mohr-coulomb yield criterion in rock plastic...

CHINESE JOURNAL OF GEOPHYSICS Vol.55, No.6, 2012, pp: 733∼741

MOHR-COULOMB YIELD CRITERION IN ROCK PLASTIC MECHANICS

WANG Hong-Cai1,2, ZHAO Wei-Hua1,2, SUN Dong-Sheng1,2, GUO Bin-Bin1,2

1 Key Laboratory of Neotectonic Movement and Geohazard, Ministry of Land and Resources, Beijing 100081, China

2 Institute of Geomechanics, Chinese Academy of Geological Sciences, Beijing 100081, China

Abstract Conventional Mohr-Coulomb strength criterion only describing the relationship between normal stress

and shear stress on the failure surface at the peak strength is used as a criterion to determine whether the beginning

of rock’s failure. This study extends the concept of the Mohr-Coulomb strength criterion by suggesting that the

Mohr-Coulomb yield criterion can express the stress relationship of strain and stress of failure surface under

various stress states after plastic deformation occurs. Based on experimental studies, an experiment approach

which employs triaxial compression testing data to determine plastic parameters c and ϕ which vary with the

internal variable κ was established, and parameter expression used to assess the accuracy of isotropic modeling

and some testing results were given. The Mohr-Coulomb yield criterion of isotropic hardening or softening, in

which stress-strain curves of whole process is simplified into a four-line model using initial, peak and remnant

parameters of rock, is more applicable than the trilinear model by Toshikazu. The results of this study provides

theoretical and experimental basis for engineering and geological numerical simulation, thus being of instructive

importance.

Key words Plastic deformation of rock, Mohr-Coulomb yield criterion, Elastic and plastic parameters, Isotropic

hardening or softening rule, Four-line model

1 INTRODUCTION

The Mohr-Coulomb strength criterion is commonly used to describe relationship between normal stressand shear stress during rock deformation. With increasing research on rock plastic deformation and strainsoftening, it is necessary to describe the relationship between strain and stress during plastic deformation[1−2].Therefore, we extend the application of the Mohr-Coulomb strength criterion by describing the relationshipbetween normal stress and shear stress of rocks at each stage of deformation, such as initial plastic yielddeformation, peak deformation and residual deformation.

In the conventional Mohr-Coulomb criterion, the internal cohesion c and internal friction angle ϕ are con-sidered constants[2], whereas at the plastic deformation stage, c and ϕ values in the Mohr-Coulomb criterion arenot constants but a function of plastic deformation internal variables. Thus to further study the Mohr-Coulombyield criterion under the condition of rock plastic deformation is an important subject in rock mechanics[3].

Beginning with the description of rock plastic deformation and the yield criterion, this study proposesa technical approach to determine c and ϕ which vary with internal variables based on triaxial experiments,and then gives a full expression of the Mohr-Coulomb yield criterion under plastic deformation condition and asimplified four-line model showing the whole process stress-strain curves.

2 PLASTIC DEFORMATION OF ROCK

Stress-strain curves of rock and concrete testers (whole process stress-strain curves) show that the residualdeformation of the rock specimen at the final failure can be very large, i.e. the specimen can elongate greatlywhen it completely fails. For such a specimen, deformation disappears partially after removal of load, while restdeformation can be retained permanently. The former which can recover by itself is called elastic deformation,while the latter is called residual deformation.

E-mail: [email protected]

734 Chinese J. Geophys. Vol.55, No.6

This study focuses on plastic deformation, i.e. the residual deformation after removal of load. It is ir-reversible, and different from metallic plasticity and mineral crystal plasticity[3−8]. Compared with plasticityof metallic materials, plasticity of rock material is much complicated[9], because both hardening and softeningexist in the same specimen during deformation. Residual deformation kept in the specimen after removal ofload is called quasi-plastic deformation so as to distinguish it from plastic deformation of metallic materials.Although residual deformations in rock and metallic materials are very different in micro-mechanism, their irre-versibility is the same on macroscale. In the macroscopic phenomenological theory, we neglect micro-mechanismof deformation, and equate plasticity with irreversibility; therefore, we name the residual deformation of rockas the plasticity deformation. Thus rock can be treated as elastoplastic material, which allows establishment ofplastic mechanic theory.

The theory of rock plastic mechanics generally deals with the issue of three-dimensional stresses. Extensionof uniaxial stress state to the state of three-dimensional stresses needs to extend the concept of yield stress tothe yield criterion in the state of three-dimensional stresses. The yield criterion of rock material is a key conceptin rock plastic mechanism[3,9]. Expression of the yield criterion in rock mechanics is more complex than thestrength criterion, because the latter just describes small damage of peak values and the former describes thechange of yield surface.

The strength criterion (failure criterion) of rock material is a curve surface in the stress space, calledstrength surface (failure surface). Material parameters (such as c and ϕ values) it contains are constants fromfailure experiments of material specimens[1].

Typical whole stress-strain curves of the rock material are shown in Fig. 1[1]. The results of microscopicanalysis show that the portion OA represents closure of primary gap; the AB portion shows a linear elasticchange and the specimen has no obvious change in structure; new crack presents in portion BC with the stressreaching the peak value; formation and extension rates of the crack speed up with increasing fracture densityin the portion CV; and in the portion VD, discontinuous cracks interconnect along the same direction and slipoccurs on microfracture surface. Cracks are pervasive prior to the peak stress value but tend to develop in aconcentrating manner after the peak value. The PQR and TUV (Fig. 1) are hysteresis loops produced due toreloading after unloading[10]. The rock material is treated as the elasto-plastic material and then comparedwith the rock specimen in the whole stress-strain curves (Fig. 1). The test results show that the yield stresses ofthe rock material consist of two types: initial yield stress (initial stage of microfractures) and subsequent yieldstress (the change of yield stress with development of microfractures), and correspondingly, there are two yield

Fig. 1 Whole stress-strain curves of

a uniaxial compression tester

criteria: initial yield criterion and subsequent yield criterion. InFig. 1, B is the initial yield point corresponding to the initial yield,and the others after B are subsequent points corresponding to sub-sequent yields. It can be seen that the strength criterion (failurecriterion) is equivalent to the peak yield criterion at the state ofthe peak value. It is a special subsequent yield criterion.

The yield criterion of rock material is one family of curvedsurfaces in the stress space, called yield surface. Initial yield sur-face refers to the limit of elastic response when the rock ma-terial does not undergo any plastic deformation. With evolv-ing of plastic deformation (microfractures), the yield surface alsochanges[11−12]. Subsequent yield surface is the limit of elasticstress response when the rock experiences to some extent plasticdeformation symbolized by change of internal microstructure.

Yield surface is no longer a single pyramid surface, but onefamily of curve surfaces varying with development of plastic de-formation. It firstly expands in response to hardening and reaches

Wang H C et al.: Mohr-Coulomb Yield Criterion in Rock Plastic Mechanics 735

the peak yield surface and then contracts in response to softening. The law of yield surface changing overdevelopment of plastic deformation is termed as the hardening-softening law.

The hardening law of yield surface of metallic material has two basic types: isotropic hardening (softening)and kinematic hardening. The kinematic hardening refers to the overall movement of the subsequent yieldsurfaces and the stress space, reflecting the Bauschinger effect of the material[3]. For rock material, there hasbeen no enough data to prove the Bauschinger effect, thus we still employ the isotropic hardening or softeningmodel. Under such a situation, describing the change of yield surface of material only needs one scalar plasticinternal variable (internal variable in short). The isotropic hardening or softening yield criterion can generallybe expressed as[3]:

f(σ1, σ2, σ3, κ) = 0. (1)

It is one set of the criteria varying with the change of internal variables, with an initial yield criterioncorresponding to κ = 0 and the subsequent criterion to κ > 0.

The internal variable is employed to describe the amount of plastic deformation (or plastic loading) of thematerial. With increasing plastic deformation, κ is a non-negative monotonously increasing scalar. If there isonly elastic response, the constant κ has dκ = 0. But choice of internal variable is not sole. For rock materials,it can be the equivalent plastic strain εp, plastic work ωp or plastic volume strain θp:

εp =∫

(dεpijdεp

ij)12

ωp =∫

σijdεpij

θp =∫

αεpkk

. (2)

3 MOHR-COULOMB YIELD CRITERION

With development of numerical methods (e.g. finite element) and popularization of computer[13−15], theyield criterion should be applied to rock material in rock engineering design involved with the plastic mechanicmethod. Many researchers resort coincidentally to the Mohr-Coulomb strength. The cohesion c and internalfriction angle ϕ are firstly treated as the constants (equivalent to those in the ideal plastic material), and thenas the parameters varying with internal variables. This practice reflects the nature of hardening and softeningof rock materials.

In rock plastic mechanics, employing the Mohr-Coulomb yield criterion is indisputable when available datais few[16−17]. Suppose that cohesion c and internal friction angle ϕ vary with internal variable κ, it is the yieldcriterion of isotropic hardening (softening) law that can apply to the strength and stability analyses of rockengineering.

For a designated internal variable κ, the Mohr-Coulomb criterion in the space of τ − σ can be expressedas[3]

f(σ, τ, c(κ), ϕ(κ)) = |τs| − c(κ) + σ tanϕ(κ) = 0. (3)

And the Mohr-Coulomb criterion in the space of principal stresses can be expressed as[3]

f(σ1, σ3, c(κ), ϕ(κ)) = σ1 −1 + sin ϕ(κ)1− cos ϕ(κ)

σ3 − 2cos ϕ(κ)

1− sinϕ(κ)c(κ) = 0, (4)

where the order of principal stresses is σ1 ≥ σ2 ≥ σ3, i.e. σ1 and σ3 are the maximum and minimum principalstresses, respectively. Eq.(4) corresponds to the section AB in the π plane (Fig. 2). If the order (σ1 ≥ σ2 ≥ σ3)is not prescribed, the symmetrical extension method is used to form an irregular hexagon in Fig. 2, named theCoulomb hexagon[3].


The hexagon is the cross section of the Mohr-Coulomb yield surface in the plane with σm = σ1 + σ2 +σ3=const (called a deviatoric plane, and the π plane is a deviatoric plane passing through the origin of coor-dinate), and reduces lineally with decreasing σm. When σ1 = σ2 = σ3=ccotϕ, the hexagon shrinks to a pointO∗. Therefore, the Mohr-Coulomb yield surface is the six surfaces of hexagonal pyramid with the Coulombhexagon as the base in the π plane and the point O∗ as the apex. The initial Mohr-Coulomb yield surfaces andsubsequent Mohr-Coulomb yield surfaces are one set of side planes of hexagonal pyramid, which varies with κ.Among the planes, the most important are the initial yield surface, the peak yield surface and the residual yieldsurface, which are respectively in response to three internal variables κin, κp and κr, which correspond to theplanes 1, 2, and 3 in Fig. 2, respectively.

The initial yield surface (initial yield criterion) is the limit of elastic response when the material under-goes no any plastic deformation (microfractures) (plane 1 in Fig. 2), and also the limit between elasticity andplasticity. When the state of stress reaches this limit, it does not apply to the elastic constitutive equation butthe plastic constitutive equation. Analyses of elastic-plastic stress and deformation of rock material do needintroduction of initial yield surface. It was, however, neglected in previous work.

The peak yield surface (or peak yield criterion) is roughly equivalent to the failure surface (or failurecriterion) of material strength theory[18−19]. Its plastic parameters c(κp) and ϕ(κp) should correspond to thestrength parameters of rock c and ϕ. From the initial yield state to the peak yield state, the faces of hexagonalpyramid continuously expand outward, thus the material is at the hardening stage. After reaching the peakstate the material enters softening stage, thus the peak yield surface represents a signature convex in which thematerial changes from hardening to softening (plane 2 in Fig. 2). For the peak yield surface corresponding tothe peak strength, its plastic internal variable is κp, it dose not need to introduce the internal variable into thestrength criterion or failure criterion.

The residual yield surface or residual yield criterion roughly corresponds to the failure strength or failurecriterion (plane 3 in Fig. 2). When the yield surface reaches the residual surface, there is κ = κr; thereafter,even if κ > κr the size of yield surface remains the same, and this is equivalent to entering an ideal plasticstage[12].

Fig. 2 Mohr-Coulomb strength criterion in the stress space and in the π plane

4 TRIAXIAL EXPERIMENT RESULTS AND FIT OF c(κ) AND ϕ(κ)

4.1 Triaxial Experiment Results

Sandstone specimens for the experiment were collected from the drilling cores in the Changqing oilfield ofGansu Province. The sampling depth is 858 m. The sandstone is the sedimentary rock, homogeneous, freshand even-sized grains, and can be regarded as isotropic rock material. Based on the Experimental Regulationof Rock Mechanic, the test specimens are cylinders, 30 mm in diameter and 60 m in height, with a ration ofheight to diameter 2:1. Verticality between the end planes and specimens’ axis is less than 0.25 degree. Twosurfaces were worn smoothly, with a roughness less than 0.5%. The experiment was carried out at the ChineseAcademy of Geological Sciences’ Institute of Geomechanics using a TAW 2000 Electro-Hydraulic Servo testing


system. The axial strain ε1 was employed to controlthe whole test process. The experiment was conductedon the specimens at different levels of confining pres-sure. The axial stress σ1 and lateral strain ε2(= ε3),collected during the test, are considered as the functionof the controllable variable ε1. The experiment resultsare shown in Fig. 3.

4.2 Fitting of c(κ) and ϕ(κ)

Fitting of c(κ) and ϕ(κ) consists of determina-tion of the initial yield point, conversion of the wholestress-strain curves into the strength curves, calcula-tion of the c and ϕ when κ is given, and fitting of c(κ)and ϕ(κ).

For the sake of simple description, σ1s is used torepresent the stress σ1 in the space of internal variable

Fig. 3 Triaxial testing curves of the sandstone

(the core collected from the Changqing oilfield)

κ. The first step is to determine the initial yield point based on test curves so as to obtain an initial yield stressσ0

1s. Determination of the initial yield point depends on whether there is plastic deformation after removal ofload. Point B in Fig. 1 is the initial yield point.

The test curves (Fig. 3) show that the initial yield σ01s are 92.98 MPa, 129.89 MPa and 145.00 MPa,

corresponding to confining pressures 10 MPa, 20 MPa and 40 MPa, respectively.The second step is to change the whole stress-strain curves into the strength curves. When the strain

exceeds the strain corresponding to the initial yield stress σ01s, the axial strain ε1 contains two parts, elastic (εe

1)and plastic (εp

1). For any given ε1, the plastic strain can be obtained from the following equationεp1 = ε1 − εe

1

εp2 = ε2 − εe

2

εp3 = ε3 − εe

3

, (5)

where the lateral strain ε2(= ε3) was directly recorded during the test. The elastic strain in Eq.(5) isεe1

εe2

εe3

=1E

1 −ν −ν

−ν 1 −ν

−ν −ν 1

σ1s

σ2

σ3

, (6)

where E and ν are the elastic constants of the material, and can be obtained from the testing curves; E is theslope of the elastic portion AB in Fig. 1 and ν is Poisson’s ratio. The axial stress σ1s can be obtained from thewhole stress-strain curves, with the confining pressure σ2(= σ3) set in advance.

The research shows that microfracture of the rock is predominated by tensional rupture[20]. Therefore,this study selected plastic strain as an internal variable. The plastic strain component can be regarded as thefunction of the control variable ε1, from which the internal variable κ can be obtained. The internal variableκ is also the function of the control variable ε1. In this way, the corresponding relation between controllingvariable ε1 and internal variables is established

κ = θp = εp1 + εp

2 + εp3 . (7)

Using this equation we can turn the whole stress-strain curves into the relation curves between strengthand internal variable κ, i.e. the hardening or softening curves, as shown in Fig. 4.

The third step is to determine c and ϕ when κ is given. The Mohr-Coulomb criterion can be expressed inthe stress space; i.e. that is Eq.(7), which can be rewritten as


Fig. 4 Curves showing change of triaxial

compression strength σ1s over internal variable κ

Fig. 5 Fitting Eq.(7) derived from the testing

data for initial yield point

σ1s =7Kσ3 + P, (8)

K =1 + sin ϕ

1− sinϕ= tan2

(π

4+

ϕ

2

), (9)

P =2c

(cos ϕ

1− sinϕ

)= 2c tan

(π

4+

ϕ

2

), (10)

where K is the slope of the σ1s−σ3 strength curves (Fig. 5),called the effect coefficient of confining pressure, and P

is the intercept of the curves, compression strength withno lateral pressure in the Mohr-Coulomb yield criterion.Eqs.(8) to (10) show that after the internal variable κ isdesignated, K and P are constant and the triaxial compres-sion strength σ1s is a linear function of confining pressureσ3. Then we use the experimental data to fit this linearfunction.

For any designated internal variable κ∗, we collect sev-eral pairs of data (σ3,i, σ1s,i), with i representing the dataunder the different confining pressures, to plot the σ1s−σ3

curves as shown in Fig. 5. The initial yield points in Fig. 5are the test data. K and P can be obtained using the leastsquare method to fit the linear Eq.(8). And then c(κ∗) andϕ(κ∗) can be yielded using Eqs.(9) and (10). Because theinternal variable κ∗ is optional, we can get the cohesion co-efficients c(κ) and internal friction angles ϕ(κ) under thedifferent values of internal variable κ. The data in Fig. 5correspond to the initial yield point (κ = 0), K = 1.594,and P = 85.42. Eqs.(9) and (10) can be used to yieldc(0) = 33.75, ϕ(0) = 13.37.

For different κ∗, a series of σ1s − σ3 curves can be plotted and the c(κ∗) and ϕ(κ∗) corresponding todifferent values of the internal variable can also be produced. The c and ϕ values of Changqing sandstonevarying with the values of the internal variable κ are listed in Table 1 and plotted in Fig. 6.

The forth step is to fit c(κ) and ϕ(κ). c and ϕ varying with the internal variable κ is far from enough andwhat needs is to solve the derivatives of c and ϕ. In addition, fitting to a derivable function is also required,

Table 1 c and ϕ values of Changqing

sandstone varying with internal variables

Plastic internal Cohesion Internal friction

variable κ c/MPa angle ϕ/(◦)

0 33.75 13.37

0.001 41.18 19.69

0.002 30.46 29.62

0.003 24.74 33.54

0.004 19.18 36.64

0.005 11.82 40.11

0.006 3.88 42.42

0.007 3.42 42.13

0.008 3.80 41.40

Fig. 6 Curves showing change of c and ϕ over κ


with avoiding the presence of inflexion points.The initial, peak and residual yield values(cin, ϕin, κin; cp, ϕp, κp; cr, ϕr, κr) can be extracted fromFig. 3 and 4 and Table 1. These data can fully describethe whole process stress-strain curves using a four-linemode, whereas the strength curves can be expressedby three-line mode, as shown in Fig. 7.

When κ is replaced by the plastic volume strainθp, the change of c(κ) can be fitted by Gaussian curves:

c = cme−(

θp−θppe

ξ

)2

+ cr, (11)

where cr is the residual internal cohesion, cm is theconstant and is the difference between peak internalcohesion cp and residual internal cohesion cr. θp

pe isthe plastic volume strain corresponding to the peakstrength. When the plastic volume strain reaches thevalue θp

pe, the internal cohesion amounts to the max-imum cp. Furthermore, cin is employed to representthe initial cohesion. ξ is a coefficient discribing the fatdegree of the curve.

The Gaussian fitting curves is shown in Fig. 8. InTable 1 there is

Fig. 7 Whole process curve (a)

and strength curve (b) of linearization

Fig. 8 Fitting curve showing the change of c over θp

c = 37.38e−(

θp−0.0010.0021

)2

+ 3.8. (12)

The internal friction coefficient f is

f = tan(ϕ) = 0.71e−(

θp−0.00650.0048

)2

+ 0.20. (13)

5 DISCUSSION AND CONCLUSIONS

5.1 Discussion

(1) The whole process curves deals with the stability of material. In order to express the change of plasticstrength of material, the strength curves are required. The x-axis of 3-directional compression strength curvesis the internal variable κ, symbolizing the change of internal structure of rock material, such as appearance anddevelopment of microfractures. Thus, the triaxial compression plastic strength of material is related to internalvariable. dσ1s/dκ > 0 indicates material is hardening whereas dσ1s/dκ < 0 indicates softening. For rockmaterials, with increasing κ, hardening is followed by softening with dσ1s/dκ = 0 at the peak state. Hardeningand softening need to be addressed in the space of strength and internal variables. The x-axis ε1 (strain) of thewhole process curves is an external variable, and has no relation with the internal change of material[21−27].

(2) Regarding the definition of internal variable κ, as the internal microfractures of rock material dominatetensile cracking[25], we employ the strain of a plastic body as a plastic internal variable. The alternative suchas equivalent plastic shearing strain εps can also be used as an internal variable[28−29]

κ = εps =1√2

√(εp

1 − εpm)2 + (εp

m)2 + (εp3 − εp

m)2, (14)

where εpm = 1

3 (εp1 + εp

3). Obviously, εps is defined based on the case of the plane strain. In the triaxial test,the second term in the radical sign should be replaced by (εp

2 − εpm)2 and εp

m = 13 (εp

1 + εp2 + εp

3), because thespecimen in the triaxial test bears strains from three directions.


(3) The effect of unloading modulus. We can calculate the elastic modulus (i.e. unloading modulus) usingthe stress-strain relation during unloading. After the triaxial compression experiment of rock specimens entersthe plastic stage, the unloading Young’s modulus decreases with the development of plastic deformation[5,21].Previous researchers called it elastoplastic coupling, but this study terms it the degradation of the elasticmodulus.

If the effect of worsening of unloading modulus is taken into consideration, and when we use Eq.(5) tocalculate the elastic strain, it is necessary to change modulus E into E(ε1), with E(ε1) representing unloadingmodulus corresponding to ε1. As E(ε1) tends to decrease, the elastic strain derived from calculation becomesbig whereas the plastic strain becomes small, leading to decreasing of the internal variable κ. This study doesnot take this into consideration.

(4) In the case of the isotropic hardening-softening model, the internal variable κ corresponding to peakvalues of different confining pressure curves (Fig. 4) should be same or close[30]. The mean value of peak internalvariables (κp)i in different loading modes (i.e. different confining pressures) can be used as the peak internalvariable (κp) of the material. The deviation between (κp)i and κp can indicate the precision of the isotropicmodel, with i representing the serial number of the confining pressures.

(5) The Drucker-Prager yield criterion is a cone, in which the Mohr-Coulomb hexagonal pyramid is ex-ternally circumscribed by or internally tangential to. Its material parameters α and κ can be expressed by thecohesion c and internal friction angle ϕ[31]. It is also an isotropic hardening-softening criterion, and thus theresults c(κ) and ϕ(κ) derived from this study can be used to describe α(κ) and ϕ(κ).

5.2 Conclusions

(1) This study gives an experimental method determining the change of cohesion c(κ) and internal frictionangle ϕ(κ) with the internal variable κ using triaxial compression data, and presents the test results of c and ϕ

for the Changqing sandstone.(2) The whole stress-strain curves of rocks can be simplified into a four-line pattern, while the strength

curves into a three-line pattern. Three sets of the key parameters, initial, peak and residual yields, (cin, ϕin, κin; cp,ϕp, κp; cr, ϕr, κr) were obtained through the experiment. Fitting these data, the strength curves can be describedas continuously differentiable function convenient to writing and developing of numerical simulation programs.

Compared with the model of the three-line whole process curves[32], the criteria proposed in this paper isof extensive applicability. The model of three-line whole process curves suggests that the initial yield point isthe peak yield point, without taking the hardening stage into consideration, and this is absolutely prohibitedin elastic-plastic calculation in some engineering. But for analysis of engineering stability only, it is completelyacceptable.

ACKNOWLEDGMENTS

We thank Prof. Yin Youquan and research fellow Wang Lianjie for their devoted instruction. Thanksalso go to reviewers for their constructive suggestions. This study was jointly funded by Ministry of Landand Resources of P.R.C. (SinoProbe-07) and Specific Funds for Public Warfare Scientific Research (201011071,201011070).

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mohr-coulomb yield criterion in rock plastic...

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