1 INTRODUCTION

Rock deformed under polyaxial loading conditions, where all principal stresses are of different magnitude, frequently exhibits an intermediate stress (2) dependence of strength (Mogi 1967, Lade 1997, Colmenares & Zoback 2002). Existing rock laboratory data hence suggest that the commonly used Mohr-Coulomb and Hoek-Brown failure criteria, which ignore the impact of the intermediate principal stress on strength, appear to be an oversimplification. Moreover, fail-ure criteria that are only functions of the first two stress invariants (mean stress and von Mises stress), such as the Drucker-Prager criterion commonly used in soil mechanics and plasticity theory, also appear to be inadequate descriptions of the strength of brittle rock. Exact definition of appropriate three-dimensional failure criteria is however difficult, since it requires lab testing that is not routinely performed.

The advantage of numerical modeling for obtaining insights into the strength of rocks is that technically demanding loading conditions can be imposed with relative ease on the model mate-rial. Moreover, results are robust, particularly if distortional periodic space is used (Thornton 2000), because boundary effects that commonly arise if loading platens are used (Albert & Rudnicki 2001) are eliminated. Existing model results on numerical granular (cohesionless) ma-terials suggest that the Lade criterion can adequately describe the intermediate stress depend-ence of strength (Thornton 2000, Suiker & Fleck 2004). A series of numerical models (cement-ed spheres) deformed under zero least compressive stress conditions suggests that also cohesive numerical rock exhibits an intermediate stress dependent strength (Fjær & Ruistuen 2002). On the basis of triaxial compression and triaxial extension tests performed on bonded particle mod-els Schöpfer et al. (2009) and more recently Schöpfer and Childs (2013) concluded that the Mo-gi criterion (Mogi 1967) or a modified Lade criterion (Lade 1997) may be used to describe the peak strength data obtained from these end-member loading conditions. However, complete en-velopes that describe yielding (onset of permanent strain), failure (peak strength) and frictional

The failure envelope of a bonded particle model for rock in three-dimensional stress space

M.P.J. Schöpfer Department for Geodynamics and Sedimentology, University of Vienna, Austria

C. Childs & T. Manzocchi Fault Analysis Group, UCD School of Geological Sciences, University College Dublin, Ireland

ABSTRACT: The complete yield, failure and residual strength envelope of a bonded particle model (BPM) for rock is determined under constant mean stress and Lode angle loading condi-tions. These loading paths are achieved using distortional periodic space. The model results il-lustrate that the emergent strengths depend on the intermediate principal stress and therefore suggest that the commonly used Mohr-Coulomb and Hoek-Brown failure criteria are an over-simplification. Moreover, two invariant failure criteria, such as Drucker-Prager, are also incapa-ble of describing the intermediate principal stress dependence of strength. Therefore a general-ized failure criterion that depends on all three stress invariants is used to represent the BPM strength envelopes in three-dimensional stress space. The stress strain behavior of these BPMs also illustrates that at high mean stress no significant stress drop occurs after peak. The transi-tion pressure at which no loss in strength occurs is a mean stress and Lode angle dependent space curve.

sliding (residual strength) in principal stress space have not previously been determined for any cohesive geomaterial, be it natural or numerical.

This paper is a summary of a recently published study (Schöpfer et al. 2013) for which the Particle Flow Code in Three Dimensions (PFC

3D; Itasca 2008) was used to determine the com-

plete yield, peak and residual strength envelopes in three-dimensional stress space.

2 METHODS

2.1 Model material

The Particle Flow Code in Three Dimensions (PFC3D

; Itasca 2008) is used to model the defor-mation of a numerical rock under polyaxial loading conditions. A complete list of particle and bond properties is provided in Table 1. The bonded particle models are generated within an ini-tially cube-shaped periodic space of edge length 0.1 m using the procedure detailed in Potyondy & Cundall (2004). These model dimensions and microproperties result in bonded particle mod-els comprised of ~5100 particles with a bulk density of 2700 kg/m

3 (Fig. 1). For each test, a

sample with a different random packing, but identical microproperties is generated; this proce-dure ensures that the mechanical response of the model material, and not a certain sample, is characterized.

Table 1. Model microproperties.* __________________________________________________________________________________________________________

Particles Parallel Bonds __________________________________________________________________________________________________________

= 4000 kg/m3

minr 0.0025 m ( 1.66 =/ minmax rr ) = 1 50cE GPa 50cE GPa

sn kk / = 2.5 5.2/ sn kk

c = 0.5 100 cc MPa __________________________________________________________________________________________________________

* Definitions of microproperties are given in Potyondy & Cundall (2004).

2.2 Boundary conditions

The bonded particle models are deformed using distortional periodic space, as implemented in PFC

3D (Emam 2009). This method has been previously applied to the deformation of numerical

granular materials (Thornton 2000) and was originally implemented for molecular dynamics simulations (Parrinello & Rahman 1981). Because loading platens are absent the average stress within the sample is determined using a measurement sphere of radius 0.04 m (Fig. 1). After ini-tial confinement to the desired mean stress, m = (xxyyzz)/3, the sample is shortened in the y-direction with a constant strain rate of 0.316 s

-1, which is low enough to ensure quasi-static

conditions. The lateral (x- and z-direction) strain rates are adjusted using a servo-control so that a constant mean stress and constant Lode angle stress path is maintained (Figure 1). The Lode angle is defined in terms of principal stresses as

(1)

where 1 > 2 > 3 and compressive stresses are negative. According to equation (1) the Lode angle ranges from 0º (axisymmetric compression, 1 =2) to 60º (axisymmetric extension, 2 =3), and is 30º for the stress state 2 = (13)/2, which is sometimes referred to as deviatoric pure shear. Bonded particle models were deformed at constant mean stress ranging from -300 to 30 MPa and with 10 MPa increments and constant Lode angle ranging from to 0º to 60º with 5º increments. This systematic variation of mean stress and Lode angle resulted in a total of 442 models which permit determination of one sextant of the yield and failure envelope, which is sufficient to characterize the entire failure envelope for an isotropic material.

Figure 1. Three-dimensional view and section of bonded particle model (bonds not shown for clarity) il-lustrating boundary conditions. The model is shortened with a constant strain rate ėyy whilst the strain rates in the x- and z-directions are adjusted in order to achieve the desired stress path, xx and zz. The av-erage stress tensor is computed within a measurement sphere (intersection shown as dashed line).

3 RESULTS AND ANALYSIS

3.1 Stress strain curves

A selection of stress-strain curves obtained from axisymmetric compression ( = 0º) and ax-isymmetric extension ( = 60º) tests is shown in Figure 2. Because of the truly three-dimensional loading conditions the curves are plotted on von Mises equivalent stress vs. octahe-dral shear strain graphs. An exact determination of the yield strength (the stress at which strain becomes permanent; yellow diamonds in Fig. 2) is achieved by comparing stress-strain curves of model runs with finite and infinite bond strength (see also Schöpfer & Childs 2013). Stress-strain curves from low pressure (m ≥ -240 MPa) tests exhibit significant stress drops, which be-come less marked with increasing pressure. On the other hand, curves from tests at greater pres-sure (m ≤ -270 MPa) do not exhibit any significant stress drops. The transition pressure at which no loss in strength occurs is a possible definition of the brittle-ductile transition (Byerlee 1968). The stress-strain curves suggest that this type of brittle-ductile transition occurs within the mean stress range m [-270,-240].

Another important feature is that at a given mean stress the peak von Mises stress (red dots in Figure 2) decreases with increasing Lode angle, an observation suggesting that a two invariant (e.g. Drucker-Prager) failure criterion is inappropriate for describing the material behavior. In the following Section 3.2 a family of yield criteria is introduced that incorporates all three stress invariants. In Section 3.3 this generalized criterion is then fitted to the data obtained from the model stress-strain curves.

3.2 Family of failure criteria

Yield and failure envelopes can be expressed in terms of the three invariants of the principal stresses translated to the origin, (i = 1, 2 and 3), given by

(2)

where a > 0 for cohesive materials. The shape of smooth yield surfaces can be varied widely us-ing a yield function of the form (Borja & Aydin 2004)

(3)

The exponent > 0 controls the shape on the triaxial plane and < 0 is a parameter in stress

units that controls the size of the surface. The function f is given by

(4) with the three invariant functions f1, f2, and f3 defined as

(5)

The positive, dimensionless constants c1, c2, and c3 control the shape of the yield surface in

the deviatoric plane (or -plane; its normal vector is the hydrostatic axis), whereas c0 deter-mines whether the surface is open or closed. Only if c0 = 3c1 + 9c2 + 27c3 then the yield surface is open to form a cone, or bullet, otherwise the surface is closed to form a teardrop, or a cap (see Borja & Aydin 2004 for details).

Failure surfaces that are linear in the triaxial plane can be expressed as

(6) where < 0 is a dimensionless parameter that controls the apex angle. Several linear failure cri-teria and their non-linear equivalents can be recovered from equations (6) and (4), respectively. The Drucker-Prager criterion is recovered using c1 = 1, c2 = c3 = 0, and c0 = 3, the Matsuoka-Nakai criterion if c1 = c3 = 0, c2 = 1, and c0 = 9, and the Lade-Duncan criterion is obtained when c1 = c2 = 0, c3 = 1, and c0 = 27 (see Borja & Aydin 2004 and references therein).

Figure 2. Stress-strain curves obtained from axisymmetric compression and extension tests. The curves of fully elastic models which were used for determining yielding (yellow diamonds) are also shown. Peak and residual strength are plotted as red and cyan dots, respectively. Labels are constant mean stress (m) in MPa. Arrows denote that curves continue beyond abscissa limit.

Figure 3. Yield, peak and residual strength data of bonded particle model and best-fit envelopes plotted on triaxial and -plane graphs (for clarity only peak data in -plane graph; sx and sy are rotated coordinates). The projected brittle-ductile transition is plotted as green curve in each graph and labeled B ↔ D. Ticks along hydrostatic axis have a mean stress spacing of 30 MPa and indicate the locations of sections shown in the -plane graph.

3.3 Best-fit failure envelopes

Before describing the properties of the envelopes that best describe the results obtained from the bonded particle models it is worthwhile to inspect the data in sections and projections that are commonly used to represent three dimensional failure criteria (Fig. 3).

Inspection of the yield data (crosses in Fig. 3) and the fact that yielding also occurs under hy-drostatic compression conditions (at a mean stress of < -300 MPa) suggests that these data are best fitted using a nonlinear criterion (Eq. 3) with a closed surface. The peak stress data (red dots in Figure 3) taken from stress-strain curves that exhibit a stress drop (Fig. 2) display non-linear mean stress dependence. Therefore a nonlinear failure envelope (Eq. 3) with an open sur-face is fitted to the low mean stress data (m ≥ -240). At higher mean stress, however, the peak strength data exhibit linear mean stress dependence and hence a linear criterion (Eq. 6) is fitted to the high mean stress data (m ≤ -270).

These three different surfaces are fitted using numerical optimization by varying the coeffi-cients c1, c2, and c3 in equation (4) and the hydrostatic translation parameter a used in equation (2). In case of the nonlinear criterion (fitted to the yield and low mean stress data) a linear equa-tion is obtained by using the logarithmic form of equation (3), from which the parameters and can be determined. More details about the fitting procedure can be found in Schöpfer et al. (2013). An important result from the fitting procedure is that none of the end-member criteria that are incorporated in equation 4 adequately describe the model data.

The best-fit envelope functions are plotted on triaxial and -plane graphs in Figure 3 and in three-dimensional stress space in Figure 4. In the -plane, the cross sectional shape of the fail-ure surface changes from triangular to become more circular with decreasing value of I1 (in-creasing pressure), similar to the non-linear Lade-Duncan criterion (Fig. 3; see also figure 7 in Lade 1997). The linear failure envelope, fitted to the high mean stress peak strength data, is tri-angular with smooth corners and has, due to its linear pressure-dependence, constant cross-sectional shape (Fig. 4). The intersection of the nonlinear and linear peak stress envelopes is the brittle-ductile transition (Figs. 3 & 4) and is, because of differences in cross-sectional shape, mean stress dependent. The locus of points that define the brittle-ductile transition is hence a space curve, meaning that for each value of Lode angle there is a particular value of mean and von Mises stress.

Figure 4. Best-fit yield and peak stress envelopes plotted in three-dimensional stress space. Peak stress da-ta (N = 442) are plotted as red spheres with a radius of 5 MPa in one sextant only.

4 DISCUSSION

The deformation of bonded particle models under polyaxial loading conditions reveals that complex stress-state dependent behavior emerges from simple particle contact laws, a fact which has already been highlighted by earlier studies on cohesionless (Thornton 2000, Suiker & Fleck 2004) and cohesive (Fjær & Ruistuen 2002) numerical materials. However, complete yield, fail-ure and residual strength envelopes have not previously been determined for any cohesive geomaterial, be it natural or numerical.

The model results are in agreement with previous rock mechanical and numerical modeling studies and show that yielding (onset of permanent strain), failure (peak strength) and frictional sliding (residual strength) depends on the third stress invariant, but none of the existing end-member failure criteria fits the complete yield and nonlinear peak stress data (the linear pressure dependent envelope can be adequately described by a Lade criterion, see also Thornton 2000 and Suiker & Fleck 2004).

Both yielding and failure exhibit nonlinear mean stress dependencies, whereas frictional slid-ing is approximately linearly pressure dependent (Figs. 3 & 4). The cross-over of the fracture and sliding envelopes is the transition pressure beyond which no stress drop occurs and is a pos-sible definition of the brittle-ductile transition (Fig. 5). This emergent behavior is consistent with the frictional hypothesis for the brittle-ductile transition (Byerlee 1968), which states that the frictional strength of a fracture may increase to such an extent that it is equal to or greater than the stress required to cause fracture, in which case no stress drop occurs after fracture (Fig. 5).

Figure 5. Schematic stress-strain curves and failure envelope for rock loaded at different confining pres-sures, illustrating the brittle-ductile transition, labeled B ↔ D (modified after Mogi 1974).

The model results may have implications for a wide range of geophysical research areas, in-cluding the strength of the crust, the seismogenic zone and slip-tendency analysis. For a more in-depth discussion and a semi-quantitative comparison of the model failure envelope with data obtained for natural rock loaded under polyaxial stress conditions the reader is referred to the full-length paper by Schöpfer et al. (2013).

Recent research on bonded particle models illustrates that the emergent behavior, such as po-rosity dependent bulk properties (Schöpfer et al. 2009), pressure dependent fracture inclinations (Schöpfer & Childs 2013) and intermediate stress dependent strength (Schöpfer et al. 2013), is, at least qualitatively, similar to that observed in natural rock. Future research should however focus on better matching the behavior of rock in a quantitative manner, since some mechanical properties (e.g. ratio of compressive to tensile strength) cannot be reproduced using the present approach in which particles offer no resistance to rolling; these shortcomings may be overcome in the near future (e.g. Potyondy 2012).

5 SUMMARY

Model results of the deformation of bonded particle models under polyaxial loading conditions suggest the following principal conclusions: 1 Complex stress-state dependent behavior including an intermediate stress dependence of

strength emerges from simple particle contact laws. 2 Commonly used failure criteria, such as Mohr-Coulomb, Hoek-Brown or Drucker-Prager, are

inadequate for describing the model strength data. 3 A generalized criterion that depends on all three stress invariants accurately describes both

the mean stress and Lode angle (or stress ratio) dependence of yield, peak and residual strength.

4 The stress drop magnitude (difference between peak and residual strength) decreases with in-creasing mean stress until eventually no loss in strength after peak occurs. The transitional pressure beyond which no stress drop occurs is a possible definition of the brittle-ductile tran-sition, which is a mean stress and Lode angle dependent space curve.

ACKNOWLEDGEMENTS

The staff of Itasca Consulting Group, in particular Peter Cundall, Sacha Emam and Dave Potyondy, is acknowledged for valuable discussions. This research was funded by Science Foundation Ireland grant 08/RFP/Geol1170.

REFERENCES

Albert, R.A. & Rudnicki, J.W. 2001. Finite element simulations of Tennessee marble under plane strain laboratory testing: Effects of sample-platen friction on shear band onset. Mechanics of Materials 33: 47-60.

Borja, R.I. & Aydin, A. 2004. Computational modeling of deformation bands in granular media: I. Geo-logical and mathematical framework. Computer Methods in Applied Mechanics and Engineering 193: 2667-2698.

Byerlee, J.D. 1968. Brittle-ductile transition in rocks. Journal of Geophysical Research 73: 4741-4750. Colmenares, L.B. & Zoback, M.D. 2002. A statistical evaluation of intact rock failure criteria constrained

by polyaxial test data for five different rocks. International Journal of Rock Mechanics and Mining Sciences 39: 695-729.

Emam, S. 2009. Distortion of the Periodic Space in PFC. Technical Memorandum, Minneapolis: Itasca. Fjær, E. & Ruistuen, H. 2002. Impact of the intermediate principal stress on the strength of heterogeneous

rock. Journal of Geophysical Research 107(B2): 2032. Itasca Consulting Group, Inc. 2008. PFC

3D - Particle Flow Code in Three-Dimensions, Ver. 4.0. Minne-

apolis: Itasca. Lade, P.V. 1997. Modelling the strengths of engineering materials in three dimensions. Mechanics of Co-

hesive-Frictional Materials 2: 339-356. Mogi, K. 1967. Effect of intermediate principal stress on rock failure. Journal of Geophysical Research

72: 5117-5131. Mogi, K. 1974. On the pressure dependence of strength of rocks and the Coulomb-fracture criterion.

Tectonophysics 21: 273-285. Parrinello, M. & Rahman, A. 1981. Polymorphic transitions in single crystals: A new molecular dynamics

method. Journal of Applied Physics 52: 7182–7190. Potyondy, D.O. & Cundall, P.A. 2004. A bonded particle model for rock. International Journal of Rock

Mechanics and Mining Sciences 41: 1329-1364. Potyondy, D.O. 2012. A flat-jointed bonded-particle material for hard rock. ARMA, Proceedings of the

46th U.S. Rock Mechanics Geomechanics Symposium (Chicago, June 2012), Paper No. ARMA 12-501. Schöpfer, M.P.J., Abe, S., Childs, C. & Walsh, J.J. 2009. The impact of porosity and crack density on the

elasticity, strength and friction of cohesive granular materials: insights from DEM modeling. Interna-tional Journal of Rock Mechanics and Mining Sciences 46: 250-261.

Schöpfer, M.P.J. & Childs, C. 2013. The orientation and dilatancy of shear bands in a bonded particle model for rock. International Journal of Rock Mechanics and Mining Sciences 57: 75-88.

Schöpfer, M.P.J., Childs, C. & Manzocchi, T. 2013. Three-dimensional failure envelopes and the brittle-ductile transition. Journal of Geophysical Research 118: 1378-1392.

Suiker, A.S.J. & Fleck, N.A. 2004. Frictional collapse of granular assemblies. Journal of Applied Me-chanics 71: 350-358.

Thornton, C. 2000. Numerical simulations of deviatoric shear deformation of granular media. Géotechnique 50: 43-53.