a reasonable simulation of the shear creep behavior of rockfills, in which particle breakage plays an important role and is influenced by the stress states. Furthermore, the accelerated creep behavior of rockfills has only been revealed in the laboratory in recent years [6], which also need to be considered in a constitutive model.

To describe the shear creep behavior of rockfills under different stress states, a rheological model is proposed in this paper. The influence of deviator stress levels and confining pressures on the shear creep characteristics of rockfills is investigated by analyzing the results of large-scale triaxial creep tests on different rockfills. A method to modify the key model parameters so that such an influence can be considered is proposed and incorporated into the proposed model. Furthermore, the accelerated creep strain of rockfills is reproduced by introducing a nonlinear elasto-damage unit. A simple calibration procedure is proposed to determine model parameters from experimental results. Compared with the results of large-scale triaxial creep tests on three types of rockfills, this simple model proves that it can predict the shear creep behavior of rockfills with fairly high accuracy.

2 Framework of this paper

The proposed model for rockfills is modified from the classical Burgers model and is schematically characterized by three modified components connected in series, including a nonlinear elasto-damage unit, a modified Newton unit and a modified Kelvin unit, as seen in Fig. 2. The total strain \(\varepsilon_{i}\) at time t is the sum of the strains of these three components:

$$\varepsilon_{i} = \varepsilon_{i}^{D} + \varepsilon_{i}^{N,M} + \varepsilon_{i}^{K,M}$$
(1)

where \(\varepsilon_{i}^{D}\) is the strain of the nonlinear elasto-damage unit, \(\varepsilon_{i}^{N,M}\) is the strain of the modified Newton dashpot unit, \(\varepsilon_{i}^{K,M}\) is the strain of the modified Kelvin unit, and subscript \(i = 1,2,3\). It should be noted that the superscripts D, N, and K denote the damage, Newton, and Kelvin units, respectively. The superscript M denotes that the unit has been modified from the original unit.

Fig. 2
figure 2

Schematic view of the shear creep constitutive model for rockfills

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This paper is structured as follows. In Sect. 3, a calibration method is proposed to determine the parameters of the classical Burgers model. In Sect. 4, the classical Burgers model is modified to consider the influence of the confining pressure and deviator stress level. In Sect. 5, based on the Kachanov damage mechanism, a nonlinear elasto-damage unit is proposed to capture the accelerated creep behavior of rockfills. In Sect. 6, a brief summary of the shear creep model is presented. Finally, in Sect. 7, the results of three sets of large-scale triaxial creep tests are used to validate the effectiveness of the proposed model.

3 Calibration method for the classical Burgers model

The classical Burgers model consists of a linear spring unit (Young’s modulus \(E\) and Poisson’s ratio \(\upsilon\)), a Newton unit (viscosity coefficient \(\eta^{N}\)), and a Kelvin unit (shear modulus \(G^{K}\) and viscosity coefficient \(\eta^{K}\)), as illustrated in Fig. 3. Thus, the classical Burgers model is a five-parameter model (\(G^{K}\), \(\eta^{K}\), \(\eta^{N}\), \(E\), and \(\upsilon\)). The Poisson ratio \(\upsilon\) has a minor influence on the simulation results and is usually set as 0.3 for rockfills.

Fig. 3
figure 3

Schematic view of the classical Burgers model

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The total strain \(\varepsilon_{i}^{B}\) at time \(t\) is expressed in Eq. (2):

$$\varepsilon_{i}^{B} = \varepsilon_{i}^{E} + \varepsilon_{i}^{N} + \varepsilon_{i}^{K}$$
(2)

where \(\varepsilon_{i}^{B}\) is the strain of the Burgers unit, \(\varepsilon_{i}^{E}\) is the elastic strain of the linear spring unit, \(\varepsilon_{i}^{N}\) is the viscous strain of the Newton unit, and \(\varepsilon_{i}^{K}\) is strain of the Kelvin unit. The superscript B denotes the Burgers unit. The constitutive equations of these three units were presented by Findley et al. [12] as:

$$\varepsilon_{i}^{E} = \frac{{\left( {1 + \upsilon } \right)\sigma_{i} - \upsilon \sigma_{kk} }}{E}$$
(3)
$$\varepsilon_{i}^{N} = \frac{{q_{i} }}{{2\eta^{N} }}t$$
(4)
$$\varepsilon_{i}^{K} = \frac{{q_{i} }}{{2G^{K} }}\left[ {1 - \exp \left( { - \frac{{G^{K} }}{{\eta^{K} }}t} \right)} \right]$$
(5)
$$\varepsilon_{i,t \to \infty }^{K} = \frac{{q_{i} }}{{2G^{K} }}$$
(6)

where \(\sigma_{i}\) is the principal stress; \(\sigma_{kk} = \sigma_{1} + \sigma_{2} + \sigma_{3}\); \(q_{i}\) is the deviator stress, \(q_{i} = \sigma_{i} - \sigma_{kk} /3\). As shown in Fig. 3, the segment OA represents the instantaneous elastic strain, which can be calculated from Eq. (3). By combining Eqs. (2)–(5), the strain of the classical Burgers model at time t is obtained as:

$$\varepsilon_{i}^{B} = \frac{{\left( {1 + \upsilon } \right)\sigma_{i} - \upsilon \sigma_{kk} }}{E}{ + }\frac{{q_{i} }}{{2\eta^{N} }}t + \frac{{q_{i} }}{{2G^{K} }}\left[ {1 - \exp \left( { - \frac{{G^{K} }}{{\eta^{K} }}t} \right)} \right]$$
(7)

Then, the creep rate can be expressed as:

$$\dot{\varepsilon }_{i}^{B} = \frac{{q_{i} }}{{2\eta^{N} }} + \frac{{q_{i} }}{{2\eta^{K} }}\exp \left( { - \frac{{G^{K} }}{{\eta^{K} }}t} \right)$$
(8)

The creep rates at \(t = 0^{ + }\) and \(t = \infty\) are presented in Eqs. (9) and (10), respectively:

$$\dot{\varepsilon }_{i}^{B} \left( {0^{ + } } \right) = \frac{{q_{i} }}{{2\eta^{N} }} + \frac{{q_{i} }}{{2\eta^{K} }} = \tan \alpha$$
(9)
$$\dot{\varepsilon }_{i}^{B} (\infty ) = \frac{{q_{i} }}{{2\eta^{N} }} = \tan \beta$$
(10)

A calibration method is proposed here to determine the four parameters (\(G^{K}\), \(\eta^{K}\), \(\eta^{N}\), and \(E\)) from the results of the creep test on redstone granular soil (\(\sigma_{3} = 100\,{\text{kPa}}\) and \(D = 0.6\)) performed by Chen and Zhang [6], as shown in Fig. 4.

Fig. 4
figure 4

Procedure to determine the parameters of the classical Burgers model for redstone granular soil (\(D = 0.6\) and \(\sigma_{3} = 100\,{\text{kPa}}\), [6])

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3.1 The Young’s modulus \(E\)

In Fig. 4, by substituting the instantaneous strain (\(\varepsilon_{1}^{E} = 1.23\%\), i.e., OA in Fig. 3) and the stress states (\(\sigma_{1} = 684\,{\text{kPa}}\), \(\sigma_{3} = 100\,{\text{kPa}}\), \(D = 0.6\)) into Eq. (3), the Young’s modulus \(E\) can be obtained.

3.2 The Newton viscosity coefficient \(\eta^{N}\)

An asymptote to the creep test curve is drawn in Fig. 4, where \(\beta\) is the incline angle of the asymptote. Thus, by substituting the value of \(\beta\) (\(5.5^{ \circ }\)) and \(q_{1} = 2(\sigma_{1} - \sigma_{3} )/3 = 390\,{\text{kPa}}\) into Eq. (10), the value of \(\eta^{N}\) can be calculated.

3.3 The Kelvin shear modulus \(G^{K}\)

By substituting the stable Kelvin strain (\(\varepsilon_{1,t \to \infty }^{K} = 1.15\%\), i.e., AB in Fig. 3) and the corresponding deviator stress value (\(q_{1} = 390\,{\text{kPa}}\)) into Eq. (6), the value of \(G^{K}\) can be obtained.

3.4 The Kelvin viscosity coefficient \(\eta^{K}\)

As shown in Eq. (9), the \(\eta^{K}\) value can be back-calculated when the initial tangent angle \(\alpha\) is determined. However, it is difficult to determine \(\alpha\) directly from the test data, as shown in Fig. 4. Thus, Eq. (5) is rewritten as:

$$t = \eta^{K} \left[ { - \frac{1}{{G^{K} }}\ln \left( {1 - \varepsilon_{1}^{K} \frac{{2G^{K} }}{{q_{1} }}} \right)} \right]$$
(11)

The axial Kelvin strain (\(\varepsilon_{1}^{K}\)) at time t can be separated from the total axial strain according to Eqs. (2)–(4). Therefore, t is plotted against \(- \frac{1}{{G^{K} }}\ln \left( {1 - \varepsilon_{1}^{K} \frac{{2G^{K} }}{{q_{1} }}} \right)\) in Fig. 5, and the slope of the fitting line equals the value of \(\eta^{K}\).

Fig. 5
figure 5

\(t - \left[ { - \frac{1}{{G^{K} }}\ln \left( {1 - \varepsilon_{1}^{K} \frac{{2G^{K} }}{{q_{1} }}} \right)} \right]\) curve and the fitting results for redstone granular soil (\(D = 0.6\) and \(\sigma_{3} = 100\,{\text{kPa}}\), [6])

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Using the above calibration method, the four parameters are obtained and summarized in Table 1.

Table 1 Parameters of the classical Burgers model for the creep test of the redstone granular soil (\(D = 0.6\) and \(\sigma_{3} = 100\,{\text{kPa}}\), [6])
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4 Modified Burgers model for rockfills without considering the accelerated creep behavior

The results of large-scale triaxial creep tests on three different types of rockfills are analyzed, and the parameters of the classic Burgers model are determined for each individual test. The influence of the deviator stress level and confining pressure on these parameters is investigated, based on which a modified Burgers model is proposed. Only the deformations of the first two creep stages (i.e., segments AB and BC in Fig. 1) are investigated here, whereas the third creep stage (i.e., the accelerated creep stage) will be further discussed in Sect. 5.

Information regarding these three sets of tests [6, 27, 36] is summarized in Table 2. By using the calibration method proposed in Sect. 3, the parameters (\(\eta^{N}\), \(\eta^{K}\), and \(G^{K}\)) of these rockfills are determined from each individual test (i.e., with different deviator stress levels and confining pressures). As shown in Tables 3, 4, and 5, these parameters vary with the deviator stress level and confining pressure. Thus, this section aims to find the mathematical relation between these parameters and the stress states. Furthermore, the parameter \(E\) is also modified to describe the nonlinear stress–strain behavior of rockfills.

Table 2 Initial properties of the specimens and testing programs of the triaxial creep tests
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Table 3 \(\eta^{N}\) values under different confining pressures and deviator stress levels for the three rockfills [6, 27, 36]
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Table 4 \(\eta^{K}\) values under different confining pressures and deviator stress levels for the three rockfills [6, 27, 36]
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Table 5 \(G^{K}\) values under different confining pressures and deviator stress levels for the three rockfills [6, 27, 36]
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4.1 Modification of the Young’s modulus \(E\)

Figure 6a, b illustrates the relationship between the deviator stress \((\sigma_{1} - \sigma_{3} )\) and the transient axial strain at different confining pressures for the rockfills from Shangzhai [36] and redstone granular soil [6], respectively. These two rockfills exhibit clear nonlinearities and stress-dependent stiffness. However, a constant Young’s modulus is adopted in the classical Burgers model, which is clearly unsuitable for rockfills. Thus, Eq. (12) is introduced to calculate the tangent Young’s modulus for rockfills, as adopted in the hyperbolic model proposed by Ducan and Chang [9]. The axial transient strain \(\varepsilon_{1}^{0}\) is formulated in Eq. (13):

$$E_{\text{t}} = \left( {1 - R_{f} D} \right)^{2} K_{1} p_{\text{a}} \left( {\frac{{\sigma_{3} }}{{p_{\text{a}} }}} \right)^{{n_{1} }}$$
(12)
$$\varepsilon_{1}^{0} = \frac{1}{{\left( {\frac{1}{{\left( {\sigma_{1} - \sigma_{3} } \right)}} - \frac{{R_{f} }}{{\left( {\sigma_{1} - \sigma_{3} } \right)_{f} }}} \right)K_{1} p_{\text{a}} \left( {\frac{{\sigma_{3} }}{{p_{\text{a}} }}} \right)^{{n_{1} }} }}$$
(13)

where \(p_{\text{a}}\) is the atmospheric pressure, \(K_{1}\) and \(n_{1}\) govern the values of the initial tangent modulus, \(R_{f} = (\sigma_{1} - \sigma_{3} )_{f} /(\sigma_{1} - \sigma_{3} )_{\text{ult}}\), where \((\sigma_{1} - \sigma_{3} )_{f}\) is the peak deviator stress and \((\sigma_{1} - \sigma_{3} )_{\text{ult}}\) is the asymptotic value of \((\sigma_{1} - \sigma_{3} )\) when the stress–strain curve approaches infinite strain. The three parameters (\(K_{1}\), \(n_{1}\), and \(R_{f}\)) can easily be obtained from a series of conventional triaxial tests under different \(\sigma_{3}\) values.

Fig. 6
figure 6

\(\left( {\sigma_{1} - \sigma_{3} } \right)\) value versus the transient axial strain: a rockfills from Shangzhai [36] and b redstone granular soil [6]

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4.2 Modification of the Newton viscosity coefficient \(\eta^{N}\)

Table 3 lists the \(\eta^{N}\) values of the three rockfills at different stress states, and Fig. 7 illustrates variations of \(\eta^{N}\) with the deviator stress levels and confining pressures. For the three materials, \(\eta^{N}\) is relatively constant at different deviator stress levels under the same \(\sigma_{3}\) (Fig. 7a, c, e). However, \(\eta^{N}\) increases almost linearly with the logarithm of \(\sigma_{3}\) (Fig. 7b, d, f), which means that the confining pressure has significant influence on \(\eta^{N}\). One possible reason for this influence may be that under a larger confining pressure, the particles are held together more tightly in the steady creep stage, which makes it more difficult to readjust their position and leads to a slower rate of the steady creep strain, i.e., a larger value of \(\eta^{N}\).

Fig. 7
figure 7

Variation of \(\eta^{N}\) with deviator stress levels and confining pressures: a, b rockfills from Shangzhai [36]; c, d redstone granular soils [6]; e, f rockfills from Dalian [27]

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Eq. (14) is proposed to approximately describe a linear relationship between \(\log (\eta^{N} )\) and \(\log (\sigma_{3} )\):

$$\frac{{\eta^{N} \left( {\sigma_{3} } \right)}}{{p_{\text{a}} \cdot t_{\text{ref}} }} = K_{2} \left( {\frac{{\sigma_{3} }}{{p_{\text{a}} }}} \right)^{{n_{2} }}$$
(14)

where \(t_{\text{ref}}\) is set to one hour, and \(p_{\text{a}}\) is the atmospheric pressure. To eliminate dimensional effects, the \(\eta^{N}\) values are all normalized by the two indices (\(t_{\text{ref}}\) and \(p_{\text{a}}\)). \(K_{2}\) and \(n_{2}\) are two new parameters of the modified Newton viscosity coefficient, which can be back-determined by the least-square fitting method according to Eq. (14) and the calibrated values of \(\eta^{N}\) in Table 3. The solid lines in Fig. 7 are obtained by substituting the values of \(K_{2}\), \(n_{2}\), and the corresponding stress states into Eq. (14). Despite some scatter, the solid lines calculated by Eq. (14) are consistent with most of the \(\eta^{N}\) points. Hereby, the strain of the modified Newton unit in Eq. (1) is written as:

$$\varepsilon_{i}^{N,M} = \frac{{q_{i} }}{{2K_{2} \left( {p_{\text{a}} \cdot t_{\text{ref}} } \right)\left( {\frac{{\sigma_{3} }}{{p_{\text{a}} }}} \right)^{{n_{2} }} }}t$$
(15)

4.3 Modification of the Kelvin viscosity coefficient \(\eta^{K}\)

The \(\eta^{K}\) values of the three rockfills are listed in Table 4 and are also plotted against the corresponding D and \(\sigma_{3}\) values (Fig. 8). The deviator stress level D also appears to have insignificant influence on \(\eta^{K}\) if \(\sigma_{3}\) remains unchanged. However, the value of \(\eta^{K}\) increases linearly with the logarithm of \(\sigma_{3}\). Thus, \(\eta^{K}\) can also be regarded as dependent only on the confining pressure. Eq. (16) is proposed to describe the approximately linear relationship between \(\log \left( {\eta^{K} } \right)\) and \(\log \left( {\sigma_{3} } \right)\):

$$\frac{{\eta^{K} \left( {\sigma_{3} } \right)}}{{p_{\text{a}} \cdot t_{\text{ref}} }} = K_{3} \left( {\frac{{\sigma_{3} }}{{p_{\text{a}} }}} \right)^{{n_{3} }}$$
(16)

where \(K_{3}\) and \(n_{3}\) are two new parameters of the modified Kelvin viscosity coefficient, which can be back-determined by the least-square fitting method according to Eq. (16) and the calibrated values of \(\eta^{K}\) in Table 4. As shown in Fig. 8, a fairly high consistency is observed between the points of \(\eta^{K}\) obtained from experimental results and the solid lines simulated by Eq. (16).

Fig. 8
figure 8

Variation of \(\eta^{K}\) with deviator stress levels and confining pressures: a, b rockfills from Shangzhai [36]; c, d redstone granular soils [6]; e, f rockfills from Dalian [27]

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4.4 Modification of the Kelvin shear modulus \(G^{K}\)

Table 5 summarizes the \(G^{K}\) values of these three types of rockfills. The relevance of \(G^{K}\) and the corresponding stress states (i.e., the confining pressures and deviator stress levels) is demonstrated in Fig. 9. \(G^{K}\) tends to decrease with the deviator stress level (Fig. 9a, c, e). A possible mechanism is that more particle breakages or grain contact crushes occur when the sample is continuously subjected to a higher deviator stress level, which leads to the development of more time-dependent deformations. This phenomenon was also observed in previous studies [1, 16, 17, 23, 34]. An increasing trend of \(G^{K}\) is observed with the confining pressure (Fig. 9b, d, f). Thus, Eq. (17) is proposed to consider the influence of both the deviator stress level and confining pressure:

$$\frac{{G^{K} \left( {\sigma_{3} ,D} \right)}}{{p_{a} }} = K_{4} (\frac{{\sigma_{3} }}{{p_{a} }})^{{n_{4} }} \left( {1 - aD} \right)$$
(17)

where the term \(\left( {1 - aD} \right)\) is used to simulate the value of \(G^{K}\) decreasing with the deviator stress level, and \(a\) represents the decreasing rate. When \(D = 0\), the modification of \(G^{K}\) has a form similar to that in Eqs. (14) and (16), which are used to simulate the effects of the confining pressure.

Fig. 9
figure 9

Variations of \(G^{K}\) with deviator stress levels and confining pressures: a, b rockfills from Shangzhai [36]; c, d redstone granular soils [6]; e, f rockfills from Dalian [27]

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The three parameters (\(K_{4}\), \(n_{4}\), and \(a\)) can be obtained using the least squares fitting method according to Eq. (17) and the calibrated values of \(G^{K}\) in Table 5. The solid lines calculated by Eq. (17) are consistent with the points of \(G^{K}\) obtained from the experimental results (Fig. 9). Hereby, the strain of the modified Kelvin unit is

$$\varepsilon_{i}^{K,M} = \frac{{q_{i} }}{{2K_{4} p_{\text{a}} \left( {\frac{{\sigma_{3} }}{{p_{\text{a}} }}} \right)^{{n_{4} }} \left( {1 - aD} \right)}}\left[ {1 - \exp \left( { - \frac{{K_{4} }}{{K_{3} t_{\text{ref}} }}\left( {\frac{{\sigma_{3} }}{{p_{\text{a}} }}} \right)^{{n_{4} - n_{3} }} \left( {1 - aD} \right)t} \right)} \right]$$
(18)

In summary, the axial strain of the modified Burgers model (\(\varepsilon_{1}^{B,M}\)) is formulated as:

$$\varepsilon_{1}^{B,M} = \varepsilon_{1}^{0} + \varepsilon_{1}^{N,M} + \varepsilon_{1}^{K,M}$$
(19)

where \(\varepsilon_{1}^{0}\), \(\varepsilon_{1}^{N,M}\), and \(\varepsilon_{1}^{K,M}\) can be calculated by substituting the corresponding stress states and model parameters into Eqs. (13), (15), and (18), respectively.

5 Nonlinear elasto-damage unit for rockfills considering accelerated creep behavior

The modified Burgers model proposed in Sect. 4 can reproduce the stress–stain and strain–time curves of rockfills in the attenuated and steady creep stages. However, the model cannot simulate the accelerated creep behavior of rockfills. To overcome this limitation, a nonlinear elasto-damage unit for rockfills is proposed. The stress–strain relation (solid line in Fig. 10a) of the nonlinear elasto-damage unit is classified into two stages: the transient deformation stage (OA) and the damage deformation stage (ABD). The deformation in the former stage (OA) was discussed in Sect. 4.1. This section aims to simulate the deformation in the latter stage (ABD). A critical threshold of the deviator stress level (\(D_{\text{thr}}\)) is first defined to determine whether the damage mechanism will be initiated after the beginning of creep (point A) by comparing D and \(D_{\text{thr}}\): (1). If \(D < D_{\text{thr}}\), the strain of the nonlinear elasto-damage unit remains unchanged: (2). If \(D \ge D_{\text{thr}}\), the damage mechanism is initiated, and the strain of this unit accelerates over time, as demonstrated by the solid line in Fig. 10b. Because only limited experience has been gained for this parameter in previous research, laboratory tests are required to determine \(D_{\text{thr}}\). Dthr is expected to be affected by a number of factors, including the mineral, gradation, and particle shape of rockfills. The following discussion is based on \(D \ge D_{\text{thr}}\); i.e., the damage mechanism is initiated.

Fig. 10
figure 10

Deformation process for a nonlinear elasto-damage unit

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Kachanov [18] proposed a damage mechanism for metal. For the cubic damage unit in Fig. 10b, \(S\) is the overall section area of the unit and \(S_{\omega }\) is the total area of the microcracks and cavities with different shapes. The damage variable \(\omega\) is defined as \(\omega = S_{\omega } /S\). Isotropic damage is assumed; i.e., the cracks and cavities are equally distributed in all directions. The strain of the unit in the damage deformation stage is calculated as:

$$\Delta \varepsilon_{i}^{D} = \varepsilon_{i}^{\text{ref}} \frac{\omega }{1 - \omega }$$
(20)
$$\varepsilon_{i}^{\text{ref}} = \frac{{\left( {1 + \upsilon } \right)\sigma_{i} - \upsilon \sigma_{kk} }}{{E_{\text{t}} }}$$
(21)

where \(\varepsilon_{i}^{\text{ref}}\) is the referenced strain to calculate the creep strain in the accelerated creep stage, and \(E_{\text{t}}\) is the tangent elastic modulus. For metal, linear elasticity is assumed, and \(E_{\text{t}}\) is the slope of the dashed straight line in Fig. 10a. For rockfills, \(E_{\text{t}}\) is the tangent modulus of the hyperbolic curve at point A, which can be calculated from Eq. (12).

Different from metal, the influence of the deviator stress level should be considered for rockfills: A modified damage evolution law for rockfills is proposed as:

$$\frac{d\omega }{dt} = C\left( {\frac{D}{1 - \omega }} \right)^{V}$$
(22)

where \(C\) and \(V\) are two parameters of the damage variables for rockfills, and D is the deviator stress level. The dimension of C is 1/hour. D and V are dimensionless. According to the initial condition, i.e., \(\omega (0) = 0\), \(\omega\) can be expressed as:

$$\omega = 1 - \left[ {1 - C\left( {1 + V} \right)D^{V} t} \right]^{{\frac{1}{1 + V}}}$$
(23)

To determine \(C\) and \(V\) from creep test results, two intermediate variables (\(t_{\text{ult}}\) and \(\gamma\)) are introduced in Eqs. (24) and (25), respectively:

$$t_{\text{ult}} = \frac{1}{{C\left( {1 + V} \right)D^{V} }}$$
(24)
$$\gamma = \frac{1}{1 + V}$$
(25)

Thus, \(\omega\) is rewritten as:

$$\omega = 1 - \left[ {1 - \frac{t}{{t_{\text{ult}} }}} \right]^{\gamma }$$
(26)

Here, \(t_{\text{ult}}\) and \(\gamma\) are not new model parameters but are functions of the model parameters \(C\) and \(V\). \(t_{\text{ult}}\) is called the “ultimate time,” as sketched in Fig. 1. \(\gamma\) controls the rate at which the damage variable increases with time. The dimension of \(t_{\text{ult}}\) is hour, and \(\gamma\) is dimensionless.

The results of the creep test on the redstone granular soil (\(\sigma_{3} = 50\,{\text{kPa}}\) and \(D = 0.8\)) performed by Chen and Zhang [6], in which the accelerated creep was observed, are used to illustrate the process to determine parameters. \(t_{\text{ult}}\) can be easily obtained from the vertical asymptote of the creep test strain. \(t_{ult}\) = 108 h is obtained based on the test results by Chen and Zhang [6]. \(\gamma\) is obtained through the following steps.

Step 1:

Once the damage mechanism is initiated, the strain of the proposed model comprises the strain of the modified Burgers model (\(\varepsilon_{1}^{B,M}\)) and the strain in the damage deformation stage (\(\Delta \varepsilon_{1}^{D}\)). \(\Delta \varepsilon_{1}^{D}\) can be separated from the measured axial strain in the test by removing the value of \(\varepsilon_{1}^{B,M}\), as demonstrated in Fig. 11a

Fig. 11
figure 11

Procedure to determine the parameters of the damage variable: a separating \(\Delta \varepsilon_{1}^{D}\) from the measured axial strain; b the \(\log \left( {\frac{{\varepsilon_{1}^{\text{ref}} }}{{\varepsilon_{1}^{\text{ref}} + \Delta \varepsilon_{1}^{D} }}} \right) - \log \left( {1 - \frac{t}{{t_{\text{ult}} }}} \right)\) curve and the fitting line for redstone granular soil (\(D = 0.8\) and \(\sigma_{3} = 50\,{\text{kPa}}\), [6])

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Step 2:

Equation (27) can be derived according to Eqs. (20) and (26):

$$\log \left( {\frac{{\varepsilon_{i}^{\text{ref}} }}{{\varepsilon_{i}^{\text{ref}} + \Delta \varepsilon_{i}^{D} }}} \right) = \gamma \log \left( {1 - \frac{t}{{t_{\text{ult}} }}} \right)$$
(27)

where \(\varepsilon_{i}^{\text{ref}}\) can be determined according to Eq. (21). By substituting the corresponding data of the last five red diamonds (Fig. 11a) into Eq. (27), \(\log \left( {\frac{{\varepsilon_{i}^{\text{ref}} }}{{\varepsilon_{i}^{\text{ref}} + \Delta \varepsilon_{i}^{D} }}} \right)\) can be plotted against \(\log \left( {1 - \frac{t}{{t_{\text{ult}} }}} \right)\), as shown in Fig. 11b. The slope of the fitting line equals \(\gamma\).

By substituting the values of \(t_{\text{ult}}\) and \(\gamma\) in Eqs. (24) and (25), the corresponding parameters \(C\) and \(V\) can be determined, respectively. Analogously, the values of \(C\) and \(V\) of the redstone granular soil at \(D = 0.8\) with the other two confining pressures (\(\sigma_{3} = 100,200\,{\text{kPa}}\)) are obtained and listed in Table 6.

Table 6 Parameters \(C\) and \(V\) of the redstone granular soil at different \(\sigma_{3}\) values [6]
Full size table

The values of \(\log (V)\) and \(\log (C)\) are further plotted against the corresponding \(\log \left( {\sigma_{3} /p_{\text{a}} } \right)\) in Fig. 12a, b, respectively. The value of \(V\) is relatively constant at different confining pressures (Fig. 12a). However, a linearly decreasing relation between \(\log (C)\) and \(\log \left( {\sigma_{3} /p_{\text{a}} } \right)\) is observed (Fig. 12b). Thus, Eq. (28) is proposed to consider the influence of \(\sigma_{3}\):

$$C = C_{\text{ref}} \left( {\frac{{\sigma_{3} }}{{p_{a} }}} \right)^{m}$$
(28)

where \(C_{\text{ref}}\) and \(m\) are the intercept and slope of the fitting line in Fig. 12b. Therefore, \(t_{\text{ult}}\) is dependent on both the deviatoric stress level and the confining pressure.

Fig. 12
figure 12

Relation between the parameters of the damage variables and the confining pressures: a \(\log (V)\) versus \(\log \left( {\sigma_{3} /p_{\text{a}} } \right)\); b \(\log (C)\) versus \(\log \left( {\sigma_{3} /p_{\text{a}} } \right)\) (redstone granular soil, performed by [6]

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In summary, the axial strain of the nonlinear elasto-damage unit proposed for rockfills is formulated as:

$$\varepsilon_{1}^{D} = \left\{ {\begin{array}{*{20}l} {\varepsilon_{1}^{0} } \hfill & {D < D_{\text{thr}} } \hfill \\ {\varepsilon_{1}^{0} + \varepsilon_{1}^{\text{ref}} \frac{\omega }{1 - \omega }} \hfill & {D \ge D_{\text{thr}} } \hfill \\ \end{array} } \right.$$
(29)

\(\varepsilon_{1}^{0}\), \(\varepsilon_{1}^{\text{ref}}\), and \(\omega\) can be calculated from Eqs. (13), (21), and (26), respectively.

6 Shear creep model for rockfills

By substituting Eqs. (15), (18), and (29) into Eq. (1), the axial strain of the shear creep model for rockfills is obtained as shown in Eq. (30):

$$\varepsilon_{1} = \left\{ {\begin{array}{*{20}l} {\varepsilon_{1}^{0} + \frac{{q_{i} }}{{2G^{K} \left( {\sigma_{3} ,D} \right)}}\left( {1 - \exp \left( { - \frac{{G^{K} \left( {\sigma_{3} ,D} \right)}}{{\eta^{K} \left( {\sigma_{3} } \right)}}t} \right)} \right) + \frac{{q_{i} }}{{2\eta^{N} \left( {\sigma_{3} } \right)}}t} \hfill & {D < D_{\text{thr}} \quad ({\text{a}})} \hfill \\ {\varepsilon_{1}^{0} + \frac{{q_{i} }}{{2G^{K} \left( {\sigma_{3} ,D} \right)}}\left( {1 - \exp \left( { - \frac{{G^{K} \left( {\sigma_{3} ,D} \right)}}{{\eta^{K} \left( {\sigma_{3} } \right)}}t} \right)} \right) + \frac{{q_{i} }}{{2\eta^{N} \left( {\sigma_{3} } \right)}}t + \varepsilon_{1}^{\text{ref}} \frac{\omega }{1 - \omega }} \hfill & {D \ge D_{\text{thr}} \quad ({\text{b}})} \hfill \\ \end{array} } \right.$$
(30)
  1. 1.

    If \(t = 0\), \(\varepsilon_{1} { = }\varepsilon_{1}^{0}\), which represents the transient deformation (i.e., segment OA in Fig. 1). As introduced in Sect. 4.1, this model can capture the nonlinear stress–strain behaviors of rockfills (i.e., red solid line OA in Fig. 10a).

  2. 2.

    If \(D < D_{\text{thr}}\), this model becomes the modified Burgers model, as proposed in Sect. 4, which can simulate the transient deformation, the deformation of the attenuated, and the steady creep stages. Eqs. (14), (16), and (17) are used to describe the variations of the parameters (\(\eta^{N}\), \(\eta^{K}\), and \(G^{K}\)) with the deviator stress levels and confining pressures.

  3. 3.

    If \(D \ge D_{\text{thr}}\), the damage mechanism is initiated, and the corresponding strain and strain rate approach infinity when time approaches \(t_{\text{ult}}\). Thus, this model has the capability to reproduce the accelerated creep behavior of rockfills.

  4. 4.

    The proposed model has a total of 14 parameters, as listed in Table 7. Ten of the parameters are used in the modified Burgers model, and the calibration methods are presented in Sect. 4. The other four parameters are used to describe the damage evolution of the damage variable \(\omega\), which can be determined by the method introduced in Sect. 5.

    Table 7 Parameters of the shear creep models for rockfills
    Full size table

7 Experimental test results and analysis

The results of the large-scale triaxial creep tests on three different rockfills [6, 27, 36] over a wide range of confining pressures and deviator stress levels are used to validate the proposed model, and their parameters are back-determined and listed in Table 8. The model-calculated lines are obtained by substituting the parameters and corresponding stress states into Eq. (30). As shown in Figs. 13, 14, and 15, the calculated results agree well with the experimental data. By using the same set of parameters for each type of rockfill, the proposed model not only can reflect the influence of \(D\) and \(\sigma_{3}\) on the shear creep behavior but also can simulate all three creep stages of the rockfill, including the accelerated creep stage.

Table 8 Parameters of the proposed shear creep models for the three rockfills [6, 27, 36]
Full size table
Fig. 13
figure 13

Comparison between the simulation results and the creep test results on the rockfills from Shangzhai [36]: a \(\sigma_{3} = 3.0\,{\text{MPa}}\), b \(\sigma_{3} = 2.0\,{\text{MPa}}\), c \(\sigma_{3} = 1.0\,{\text{MPa}}\), and d \(\sigma_{3} = 0.5\,{\text{MPa}}\)

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Fig. 14
figure 14

Comparison between the simulation results and the creep test results on the rockfills from Dalian [27]: a \(\sigma_{3} = 1.2\,{\text{MPa}}\), b \(\sigma_{3} = 0.8\,{\text{MPa}}\), and c \(\sigma_{3} = 0.4\,{\text{MPa}}\)

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Fig. 15
figure 15

Comparison between the simulation results and the creep test results on the redstone granular soils [6]: a \(\sigma_{3} = 0.2\,{\text{MPa}}\), b \(\sigma_{3} = 0.1\,{\text{MPa}}\), and c \(\sigma_{3} = 0.05\,{\text{MPa}}\)

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8 Conclusions

Rockfills have been used extensively for the construction of high-fill embankments and dams, which exhibit complex time-dependent behavior. This paper presents a rheological model to simulate the shear creep behaviors of rockfills based on the classic Burgers model, in which a nonlinear elasto-damage unit, a modified Newton unit, and a modified Kelvin unit are connected in series. By analyzing the results of large-scale triaxial creep tests on different rockfills, the influence of the stress state (i.e., the deviator stress level and confining pressure) on the shear creep behavior of rockfills is revealed. A method to modify the key model parameters (i.e., the transient elastic modulus, the Newton viscosity coefficient, the Kelvin shear modulus, and the Kelvin viscosity coefficient) is proposed to consider such influence and is incorporated into the proposed model. Furthermore, the accelerated creep behavior of rockfills is reproduced by a nonlinear elasto-damage unit, which is developed by modifying the damage mechanism proposed by Kachanov [18]. A critical threshold of the deviator stress level (\(D_{\text{thr}}\)) is defined to determine whether the accelerated creep stage will be initiated.

Despite its simplicity, the model can simulate the essential aspects of the shear creep behaviors of rockfills, and fairly high consistencies are found between the simulation results and the large-scale triaxial creep test results. However, this model is developed based on limited tests because only a few large-scale triaxial apparatuses exist that are capable of performing such long-term tests. Further laboratory studies are required to improve our understanding of the shear creep behavior of different rockfills and to improve this model.