J. Cent. South Univ. (2018) 25: 1665-1677

DOI: https://doi.org/10.1007/s11771-018-3858-9

A nonlinear rheological damage model of hard rock

HU Bo(胡波)¹, YANG Sheng-qi(杨圣奇)^{1, 2}, XU Peng(徐鹏)¹

1.State Key Laboratory for Geomechanics and Deep Underground Engineering,China University of Mining and Technology, Xuzhou 221116, China;

2. School of Mechanics and Civil Engineering, China University of Mining and Technology,Xuzhou 221116, China

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Abstract: By adopting cyclic increment loading and unloading method, time-independent and time-dependent strains can be separated. It is more reasonable to describe the reversible and the irreversible deformations of sample separately during creep process. A nonlinear elastic-visco-plastic rheological model is presented to characterize the time-based deformational behavior of hard rock. Specifically, a spring element is used to describe reversible instantaneous elastic deformation. A reversible nonlinear visco-elastic (RNVE) model is developed to characterize recoverable visco-elastic response. A combined model, which contains a fractional derivative dashpot in series with another Hook’s body, and a St. Venant body in parallel with them, is proposed to describe irreversible visco-plastic deformation. Furthermore, a three-stage damage equation based on strain energy is developed in the visco-plastic portion and then nonlinear elastic-visco-plastic rheological damage model is established to explain the trimodal creep response of hard rock. Finally, the proposed model is validated by a laboratory triaxial rheological experiment. Comparing with theoretical and experimental results, this rheological damage model characterizes well the reversible and irreversible deformations of the sample, especially the tertiary creep behavior.

Key words: Hard rock; multi-step loading and unloading cycles; nonlinear; damage; rheological model

Cite this article as: HU Bo, YANG Sheng-qi, XU Peng. A nonlinear rheological damage model of hard rock [J]. Journal of Central South University, 2018, 25(7): 1665–1677. DOI: https://doi.org/10.1007/s11771-018-3858-9.

1 Introduction

The stability of rock engineering is governed by the mechanical behaviors of surrounding rocks. There are a number of experimental studies on the basic deformational and failure behaviors of different rocks under complicated stress states [1–4]. However, the stability of rock engineering during a long period is related to the time-related properties of rocks, so rheological characteristic of soft and hard rock materials must be considered [5–11]. Especially, excavation activities in the deep underground always perturbed the stress state and affected the stability of surrounding rocks frequently. Moreover, the perturbed surrounding rocks subjected loading and unloading alternatively due to the excavation, whose rheological characteristics could not be ignored [12–15].

The creep deformation of rock consisted of reversible and irreversible deformations, and the time-related instability of rock engineering was closely related with the irreversible deformation [13]. Therefore, the elasto-visco-plastic behaviors of rock should be investigated, especially for the hard rock subjected higher geo-stress. MALAN [7] used a viscoelastic and developed viscoplastic theory to analyze the time-dependent deformational behavior of hard rock. To construct suitable creep models, the creep experiments are indispensable.DEBERNARDI et al [9] developed a new visco-plastic constitutive law and described the deformation features in tunnel excavations. MARANINI et al [14] presented a creep constitutive equation to characterize the rheological deformational behavior of granite. ZHAO et al [15] performed cyclic loading–unloading creep experiments on rocks to study the elasto- viscoplastic characteristics of iherzolites, and presented appropriate model to describe their creep deformations. XU et al [16] developed a nonlinear creep model to characterize the time-related deformational behaviors of green schists. CAO et al [17] presented a coupling model to calculate the relaxation loss of concrete. A CVISC model and a VIPLA model were developed to describe the creep behaviors of Apennines Clay Shale specimens by BONINI et al [18]. YANG et al [19] introduced a non-linear viscoplastic body to build a new creep equation to characterize the non-linear time-related deformational behaviors of diabase samples. SCIUM et al [20] deduced a damage equation based on a recoverable elastic-viscosity element to describe the time-dependent deformation and crack width of concrete materials. MIURA et al [8] proposed a micro-crack propagation model to predict the long period deformational result of hard rock and this model can reflect the creep rupture mechanism. SHAO et al [21] developed an anisotropic constitutive model to explain the anisotropic damage in brittle rocks, and applied the model to characterize the creep deformation of Lac du Bonnet granite. LI et al [22] presented a micro- mechanical constitutive equation to investigate the relation between macro rheological mechanical behavior and micro-crack extension of brittle rock. WEN et al [23] considered the discontinuity in natural rocks and then presented a simple multi axial damage creep constitutive equation. KACHANOV [24] presented damage concept in materials groundbreaking, then MURAKAMI [25] developed a damage model considering anisotropy in materials and KYOYA [26] used the presented damage concept to research rock materials. LIU et al [27] constructed a rate-damage creep model based on Kachanov’s report and detailed how to determine the creep parameters. YANG et al [28] performed triaxial compressive tests to explore the traditional triaxial and creep mechanical behaviors of marbles under cyclic loading and unloading, and proposed a visco-elastic-plastic rheological model to explain time-based deformational behavior of sample. HU et al [29] developed a damage model base on CVISC model to characterize tertiary creep of argillaceous siltstone. BUI [30] proposed a constitutive model extended from the original Lemaitre’s model for creep behavior of partially saturated rocks.

Moreover, fractional calculus was applied to describe materials’ visco-elastic and visco-plastic deformational behaviors [31–33]. WELCH et al [34] presented a constitutive model based on fractional calculus for describing the elastic- viscosity time-related deformation of polymeric material. CELAURO et al [35] deduced a simple creep constitutive equation to predict creep/recover behavior of asphalt mixtures. ZHOU et al [36, 37] used fractional calculus to deduce a nonlinear viscosity damage growth equation, which can reflect creep deformation of salt rock. KANG et al [38] presented an elastic-viscoplastic rheological equation to describe coal’s nonlinear time-based deformation.

However, there is no much report about a kind of rheological model to characterize the loading and unloading deformation behavior of hard rock. Especially, it is able to describe the reversible and irreversible characteristics of the sample. Consequently, in order to explain and predict the rheological deformation for hard rock during a long period, and explore its time-based damage evolution, a rheological constitutive model should be developed. A non-linear rheological damage model based on component model composing of the linear-elastic element, the plastic slider element, a new RNVE model, and a fractional derivative dashpot with damage, was presented. Finally, the proposed model was verified by cyclic loading and unloading experiment.

2 Non-linear rheological damage model

During creep experiment, specific strains can be separated by cyclic loading and unloading method, which include reversible strain and irreversible strain. The reversible strain can be divided into instantaneous elastic strain (ε_me) and visco-elastic strain (ε_ve), whereas the irreversible strain can be divided into instantaneous plastic strain (ε_mp) and visco-plastic strain (ε_vp), as shown in Figure 1. Moreover, total strain (ε) also can be divided into time-independent strain (ε_m) and time-dependent strain (ε_c) [39, 40], which can be expressed as

ε_m=ε_me+ε_mp                                               (1)

ε_c=ε_ve+ε_vp                                                 (2)

Figure 1 A typical sketch of different strains during creep test

XIA et al [39] supposed that the shape of reversible curve during loading was the same to that of delayed reversible curve, and the visco-plastic curve under loading can be obtained by creep curve and visco-elastic curve. Therefore, to characterize the different time-based curves during rheological experiment, a rheological model is studied as following parts.

2.1 Recoverable deformation model

2.1.1 Instantaneous elastic model

Lots of creep experiment results showed that instantaneous elastic deformation has a linear correlation with deviatoric stress [13–15, 40], therefore, it can be described by a Hooke body, as shown in Figure 2. The three dimensional state formulation can be written as Eq. (3).

                           (3)

where S_ij and σ_m represent the deviatoric stress tensor and the mean stress tensor, correspondingly, e_me and ε_m are respectively the corresponding deviatoric strain tensor and mean strain tensor, K_e and G_me are respectively the bulk moduli and the shear moduli. When confining pressures are same (σ₂=σ₃), the mean stress (σ_m) can be calculated by Eq. (4), and Eq. (5) can be used to calculate the mean strain (ε_m). Eq. (6) can be used to calculate the axial deviatoric stress (S₁₁) [41].

          (4)

          (5)

                  (6)

Figure 2 Instantaneous elastic model (Hooke body)

The bulk modulus (K_e) of material can be calculated by Eq. (7).

                            (7)

where E represents the elastic modulus and μ represents Poisson ratio of rock sample.

Assuming that the mean stress tensor just makes rock yield volume deformation and the deviatoric stress tensor just induces rock to adjust shape including time-dependent deformation [42]. Considering the hydrostatic pressure is unable to be unloaded during triaxial compressive creep experiment, so the axial strain produced by hydrostatic pressure can be recognized as a part of irrecoverable plastic strain. Instant elastic strain can be calculated by Eq. (8), and G_me can be calculated by Eq. (9).

          (8)

                     (9)

2.1.2 Nonlinear viscoelastic rheological model

As well known, Kelvin body, as shown in Figure 3, is able to describe the viscoelastic recoverable strain for rock [42], whose constitutive and creep equations can be written as Eqs. (10) and (11).

                     (10)

               (11)

where S_ij is the deviatoric stress tensor; e_ve represents corresponding recoverable deviatoric strain tensor; η_ve and G_ve represent Kelvin element’s viscosity parameter and shear modulus.

Figure 3 Viscoelastic model (Kelvin body)

However, the Kelvin model is unable to characterize well the delayed recoverable strain of the hard rock under high pressure condition (e.g.,the results of sample after unloaded deviatoric stresses in the later validation portion). YANG et al [43] reported that the time scale had a great influence on the creep parameters of model and the shear modulus and viscosity coefficient showed nonlinear correlations with time. Inspired by XU et al [16], we proposed RNVE model, as shown in Figure 4, whose one-dimensional stress–strain relation can be expressed as Eq.(12).

              (12)

where A and B represent intrinsic properties of rock; m is time index (m＞0); C is an integration constant; t₁ and t₀ are initial time and unit time, which can be determined by initial conditions.

Figure 4 Recoverable nonlinear viscoelastic model

Under loading condition, when σ=σ^l, t=t₁=0, ε=0, C can be worked out and creep constitutive expression is

               (13)

When the loading is unloaded to some value at time t₁, where t=t₁≠0, σ=σ^u, ε=ε₁, the unloading constitutive expression can be written as

                    (14)

When t≥t₁≠0 and σ^u=0, the unloading equation can be simplified as

(15)

The three-dimensional creep equations under loading and unloading conditions can be written as Eqs. (16) and (17).

          (16)

                  (17)

where G_nve and η_nve are material parameters in three- dimensional condition which are similar to the shear modulus and viscosity coefficient of Kelvin model in unit time.

Figure 5 plots the comparison results between different models and test result. According to the fitting result, the RNVE model agrees well with the test results comparing with Kelvin component by assuming that the shape of reversible visco-elastic strain curve under unloading stage is the same to that under loading phase.

Figure 5 Comparison with Kelvin and RNVE model

Table 1 Model parameters of Kelvin and RNVE model

Therefore, the axial recoverable strain, which includes instant elastic and recoverable visco-elastic stains, can be expressed as

                (18)

2.2 Irrecoverable deformation model

2.2.1 Instantaneous plastic model

The instantaneous plastic strain can be characterized by the parallel combination of Hooke and plastic slider bodies [13, 15, 40], as shown in Figure 6. Similarly, instantaneous plastic strain can be expressed as

      (19)

where G_mp is shear modulus; K_p is bulk modulus; S₀ is the stress threshold above which irrecoverable plastic strain will be produced. K_p also can be calculated by Eq. (7), and G_mp can be calculated by

           (20)

Figure 6 Instantaneous plastic model

2.2.2 Visco-plastic rheological model

Figure 7 plots a widely used basic fractional derivative component substituted conventional dashpot [44, 45], which was able to describe the deformation between ideal fluid and solid. The model expression of fractional derivative dashpot is

0≤n≤1                   (21)

where η represents viscosity parameter; n represents fractional derivative. When n=0, this model represents the ideal Hooke body, of which η means the elastic modulus; whereas n=1 represents the ideal Newton body. Therefore, when 0 material exhibits a combination of the two behaviors of both a spring and a Newton dashpot.

Figure 7 Fractional dashpot

The form of the Caputo’s fractional derivative equation is expressed as [45]

, 0a(22)

where Γ(·) means the gamma function.

The Laplace transform equation of Caputo’s fractional derivative is written as [45]

0≤n≤1       (23)

Equation (21) can be transformed using Eq. (23), we obtain:

0≤n≤1             (24)

i.e.,

0≤n≤1              (25)

where ε(0)=0, and Eq. (25) can be transformed using Laplace inverse transform, we obtain:

0≤n≤1     (26)

The visco-plastic strain is irrecoverable, supposing that it is yielded after the instantaneous plastic strain is generated. Therefore, the visco- plastic model contains a fractional derivative dashpot paralleled with a plastic slider body, as shown in Figure 8. However, strictly speaking, the unit of η is GPa·hⁿ, non-dimensional processing is used to solve this problem. Therefore, the final one- dimensional and three-dimensional creep equations can be expressed as Eq. (27) and Eq. (28).

0≤n≤1         (27)

0≤n≤1    (28)

where S₀ is the same to that in instantaneous plastic model; t₀ is unit time.

Figure 8 Irrecoverable visco-plastic model

Moreover, the axial irreversible strain, which includes instant plastic strain and irreversible visco-plastic stain, can be expressed as

            (29)

Therefore, the presented creep model can describe instantaneous elastic and instantaneous plastic deformations and nonlinearly time- dependent deformations, as shown in Figure 9, whereas it is unable to characterize tertiary creep. Next, a creep model considering damage variation will be constructed.

Figure 9 Nonlinear elastic-visco-plastic rheology model

2.3 Construction of creep damage equation

Various scales of micro- and macro-defects are contained in natural rock; therefore, its deformational behavior and cumulative damage cannot be ignored influenced by these defects, which show that cracks experience different degree of interaction. Therefore, rock rheological constitutive model should include damage evolution rule. XIE et al [46] reported that damage evolution in rock was basically a process of energy accumulation and dissipation. And acoustic emission (AE) results of rocks during creep showed that the energy increased remarkably when accelerating creep occurred [11, 47–49], and cumulative strain energy released sharply. XIE et al [46] developed a damage constitutive equation linked between energy dissipation and damage variable:

                 (30)

where U^d represents dissipation energy for cumulative damage in rock; Ud 0 represents dissipated energy for initial damage in rock; α and β represent material intrinsic properties.

According to the previous research [41, 50], during creep process the calculated strain energy is

      (31)

where

                (32)

where σ₂ equals σ₃, we get

       (33)

                              (34)

where k is the ratio of the lateral creep response to the axial creep response in experiments, we get

        (35)

Equation (35) gives the calculation method for strain energy.

Equation (36) represents cumulative damage in rock. Figure 10 plots its internal mechanism.

          (36)

where U represents cumulative strain energy; U₀ means the critical value of strain energy cumulated in rock; ζ is always bigger than 0, which represents creep characteristic parameter; D represents damage variable in rock. As D equals 0, there is damage in rock creep deformation. When D equals 1, it represents completely creep failure for materials. Therefore, 0onset is the start time of damage.

During creep process, the inner damage of rock increases gradually and the mechanical parameter decreases gradually. Assuming that rock material just shows isotropy damage in creep process (for visco-plastic deformation), as shown in Figure 11, the loading rheological damage equation can be expressed as

                  (37)

               (38)

Figure 10 Rock creep damage model

Figure 11 Nonlinear elasto-viscoplastic rheology damage model

Correspondingly, when the loading deviatoric stress (σ^l=σ₁–σ₃) is unloaded to σ^u at t=t₁>0, the unloading rheological equation can be written as

                        (39)

Specially, when σ^u=0 MPa, Eq. (39) can be simplified as

             (40)

3 Model validation

3.1 Experiment introduction

In order to validate the rheological damage model, experimental work is necessary. In order to circumvent the influence from sample difference, stress-stepping methodology is adopted on a single sample, until the sample ruptured eventually [48]. The triaxial rheological experimental equipment is shown in Figure 12(a), whose maximum capacities of deviatoric stress, confinement and permeability pressure governed by three pumps with high- precision were 400, 70 and 60 MPa, respectively. The peak strength (σ_p) of red sandstone sample was 234.32 MPa under confining pressure of 25 MPa according to traditional triaxial compressive test. According to the previous research, the first applied stress level was larger than 50% of σ_p [15, 16, 19, 40]. Considering that the sandstone sample is one kind of hard rock, whose creep deformation is not obvious, it is reasonable for the first deviatoric stress was applied to 160 MPa, which was about 68% of σ_p. Then, long-term experiments with cyclic loading and unloading have been performed on red sandstone under confinement of 25 MPa. Loading axial differential stresses equal 160, 175, 190, 205 and 215 MPa, and unloading deviatoric stress was 0 MPa. The step-loading procedure is illustrated in Figure 12(b).

3.2 Validation results

It is apparently shown in Figure 13 that the axial strain increases with the progressively increasing deviatoric stress and time in successive creep steps. The time-dependent deformation is not very noticeable under relative low stress, which includes attenuation creep and secondary creep stages. For the final loading, creep rupture occurrs in short time when accelerating creep starts. However, it is unable to separate reversible strain and irreversible strains. Hence, the critical deviatoric stress (S₀) can be obtained according to the fitting result between instantaneous plastic strain and differential stress, of which ε_mp=0, as shown in Figure 13. Moreover, the value of instantaneous elastic strain (ε_me) under deviatoric stress of 215 MPa also can be obtained by fitting result, and the corresponding instant plastic response (ε_mp) equals the difference value between the total instant response (ε_m) and the fitted instant elastic response (ε_me). The results of test and model are listed in Table 2.

Figure 12 Rock triaxial rheological testing equipment (a) and loading procedure for tests (b)

Figure 13 Relationships between instantaneous strains and deviatoric stress

                           (41)

                      (42)

Similarly, the visco-elastic and the visco- plastic response are also unable to be divided under deviatoric stress of 215 MPa. Therefore, the nonlinear visco-elastic parameters (e.g., G_nve, η_nve and m) under the final stress level are determined by the fitting relationships between the parameters and deviatoric stress, whose fitting equations are expressed as Eqs. (43), (44) and (45), as shown in Figure 14. And the ratio of lateral time-based strain to axial creep strain (k) under deviatoric stress of 215 MPa can be calculated by fitting experimental result, as shown in Figure 15. It should be noted that the minus sign represents lateral dilatancy. Finally, the model parameters can be obtained by the nonlinear least square method for the recoverable strain and irrecoverable strain results. The calculated results are provided in Table 3, and the comparing results between the experiment and the presented model are illustrated in Figure 16. It is clear that the proposed constitutive model agrees well with the test data.

Table 2 Experimental results and model parameters of instantaneous deformation

Figure 14 Fitting results of nonlinear visco-elastic parameters

Figure 15 Relationship between axial and lateral time- dependent strains

              (43)

(44)

     (45)

From Figure 16, the axial total strain curve of red sandstone shows decelerating creep, secondary creep under each stress levels other than the final stress level, in which tertiary creep occurred in short time under loading stage. However, under unloading conditions, the instantaneous elastic strain recovers quickly after unloading deviatoric stress, and the visco-elastic strain recovers gradually with time. From Figure 16(e), under differential stress of 215 MPa, the variation in visco-plastic response is more obvious than the visco-elastic response. Moreover, the damage evolution during creep stage under creep failure stress is also plotted in Figure 16(f), in which the damage variable is 0 during primary and steady-state creep stages, whereas it becomes active at the onset of accelerating creep stage and then increases rapidly.

4 Conclusions

A nonlinear rheological damage model of hard rock under loading and unloading condition is investigated and validated using experimental results. Conclusions are drawn as follows:

1) The instantaneous strains (elastic and plastic response), and the viscosity strains (visco-elastic and visco-plastic response) of red sandstone are separated using cyclic increment loading and unloading method. It is more reasonable to describe the reversible and the irreversible deformations of sample separately.

Table 3 Model parameters of rheological experiments

Figure 16 Present damage creep equation fitted over tests results

2) A nonlinear rheological model is presented, in which a Hooke body is used to characterize instantaneous elastic deformation, and a RNVE model is presented to describe visco-elastic deformation, whose fitting result is better than the Kelvin body. A combined model, which contains a fractional derivative dashpot and another spring element in series, and a plastic slider element in parallel with them, is designed to describe instantaneous plastic and visco-plastic deformations.

3) A creep-damage model based on strain energy is introduced in the visco-plastic model to characterize the accelerating creep behavior of sample. The damage model can be divided into three portions: Firstly, no damage occurs before tertiary creep stage; secondary, linear-increasing damage with strain energy occurs during accelerated creep phase, in which strain energy exceeds the critical strain energy; finally, damage value equals 1, meaning material ruptured.

4) The proposed rheological damage model characterizes well the reversible and the irreversible deformations and tertiary creep deformation of sample under loading and unloading condition comparing with the tested and theoretical calculation results.

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(Edited by YANG Hua)

中文导读

硬岩非线性流变损伤模型

摘要：通过循环增量加–卸载的方法，可以将岩石在恒定载荷下的时效变形进一步分离成瞬时弹性应变、瞬时塑性应变、黏弹性应变和黏塑性应变。在蠕变过程中分别描述试样的可恢复和不可恢复的变形是比较合理的。因此，本文提出了一种可以描述硬岩时效变形的非线性流变模型。其中该模型采用一个弹簧元件来描述可恢复的瞬时弹性变形；一个改进的非线性黏弹性模型（RNVE）可以描述可恢复的黏弹性变形；一个分数阶黏壶和弹簧元件串联的组合模型与另一个塑性摩擦块并联形成一个新的组合模型来描述不可恢复的瞬时塑性变形和黏塑性变形。此外，在黏塑性元件中引入一个基于应变能的含三阶段损伤模型来描述硬岩的加速蠕变变形。最后，通过流变试验验证提出模型的合理性。对比试验结果和理论结果，本文提出的流变损伤模型可以很好地描述硬岩的可恢复和不可恢复变形，尤其是描述加速蠕变变形。

关键词：硬岩；循环加–卸载；非线性；损伤；流变模型

Foundation item: Project(BK20150005) supported by the Natural Science Foundation of Jiangsu Province for Distinguished Young Scholars, China; Project(2015XKZD05) supported by the Fundamental Research Funds for the Central Universities, China

Received date: 2017-07-13; Accepted date: 2017-12-19

Corresponding author: YANG Sheng-qi, PhD, Professor; Tel: +86-516–83995856; E-mail: yangsqi@hotmail.com; ORCID: 0000-0003- 1493-6136