J. Cent. South Univ. Technol. (2010) 17: 394-399

DOI: 10.1007/s11771-010-0058-7                                                                                                                   

Effect of volume changes on complete deformation behavior of rocks

ZHAO Heng(赵衡), CAO Wen-gui(曹文贵), LI Xiang(李翔), ZHANG Ling(张玲)

Institute of Geotechnical Engineering, Hunan University, Changsha 410082, China

? Central South University Press and Springer-Verlag Berlin Heidelberg 2010

Abstract: For the purpose of describing the deformation characteristics of rocks, the effect of volume changes on mechanical properties of rocks should be taken into account with relation to the development of constitutive model. Firstly, rocks are divided into three parts, i.e., voids, a damaged part and an undamaged part in the course of loading. The void ratio was applied to describing the changes of voids or pores during the deformation process. Then, using statistical damage theory, a constitutive model was developed for rocks to describe their strain softening and hardening on the basis of investigating the relationship between the net stress and apparent stress, in which the influence of volume changes on rock behavior was correctly taken into account, such as the initial phase of compaction and the latter stage of dilation. Thirdly, a method of determining model parameters was also presented. Finally, this model was used to compare the theoretical results with those observed from experiments under conventional triaxial loading conditions.

Key words: rock mechanics; constitutive model; statistical damage theory; volume change; void ratio

1 Introduction

Simulation of the complete deformation process of rocks is a fundamental issue in many fields of rock mechanics. With the development of the geotechnical projects, the knowledge of deformation behavior of rocks becomes increasingly important in design and construction of rock engineering. Many investigations have been performed during the past two decades on constitutive relationship for rocks [1-8]. Under the external loads, the deformation characteristics of rocks, such as the initial state of compaction and the latter stage of dilation, are significant. Then, the macroscopic volumes of rocks are changed. However, the effect of volume change on rock deformation fails to be considered in the current models, and there is a great gap in describing the stress-strain relationship between calculated results and measured data.

By establishing the damage model to simulate the deformation process of rocks, LEMAITRE [9] proposed a constitutive law of strain softening type, in which the damaged part in rocks cannot resist any possible load. In China, CAO and ZHANG [10] and SHEN [11] indicated that besides undamaged part, the damaged part can still resist the external loading until the loss of rock strength, and further developed a damage model to describe softening and hardening behavior by incorporating the breakage mechanics on the basis of brittle-ductile transition property. Apparently, the above models are not applicable to describing the influence of volume changes on the deformation behavior of rocks.

In order to develop a more reasonable constitutive relation for rocks, it is important to emphasize the influence of volume changes on the damage behavior of rocks. To do so, it was assumed that rock materials consisted of three parts, i.e., voids, a damaged part and an undamaged part, and the void ratio was used to describe the characteristics of volume changes. Introducing a new type of damage concept [10] and the strain equivalence hypothesis [9], a damage model was then proposed based on the correlation between the net stress and apparent stress acting on each part of rocks, respectively. Furthermore, a damage statistical constitutive model was developed by employing the statistics-based theory and the damage threshold concept [12], in which the important behavior of volume changes in rocks can be well embodied in the process of loading conditions. At last, a method to determine model parameters was also presented.

2 Damage model

2.1 Damage model considering volume changes

As the composition of rock materials includes voids or pores, the mass density of rocks varies with porosity.

The void ratio is adopted to describe the effect of volume changes on the damage process. Suppose that rock is divided into three parts, i.e., voids, an undamaged part and a damaged part under apparent stress σ_i (i=1, 2, 3). Here, voids or pores can no longer support stress, and net stresses  and  are assumed to be supported by the undamaged part and damaged part, respectively (see Fig.1).

Fig.1 Damage model of rocks

Clearly, according to the static equilibrium condition, the following relationships are given:

A₀=nA                                      (1)

A₁+A₂=(1-n)A                                (2)

                            (3)

where A is the total area of cross-section; A₀ is the area of voids; A₁ is the area of undamaged part; A₂ is the area of damaged part; and n is the void ratio. If the damage variable D is defined as

D=A₂/(A₁+A₂)                                (4)

the apparent stress and net stress will have the following relationship from Eqs.(1)-(4):

                    (5)

It should be noted that the two types of damage models mentioned above are both particular cases of Eq.(5) when n=0,  and n=0, respectively.

It is further observed that variables   and n in Eq.(5) cannot be gained by experiment directly. Therefore, the correlation between these variables and general mechanical parameters should be developed.

2.2 Calculation of net stress

For the sake of simplicity, the aim of this work is to discuss the deformation process under conventional triaxial loading conditions, i.e.,   or  According to Ref.[10], can be obtained as

                        (6)

                                     (7)

                              (8)

                                (9)

                        (10)

                        (11)

                                 (12)

where c and φ are the cohesion and internal friction angle of intact rock (undamaged part), respectively; σ_c denotes the uniaxial compression strength; E′ and μ are elastic modulus and Poisson ratio of undamaged part, respectively;  is computed from Eqs.(5), (7)-(8):

                          (13)

Assuming that the intact rock is considered to be isotropically elastic, effective stress  acting on this part observes Hooke’s law, then

                       (14)

Combining this result with Eqs.(12)-(14),  can be estimated as

                    (15)

2.3 Calculation of void ratio

As mentioned earlier, the effect of volume changes on deformation behavior varies with porosity because there exist voids or pores in natural rocks.

A cube is considered as cell element in rocks, as shown in Fig.2. Under stress, if the initial void ratio is n₀, the initial volume and the final volume are V₀ and V, respectively, the following relation can be obtained:

V₀(1-n₀)=V(1-n)                            (16)

where V₀=abc and V=(a-Δa)(b-Δb)(c-Δc); a, b and c are the length, width and height of the cell element, respectively; Δa, Δb and Δc are the changes in length, width and height, separately. Since ε₁=Δa/a, ε₂=Δb/b and ε₃=Δc/c, Eq.(16) can be rewritten as

1-n₀=(1-ε₁)(1-ε₂)(1-ε₃)(1-n)                  (17)

Fig.2 Schematic diagram of cell element in rocks

Let ε_V=(1-ε₁)(1-ε₂)(1-ε₃), and omit the high-order perturbation terms in Eq.(17), ε_V can also be simplified as

                    (18)

where ε_V is the volumetric strain.

In view of Eq.(14), ε₃ can be estimated as

               (19)

Then, void ratio n can be obtained from Eqs.(17) and (18):

                                (20)

Eq.(20) has term n₀ that is very difficult to be measured experimentally. Herein, a method to determine n₀ indirectly under uniaxial loading conditions will be discussed as follows.

Suppose that the composition of rocks contains intact part and voids (Fig.3),  is the apparent strain of rocks due to a vertical load P, and  is the net strain of intact body in rocks, the following equation is expressed from the strain equivalence hypothesis [9]:

 (i=1, 2, 3)                           (21)

Fig.3 Schematic diagram of determination of initial void ratio n₀

If μ and μ′ are respectively Poisson ratios of the entire rock and intact part, μ=μ′ can be readily obtained from Eq.(21).

In addition, the apparent stress and net stress can be separately calculated by

                              (22)

                          (23)

where E and E′ denote the elastic moduli of rock and intact part, respectively.

Combining Eq.(2) with Eqs.(21)-(23), E can be computed by

E=(1-n)E′                                  (24)

Furthermore, let E₀ be the initial elastic modulus of rocks. It can be measured directly from experiment. Therefore, n₀ is obtained from Eq.(24) as follows:

                               (25)

2.4 Development of damage model

Substituting Eqs.(15) and (19) into Eq.(20) gets:

                               (26)

where

P₁=P₂₄ε₁-P₂₃σ₃+n₀                           (27)

                     (28)

                      (29)

                               (30)

                         (31)

                                (32)

                                (33)

                                (34)

Substituting Eq.(26) into Eq.(5), a damage model is proposed:

      (35)

where

                        (36)

               (37)

               (38)

3 Damage statistical constitutive model

For the purpose of developing damage constitutive model, besides the damage model proposed above, a damage evolution equation should also be formulated. At present, the statistics-based approach is effective for describing such an evolution equation. A key challenge for it is to determine the mesoscopic element strength (F) and the damage variable (D).

3.1 Mesoscopic element strength

TANG [13] provided a formula for application of axial strain for mesoscopic element strength. However, mesoscopic element strength is in fact associated with stress state [10]. Thus, in this work, a formula for the determination of mesoscopic element strength is proposed based on Drucker-Prager criterion, which is expressed as

                          (39)

where  is the material constant related to the internal friction angle; I₁ is the first stress invariant; and J₂ is the second deviatoric stress invariant. Under conventional triaxial loading conditions, they are calculated as follows:

                           (40)

                               (41)

                             (42)

                           (43)

Substituting Eqs.(40)-(43) into Eq.(39) produces

                  (44)

where

3.2 Statistical evolution equation

Using damage statistical method, the mechanical properties of mesoscopic elements in rocks can be well described. CAO et al [12] developed an expression for the damage statistical evolution equation by incorporating the concept of damage threshold, which is expressed in the form:

                  (45)

where m and F₀ are parameters of Weibull distribution function.

According to Eqs.(35) and (44)-(45), the damage statistical constitutive model for rocks in conventional triaxial stress conditions is therefore formulated.

4 Determination of model parameters

To determine the model parameters m and F₀, an extremum approach [14] is used for the style of strain softening. However, in the case of the style of strain hardening, it can be considered as an empirical method, whose feasibility should be checked through experimental data.

Assuming that σ_sc and ε_sc are the axial stress and the corresponding strain at the peak point in the complete stress-strain curve, we obtain a form of the relation:

                          (46)

The model parameters m and F₀ are then obtained after rearranging of Eq.(46).

 (47)

                     (48)

where

             (49)

          (50)

             (51)

     (52)

                     (53)

              (54)

         (55)

          (56)

(57)

                    (58)

                   (59)

      (60)

                           (61)

                      (62)

                (63)

                 (64)

                             (65)

           (66)

              (67)

                   (68)

                          (69)

               (70)

     (71)

             (72)

                 (73)

               (74)

              (75)

                  (76)

                               (77)

where a and b are both constants, which are determined through regression analysis according to a set of test data [15].

5 Example

5.1 Comparison of theoretical results with experimental data

To check the validity of the proposed constitutive model in this work, a practical problem [16] is adopted. The corresponding calculation parameters are given as follows: E=35.34 GPa, μ=0.25, φ=26.82?, c=22.7 MPa, σ_c=65.0 MPa, a=2.400 6 kPa^-1 and b=2.212 2×10^-3. The theoretical curves at different confining pressures are shown in Fig.4, and the experimental curves are also illustrated for comparison. It is indicated that the test data and the calculated results are very close.

5.2 Characterization of deformation behavior of rocks

The test data in Fig.4 are obtained from intact rock specimens, in which there exist the initial voids or pores in rocks. It is shown that there is not obvious difference between theoretical results of the proposed model and those of the previous model [9]. However, the current model is absolutely applicable to the case of variation of voids.

To characterize the influence of initial voids on the deformation behavior of rocks correctly, the initial void ratio n₀ is assumed to be changed as 0, 5%, 10% and 15%, respectively. Taking an example for σ₃=40 MPa, the curves of the vertical differential stress vs axial strain are plotted, which are shown in Fig.5. It is observed that when the axial strain is small, the curves of (σ₁-σ₃)-ε₁ are concave. As a result, the deformation modulus (stiffness) is increased gradually at the early stage, because the voids or pores are compressed at this stage. With further increase of the axial strain, the curves show a general tendency of convex. This is because the propagation condition of the damage results in new voids or pores. Thus, the slopes of the curves become smaller, suggesting that rocks tend to show dilation and the deformation modulus also reduces with increasing axial strain.

Fig.4 Comparison of experimental and theoretical curves at different confining pressures: (a) σ₃=20 MPa; (b) σ₃=40 MPa; (c) σ₃=  60 MPa; (d) σ₃=120 MPa

Fig.5 Curves of vertical differential stress vs axial strain at σ₃=40 MPa and different initial void ratios

6 Conclusions

(1) Considering that the deformation behavior of rocks is strongly affected by the volume changes during the loading process, a new damage model is developed, in which particular concern is paid to the effect of the amount of voids, and the variation of volume is therefore characterized by the void ratio.

(2) A constitutive model of rocks is established through damage statistical theory. This model takes into account the influence of volume changes in the complete process of rock deformation, such as the initial state of compaction and the latter stage of dilation through the corresponding variations of void ratio. The theoretical results agree with the experimental results. In addition, the mesoscopic element strength and the damage threshold are both correlated in the proposed model.

(3) A method for determining model parameters is presented. It is suitable not only for the strain softening case, but also for the type of strain hardening in a quite accurate way. Furthermore, there are a small number of model parameters, whose mechanical properties are explicit and application may be convenient.

References

[1] KAWAMOTO T, ICHIKAWA Y, KYOYA T. Deformation and fracturing behavior of discontinuous rock mass and damage mechanics theory [J]. International Journal for Numerical and Analytical Methods in Geomechanics, 1988, 12(1): 1-30.

[2] KAISER P K, TANG C A. Numerical simulation of damage accumulation and seismic energy release during brittle rock failure: Part II. Rib pillar collapse [J]. International Journal of Rock Mechanics and Mining Sciences, 1998, 35(2): 123-134.

[3] BLAIR S C, COOK N G W. Analysis of compressive fracture in rock using statistical techniques: Part II. Effect of microscale heterogeneity on macroscopic deformation [J]. International Journal of Rock Mechanics and Mining Sciences, 1998, 35(7): 849-861.

[4] AUBERTIN M, LI L, SIMON R, KHALIF S. Formulation and application of a short term strength criterion for isotropic rocks [J]. International Journal of Rock Mechanics and Mining Sciences, 2000, 37(8):1169-1193.

[5] PAN Y W, WEN B H. Constitutive model for the continuous damage of brittle rock [J]. Geotechnique, 2001, 51(2): 155-159.

[6] COLMENARES L B, ZOBACK M D. A statistical evaluation of intact rock failure criteria constrained by polyaxial test data for five different rocks [J]. International Journal of Rock Mechanics and Mining Sciences, 2002, 39(6): 695-729.

[7] LIU H Y, KOU S Q, LINDQVIST P A, TANG C A. Numerical studies on the failure process and associated micro-seismicity in rock under triaxial compression [J]. Tectonophysics, 2004, 38(4): 149-174.

[8] BAZANT Z P, PANG S D. Activation energy based extreme value statistical and size effect in brittle and quasibrittle fracture [J]. Journal of Mechanics and Physics of Solids, 2007, 55(1): 91-131.

[9] LEMAITRE J. A continuous damage mechanics model for ductile materials [J]. Journal of Engineering Materials and Technology, 1985, 107(1): 83-89.

[10] CAO Wen-gui, ZHANG Sheng. Study on random statistical method of damage for softening hardening constitutive model of rock [C]// Proceedings of the 2nd China-Japan Geotechnical Symposium. Shanghai, 2005: 269-273.

[11] SHEN Zhu-jiang. Breakage mechanics for geological materials: an ideal brittle-elasto-plastic model [J]. Chinese Journal of Geotechnical Engineering, 2003, 25(3): 253-257. (in Chinese)

[12] CAO Wen-gui, ZHAO Heng, ZHANG Lin, ZHANG Yong-jie. Damage statistical softening constitutive model for rock with consideration about damage threshold value and its parameter determination method [J]. Chinese Journal of Rock Mechanics and Engineering, 2008, 27(6): 1148-1154. (in Chinese)

[13] TANG Chun-an. Catastrophe in rock unstable failure [M]. Beijing: The Coal Industry Press, 1993. (in Chinese)

[14] CAO Wen-gui, LI Xiang, LIU Feng. Discussion on strain softening damage constitutive model for fissured rock mass [J]. Chinese Journal of Rock Mechanics and Engineering, 2007, 26(12): 2488-2494. (in Chinese)

[15] CAO Wen-gui, LI Xiang, ZHAO Heng. Damage constitutive model for strain-softening rock based on normal distribution and its parameter determination [J]. Journal of Central South University of Technology, 2007, 14(5): 719-724.

[16] LIN Zhuo-ying, WU Yu-shan, GUAN Lin-li. Research on the brittle-ductile transition property of rocks under triaxial compression [J]. Chinese Journal of Rock and Soil Mechanics, 1992, 13(2): 45-51. (in Chinese)

Foundation item: Project(2006AA11Z104) supported by the National High-Tech Research and Development Program of China

Received date: 2009-03-26; Accepted date: 2009-07-09

Corresponding author: CAO Wen-gui, PhD, Professor; Tel: +86-731-88821659; E-mail: cwglyp@21cn.com

(Edited by CHEN Wei-ping)