Z. Li et al.: Loading and unloading test of hard rock and its elastoplastlc damage coupling model LOADING AND UNLOADING TEST OF HARD ROCK AND ITS ELASTOPLASTIC DAMAGE COUPLING MODEL OBREMENILNI IN RAZBREMENILNI PREIZKUS TRDNIH KAMNIN IN NJIHOV ELASTOPLASTIČNI MODEL POŠKODBE KONTAKTA Zhen Li (corresponding author) Henan Polytechnic University, School of Civil Engineering, Jiaozuo, Henan 454000, China E-mail: zhenli@hpu.edu.cn Dongdong Zhang Yankuang Group Company Limited, Yanzhou Coal Ordos Neng Hua Company Limited, Ordos, Inner Mongolia Autonomous Region 017000, China Shuang Zhao Yankuang Group Company Limited, Yanzhou Coal Ordos Neng Hua Company Limited, Ordos, Inner Mongolia Autonomous Region 017000, China https://doi.org/10.18690/actageotechslov.15.2.38-46.2018 DOI Keywords high stress; hard rock; loading and unloading test; coupling model Ključne besede visoka napetost; trdna kamnina; obremenilni in razbremenilni preizkus; model kontakta Abstract Deep-buried engineering and test results show that hard rocks behave as part of an elastoplastic damage coupling process. The coupling effect can contribute to the weakness of the surrounding rocks and the extension of the water channels. As a result, the coupled elastoplastic damage model is the basis for a stability analysis in deep engineering. In this paper loading and unloading tests were conducted on T2b marble in the Jinping II hydropower station. Based on the tests the effects of the confining pressure on the strength, the failure strain and the dilation were analyzed. According to the plastic shear failure and the parameters weakness mechanism, the damage-evolution function reflecting the weakness character, the loading function and the plastic potential function regarding plastic hardening were proposed. The activation of the damage and plastic process was then studied. The coupled elastoplastic damage model was finally established. Through simulating the test curve, the proposed model was verified. This model could play an important role in the stability analysis of deep-buried hard-rock engineering. Izvleček Globoko vkopani inženirski objekti in rezultati preizkusov kažejo, da se trdne kamnine obnašajo kot del elastoplastičnega procesa poškodbe kontakta. Vpliv na intaktnost kontakta lahko prispeva k šibkosti okoliških kamnin in razširitvi vodnih kanalov. Rezultat tega je, da je elastoplastični model poškodbe kontakta osnova za analizo stabilnosti pri globoko vkopanih inženirskih objektih. V prispevku smo na marmorju T2b na hidroelektrarni Jinping II izvedli obremenilne in razbremenilne preizkuse. Na podlagi preizkusov smo analizirali učinke hidrostatskega tlaka na trdnost, specifično deformacijo pri porušitvi in dilatacijo. Glede na plastično strižno porušitev in mehanizem slabosti parametrov smo predlagali funkcijo razvijanja poškodb, ki odraža značilnost šibkosti, obremenilno funkcijo in funkcijo plastičnega potenciala pri plastičnem utrjevanju. Nato smo proučevali aktivacijo poškodb in plastičnih procesov. Končno je bil ugotovljen povezani model elastoplastične poškodbe. Predlagan model je bil preverjen s simulacijo preizkusne krivulje. Ta model bi lahko imel pomembno vlogo pri analizi stabilnosti globoko vkopanih inženirskih objektov v trdnih kamninah. 38. Acta Geotechnica Slovenica, 2018/2 Z. Li et al.: Loading and unloading test of hard rock and its elastoplastlc damage coupling model 1 INTRODUCTION To deal with the potential problems associated with a shortage of energy supply, many hydropower stations and transportation tunnels are being bult in China, implementing the western development strategy project. The huge rock engineerings located in high mountains in the west of China, have characters such as deep buried, high stress and long span. For example, the diversion tunnel in the Jinping II hydropower station has a maximum length of 16.6km, an excavation diameter of 13m, a general buried depth of 1500~2000m and a maximum buried depth exceeding 2500m. In this case its high initial geostress and water pressure add difficulties to the construction. Besides these, the diversion tunnel goes through the Jinping mountain and various rock layers. The rock mass has many fractures and water flowing structures. The maximum water pressure exceeds 10MPa[1]. During the excavation of the auxiliary tunnels, the rockbust and water gushing problems resulted in a lot of loss and damage (see Fig.1) [2]. Therefore, the disaster prevention and control in the high geostress and water presure are the key problems in the construction of deep buried tunnels. Figure 1. Rock-burst and water-gushing problems in deep buried hard rock in the Jinping II hydropower station: (a) strong rock burst (b) high-pressure and large-volume groundwater infiltration.[2]. In the case of high geostress, the behavior of hard rock has its special characters: a) it has the ductility characters in high confining pressure in contrast to the brittleness in the low confining pressure[3]; b) the Mohr-Coulomb criterion applying to the shallow buried rock cannot be used in the deep buried rock with non-linear charac-ters[4]; and c) the elastoplastic deformation becomes more complex. Under the applied force, the microfissures extend and penetrate. On one hand, the strength of the surrounding rock weakens with additional strain generation. On the other hand, the connected fissures act as the water-gutting channels. Therefore, the plastic deformation and damage to the rock have a coupling effect. The coupled elastoplastic damage analysis is the basis for solving the stability problem of deep buried tunnels. In the triaxial test the microfissures propagate with the increasing load. Material damage accumulates due to the plastic process. Meanwhile, the damage leads to further irreversible plastic deformation, a non-linear constitive relationship, a strain-softening material and dilation. These mechanical phenomena can be viewed as the coupling effect of damage and plastisity, which is described by continuum damage mechanics and plasitic theory, respectively[5]. Salari et al.[6] formulated a constitutive model in the framework of continuum thermodynamics using internal variables. Chiarelli et al.[7] proposed that the plastic deformation and anisotrophy damage are coupled using a triaxial test with claystones. Zhu et al.[8] formulated a micromechanics-based constitutive model for granular materials like sands under a relatively low confining pressure. Zhou and Zhu[9] developed an elasto-plastic damage constitutive model with double yield surfaces for the soft rock. The existing research was mostly focused on the elas-toplastic damage coupling problems in soft rock, while research on the hard rock with deep buried high stresses is still not sufficient. In this paper we made an analysis of the mechanical behavior of the hard rock based on the test results. The coupled elastoplastic damage model was proposed and validated. The work acts as the basis for a study on the weakening process of the surrounding rocks and the formulation of water-gushing fissures. 2 TEST ANALYSIS 2.1 Test methods and results The standard cylinder marble samples with sizes of 50x100 mm were made from the marble core buried 2500 m underground in the Jinping II hydropower station diversion tunnel in the Sichuan province of Acta Geotechnica Slovenica, 2018/2 39. Z. Li et al.: Loading and unloading test of hard rock and its elastoplastlc damage coupling model Distance (m) Figure 2. Location of the T2b marble in the Jinping II hydropower station. China (see Fig.2). For the purposes of getting the elastic parameters at a different loading stage, a loading-unloading triaxial test was conducted on the MTS815.03 test machine. In order to avoid the sudden energy release and brittle failure caused by post-peak unloading of rock in an unstable state, a special control method was used (see Fig.3). The confining pressure was increased by 1 MPa, followed by an unloading process during which the extra confining pressure was again decreased. The loading process is controlled by deformation and the loading rate is 0.06 mm/min. The unloading process is controlled by the axial stress and the unloading rate is 26MPa/min. At the end of each unloading the deviatoric pressure is close to 5MPa. Fig.4 shows the stress-strain =5MPa C >/\D Yielding surface A/ Fa / / Loading / /Unl0ading --/---r I P -W B G Figure 3. Loading and unloading stress path in the p-q plane. Oj -a3/MPa -0.012 -0.008 -0.004 0 0.004 0.C Figure 4. Loading and unloading stress-strain curve under the confining pressure of 5MPa. 40. Acta Geotechnica Slovenica, 2018/2 Z. Li et ai.: Loading and unloading test of hard rock and its eiastopiastic damage coupling model curve under the 5MPa confining pressure. The elastic modulus decreases, while the plastic deformation increases with the loading process. As is well known, the weakness of the elastic modulus is related to the evolution of the damage. Thus, the eiastopiastic damage evolution is accompanied by the loading process of the T2b marble. Fig.5 shows the envelope line for the loading-unloading test curve. The envelope line also stands for the yield surface before the yield and the subsequent yield surface after the yield. (a) Figure 5. Envelope line of the loading-unloading test curve of the T2b marble: (a) axial stress-axial strain curve, (b) volumetric strain—axial strain curve. 2.2 Test result analysis The plastic characters in the loading process can be seen from Fig.5: a) The stress-strain test curve is approximately linear under a low confining pressure. The elastic deformation at failure increases with the increasing confining pressure. After a threshold the test curve shows obvious non-linearity and plasticity, b) The confining pressure influences the strength and the deformation. With the increasing confining pressure, the strength and axial strain increases, while the volumetric strain seems restrained, c) In the case of a low confining pressure, the sample fails in a strain softening character, while strain hardening shows for the high confining pressure, d) There is a transition from volumetric compression to dilation. At first, the micro-fissures were compressed. Then the loading propagates the cracks after a threshold and dilation occurs. However, with the increasing confining pressure, the dilation is slow to happen and appears to be suppressed. In the meantime, the fissures in the T2b marble have such a sequence of evolution due to damage: a) Under low applied loading stress, the microcracks are compressed and the volume decreases, b) With the increasing loading stress, local instability leads to meso-scopic fracture, c) Mesoscopic fracture propagates and extends, d) The mesoscopic fracture develops into the penetrating crack and the sample fails. To sum up, there is a coupling eiastopiastic and damage effect in the loading process of the T2j, marble. When the load is not large enough, the microfissures are compressed and the material is elastic. With the increasing loading pressure, cracks propagate and dislocation occurs. Plastic deformation induces damage and, conversely, there is a further development of the plastic deformation. Therefore, it is necessary to use the coupled eiastopiastic damage theory in the loading analysis of the T2b marble [10], 3 ELASTOPLASTIC DAMAGE-COUPLING MODEL There is no obvious anisotropic character in T2b marble. So the isotropic theory is used in the analysis. It is presumed that the weakness due to the mesoscopic fissures can be described by the macroscopical isotropic damage. The damage process is defined by the damage variable in a scalar form. Under isothermal conditions, the state variables in the loading process are listed as the total strain tensor e, the damage variable a>, the elastic strain tensor ee and the plastic strain tensor ep. The full variable form and incremental form of strain can be expressed £=£e + ep ,de = dee +dep (1) It is assumed that there is a thermodynamic potential coupling the damage and plasticity. The thermodynamic potential can be expressed as :C(co):s!+y/p(K,co) (2) 41. Acta Geotechnlca Slovenlca, 2018/2 Z. Li et ai.: Loading and unloading test of hard rock and its eiastopiastic damage coupling model where fp is the thermodynamic potential describing the strain hardening of the damaged material, k is the internal variable for plastic hardening, which can be described by the plastic deformation, such as the plastic strain or the equivalent plastic shear strain. C(co) is the fourth order of the elastic tensor. For isotropic material, C(co) is expressed as [11] C(©) = 2G(©)K + 3Är(©)j (3) where G and K are the shear modulus and bulk modulus, respectively. IK and J are the isotropic fourth tensor, which can be expressed as K = I-JU=-5®5 (4) 3 where S is a second-order unit tensor. I is a symmetric fourth-order unit tensor. For a second-order tensor E, we have JJ: E = , K: E = E . The partial derivative operation is done with equation (2) and we have dee ^ ' (5) Considering the damage to the elastic parameters, thus equation (5) can be rewritten in incremental form as da = C((d):dee 3.1 Damage description © 0(0 : 8 dco (6) In the framework of the irreversible energy, many damage models have been established. In these models, the damage evolution is determined by the evolution function with a damage variable. The test shows that there is a shear bond penetrating the sample and particles slide along the microcrack with damage evolution. Thus, in this paper the damage driving force Ya is Ymax is the maximum damage variable. The parameter Bc controls the kinetics of the compressive damage and can be determined by using a uniaxial compression test. The parameter Ya)0 is the damage threshold. To sketch the damage evolution during the loading process, the damage-evolution curve for the T2b marble under a confining pressure of 40MPa is shown in Fig.6. 300 -, 250 - 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 E1 Figure 6. Damage-evolution curve for T2b marble under confining pressure of 40MPa. 3.2 Plasticity description According to the test research, rock has the characteristic of pressure sensitivity[13]. In other words, the strength increases with the increasing confining pressure or hydrostatic pressure. In this paper, the value of the variable is positive when the stress is tensile. Inspired by Chen et al, the strength criterion is adopted as r r \ q - + c g(8)/c0 "U(0)/CoJ q -- fcO = 0 (9) 1 wherep is the hydrostatic pressure expressed as p—fr (o). q is the equivalent shear stress expressed as a = J—S :S . S is the deviatoric shear tensor V 2 i expressed as S ~a--[tra)8 ,fc0 is the uniaxial strength determined by the uniaxial test. The parameters q, c2 and c3 define the curvature of the failure surface and can be determined by plotting the failure surface. g(9) determines the effect of the Lode angle on the strength. In the loading path in this paper, g(9) is simplified as g(0)=l. It is noted that equation (9) is the Drucker-Prager criterion in the case of c2=0 for the simplified calculation. The loading equation similar to equation (9) is /s=?-asg(8)ac=0 (10) c1+ 0, as ■ an • For Yp ■co, a. •1. 3.3.1 (a) da»0, fs=0 In this case, the plastic flow and the damage evolution are activated. The damage variable increment is determined by 5/co dco = d\,„ = dX„ dZ, -co. D max <: (16) o)] where d\a is the damage multiplier, which can be determined by the damage-consistency condition The generation of microcracks is accompanied by the plastic deformation. To express the weakness the loading function is expressed as fs=q-asg(Q) ("cl + Jci+4 cl{c2-pl fco) Ico /co=0 (13) L-(ö)-ö)y^, ü)f is the threshold of where fc0 =/co(1~(Cl the damage to the plasticity. When the damage variable is larger than ay, the loading curve shrinks and the strength decreases. The symbol (x) is expressed as: (x) = 0 when x<0 and (x) = x when x>0. The test shows that the volumetric deformation changes from compression to dilation and this effect decreases with the increasing confining pressure. To express the deformation character, inspired by the work[14, 15], the plastic potential function is gs + fl' yhj = 0 (14) where I = -p + c^fc0 . The parameter I0 determines the intersection point of the plastic potential surface and the coordinate/). The threshold of the volumetric compression to dilation can be determined by ^gs _ q ( expressed as SI /™=^-^(0)(-jP + c3/co) = ° (15) where the parameter ¡¿s defines the slope of the boundary between the compressibility and dilatancy domains. Its value can be obtained by plotting the transition boundary line in the p-q plane. 3.3 The damage and plasticity coupling numerical process In a general loading path, the plastic flow and damage evolution can be activated regarding the loading law, respectively. There are two kinds: a) da)>0,fs=0. b) da)>0,fs<0. oYm oco ®max^c dZ, dZ, exP \ßc (^co ~ ^coo )] V de From equations (16) and (17) we have — : dze + —: dzp + c/co = 0 dzp (17) dXm=-\^:dEe+^:dEp I (18) ,8ee 8ep For plasticity, the plastic strain increment is dep =dl^ s 8a (19) where dXs is the plastic multiplier, which can be determined by the plastic consistency condition as dfsJA:da+ÈL^:dEP + dAd^ o da dyp dep dco (20) From equations (19) and (20), the plastic multiplier is determined as — :da + — dco dX, - - da dco dyp dsp ' da (21) Substitute (16) into (21) and (19) into (18), the simultaneous equations are obtained and thus we get the plastic multiplier and the damage multiplier. The constitutive function in incremental form can be expressed as da - C(co): I dz-d\s^~ —d À,,. co, z> 'max nc 3C( (22) co exp[5c(rffl-rffl0)] ÔCO - : e 3.3.1 (b)flk»0, fs<0 For/s<0, the stress point is in the elastic range and no plastic flow happens. da)>0 and there is only damage evolution. Accordingly, the plastic strain increment is 43. Acta Geotechnlca Slovenlca, 2018/2 Z. Li et ai.: Loading and unloading test of hard rock and its eiastopiastic damage coupling model 0. Therefore, dep=0. The damage-variable increment is expressed as equation (16). Similar to equation (17), the damage-consistency condition is expressed as oYm öco (23) exP \ßc (Y