Rock Engineering and Rock Mechanics: Structures in and onRock Masses – Alejano, Perucho, Olalla & Jiménez (Eds)

© 2014 Taylor & Francis Group, London, 978-1-138-00149-7

A new rheological hardening model for prediction of creepdeformation of rock samples

M. Karami & A. FahimifarDepartment of Civil and Environmental Engineering, Amirkabir University of Technology, Tehran, Iran

E. PakniatDepartment of Civil Engineering, Shahid-Bahonar University, Kerman, Iran

ABSTRACT: Time-dependent deformation may be emerged in rocks and rock masses under different stresslevels from relatively low stresses up to stresses near peak strength. Most type of rocks, particularly soft rocks,have both reversible and irreversible creep strains under a wide range of stress levels.Thus, a desirable constitutivemodel should calculate both elastic and plastic creep strains, simultaneously.

In this paper, the concept of hardening behavior of geomaterials is adopted to develop a new elasto-plasticvisco-plastic hardening model. The model has a non-linear elasto-plastic hardening behavior for the short terminstantaneous deformation.The non-linearity is related to the current confiding pressure and the hardening featureis based on Mohr-Coulomb failure criterion. For the delayed deformations, on the other hand, a viscoplastichardening unit is used consisting of a frictional hardening Mohr-Coulomb slider and of a viscous dashpotelement, connected in parallel. The viscoplastic hardening unit is connected to the ordinary viscoelastic Kelvinmodel in series. As a result, an elasto-viscoplastic hardening model is obtained for long term deformations.

The governing equations of the proposed model are implemented in numerical finite difference code (FLAC)using its built-in FISH language for constitutive models, and then applied to a series of triaxial and uniaxiallaboratory tests presented in literature. The potential applicability of the model is examined predicting bothinstantaneous and creep deformations under different stress levels. A discussion is also presented comparing thenumerical results and the test data of the samples.

1 INTRODUCTION

Time-dependent deformation behavior of rock masseshas a significant impact on the stability of rock slopesand underground structures (Tsai, 2007). Regardingthe behavior of geomaterials, various kinds of con-stitutive equations have been developed following dif-ferent assumptions and principles. Elastic or reversibledeformation is one of the assumptions which is con-sidered in classic models. Many researchers have usedsuch assumption calculating tunnel face convergence(Sakurai, 1978; Zhifa, 2001; Kontogianni, 2006, Dai,2004; Fahimifar et al., 2010).

In contrast with the visco-elastic models and assum-ing fully irreversible strains for the primary, secondaryand tertiary stages of the creep curve, some researchersproposed visco-plastic models to evaluate the longterm stability of rocks, mainly rock salts, in under-ground excavations (Malan, 1999; Wallner, 1983; Liaoet al., 2004; Erichsen, 2003). However, the weaknessof fully vicso-plastic models have been investigatedthrough the experimental creep tests including unload-ing and reloading tests at which the reversible creepdeformation of rocks occurred (Tomanovic, 2006;Hoxha, 2005; Shao, 2003).

Besides the fully elastic or plastic models, a numberof rheological models have been introduced consider-ing both elastic and plastic time-dependent featuresof rocks (Tomanovic, 2006; Sterpi and Gioda, 2007;Karami, 2013). Generally, the purpose of such visco-elastoplasic models is developing an exact descrip-tion of time-dependent creep behavior of soft rocksunder different stress levels, which is particularly sig-nificant for the stress conditions in the rock masssurrounding tunnels. Regarding this fact, this paperintroduces a non-linear elasto-visco-palstic constitu-tive model. Potential applicability of the model isalso examined simulating the creep behavior of marlsamples, presented in literature, under different stresslevels.

The governing equations of the models are workedout and implemented in numerical finite differencecode (FLAC) using its built-in FISH language for con-stitutive models. To verify the models, they are appliedto a series of experimental data (presented in literature)accounting for the creep deformations in two phasesof loading and unloading. A comparison between pre-diction of the models and the experimental data isprovided.

95

2 MODEL DESCRIPTION

Hardening creep behavior is observed in most rocksand geomaterials particularly soils and soft rocks, andinvestigated by researchers (Critescu, 1998; Wallner,1983; Shao, 2003; Hoxha, 2005). Generally, harden-ing models give an initial envelope which hardens afterthe onset of plastic yield, resulting in producing plasticstrains even at low stress levels. The hardening behav-ior of envelop is usually governed by the hardeningof its associated parameters. Here, the Mohr-Coulombplastic envelop is used for both instantaneous and creepirreversible deformations. The total deformation is thesum of these two types of deformations as Eq. (1).

In Eq. (1), superscripts T , I and C denotetotal, instantaneous and creep deformations of rock,respectively.

2.1 Instantaneous behavior

A non-linear elasto-plastic constitutive model for theshort term behavior of rocks is proposed. The elasticresponse is assumed to be depended upon the con-fining pressure through the hyperbolic formulationproposed by Duncan and Chang (1970) presented inEq. (2).

In Eq. (2), the elastic modulus E depends on theconfining pressure σ3. Pa is the atmospheric pres-sure equal to 0.1 MPa and is used for normalization.K and n are two constant parameters which can beeasily obtained from triaxial tests. Poisson’s ratio υis assumed to be constant in the elastic domain andis used to calculate the shear (GM ) and bulk (BM )modules according to the elastic modulus as:

The elastic strain rate is then calculated by theelastic Hooke’s law:

where σm is the mean stress and Sij is the component ofdeviatoric stress tensor. δij is the kronocker parameterwhich is equal to 1 in a condition of i = j, otherwise itis equal to zero.

On the other hand, for the irreversible deformation,Mohr-Coulomb model defined by failure criterionfs and a non-associated plastic potential Q is used.The failure criterion and plastic potential function aregenerally expressed as:

Table 1. Parameters of the elasto-plastic hardening model.

Parameter K n υ C (Pa) φ (◦) ψ (◦) εf

Value 800 0.26 0.3 1.5e6 40 4 0.14

where σ1 and σ3 are the major and minor princi-pal stresses, and σ1 < σ2 < σ3 for an assumption ofnegative compressive stresses, and C, φm and ψ arecohesion, mobilized internal friction angle and dilationangle, respectively. The hardening feature of the plas-tic criterion is described by the frictional hardeningformulation presented byVermeer (1984):

where εf is the plastic strain at failure and εp is thecurrent plastic strain which is the sum of plastic strainprincipals:

In Eq. (7), ϕm is the mobilized friction angle whichgradually increases as current plastic strain propa-gates, and reaches the limit friction angle ϕ whenthe current plastic strain is equal to εf . Putting theparameters C and ϕm in Eq. (6) for the condition offailure (fs = 0, the amount of plastic strain rate can becalculated using the flow rule:

where Q(σ) is the plastic potential function describedin Eq. (6) and λ is a plastic multiplier that is calculatedstepwise during the analysis (Itasca consulting GroupFLAC, 2004).

The model is implemented in numerical finite dif-ference code FLAC using its built-in FISH languagefor constitutive model and then applied to a series oftiaxial test on marl samples presented by Tomanovic(2012). The parameters associated with the model arepresented in Table 1.

The results of the analysis is presented in Fig. 1for three confining pressure. Proper agreements areobtained between experimental data and analyticalresults.

The model is also applied on the samples havingthe dimensions of 15×15×45cm under uniaxial load-ing to account for the stress-strain diagram. Since theconfining pressure in uniaxial loading is equal to zero,elastic modulus cannot be calculated by Eq. (2). Thus,the elastic modulus is directly set equals to 145 MPa.The diagram obtained from the analysis is presentedin Fig. 2 accounting for the axial, lateral and volumet-ric strains. Acceptable results are also obtained for theuniaxial loading.

96

Figure 1. Stress-strain diagram of triaxial test on cylindri-cal marl samples (h/d = 10.8/5.4 cm) in comparison with thenumerical results (solid lines).

Figure 2. Comparative stress-strain curves between exper-imental data and numerical results (solid lines) of marlsamples performed under uniaxial loading.

Figure 3. Schematically illustration of deviatoric behaviorof the rheological hardening model.

2.2 Time-dependent behavior

The deviatoric behavior of the creep model is schemat-ically illustrated in Fig. 3, where the visco-elasticKelvin unit is characterized by Kelvin shear modulusGk and Kelvin viscosity ηk , and the visco-plastic hard-ening unit is identified with its viscosity ηvp and Mohr-Coulomb hardening slider. As a result, the relationshipbetween deviatoric strain rates can be formulated asEq. (10), where eij is the deviatoric component derived

from the strain tensor. The superscript K refers to theKelvin components and the superposed dot denotestime derivative. Eqs. (11) and (12) represent the con-stitutive laws of the deviatoric behavior of Kelvin andviscoplastic slider, respectively. Sij is the deviatoriccomponent of stresses derived from the stress tensor.

In Eq. (12), Q(σ) is the plastic potential functionwhich was described in Eq. (6). λ is a plastic multiplierrate which will be calculated later.

In Eq. (12), evpij and evp

kk are the deviatoric and volu-metric plastic strain rates respectively. σ ′

11 is the totalstress carried by the slider.

The viscous element is described by the constitutiverelation as Eq. (13) where evpd

ij is the deviatoric strain

rate contributed in the visco-plastic unit, Svpdij is the

portion of the deviatoric stress carried by the viscouselement, and ηvp is the viscosity of this element.

Using three equations of the visco-plastic unit,namely: equating deviatoric strain rates, stress equilib-rium and the failure criterion, the unknown parameterλ is obtained as:

In Eq. (18), the parameter ϕm is the mobilized fric-tion angle calculated through Eq. (7). With calculationof λ, the deviatoric components of visco-plastic strainrate can be computed using Eq. (12):

Viscoplastic deviatoric strain with visco-elasticKelvin strain are added to the non-linear elastic-plastic model for instantaneous response resulting arheological hardening model as illustrated in Fig. 4.

97

Figure 4. Non-linear elastoplastic viscoplastic model withhardening behavior in both instantaneous and delayeddeformatios.

Table 2. Parameter values of the viscoplastic hardeningmodel used for creep analysis.

E C f � ηvp

Parameter (MPa) υ (MPa) (◦) (◦) εf (MPa.day)

Value 145 0.3 0.6 40 4 0.14 81

Figure 5. Uniaxial loading creep tests on marl samples;experimental data (dots) (Tomanovic, 2006), numericalresults (solid lines).

Verification of the proposed model is con-ducted through a comparative discussion between thecalculated deformation using the model, and the exper-imental results based on laboratory tests presentedin the literature. For this purpose, the uniaxial creeptest results on marl samples which were examined fortheir instantaneous deformation in the previous sectionare used. The uniaxial tests were performed in twogroups; each one consists of three samples having thedimensions of 15 × 15 × 45cm, and uniaxial strengthof about 8.8 MPa. The first group was loaded upto2.0 MPa in an hour, and kept under constant stressfor 180 days, then unloaded from 2.0 MPa to zero.The current stress state was maintained for 30 days,and then reloaded up to 4.0 MPa. While, the secondgroup was loaded upto 4.0 MPa, unloaded to 2.0 MPaand reloaded upto 6.0 MPa (Tomanovic, 2006). Theparameters used for the creep analysis are presentedin Table 2.

Fig. 5 includes the experimental data of loadingphase and the numerical results of the proposed model.

A proper agreement is observed between the experi-mental data and the results of the proposed model inboth instantaneous and delayed deformations. It canbe seen in Fig. 5 that strain rate of the proposed modelconsiderably increases as axial stress level rises. Thiseffect can be attributed to the development of plas-tic domain in the rock specimens under higher stresslevels at which the creep deformations are more conve-nient. In addition, creep plastic strains can be estimatedwell for the specimen under low stress level of 2 MPa,which indicates that applying the hardening behaviorcan compensate the weakness of viscoplastic modelsin predicting plastic deformations at low stress levels.

3 SUMMARY AND CONCLUSION

To develop a time-dependent rheological model thatcan accurately predict creep deformation of rocks, avisco-plastic hardening model was developed throughadopting a frictional hardening behavior for its plasticslider element. A non-linear elasto-plastic hardeningmodel was also proposed for the short term responseof rock specimens.

Comparative analysis between the viscoplastichardening model and the laboratory data measured inthe creep tests on marl samples indicates that the modelcan properly predict delayed deformation of samplesunder different stress levels. In fact, viscoplastic defor-mations, which can occur under relatively low stresslevels, ware accurately simulated. This applicability iscrucial in modeling creep behavior of rock surround-ing a tunnel where the rock mass experiences a widerange of stress levels.

REFERENCES

Cristescu N.D., Hunsche U. 1998. Time effects in rockmechanics. New York, Wiley

Dai, H. L. 2004. Theoretical model and solution for the rhe-ological Problem of anchor-grouting a soft rock tunnel.International Journal of Pressure Vessels and Piping 81,739–748

Duncan, J.M. and Chang, C.Y. 1970. Nonlinear analysis ofstress and strain in soils. Journal of Soil Mechanics andFoundation Division, ASCE, 96. No. SM5

Erichsen, C., Werfling, J. 2003. Stability of undergroundopenings in rock salt. ISRM– Technology roadmap forrock mechanics, South African Institute of Mining andMetallurgy

Fahimifar, A., Monshizadeh Tehrani, F., Hedayat, A.,Vakilzadeh, A. 2010. Analytical solution for the excava-tion of circular tunnels in a visco-elastic Burger’s materialunder hydrostatic stress field.Tunneling and UndergroundSpace Technology, vol. 25, no. 4, pp. 297–304

Hoxha, D., Giraud, A., Homand, F. 2005. Modelling long-term behaviour of a natural gypsum rock. Mechanics ofMaterials 37, 1223–1241

Itasca Consulting Group. 2004. FLAC, Fast Lagrange Anal-ysis of Continua in 2D Dimensions, Version 4, UserManual. Minneapolis

Karami, M., and Fahimifar,A. 2013.A New Time-DependentConstitutive Model and its Application in Underground

98

Construction. ISRM International Symposium on RockMechanic for Resources, Energy and Environment, Wro-claw, Poland, 21–26 September

Kontogianni, V., Psimoulis, P., Stiros, S. 2006. What isthe contribution of time-dependentdeformation in tunnelconvergence?. Engineering Geology 82, 264–267

Liao, H. J. et al. 2004. Numerical Modeling of the Strain RateEffect on the Stress-Sstrain Relation for Soft Rock Usinga 3-D Elastic Visco-Plastic Model. International Journalof Rock Mechanics & Mining Sciences. Vol.41, No. 3

Malan, D. F. 1999. Time-dependent Behaviour of Deep LevelTabular Excavations in Hard Rock. Rock Mech. RockEngng. 32 (2), 123–155

Sakurai, S. 1978. Approximate Time-Dependent Analysis ofTunnel Support Structure Considering Progress of TunnelFace. International Journal for Numerical and AnalyticalMethods in Geomechanics. Vol. 2

Shao, J.F., Zhu, Q.Z., Su, K. 2003. Modeling of creep in rockmaterials in terms of material degradation. Computers andGeotechnics 30 549–555

Sterpi, D., Gioda, G. 2007. Visco-Plastic Behaviour aroundAdvancing Tunnels in Squeezing Rock. Rock Mech. RockEng. DOI 10.1007/s00603-007-0137-8

Tomanovic, Z. 2006. Rheplogical model of soft rock creepbased on the tests on marl. Mech Time-Depend Mater10:135–154

Tomanovic, Z. 2012.The stress and time dependent behaviourof soft rocks. GRAÐEVINAR 64 12, 993–1007, 993

Tsai, L.S. 2007. Time-dependent deformation behaviors ofweak sandstones. International Journal of Rock Mechan-ics & Mining Sciences 45, 144–154

Vermeer, P. A., de Borst, R. 1984. Non-Associated Plasticityfor Soils, Concrete and Rock. HERON vol. 29, No. 3

Wallner, M. et al. 1983. Stability calculation concerning aroom and pillar design in rock salt. 5th ISRM Congress,Melbourne, Australia. April 10–15

Zhifa, Y. et al. 2001. Back-analysis of viscoelastic displace-ments in a soft rock road tunnel. International Journal ofRock Mechanics & Mining Sciences 38, 331–341

99