1 Introduction

Over the last decade, the part of renewable energy has considerably grown in the energy mix. However, solar and wind energy particularly suffers from a crucial drawback, their intermittent nature. The period of maximum production does not necessarily coincide with the period of maximum consumption (Førsund 2007). Two systems allow a very large energy storage deliverability (>100MW with a single unit): pumped hydroelectric storage (PHS) and compressed air energy storage (CAES; Chen et al. 2009).

PHS stores energy as gravitational potential energy. Water is pumped from a lower reservoir into a higher reservoir when electric power is available at a low cost (off-peak of consumption; Steffen 2012). During the peak of energy consumption, turbines produce energy at a higher cost, ensuring the economic viability of the system. This system is already widely used all over the world with more than 200 sites and 100 GW of installed power in 2005. Reservoirs used may be created artificially for that purpose. However, abandoned mines or open pits may be used as investigated by Pujades et al. (2016).

CAES working is a bit more complex and works on the basis of conventional gas turbine (Chen et al. 2009). During low demand air is stored and compressed to 4–8 MPa. To recover stored energy, air is heated and expanded through a high-pressure turbine first. It is then mixed with fuel and combusted with the exhaust expanded through a low-pressure turbine. This system is much less widespread. The first plant was built in 1978 in Huntorf, Germany, in an abandoned salt mine (Raju and Kumar Khaitan 2012; Mahmoudi et al. 2016). A second unit is in operation in Alabama since 1991.

The opportunity to build such systems is strongly dependent on the geography (height difference between reservoirs), space available to build reservoir, environmental concerns and capital costs (Chen et al. 2009). Reusing abandoned mines as reservoir solves some of these issues. However ensuring the overall stability of the mine galleries and walls becomes a challenge. While storage of natural gas has an annual period, PSH and CAES are operated in daily cycles or faster. Therefore variations of air or water pressures induce a cyclic loading of rock walls.

Experimental investigations of the fatigue behaviour of rock material are scarce and complex to carry out (Attewell and Farmer 1973; Martin and Chandler 1994; Gatelier et al. 2002; Royer-Carfagni and Salvatore 2000; Erarslan and Williams 2012; Liu and He 2012; Wang et al. 2013; Liu et al. 2014; Ghamgosar and Erarslan 2015). However it clearly appears that failure of the material may be reached upon cyclic loading for an amplitude significantly lower than the monotonic resistance. This result is of paramount importance for design. Given the life span of the rock structure, the maximum number of cycles or the maximum amplitude that could be sustained must be assessed.

Models able to reproduce evolution of rock strength with cyclic loading are very limited (Li et al. 2003; Xiao et al. 2009, 2010; Bastian et al. 2014; Liu et al. 2014). They mostly consist in predicting strength or strain evolution of triaxial or uniaxial laboratory tests for a given material with respect to the number of cycles. General constitutive laws (stress–strain relation) able to reproduce this cyclic behaviour numerically in a finite element code have deserved even less attention. A Ramberg–Osgood-based model is proposed by Hueckel (1991) using discretised kinematic hardening concept, while a bounding surface approach is developed by Wang et al. (2015) for granite. Two models dedicated to more ductile salt rocks are developed, respectively, based on elastoviscoplasticity (Khaledi et al. 2016a, b) or a micro–macro viscous approach (Pouya et al. 2016).

On the contrary, much more work is devoted to the cyclic behaviour of plain concrete material or concrete with fibres, which have a behaviour similar to natural rocks. Damage mechanics is the mostly used framework (Peng and Meyer 2000; Alliche 2004; Wu et al. 2006; Mazars et al. 2015; Breccolotti et al. 2015). Another model couples damage evolution and bounding surface model (Suaris et al. 1990). Separate compliance matrices are introduced in tension and compression since crack growths mechanisms are different. It was successfully applied to the modelling of low-cycle experiments.

The objective of this paper is to develop, formulate and validate a constitutive law able to reproduce the cyclic behaviour of brittle rock materials. The paper aims at developing a physically based model with a reasonable amount of parameters that must be calibrated. It is validated against results published in the literature for three different materials.

Contrary to rock materials, different models exist for the cyclic behaviour of sands such as the Prevost model (Prevost 1985; Cerfontaine et al. 2016), generalised plasticity (Mira et al. 2009) or subloading surface (Hashiguchi 2009). The formulation proposed hereinafter is based on the bounding surface plasticity (Dafalias 1986). The pioneering work on its application in form of a two surface plasticity model for sands (Manzari and Dafalias 1997) was later extended further (Dafalias and Manzari 2004; Dafalias et al. 2004; Taiebat and Dafalias 2008; Li and Dafalias 2012; Dafalias and Taiebat 2016) and adopted the title of SANISAND, an acronym term for Simple ANIsotropic SAND. This framework is mainly chosen for its ability to reproduce accumulation of plastic strain upon constant amplitude cycles, for its physically based concept and relative simplicity. In addition, review of results published in the literature highlights that damaging of elastic properties is not always observed or may be neglected during fatigue experiments. Therefore the accumulation of plastic strain is of greater importance.

The model is composed of a yield surface subject to kinematic hardening, allowing accumulation of plastic strain due to cyclic loading. A limit surface represents the failure locus of the material. Cohesion is the main internal variable and is degraded cycle after cycle, finally leading to failure. A bounding surface is introduced to compute the stiffness of the material. Its slope is larger than or equal to the limit surface in order to reproduce both ductile (tangent modulus tends to zero at failure) or brittle failure (limited decrease of tangent modulus with respect to elastic stiffness).

First section gathers main features of the cyclic behaviour of rock materials observed in laboratory. The triaxial formulation of the model is then derived, and constitutive equations are detailed. Finally physical parameters are calibrated with respect to three different types of rocks cyclically loaded in simple compression or triaxial conditions.

2 Experimental Review

The objective of this section is to highlight main features of monotonic and cyclic behaviours of brittle rock materials. These mechanisms must be correctly reproduced by any constitutive law. Confining pressures investigated in the following remain limited (with respect to their simple compression strength) since the application of the model targets the modelling of mine galleries or open pits. In these cases, rock material around galleries or close to the pit’s wall is close to simple compression conditions.

Brittle rock materials, by definition, exhibit only small plastic strain before failure, although ductility may increase with confining pressure (Liu and He 2012). The strength increases with confining pressure, which is a fundamental ingredient of any constitutive law. Many other features could be relevant such as: material (Attewell and Farmer 1973) anisotropy, load-induced anisotropy (Benz and Schwab 2008) or creep behaviour (Brantut et al. 2013). However they are not considered in the following.

A constitutive law should be consistent with the failure mode of the rock sample it represents. Failure of brittle rock is mainly driven by the generation of microcracks that finally coalesce within a single macrocrack. Relations between macrocrack behaviour (vertical and volumetric deformation) and microcracking were described in pioneering work of Bieniawski (1967a, b). However as described in the following, cyclic mechanisms are slightly different.

Cyclic behaviour of rock depends on the material investigated but also on the type of cyclic loading applied. However it results in a weakening of the tested sample, and finally a failure occurs for a maximum stress lower than the monotonic resistance.

The typical behaviour of a Lorano marble is presented in Fig. 1. Cyclic loading generates accumulation of plastic strain within the material cycle after cycle. The effect on lateral strain \(\epsilon _3\) is more marked than on axial strain \(\epsilon _1\) and results from dilatancy and microcracks. The hysteretic behaviour of the cycles tends to increase progressively.

Fig. 1
figure 1

Monotonic and cyclic simple compression (constant amplitude) tests on Lorano marble, data traced from Royer-Carfagni and Salvatore (2000)

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The first cycle is very different from the following ones. It is more open and generates more residual strain. This is generally observed in cyclic results but not totally explained. It is mainly due to hardening of the material but also to experimental set-up (non-parallel faces of the sample, sliding, etc.).

Figure 1 also highlights the inherent heterogeneity related to rock materials. While two simple compression tests are carried out on a Lorano marble, it is clear that the response corresponds to two samples having different strength parameters. If the first compression and monotonic results are compared, it appears that the stiffness and strength are different. For an identical deviatoric stress q, more plastic strains are generated in monotonic results. This may be due to different rock blocs, weathering of the material, loading rate or simply damages due to sampling.

Constant amplitude cyclic loading consists in imposing variations of stress between a minimum and a maximum value (lower than estimated monotonic peak strength). Such a loading is reported to generate plastic strain cycle after cycle as described in Fig. 1. However this accumulation rate is nonlinear and increases when sample approaches failure. Plastic strain accumulated is often represented with respect to the number of cycles applied as depicted in Fig. 2. In this figure axial residual strain for a sandstone is compared at six confining pressures. Three phases are often identified (Liu and He 2012): a decelerating accumulation of plastic strain, a stationary phase and finally an accelerating phase.

Fig. 2
figure 2

Cyclic compression triaxial tests at six confining pressures, plastic axial strain \(\epsilon _1^\mathrm{p}\) versus cycle number N, sandstone (Liu and He 2012)

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Residual strain at the end of the cyclic test increases with confining pressure. However there is no clear tendency related to the number of cycles to failure. Some studies (Haimson and Kim 1971; Burdine 1963; Erarslan et al. 2014) report that the number of cycles to failure depends on the ratio of the maximum cyclic amplitude to the monotonic resistance \(q_\mathrm{max}/ q_\mathrm{peak}\). Therefore a rigorous comparison of behaviour at different confinements would require to impose the same ratio to all samples, since \(q_\mathrm{peak}\) increases with confinement. However most of the time the peak strength is unknown a priori because rock material is inherently heterogeneous. Therefore it is very difficult to compare results.

Figure 3 presents an indirect tensile cyclic test on a Brisbane tuff. The cyclic amplitude is constant, but the mean stress increases linearly (Erarslan and Williams 2012). Cyclic maximum stress is significantly lower than the monotonic resistance, revealing that cyclic load has a progressive weakening effect, even for moderate amplitudes. This is also marked in Fig. 1, although monotonic and cyclic results are not easy to compare (first cycle and monotonic test have a different behaviour).

Fig. 3
figure 3

Comparison of monotonic and cyclic indirect tensile strength results of Brisbane tuff (Erarslan and Williams 2012)

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Failure modes are shown to be quite different in both cases. The monotonic samples present a straight crack, while a larger and branching fractured zone is observed upon cyclic loading (Ghamgosar and Erarslan 2015). In addition these samples are surrounded by debris and crushed rock material. This suggests that much more microcracks are generated (Erarslan and Williams 2012). It is then concluded that fatigue is generated by a progressive debonding, loosening and decohesion of rock matrix.

For a given confining pressure and loading type (indirect tensile test, simple compression or triaxial), it is shown that a threshold exists on the cyclic amplitude below which there is no failure of the material even upon a large number of cycles (Burdine 1963; Haimson and Kim 1971; Prost 1988; Erarslan and Williams 2012). Figure 4 presents the number of cycles as a function of the amplitude applied for indirect tensile test on Brisbane tuff (Erarslan and Williams 2012). When the amplitude is below 70% of the monotonic resistance, no failure was observed even after more than 500,000 cycles. This threshold is defined as the fatigue resistance of the material.

Fig. 4
figure 4

S–N curve for indirect tensile strength of Brisbane tuff (Erarslan and Williams 2012)

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A sketch of a damage-controlled test is represented in Fig. 5. In such a test, the maximum axial strain is increased cycle after cycle before unloading to a minimum stress \(q_\mathrm{min}\). Therefore pre-peak and post-peak behaviour are both investigated. Subsequently the stress amplitude increases before peak and decreases after.

Fig. 5
figure 5

Sketch of a damage-controlled test

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An example proposed by Martin and Chandler (1994) consists of a triaxial cyclic test where the maximum cyclic stress varies, as depicted in Fig. 6. Up to 75% of the peak strength, it is incremented of 40 MPa before reversal to unloading phase. After this threshold, the load cycles are performed by increments of circumferential strain equal to 0.063 mm, before reversal.

Fig. 6
figure 6

Damage-controlled test on Lac du Bonnet granite, \(\sigma _3=15\) MPa (Martin and Chandler 1994)

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Similarly to constant amplitude loading, cycles are prone to open progressively and especially in the post-peak zone. Plastic strain is accumulated cycle after cycle. However it is shown by Martin and Chandler (1994) that elastic properties are prone to damaging, especially in the post-peak domain and at low confinement. In addition it is shown that the monotonic results act almost as a failure locus in the \(q-\epsilon _1\) plane for the cyclic results at identical confinement.

A conclusion of the study presented by Martin and Chandler (1994), Martin (1997) is that cohesion is progressively decreased cycle after cycle as a consequence of sample damaging. This is correlated with the opening and propagation of microcracks.

3 Triaxial Formulation of a Constitutive Law

The objective of this section is to formulate a constitutive law in triaxial notations. Indeed in the following only simple compression and triaxial tests are modelled. In addition these notations simplify the formulation of the model and better highlight how it works.

This work is derived from the bounding surface plasticity concept (Dafalias 1986) and its application in the framework of SANISAND models (Manzari and Dafalias 1997; Dafalias and Manzari 2004; Dafalias et al. 2004; Taiebat and Dafalias 2008; Li and Dafalias 2012; Dafalias and Taiebat 2016). This choice is justified by the capability of these models to reproduce accumulation of plastic strain for constant cycle amplitude. In this work, creep and dynamic effects are not taken into account.

3.1 Triaxial Notations

In triaxial conditions, only the diagonal components of the strain \(\varvec{\epsilon }\) and effective stress \(\varvec{\sigma }\) tensors are different from zero. Two stress invariants describe the stress state of the material

$$\begin{aligned} p= & {}\, \left( \sigma _1 + 2\, \sigma _3\right) /3, \end{aligned}$$
(1)
$$\begin{aligned} q= & {}\, \sigma _1 - \sigma _3, \end{aligned}$$
(2)

where \(\sigma _1\) and \(\sigma _3\) are, respectively, the vertical and lateral effective stresses applied to the sample. The stresses are gathered into the vector \({\mathbf {s}}^\mathrm{T} = [q , \quad p]\) where \(^\mathrm{T}\) is the transpose operator. A reduced deviatoric stress \(\xi\) is defined according to

$$\begin{aligned} \xi = \dfrac{q}{p + p_\mathrm{c}}, \end{aligned}$$
(3)

where \(p_\mathrm{c}\) is further defined in Eq. (6). Similarly strain invariants are defined according to

$$\begin{aligned} \epsilon _v&= \epsilon _1 + 2\, \epsilon _3, \end{aligned}$$
(4)
$$\begin{aligned} \epsilon _q&= \epsilon _1 - \epsilon _3 , \end{aligned}$$
(5)

where \(\epsilon _1\) and \(\epsilon _3\) are the vertical and lateral strains. They are assembled into the vector \({\mathbf {e}}^\mathrm{T} = [\epsilon _q, \quad \epsilon _v]\).

3.2 Surfaces

The model is defined by different surfaces in the p–q plane. A yield surface \(M^\mathrm{y}\) delineates the elastic range. The plastic modulus is proportional to the distance between the stress state and the bounding surface \(M^\mathrm{b}\). Subsequently when the stress state approaches the bounding surface, this distance tends to zero, leading to a zero plastic modulus and a ductile behaviour.

In this work, brittle material is investigated. Therefore upon monotonic loading, a peak strength must be encountered followed by softening, after a limited plastic strain. Therefore a new surface \(M^\mathrm{l}\le M^\mathrm{b}\) but homothetic to \(M^\mathrm{b}\) is introduced. The stress state is not allowed to overcome \(M^\mathrm{l}\), which is termed limit surface. It will be shown in the following that reaching this surface will be synonym of failure, namely reaching a peak resistance.

Figure 7 summarises differences between original and modified concepts through a uniaxial compression, if critical state effect is neglected. If the slopes of limit and bounding surfaces are identical \(M^\mathrm{b} = M^\mathrm{l}\), distance between the stress state and bounding surface comes very close to zero leading to a ductile behaviour. The stress state tends asymptotically to the bounding surface but never reaches it. On the other hand if those surfaces are distinct \(M^\mathrm{b} > M^\mathrm{l}\), the stress state reaches the limit surface, while the distance to the bounding surface is different from zero, namely the plastic modulus is non-null. The behaviour is brittle.

Fig. 7
figure 7

Difference between ductile and brittle failure, solid line: stress path, square markers: start of plastic strain, circle markers: peak strength

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All surfaces are represented in Fig. 8, only on the compression side (\(q>0\)) for clarity. All surfaces are homothetic and share a common apex located on the p axis at coordinate -\(p_\mathrm{c}\) which is a function of the cohesion c and the friction angle \(\phi\)

$$\begin{aligned} p_\mathrm{c} = \dfrac{c}{\tan \phi }, \end{aligned}$$
(6)

where \(\phi\) is the friction angle.

Fig. 8
figure 8

Description of the different surfaces of the model (compression side \(q>0\)): \(\alpha \) back-stress ratio, \(M^\mathrm{y}\) yield surface, \(M^\mathrm{l}\) limit surface, \(M^\mathrm{b}\) bounding surface, \(M^\mathrm{pc}\) decohesion surface

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The yield surface delineates the elastic domain and has an opening of \(2\, M^\mathrm{y}\). The opening of this surface is constant, but the position of its axis evolves from its initial position \(\alpha =0\), namely an axis centred on the p axis. The initial stress state must lie inside or on this surface. Its mathematical formulation reads

$$\begin{aligned} f^y \equiv \left( q - \left( p + p_\mathrm{c}\right) \, \alpha \right) ^2 - \left( p + p_\mathrm{c}\right) ^2\,\left( M^\mathrm{y}\right) ^2 \end{aligned}$$
(7)

where \(\alpha\) is termed the back-stress ratio of the surface. This surface is subject to kinematic hardening inducing variations of \(\alpha\) within the following limits

$$\begin{aligned} \alpha + M^{\mathrm{y}}\le M^{\mathrm{l}}&\quad {\mathrm {in \, compression,}} \end{aligned}$$
(8)
$$\begin{aligned} \alpha - M^\mathrm{y}\ge M^\mathrm{l}/\delta ^l&\quad {\mathrm {in \, extension,}} \end{aligned}$$
(9)

where \(M^\mathrm{l}\) is the slope of the limit surface and \(\delta ^l\) is the ratio between the compression and extension openings of the limit surface. The slope \(M^\mathrm{l}\) is related to the friction angle through

$$\begin{aligned} M^\mathrm{l} = \dfrac{6 \, \sin \phi }{3 - \sin \phi }. \end{aligned}$$
(10)

The yield surface is then a function of two internal variables: \(\alpha\) and \(p_\mathrm{c}\). The limit surface is the failure locus associated with the material, namely it describes the peak strength of the material. It is given by

$$\begin{aligned} f^l \equiv q^2 - \left( p + p_\mathrm{c} \right) ^2 \, \left( M^\mathrm{l}\right) ^2 \end{aligned}$$
(11)

and depends only on the internal variable \(p_\mathrm{c}\).

Experimental results evidence that cohesion loss is the main strength degradation mechanism of rock samples as reported in Sect. 2. Therefore limit surface should be subject to softening (cohesion \(p_\mathrm{c}\) decreases) during loading. In addition cyclic loading seems to have no effect beyond a given threshold, namely there is no strength reduction. Therefore we introduce a fatigue surface \(M^\mathrm{pc}\), homothetic to the bounding surface Eq. (11), which is a threshold below which no strength degradation occurs. If the stress path lies within this surface, there is no cohesion weakening. If the stress path lies outside this surface, cohesion is degraded.

3.3 Flow Rule

Plastic strain is assumed to be proportional to the reduced deviatoric stress. In addition the flow rule is non-associated with take into account the volumetric dilatancy due to crack opening. The variations of the plastic components of strain are defined such that

$$\begin{aligned} \dot{\mathbf {e}}^\mathrm{p} =\begin{bmatrix} \dot{\epsilon }_\mathrm{q}^\mathrm{p} \\ \dot{\epsilon }_\mathrm{v}^\mathrm{p} \end{bmatrix} = \dot{\lambda } \begin{bmatrix} A_\mathrm{q} \, \xi \\ A_\mathrm{d} \, \xi \end{bmatrix} = \dot{\lambda } \, {\mathbf {G}} \end{aligned}$$
(12)

where \(\dot{\lambda }\) is the variation of the plastic multiplier, \(A_\mathrm{q}\) is a material parameter, and \(A_\mathrm{d}\) is a parameter of dilatancy.

3.4 Hardening of Internal Variables

There are two main internal variables in this model: the back-stress ratio \(\alpha\) and the cohesion \(p_\mathrm{c}\). Two different cases are considered in the following. Firstly if the stress state lies below the limit surface \(M^\mathrm{pc}\), the yield surface is subject to kinematic hardening, namely the back-stress ratio increases or decreases, \(\dot{\alpha } \ne 0\). This is referred as mechanism 1. If the reduced deviatoric stress is greater than the fatigue threshold, namely \(\xi > M^\mathrm{pc}\), mechanism 1 also involves isotropic softening of the cohesion, namely \({\dot{p}_c}<0\).

On the other hand, if the stress state lies on the limit surface \(M^\mathrm{l}\), it is not allowed to overcome it. Therefore there is no kinematic hardening, \(\dot{\alpha } = 0\). However isotropic cohesion softening continues, \(\dot{p}_\mathrm{c} <0\). This is referred as mechanism 2 in the following. In both cases all surfaces are subject to the same isotropic hardening on \(p_\mathrm{c}\), namely their apex remains common.

The kinematic hardening law of the yield surface reads

$$\begin{aligned} \dot{\alpha } = \dot{\lambda } \, h \, \left( \alpha ^\mathrm{b} - \alpha \right) \, \vert \alpha ^\mathrm{b} - \alpha \vert ^{n_{\alpha }} = \dot{\lambda } \, h_1, \end{aligned}$$
(13)

where \(n_{\alpha }\) is a parameter that rules the stiffness and brittleness of the material, and h is defined as Taiebat and Dafalias (2008)

$$\begin{aligned} h = \dfrac{b_0}{\left( M^\mathrm{b}\, (1+1/\delta ^l) - 2\, M^\mathrm{y} - \left( \alpha ^\mathrm{b} - \alpha \right) \right) ^2}, \end{aligned}$$
(14)

where \(b_0\) is a material parameter. The definition of the \(\alpha ^\mathrm{b}\) variable depends on the sign of deviatoric stress increment \(\mathrm{d}q\)

$$\begin{aligned} \alpha ^\mathrm{b}&= M^\mathrm{b} - M^\mathrm{y}& \quad {\mathrm{d}} q >0,\end{aligned}$$
(15)
$$\begin{aligned} \alpha ^\mathrm{b}&= -M^\mathrm{b}/\delta ^l + M^\mathrm{y}& \quad {\mathrm{d}} q <0. \end{aligned}$$
(16)

Such a definition allows different stiffness in loading and unloading.

The isotropic softening rule simply reads

$$\begin{aligned} \dot{p}_\mathrm{c} = \dot{\lambda } \, A_{c1} \, \left( p_\mathrm{res} -p_\mathrm{c}\right) \, \left\langle \vert \alpha \vert - \vert \alpha ^\mathrm{pc}\vert \right\rangle \,\vert \alpha - \alpha ^\mathrm{pc}\vert ^{n_\mathrm{pc}} = \dot{\lambda } \, h_2, \end{aligned}$$
(17)

where \(A_\mathrm{c1}\) and \(n_\mathrm{pc}\) are parameters ruling the rate of decohesion, \(p_\mathrm{res}\) is the residual cohesion after full damage, the operator \(\left\langle X \right\rangle = 0.5 \, \left( \vert X\vert + X \right)\) is the Macauley brackets and

$$\begin{aligned} \alpha ^{\mathrm{pc}}&= M^{\mathrm{pc}} - M^{\mathrm{y}}& \quad q >0, \end{aligned}$$
(18)
$$\begin{aligned} \alpha ^{\mathrm{pc}}= -M^{\mathrm{pc}}/\delta ^l + M^{\mathrm{y}} \quad q < 0. \end{aligned}$$
(19)

When the yield surface reaches the limit one, namely \(M^\mathrm{y} + \alpha = M^\mathrm{l}\), \(\alpha\) has reached its maximum value, \({\mathbf {\dot{\alpha }}}= 0\). Parameter \(A_\mathrm{c2}\) is used instead of \(A_\mathrm{c1}\) in order to control the rate of cohesion decrease.

The hardening laws are written in compact form such that

$$\begin{aligned} \dot{\mathbf {v}} = \begin{bmatrix} \dot{\alpha } \\ \dot{p}_\mathrm{c} \end{bmatrix} = \dot{\lambda } \begin{bmatrix} h_1 \\ h_2 \end{bmatrix} = \dot{\lambda } \, {\mathbf {H}}. \end{aligned}$$
(20)

3.5 Plastic Multiplier

Assuming the classical additive decomposition of strain between plastic and elastic components (Simo and Hughes 1998), the triaxial stress–strain relation is written

$$\begin{aligned} \dot{\mathbf {s}} = {\mathbf {E}} \cdot \left( \dot{\mathbf {e}} - \dot{\mathbf {e}}^\mathrm{p} \right) \end{aligned}$$
(21)

where

$$\begin{aligned} {\mathbf {E}} = \begin{bmatrix} 2\, G&0 \\ 0&K \end{bmatrix} \end{aligned}$$
(22)

where G is the shear modulus and K the bulk modulus. The consistency condition is then written

$$\begin{aligned} {\dot{f}}^{\mathrm{y}} \equiv \left[ \dfrac{\partial f^{\mathrm{y}}}{\partial {\mathbf{s}}}\right] ^{\mathrm{T}} \cdot \dot{\mathbf {s}} + \left[ \dfrac{\partial f^{\mathrm{y}}}{\partial {\mathbf {v}}}\right] ^{\mathrm{T}}\cdot \dot{\mathbf {v}} = 0 \end{aligned}$$
(23)

Inserting Eqs. (12), (20) and (21) into Eq. (23) leads to the following expression of the plastic multiplier

$$\begin{aligned} \dot{\lambda } = \dfrac{\left[ \dfrac{\partial f}{\partial {\mathbf {s}}}\right] ^\mathrm{T} \cdot {\mathbf {E}} \cdot \dot{\mathbf {e}}}{\left[ \dfrac{\partial f}{\partial {\mathbf {s}}}\right] ^\mathrm{T} \cdot {\mathbf {E}} \cdot {\mathbf {G}} - \left[ \dfrac{\partial f}{\partial {\mathbf {v}}}\right] ^\mathrm{T} \cdot {\mathbf {H}}} \end{aligned}$$
(24)

where

$$\begin{aligned} \left[ \dfrac{\partial f}{\partial {\mathbf {s}}}\right] ^\mathrm{T} = \begin{bmatrix} \dfrac{\partial f^y}{\partial q}&\dfrac{\partial f^y}{\partial p} \end{bmatrix} \end{aligned}$$
(25)

and

$$\begin{aligned} \left[ \dfrac{\partial f}{\partial {\mathbf {v}}}\right] ^\mathrm{T} = \begin{bmatrix} \dfrac{\partial f^y}{\partial \alpha }&\dfrac{\partial f^y}{\partial p_\mathrm{c}} \end{bmatrix}. \end{aligned}$$
(26)

3.6 Implementation

The equations are implemented in a set of MATLAB routines using a forward Euler scheme. This scheme is not the most efficient but is accurate enough provided the strain steps are sufficiently small. In addition a correction step to ensure the stress state lies on the yield surface is performed at the end of each iteration.

3.7 Extension to Multiaxial Formulation

The constitutive law is presented in triaxial notations, which is suitable for simulation of uniaxial or triaxial tests. However modelling of real case studies requires the development of a 3D stress state model. The full development of a multiaxial model is out of the scope of this paper. However generalisation of the model to a multiaxial formulation can be carried out following methodology proposed in Taiebat and Dafalias (2008). All the equations are systematically rewritten as function of tensors. All surfaces are expressed as a function of \({\mathbf {s}}\), the deviatoric stress tensor, and the back-stress variable \(\alpha\) becomes a second-order tensor. The integration of the constitutive law requires a more complex integration scheme. We recommend the explicit integration scheme with automatic control error presented in Sloan et al. (2001).

4 Validation of the Model

The objective of this section is to prove the capability of the model to reproduce different types of experimental results. In the following, three different materials are investigated: Lorano marble, a sandstone and the Lac du Bonnet granite. The choice of these three materials is mainly based on the availability of results published in the literature. Different types of experiments are calibrated: simple compression, triaxial or damage controlled. The focus is mainly placed on the reproduction of cyclic rather than monotonic results. In addition, simulations consider only a static loading and viscous effects are not taken into account. A final parametric study exhibits the effect of the three main parameters affecting cyclic loading.

4.1 Calibration Process

Calibrating material parameters for such a constitutive law is not straightforward. Many experiments are necessary for that purpose but are not always provided all together in the literature. Some model parameters are interrelated and may have similar or opposite effects on results. Therefore the optimisation process may be carried out manually in a trial-and-error process. All parameters are listed in Table 1 together with experiments necessary to calibrate them.

Table 1 Name and function of parameters and internal variables, experimental tests required for their calibration (monotonic test if not specified)
Full size table

The full determination of parameters requires many uniaxial, triaxial, monotonic and cyclic experiments. Uniaxial monotonic experiment is used to determine classical elastic (E,\(\nu\)) parameters and slope of the failure criterion (\(M^\mathrm{l}\)). The opening of the elastic zone depends on \(M^\mathrm{y}\) and may be defined within the range 0.0.5–0.1. Cohesion \(p_\mathrm{c}\) is estimated as an average parameter but is rigorously an internal variable, representative of the state of damage of the sample. Therefore it could be modified depending on the sample for an identical material. Similarly \(\alpha\) is another internal variable representing a load-induced anisotropy. It is assumed equal to zero initially.

The elastic parameters are assumed constant during loading. Therefore results obtained are assumed to be valid only in a limited range of confining pressure. In the following the hypothesis of identical compression and traction strength is also assumed, namely \(\delta _\mathrm{l} =1\), because there were no extension tests available for the considered rock materials. However it should be noted that results are not influenced since only compression tests are investigated. If extension triaxial tests are available, it is possible to define a different \(\delta _\mathrm{l}\) value. Finally a purely associated deviatoric response is assumed, leading to \(A_\mathrm{q} = 1\) in Eq. (12). On the contrary the volumetric plastic behaviour is non-associated and depends on the dilatancy parameter \(A_\mathrm{d}\).

Kinematic hardening of the internal variable \(\alpha\) depends on the slope of the bounding surface \(M^\mathrm{b}\), the exponent \(n_{\alpha }\) and a stiffness parameter \(b_0\). These parameters rule the nonlinear behaviour of the material. Similarly parameters \(n_\mathrm{pc}\), \(A_\mathrm{c1}\) and \(A_\mathrm{c2}\) (post-peak) rule the cohesion degradation (softening of \(p_\mathrm{c}\)). All of these parameters must be calibrated as a best fit of experimental results.

The \(M^\mathrm{pc}\) parameter represents the fatigue limit. It indeed requires many constant cycles experiment to be accurately captured. However results provided in the literature indicate that the fatigue limit may lie between 60 and 70% of the peak stress. Therefore a good starting point for this value is equal to 0.7 \(M^\mathrm{l}\).

Parameters calibrated for all materials are provided in Table 2.

Table 2 Calibrated parameters for three different materials: Lorano marble (Royer-Carfagni and Salvatore 2000), sandstone (Liu and He 2012) and granite (Martin and Chandler 1994)
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In addition experimental results may be highly variable due to the heterogeneity, damaging of samples during drilling and/or anisotropy of the material. Therefore results are not very reproducible, and it is difficult to predict a priori the peak strength of a given rock sample. Subsequently the cyclic amplitude to monotonic peak stress ratio is not easily obtained, and cyclic results are difficult to compare. A method to overcome this problem was recently proposed by Taheri et al. (2016).

Due to the experimentally observed complexity, heterogeneity and variability of rock materials, it is not possible to reproduce perfectly all the results of monotonic and cyclic tests for a single set of parameters. In this work we decided to focus on the cyclic experiments for the determination of a set of parameters.

4.2 Characterisation of Rock Materials

Lorano marble belongs to the family of white Carrara marble and is one of the most widely used varieties. It is composed of over than 99% of calcium carbonate. It has a homoblastic arrangement and is composed of regularly shaped grains with straight or gently curbing boundaries (Royer-Carfagni and Salvatore 2000). Tested cylindrical specimens reported in this paper are 10 cm long and 4 cm in diameter. The loading rate of the test is about 1–2 MPa/s and of the order of one cycle per minute. The experiment investigated is a constant amplitude cyclic simple compression.

Characteristics of the tested sandstone are not reported in the reference paper (Liu and He 2012). The samples have a diameter to length ratio of 1:2 with an average diameter of 48.9 mm. Cycles are applied at a frequency of 1 Hz. The investigated experiment is a constant amplitude cyclic triaxial test with a confining pressure equal to 10 MPa.

The Lac du Bonnet granite is medium to coarse grained and composed of approximately 30% K-feldspar, 30% plagioclase, 30% quartz and 10% mafic minerals, mainly biotite. The average grain size of the medium-grained granite is about 3–4 mm. Results reported in this paper were obtained from the 420 level of AECL’s Underground Research Laboratory (Martin and Chandler 1994). Uniaxial and triaxial post-failure tests were carried out on 63-mm-diameter grey samples. The stress rate is equal to 0.75 MPa/s. The experiment investigated is a damage-controlled triaxial test with a confining pressure equal to 15 MPa.

4.3 Lorano Marble

Results of a monotonic simple compression simulation are firstly compared with respect to published data (Royer-Carfagni and Salvatore 2000). The objective is twofold: validating the model and illustrating its basic working. The reference cyclic paper (Royer-Carfagni and Salvatore 2000) does not provide monotonic triaxial tests at different confining pressures. Therefore friction angle, cohesion and elastic parameters are estimated and compared with the literature (Dal Pino et al. 1999; Royer-Carfagni and Salvatore 2000; Cattaneo and Labuz 2001; Papamichos et al. 2004; Stavropoulou et al. 2004; Ferrero et al. 2009).

Results are in good agreements with experimental results as shown in Figs. 9 and 10. This material presents a relatively large amount of plastic strain before failure, with respect to the other investigated rocks, that is correctly reproduced. Brittle failure is also captured in terms of both peak resistance (\(\sim 85\) MPa) and strain (\(\sim 0.22\%\)). No data are available on the post-peak behaviour; therefore, it is simulated for three different values of \(A_\mathrm{c2}\). The brittleness of failure increases with this parameter.

Fig. 9
figure 9

Comparison of monotonic simple compression simulation and experimental results, deviatoric stress q versus axial strain \(\epsilon _1\), Lorano marble, \(A_\mathrm{c2}=[500, 750, 1000]\), experimental results from Royer-Carfagni and Salvatore (2000)

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Volumetric behaviour before peak is depicted in Fig. 10. Behaviour is mainly dilative due to crack opening. Data and simulations are in good agreement.

Fig. 10
figure 10

Comparison of monotonic simple compression simulation and experimental results, volumetric \(\epsilon _\mathrm{v}\) versus axial \(\epsilon _1\) strain before peak, Lorano marble, experimental results from Royer-Carfagni and Salvatore (2000)

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The evolution of internal variables is reported in Fig. 11. During the first part of the simulation, \(\alpha\) nonlinearly increases, indicating kinematic hardening. In the meantime, the cohesion internal variable \(p_\mathrm{c}\) is only slightly affected. When \(\alpha = M^\mathrm{l} - M^\mathrm{y}\) the limit surface is reached and \(\alpha\) remains constant. However \(p_\mathrm{c}\) strongly decreases indicating failure. The slope discontinuity is due to the change from mechanism 1 to mechanism 2. In this case, \(h_1\) changes from a value different from zero to zero, affecting plastic multiplier computation in Eq. (24) and generating this sharper cohesion decrease.

Fig. 11
figure 11

Evolution of internal variables \(p_\mathrm{c}\) and \(\alpha \) during the simple compression test, Lorano marble

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The evolution of the internal variables affects the position of the yield surface as represented in Fig. 12. The elastic zone (in grey), the trace of the limit surface (solid black line), the trace of \(\alpha\) (dashed line), the stress path (dashed and pointed line) and the current stress state (circle marker) are represented in this figure for three different vertical strain \(\epsilon _1\) of the monotonic simple compression simulation. In the top figure, only mechanism 1 is activated; \(\alpha\) increases (kinematic hardening, Eq. 13). In the meantime cohesion \(p_\mathrm{c}\) has only slightly decreased (isotropic softening, Eq. 17). In the second figure, the stress states get into contact with the limit surface. Changes in cohesion \(p_\mathrm{c}\) are still very limited since the stress states lied below the limit surface. The last figure represents the post-peak behaviour, \(\alpha\) is constant but \(p_\mathrm{c}\) decreases strongly, causing softening.

Fig. 12
figure 12

Evolution of yield and limit surfaces during the simple compression test for three different vertical strain \(\epsilon _1\) (0.1% top, 0.2% middle, 0.3% bottom), Lorano marble

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According to this model, the experimentally measured cohesion at peak is different from the initial cohesion used for \(p_\mathrm{c}\). This could complicate the calibration process. However since cohesion is only slightly modified during monotonic loading, the measured value at peak may serve as a good guess for its initialisation.

The best fit for cyclic results is provided in Figs. 13 and 14. The first figure represents the stress–strain evolution and could be compared with Fig. 1. It is impossible to superpose both figures (it would be impossible to read) or to compare each cycle independently (it would need a lot of figures). Qualitatively the physical trends are correctly reproduced. The total number of cycles and final plastic strain is correctly captured by the simulation. Lateral strain is larger than the axial one as observed experimentally. Finally the cycles are progressively opening, indicating a larger amount of plasticity as the stress path approaches failure.

Fig. 13
figure 13

Deviatoric stress q versus axial \(\epsilon _1\) or lateral \(\epsilon _3\) strain, cyclic simple compression test, Lorano marble

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Fig. 14
figure 14

Indicators of maximum \(\varDelta _\mathrm{max}\) and minimum \(\varDelta _\mathrm{min}\) volumetric strain with the number of cycles N, Lorano marble (Royer-Carfagni and Salvatore 2000) for experimental results

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Accumulation of strain can be compared through a strain indicator computed for each cycle. Maximum volumetric strain reached over a cycle N may be quantified by Royer-Carfagni and Salvatore (2000).

$$\begin{aligned} \varDelta _\mathrm{max}(N) = \max \limits _{\mathrm{cycle} ~N} \left[ \dfrac{\epsilon _1 + \epsilon _3}{2}\right] , \end{aligned}$$
(27)

Similarly the minimum value \(\varDelta _\mathrm{min}(N)\) may be computed. Indicators evolution with respect to the number of cycles is provided in Fig. 14. Simulated and experimental results are in relatively good agreement, with respect to final cycle number (45 numerically and 51 experimentally) and strain. The general trend is accumulation of negative strain, due to microcrack opening. A shear strain indicator could be computed considering \(\vert \epsilon _1 - \epsilon _3\vert\) in Eq. 27. Experimental and simulated results match similarly and are not presented. Therefore it could be concluded that the accumulation of plastic strain cycle after cycle is correctly captured.

The evolution of cohesion variable \(p_\mathrm{c}\) with axial strain is provided in Fig. 15. It shows that the cyclic loading progressively weakens the material and degrades cohesion. Finally the imposed stress path reached the limit surface and failure occurs. By comparison with Fig. 11, cohesion is significantly decreased before the second mechanism is activated.

Fig. 15
figure 15

Evolution of cohesion internal variable \(p_\mathrm{c}\) with the number of cycles N, Lorano marble

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Optimisation of parameters for the cyclic experiment requires modification of three values of the initial set of parameters provided in Table 2. Firstly as mentioned in Sect. 2, rock samples used for monotonic and cyclic tests do not have the same strength properties. Therefore the cohesion parameter \(p_\mathrm{c}\) reflecting indirectly the initial damage is set up to 34 MPa.

Stiffness \(b_{0}\) and degradation \(A_{c1}\) parameters are modified to \(4\,10^5\) and 400, respectively. Calibrating these parameters on monotonic tests only always leads to overestimation of plastic strain during cyclic loading. Physically it is assumed that the rock material hardens during the first or few first cycles and softens afterwards. In order to simplify the model formulation, this fast hardening is neglected and only final hardened parameters are calibrated. This does not affect comparison of the global tendency.

4.4 Sandstone

Cyclic triaxial experiments on a sandstone are provided below from the original paper (Liu and He 2012). Monotonic results characterising the material are provided in Fig. 16. Therefore elastic and strength parameters could be calibrated directly from data provided in the paper. Young's modulus only slightly varies with confining pressure. The post-peak behaviour is very sharp at low confining pressure but is smoother at higher confining pressure. Calibrated parameters on the cyclic triaxial test with a confining pressure equal to 10 MPa are provided in Table 2.

Fig. 16
figure 16

Monotonic compression triaxial tests at six confining pressures, deviatoric stress q versus axial strain \(\epsilon _1\), sandstone (Liu and He 2012)

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The reference paper does not detail how the cyclic amplitude was chosen for each presented experiment (Liu and He 2012). Moreover it seems that the cyclic load is always larger than the reported monotonic peak strength. Therefore it is assumed the cyclic load is 15 MPa lower than the peak strength in this section.

Simulation of monotonic and cyclic triaxial test (confining pressure equal to 10 MPa) and an experimental monotonic test are compared in Fig. 17. Once again, parameters are calibrated to reproduce the cyclic behaviour rather than the monotonic test. Therefore monotonic simulation is stiffer than observed experimentally.

Fig. 17
figure 17

Triaxial compression test (confining pressure = 10 MPa), deviatoric stress q versus axial strain \(\epsilon _1\), sandstone, experimental results from Liu and He (2012)

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Once again, it is difficult to compare the effect of each cycle independently. Therefore simulated axial and volumetric residual strain corresponding to each cycle is compared with experimental ones in Figs. 18 and 19. Experimental results are corrected to start from 0 at the beginning of the test. This neglects the effect of the first cycle which is not reproduced by the model as previously discussed. Subsequently comparison is focused here on the following accumulation.

Fig. 18
figure 18

Cyclic triaxial compression test, axial residual strain \(\epsilon ^\mathrm{p}_\mathrm{1}\) versus number of cycles N, experimental results from Liu and He (2012)

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Fig. 19
figure 19

Cyclic triaxial compression test, volumetric residual strain \(\epsilon ^\mathrm{p}_\mathrm{1}\) versus number of cycles N, experimental results from Liu and He (2012)

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Comparison of simulation and experiment can be based on different criteria: slope of the stationary phase, number of cycle to failure, final residual strain, axial or volumetric strain. In this work we focus on the slope of the stationary phase and number of cycles to failure of both axial and volumetric residual strain

They are both well captured by the simulations. Final strain at the end of the experiment and activation of the second mechanism are different. For instance in Fig. 18, \(\epsilon _{1,\mathrm{residual}}\) is equal to 0.15% when the stress path crosses the limit surface, while it reaches 0.3% at the end of the experiment. For the simulation, further accumulated strain is due to the post-peak behaviour. For the experiment, half of the final strain is accumulated during the last few cycles. It is assumed that they may be assimilated to failure.

4.5 Lac du Bonnet Granite

Lac du Bonnet granite is a well-studied rock material (Martin and Chandler 1994; Chow et al. 1995; Martin 1997; Eberhardt et al. 1999a, b), exhibiting clearly a brittle behaviour as shown in Fig. 20. Elastic and failure parameters are calibrated with respect to literature. However samples presented by Martin and Chandler (1994) are known to be initially damaged due to sampling, which modifies their properties with respect to literature. It is shown that the failure locus is described by a Hoek-Brown criterion (Hajiabdolmajid and Kaiser 2002). However it is assumed that a Drucker–Prager limit surface will be a reasonable approximation in the considered range of confining pressures. Calibrated parameters are provided in Table 2.

Fig. 20
figure 20

Monotonic compression triaxial tests at six confining pressures, deviatoric stress q versus axial strain \(\epsilon _1\), Lac du Bonnet granite (Martin 1997)

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Monotonic simulation and experiment at a confining pressure equal to 15 MPa are compared in Fig. 21. The behaviour is relatively stiff, and only few plastic strains are observed before brittle failure. The first part of the post-peak behaviour is well reproduced although the residual strain is lower experimentally. This may result from a degradation of the friction angle. It should also be noted that a too high \(A_\mathrm{c2}\) parameter may lead to snapback phenomenon which cannot be captured with the integration scheme used in this work.

Fig. 21
figure 21

Triaxial compression test (confining pressure = 15 MPa), deviatoric stress q versus axial strain \(\epsilon _1\), Lac du Bonnet granite, experimental results from Martin (1997)

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Driving of the damage-controlled test presented in Fig. 6 is complex. Achieving such a special driving numerically is not easy. The numerical integration of the constitutive equation is strain driven and divided into \(\varDelta \epsilon _1\) increments. Before the peak it is assumed the stress reversal occurs every \(N \, \varDelta \epsilon _1\) increments. After the peak, it is assumed that only two time steps are allowed when the limit surface is reached.

Stress–strain results of the damage-controlled tests are presented in Fig. 22 along with a post-peak monotonic simulation. They are compared with experimental results (Martin and Chandler 1994) provided in Fig. 6. The trend of the results is well reproduced. The pre-peak cycles generate plastic strain, and the peak strength is decreased. Maximum stress reached after the peak fast decrease due to the cohesion’s weakening. The monotonic loading is almost an envelope for cyclic results, as expected. The small shift is due to the amount of plastic strain accumulated before peak.

Fig. 22
figure 22

Damage-controlled test and monotonic tests, deviatoric stress q versus axial strain \(\epsilon _1\)

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A damage indicator \(\omega\) is defined as the volumetric residual strain \(\epsilon ^\mathrm{p}_\mathrm{v}\) reached at the end of the unloading. A comparison of this indicator evolution for both numerical and experimental results is made in Fig. 23. Numerical results do not fit perfectly the experimental points but are in reasonable agreement. The difference in the post-peak domain may be mainly due to the driving process which is different numerically and allows less plastic strain before reversal.

Fig. 23
figure 23

Damage-controlled test, deviatoric stress q versus damage \(\omega \), experimental data from Martin and Chandler (1994)

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It should be noted that comparison in the post-peak zone is subject to caution. Indeed, experimentally cracks are generated and coalesce. Therefore measured results are representative of the rock sample in which stresses and strain are not homogeneous. On the contrary numerical results assume homogeneity over a sample.

However the post-peak behaviour must be numerically defined for several reasons in the constitutive law. Firstly a softening regime is necessary to reproduce numerically fracture initiation or strain localisation. Secondly if the constitutive law is applied to the modelling of a real case study, for instance of a cavern or rock pillar, a local failure may happen without leading to a global collapse of the studied problem. Therefore the constitutive law may be defined for such a post-peak state.

4.6 Parameters Influencing Cyclic Response

A parametric study is provided in the following, based on the Lorano marble cyclic test, in order to illustrate the influence of three parameters affecting cyclic results: \(b_0\), \(A_\mathrm{c}\), \(M^\mathrm{pc}\). The evolution of \(\varDelta _\mathrm{min}\) with the number of cycles up to failure is reported for each parameter and compared with the reference value. The influence of each parameter is summarised in Table 3.

Table 3 Influence of increasing one of the three main parameters on the number of cycles to failure (\(N_\mathrm{fail}\)), the plastic strain at failure (\(\epsilon ^\mathrm{p}_\mathrm{fail}\)) and the degradation rate/cycle (\({\mathrm dp_c}\)/cycle)
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The influence of the fatigue limit \(M^\mathrm{pc}\) is obvious, as reported in Fig. 24. By definition of this model, cohesion is only degraded when the stress path lies outside the fatigue surface \(q/(p + p_\mathrm{c})>M^\mathrm{pc}\). Therefore increasing the size of this surface reduces the degradation rate of each cycle and increases the number of cycle to failure as well as final strain.

Fig. 24
figure 24

Indicators of minimum \(\varDelta _\mathrm{min}\) volumetric strain with the number of cycles N, Lorano marble, influence of \(M^\mathrm{pc}\) parameter

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Parameter \(b_0\) affects the amount of plastic strain that is generated during loading, as reported in Fig. 25. Therefore increasing this value decreases the amount of plastic strain generated during each cycle. Consequently, the number of cycle at failure also increases since the degradation of cohesion is a function of the accumulated plastic strain. The final strain is larger in absolute value since there are more cycles.

Fig. 25
figure 25

Indicators of minimum \(\varDelta _\mathrm{min}\) volumetric strain with the number of cycles N, Lorano marble, influence of \(b_0\) parameter

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Finally the parameter \(A_\mathrm{c}\) affects the rate of cohesion degradation as a function of the plastic strain accumulation. Therefore increasing this value directly leads to a faster degradation of each cycle, a lower number of cycles to failure and final strain, as depicted in Fig. 26.

Fig. 26
figure 26

Indicators of minimum \(\varDelta _\mathrm{min}\) volumetric strain with the number of cycles N, Lorano marble, influence of \(A_\mathrm{c}\) parameter

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5 Conclusion

This paper presents the mathematical formulation of a new constitutive law able to reproduce the cyclic behaviour of brittle rock materials at low confining pressure. It lies on the elastoplastic framework and is based on a bounding surface model developed initially for sands. It is able to reproduce salient features of the monotonic and cyclic behaviour of rock materials

  • brittle failure of the material,

  • post-peak behaviour,

  • accumulation of plastic strain upon cyclic load,

  • progressive opening of the cycles,

  • degradation of the cohesion,

  • fatigue resistance of the material.

The basic idea of this model assumes that cyclic loading progressively degrades cohesion of a rock material even if the load applied is lower than its monotonic resistance, namely if the loading lies within the limit surface. A fatigue limit is incorporated within the formulation to take into account the fatigue resistance observed experimentally, namely the maximum load below which there is no degradation of the resistance. The isotropic mechanism (cohesion degradation) is coupled with a kinematic mechanism to reproduce accumulation of plastic strain during cycling. The model is formulated in triaxial notations and implemented in a MATLAB set of routines using a forward Euler scheme.

The new model is validated against three different rock materials (Lorano marble, a sandstone, Lac du Bonnet granite) and diverse monotonic (simple compression, triaxial compression) and cyclic tests (simple compression, triaxial compression, constant amplitude, damage controlled). It is shown that a set of parameters can be obtained in any case to reproduce the trend of cyclic experiments, namely to capture the residual strain or number of cycles to failure. Therefore the model is proven to be suitable to reproduce such a behaviour.

However it is also highlighted that calibration process is not straightforward. The first issue is mainly the lack of data related to a given material. Due to the relative complexity of the model, numerous triaxial tests in compression and also in extension should be necessary to calibrate failure parameters. Similarly sufficient cyclic tests should be carried out to derive the fatigue strength and cyclic degradation of the material. The second major issue is the heterogeneity and variability of rock samples. Therefore it is often uneasy to compare results from different samples, between monotonic and cyclic or simply between two cyclic tests.

This model incorporates many features of the monotonic and cyclic behaviours of rock materials. However it does not reproduce all physical phenomena and is mainly applicable under some hypotheses. Elastic properties of the material are considered isotropic as well as cohesion of the material, while some anisotropy is likely. Rate and viscous effects are not considered; therefore, parameters correspond to a given strain rate.

The quality of the model may be improved in several ways, at the cost of additional parameters to be calibrated or implementation complexity increase. Firstly the shape of the limit surface may be modified. Indeed it is shown that the Hoek–Brown criterion is more relevant than a Drucker–Prager one to represent failure locus (Hoek and Brown 1980; Hajiabdolmajid and Kaiser 2002). However the implementation of such a criterion would be more complex since the yield surface should be homothetic to this surface in order to avoid overlap. In addition a cap should be applied to close the surface and allow plasticity upon constant stress ratio loading. Different types of yield surfaces are investigated by Taiebat and Dafalias (2010) and may be adopted for improving the present model.

In this model it is assumed cohesion degrades progressively, but friction angle remains constant. However it is shown that friction is progressively hardening as cohesion decreases (Martin and Chandler 1994; Hajiabdolmajid and Kaiser 2002). This could be included within the model by introducing a hardening law of the limit surface. This mechanism could allow a larger plastic strain accumulation for monotonic and first cycle of cyclic loading. In addition damaging of elastic properties, despite not observed in all cyclic experiments, may represent another major improvement.

In the opinion of the authors, the last very important issue is modification of parameters from intact rock sample to rock mass, as studied by Hoek and Brown (1997). However the fundamental objective of a constitutive law is to study larger structures such as caverns, tunnels or pillars. Therefore parameters should be scaled to correspond to the rock mass, which has a lower strength due to heterogeneities and discontinuities. The use of more complex constitutive laws using more parameters makes this process more difficult.