JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 96, NO. BS, PAGES 8337-8349, MAY 10, 1991

Modelling Plastic Deformation of Peridotite With the Self-Consistent Theory

H.-R. WENK AND K. BENNETr

Department of Geology and Geophysics, University of California, Berkeley, California

G. R. CANOVA AND A. MOLINARI

Laboratoire de Physique et Mdcanique des Matdriaux, Facultd des Sciences, Metz, France

Theories for deformation of Polycrystals have been substantially refined, enabling us to model deforma- tion of metals and minerals with considerable sophistication. So far, most modelling has been confined to single-phase aggregates such as quartzite and limestone. We present the first results for a polyphase aggregate, peridotire, consisting of 70% olivine and 30% enstatite. The problem is approached with a viscoplastic self-consistent theory satisfying stress equilibrium and strain compatibility for the average polycrystal and taking account of anisotropic neighbor interactions. It is assumed that olivine deforms by (010)[100], (001)[100], and (010)[001] slip and enstatite deforms by (100)[001] slip. Simulated tex- tures for olivine and enstatite in peridotire resemble simulated textures in the pure phases, indicating that for this system and for these volume fractions there is little influence of the different phases upon each other. In our model the harder mineral enstatite deforms at a slower rate than olivine. Interaction between

neighboring grains appears to be minimal, which may be due to model assumptions. Predicted pole fig- ures with olivine (010) axes and enstatite (100) axes aligning with the direction of shortening are in good agreement with preferred orientations in naturally and experimentally deformed peridotires.

INTRODUCTION

Considerable efforts have been made to apply polycrystal plasticity theory to rocks. The Taylor [1938] theory, which assumes homogeneous deformation, has been used to simu- late texture development for quartz [e.g., Lister et al., 1978], calcite [e.g., Lister, 1978; Wagner et al., 1982], and halite [e.g., Chin and Mammel, 1973; Siernes, 1974]. However, these minerals differ considerably from fcc metals for which this approach was highly successful. Many minerals have fewer and asymmetrically disposed slip systems, resulting in a high plastic anisotropy. Some orientations are much more favorably oriented for slip than others. Also, minerals with stress exponents between three and nine rather than between 50 and 90 in fcc metals are more rate sensitive than metals; i.e., slip occurs at a lower stress than the critical stress, and at a lower rate. Both problems can be approached with a viscoplastic self-consistent theory [Molinari et al., 1987]. This theory, which compromises between stress equilibrium and strain continuity, has provided useful results for halite [Wenk et al., 1989a], quartz [Wenk et al., 1989b], olivine [Takeshita et al., 1990], and calcite [Tomd et al., 1991]. A particularly interesting aspect is that the model can be rela- tively easily modified for polyphase systems which are, of

has been a limiting factor for applying plasticity theory to rocks. Quartzites and limestones may be useful local strain indicators but are not representative of macroscopic deforma- tions in the crust or in the mantle.

In this report we introduce the viscoplastic self-consistent theory to model deformation of a two-phase polycrystal. We choose peridotire consisting of 70% olivine and 30% en- starire because of its significance for upper mantle convec-

Copyright 1991 by the American Geophysical Union.

Paper number 91JB00117. 0148-0227/91/91JB-0011755.00

tion [e.g. Carter and Ave'Lallemant, 1970]. It has to be em- phasized that at this stage our model is for deformation by slip only which is a significant mechanism in peridotites of the upper mantle [e.g., Stocker and Ashby, 1973]. Previously, deformation of pure olivine has been modelled with the relaxed Taylor theory [Takeshita, 1987; Takeshita et al., 1990], the constrained hybrid theory [Parks and Ahzi, 1990], and purely kinematic models which started out as two- dimensional [e.g., Etchecopar, 1977; Ribe, 1989] and were more recently generalized to a third dimension [e.g., Etchecopar and Vasseur, 1987; Ribe and Yu, this issue]. The kinematic models minimize misfits between grains. The effects of different plasticity model assumptions on texture development is discussed by Tomd et al. [1991], and Wenk and Christie [1991] review the general problem of stress equilibrium and strain compatibility in geological ap- plications. Since Ribe and Yu [this issue] have done so in detail for olivine, we will not compare results from different models for pure olivine; instead, we emphasize polycystal plasticity in a two-phase system, explain the self-consistent theory with neighbor interactions as a generalization of the Taylor theory, and illustrate results for peridotite.

Tim MODEL

In polycrystal plasticity we assume that we know all the potentially active slip systems and for each slip system the critical resolved shear stress, the rate sensitivity, and the rate of hardening. We also need to know the initial orienta- tion and grain shape distribution and the strain path. With this information we simulate the evolution of the yield strength, the activity of slip systems, and the development of preferred orientation based on plasticity theory. It is as- sumed that all deformation occurs by slip.

For slip systems we choose those which have been estab- lished in single-crystal experiments at high temperature (Durham and Goetze [1977], Mackwell et al. [1985], and Bai et al. [1991] for olivine and Raleigh et al. [1971] for statite) (Table 1). For olivine,' approximate critical shear

8337

8338 WENK ET AL.' MODELLING PLASTIC DEFORMATION OF PERIDOTITE

TABLE 1. Slip Systems and Critical Resolved Shear Stresses Used in Peridotite Plasticity Models

• (MPa)

Olivine

(OlO)[lO0] 18 (001)[lO0] 18 (010)[001] 45

Enstatite

(100)[0011 15 (010)[100] 100'

For olivine approximated fore Bai et al. [1991], for enstatite esti- mated from Raleigh et al. [1971].

*Entered only to provide more numerical stability.

stresses are from single-crystal experiments at 1400øC, /• = 10 -5 s -1 orthopyroxene buffer with an oxygen fugacity of 10 -4'5 arm [Bai et al., 1991]. Information is less satisfac- tory for enstatite, and we estimated the critical shear stress from experiments on polycrystals [Raleigh et al., 1971]. Since in both minerals all slip directions are parallel to crys- tallographic axes, normal strain components ell, e22, e33 cannot be accommodated by existing slip systems, and there- fore the full constraint Taylor model, which assumes homo- geneous deformation, is not applicable. It is assumed that crystals behave as a viscoplastic material. At the micro- scopic level the shear rate ,•s on a slip system s is related to the resolved shear stress •s by a viscoplastic law. We use a power law

(1)

where n is the stress exponent and •'0 and x[ are reference shear rates and reference resolved shear stresses, respec- tively. In the case of olivine and peridotire a value for n = 3.5 has been established experimentally [e.g., Bai et al., 1991]. We used n = 3.0 in this study. In our computer code, n has to be an odd interger because it is used to trans- fer information on positive and negative shear sense. Small deviations in n have no significant effect on the simulations [e.g., Wenk et al., 1989b ].

We assume an initially random orientation distribution of 200 grains and modelled three different and independent de- formation paths. The strain history is imposed by prescrib- ing a macroscopic velocity gradient tensor L for axial com- pression (2a), pure shear (2b) and simple shear (2c).

1/4 0 • ] 0 -1/2

0 0 1/4

+i/2 0 -1/2

0

(2a)

(2b)

(2c)

where the time increment is set so as to achieve an equiva-

lent strain increment of 5% or 10% for each deformation

step. We found that the larger step size had m'mimal effect on the results but still did most of our calculations with 10

steps at 5% to obtain an equivalent strain of 0.5 (50%). The equivalent strain eeq is defined as [Molinari et al., 1987]

-i_* (3)

where the Von Mises equivalent strain rate Deq is

=,/2/3 For a total equivalent strain eeq of 0.5 we obtain a deforma- tion tensor F [e.g., Shrivastava et al., 1982].

e '25 0 e -0.5

e 4•'ø'25 0 0 e -•fj'0'25 0 0

I ?=x/'•-0.5 0

0

o

o

1

(Sa)

(Sb)

(5c)

where og and • are deviatoric stress tensors of the grain g and the average polycrystal and •:g and • strain rate tensors,

0 g -a = IIA(•: g -•) (6)

The microstructure was described with a three-dimensional

cell pattern based on Voronoi [Gilbert, 1962; Frisch, 1965]. Within a three-dimensional (3-D) cubic subspace, the posi- tions of grain centers were generated at random. Subse- quently, bisecting planes between pairs of centers were con- sumcted. The inner envelope of these planes around each point establishes the grain geometry. This scheme, which is based on the principle that bisecting planes of a tetrahedron intersect in a point, produces a microstructure with triple junctions in two-dimensional (2-D) sections, only convex grain boundaries, and a Poisson grain size distribution with an average of 14.2 nearest neighbors. Individual polyhedra can be assigned phases at random in a selected volume pro- portion. Figure l a shows a 2-D section through the cube with 70% olivine (light grains) and 30% enstatite (dark grains). In this 2-D section through the center of the mi- crostincture there appears to be a layering of pyroxene which is a result of the random rather than regular distribution of phases. Also the apparent grain sizes in this section are generally smaller than the ume sizes.

A self-consistent approach satisfying both stress equilib- rium and strain continuity for the average polycrystal was used to model the deformation behavior (see Molinari et al. [1987] for details). In the simplest form of the self-consis- tent scheme we consider each grain as an inclusion in a ho- mogeneous isotropic medium representing the weighted "average" of all grains. We use a KrOner [1961]-like equa- tion

WENK ET AL.: MODELLING PLASTIC DEFORMATION OF PERIDOT1TE 8339

b

T

Fig. 1. Grain morphology used to define the topology of each grain and its neighbors. The microstructure was generated at random as 200 Voronoi cells which are repeated by translation to avoid boundary effects. The illustrated xy section at z--0 shows four unit cells: 70% olivine (light), 30% enstatite (dark). (a) Initial mi- crostructure; (b) after 50% equivalent strain in axial compression; (c) after 50% pure shear;, (d) after 50% simple shear.

respectively. The scalar IX representing the interaction be- tween the grain and its environment and the fourth rank iden- tity tensor A describe the grain shape evolution. A solution for Ix is obtained by minimizing the deviation in stress 8o and strain rate 8/• in equation (6):

-•- IXA =min (7) If Ix is chosen to be 0, stress equilibrium (o g = if) results, but deviations in strain rate are large. On the other hand, if Ix is large, deviations in strain rate become small and the Taylor [1938] solution prevails. The self-consistent Ix is be- tween these two extremes. In the self-consistent scheme, each grain is allowed to deform differently, from a spherical to an ellipsoidal shape, depending on its orientation with re- spect to slip systems and stress. We have applied this isotropic model previously to halite [Wenk et al., 1989a] and quartz [Wenk et al., 1989b] and were able to match closely texture patterns observed in experiments and in nature;

The isotropic approximation is rather crude. It assumes that the anisotropy of the yield surface of a grain is much larger than that of the matrix which may not be true even for strongly rate sensitive materials with a rounded single- crystal yield surface. Furthermore, the different local en- vironment of each grain is not accounted for. The deforming polycrystal develops local deviations from an average stress and an average strain rate, and these fluctuations axe mini- mized with the self-consistent theory. To analyze the local fluctuations, the neighborhood r' of each point r (Figure 2a) is evaluated. According to Hooke's law, the stress influence on strain rate decays linearly with distance r'. Green's func- tions axe a useful description of integral heterogeneity, and axe incorporated in the self-consistent scheme [e.g., Molinari et al., 1987; Adams et al., 1989]. We can write an integral equation to describe the influence of neighborhood on the local strain rate /•

•;(r)=•+ J F(r-r') '[o(r')-•-Aø:(•;(r')-•)] dr' (8) where F is an influence tensor which contains information on interaction of different regions within each grain and be- tween neighbors ("n site" scheme). The continuous volume is divided into a set of discrete, homogeneous polyhedra g surrounded by neighbors g' and equation (8) is rewritten as

g'..• g

(9)

(b)

Fig. 2. Evaluation of neighborhood in the anisotropic n site vis- coplastic serf-consistent scheme. (a) The microstructure is scanned with a vector ß which defines the location of a point. Around each point a vector r' evaluates the neighborhood both within the same grain and in neighboring grains. (b) The integration of r' is difficult to perform. Instead, we use a cell scheme to evaluate numerically for each cell the grain affinity and that of its 26 neighboring cubes (shaded area).

8340 WENK ET AL.' MODELL•O PLASTIC DEFORMATION OF PERIDOTITE

In ,this equation, term I is equivalent to the Taylor model; that is, the local strain rate /•s equals the macroscopic strain

o

rate •. Term II is a first-order correction for self-consistency in which the tensor A ø is set to fulfill self-consistency in the average poly•ryst:•. Te•? .I•I• describes the neighbor in- teraction betwte n .grains g ai•d the'ff neighbors g' which is a seqpn, d-order correction.

•uation (9) can be rearranged as

g'•g

In this form it resembles the Kr6ner equation (6), except that A ø is now a fourth-rank symmetrical tensor and a sum has, been added to account for neighbor interaction. The F gg tensors are volume integrals of the Greeds function deriva- tives of the type

Vg where Vg is the volume of the grain g, Vg' is the volume of a neighbor g', and r ao•d r' are vectors scanning the two vol- umes (Figure

It is very difficult to perform integrals on polyhedral vol- umes, and we desiõn9d a numerical. procedure to estimate the coupling terms Fog. The large cube containing all grain centers is divided into 8000 cells (20 divisions along each axis), and •he number of centers of these cells which fall into each grain are counted. This provides a volume for each grain which is used as a weight. In order to get the interac- tion terms F, a volume element consisting of 3x3x3=27 lit- tle cubes is translated through all cells (Figure 2b). A cou- pling term F gg is entered whenever the center of the test cube and one of the surrounding cells are in the same grain. F gg coupling accounts for influence of anisotropic grain shape. A coupling term F gg' is counted whenever'the center of the test cube and a neighboring cube fall into different grains. These neighbors g' can belong to the same or to different phases than grain g.

c•g c'•g

Fgg, = 1 • N½ i•"• jceu c•g c'•g'

(2b)

where N c is the number of grains. Interaction with the far- ther environment is neglected since the interaction decays with 1/r'. With such a scheme, strong interactions for grains with large common faces result.

After each deformation step the theory predicts the activity of slip systems, the grain shape, the stress state, and the new orientation. The grain shape has a double role. On one hand, within the self-consistent scheme, grains with favor- ably oriented slip systems deform fast, others more slowly. This leads to incompatibility at grain boundaries, which is difficult to handle with the polyhedra scheme. We therefore prescribe that the microstructure which only determines the topology of the neighbors deforms homogeneously with no gaps or overlaps developing between polyhedra. Sections for compression, pure shear and simple shear corresponding to five strain increments described by (2) and (3) are shown in Figures lb-ld. The topology of neighbors remains con- stant, but the shapes used to calculate interaction terms F change and are updated after each step by convecting polyhe- dral vertices as prescribed by the macroscopic velocity gradi- ent. The actual grain shapes (described as ellipsoids) do not correspond to grains in the microstructure (described as polyhedra).

In our deformation model of peridotite we neglect micro- structural hardening which has been considered in halite [Wenk et al., 1989a] and quartz [Wenk et al., 1989b]. We assume that at upper mantle temperatures, recovery is fast and dislocation interaction minimal.

Calculations were done with Apollo and PC 486 computers using a 32-bit Fortran compiler. For 200 grains each defor- mation step takes less than 1 hour with a 25-MHz 486 pro- cessor.

RESULTS

Preferred orientations of olivine and enstatite are repre- sented in (100), (010), and (001) pole figures (Figures 3-7). In compression of a pure olivine polycrystal a distinct tex- ture pattern develops between 20% and 30% equivalent strain with (010) poles rotating towards the compression direction (Figure 3). Some grains are barely deformed (small sym- bols), whereas others are strongly deformed (large symbols), but in pole figures the patterns of deformed and undeformed grains are not very different. After 50% equivalent strain, pole figures of olivine and enstatite display statistically ax- ial symmetry for compression (Figure 4), orthorhombic symmetry for pure shear (Figure 5), and monoclinic symme- try for simple shear (Figure 6). In these model calculations the olivine orientation pattern in pure olivine is similar to

•q = 10% 20% 30% 40% 50%

Fig. 3. Evolution of preferred orientation in an olivine polycrystal as predicted by the viscoplastic self-consistent theory. Axial compression, (010) pole figures, equivalent strain is indicated. Equal-area projection, 200 grains.

100% Olivine

70% Olivine

(100) (010) (0•)1) x xs< xX= '=

œ.'•x•'.--; •'?, •

.

30% Enstatite

100% Enstatite

Fig. 4. (100), (010), and (001) pole figures after 50% equivalent strain in axial compression for pure olivine, 70% olivine-30% enstatite, and pure enstatite. Equal-area projection, 200 grains total. The symbol size in this and the following figures is a measure for the relative deformation of grains. Large symbols indicate grains which are strongly deformed.

100% Olivine

70% Oilvine

30% Enstatite

100% Enstatite •< .....' x•... . '. • :.•,,,.Xx.. •

Fig. 5. (100), (010), and (001) pole figures after 50% equivalent strain in pure shear for pure olivine, 70% olivi•e'30% enstatite, and pure enstatite. Equal-area projection, 200 grains total.

8342 WENK ET AL.: MODELLING PLASTIC DEFORMATION OF PERIDOT1TE

100% Olivine

70% Olivine

(100) (010) (001)

-½

F •

30% Enstatite

x

x x •,, 100% Enstatite .... ..;,•.:• ' ' "?

ß •.x. •.•

Fig. 6. (100), (010), and (001) pole figures after 50% equivalent strain (y=0.5•'•=0.866) in simple shear for pure olivine, 70% olivine-30% enstatite, and pure enstatite. Equal-area projection, 200 grains total. The macroscopic shear plane C, the shear sense, and the foliation plane F are indicated.

that of the 70% olivine-30% enstatite mixture. The same is

the case for enstatite.

In detail the olivine texture in the mixture is stronger than the pure olivine texture for the same amount of overall strain. The enstatite texture in the mixture appears weaker than that in pure enstatite, but it is difficult to evaluate en- statite in the mixture quantitatively because of poor statistics (only 60 grains). We attribute this difference to the concen- tration of strain in the plastically weaker olivine which has more slip systems available than the enstatite which remains more rigid. Except for this overall strain partitioning there is not much evidence for interaction between the two phases or between neighbors in general.

We find empirically that in compression and pure shear the slip plane normal of the easiest slip system ((010) in the case of olivine and (100) in the case of enstatite) rotates to- ward the direction of maximum compression. In simple shear the easiest slip system is inclined to the macroscopic shear plane, against the sense of shear. The behavior in simple shear is particularly interesting because of recent discussions in the geological literature about the relationship between the macroscopic shear plane and the microscopic slip plane [e.g., Etchecopar and Vasseur, 1987; Schmid and Casey, 1986; Takeshita et al., 1990; Wenk and Christie, 1991]. Figure 7 illustrates in (010)pole figures the evolution of the simple shear texture for olivine. The macroscopic shear plane and the orientation of the foliation defined as the plane normal to the direction of maximum to-

tal shortening are indicated. This plane rotates with increas- ing deformation toward coincidence with the shear plane (tan 20 = 2/T, where T is the shear and O is the angle between the shear plane and the foliation). The (010) maximum is also rotating slightly. This is best understood by consider- ing rotation trajectories of 14 selected individual grains (Figure 8). Figure 8 illustrates that some orientations barely rotate, whereas others undergo large rotations; some undergo large shape changes, whereas others remain equiaxed. Most grains rotate into harder orientations (symbol size in- creases); a few rotate into softer orientations (symbol size decreases). In general, grains rotate with the sense of shear, but there are exceptions due to the relaxed condition for ho- mogeneous strain in the self-consistent model. Incremental rotations for (010) poles are minimal along a NNW-SSE di- agonal, and this corresponds more or less to the texture max- imum in Figure 7. As we have pointed out for quartz [Wenk et al., 1989b], texture concentrations in simple shear, are regions of slow rotations and are not stable orientations; the latter cannot exist in simple shear, but in the case of olivine (010) rotations are very small in the NNW-SSE diagonal so that the orientation seems stable. The reason for the slight rotation of the (010) maximum in Figure 7 is that at moder- ate strains, grains move into the texture maximum with the sense of shear. Orientations which are clockwise from the

predicted locus of slow rotations get rapidly depleted; those which are counterclockwise slowly move toward the maxi- mum. At high strains a majority of orientations is collected

WENK ET AL.: MODELLING PLASTIC DEFORMATION OF PERIDOTrrE 8343

r•q = 10% 20% 30% 40% 50% 7 = 0.173 0.346 0.520 0.693 0.866 Fig. 7. Texture evolution for olivine polycrystal deformed in simple shear. (010) pole figures; equivalent strain, ¾, and corresponding traces of the foliation F, and the macroscopic shear plane C are indicated. Equal-area projection, 200 grains.

(100) (010) (001)

Fig. 8. Rotation trajectories for 14 selected grains in simple shear. Increment is 5%, 10 steps. Square indicates the starting orientation. The symbol size is proportional to the effective stress. Equal-area projection.

in the maximum with very slow rotations, and a few grains ultimately begin to move out of it. These trends result in an asymmetry and a slight overall clockwise rotation of the maximum toward the locus of slow rotation, but the maxi-

mum never fully rotates into the shear plane. It should be pointed out that the trajectories in Figure 8 represent only those for single-crystal directions and not the total lattice rotation.

Orientation changes and texture development are directly linked to slip system activity. Figure 9 displays the distri- bution of shear over the different families of slip systems. For olivine the average number of active systems which con- tribute more than 5% to the shear in each grain varies be- tween 1.9 and 2.6. There is no major change with texture evolution for pure and simple shear. The soft systems (010)[100] and (001)[100] are equally active. The harder system (010)[001] contributes less than 20%. In axial com- pression the geometry is more restrictive, and as texture de- velops, it becomes necessary to activate the hard system even for self-consistent conditions. The geometrical con- straints are more restrictive for pure enstatite, with only a single easy slip system (100)[001]. At the onset of defor- mation all slip occurs on (100)[001], but as texture devel- ops, single slip becomes impossible and in the framework of our model the 6 times harder artificial (010)[100] system be-

comes active. This is illustrated in Figure 9 by the dashed curves with decreasing values.

The decrease of activity for (100)[001] slip in enstatite is most pronounced for simple shear which is plausible if we think of a textured polycrystal as an obliquely oriented sin- gle crystal which, of course, can only be sheared in one sense. This is possible with the self-consistent scheme only as long as the orientation distribution is close to random and at least some grains are favorably oriented. In simple shear we were unable to model enstatite deformation beyond 45% equivalent strain. This shows that polycrystalline aggregates, even under favorable assumptions of self- consistency, can not deform by single slip to appreciable strain. Note also that even with single slip the "easy glide" orientation, with the slip plane (100) in the macroscopic shear plane and [001] in the shear direction, is not stable. Instead, oblique (100) and (001) texture maxima develop. The incompatibility problem does not occur in the case of the olivine-enstatite mixture, where enstatite can simply stop deforming and strain accommodation is transferred to the plastically softer and less anisotropic olivine.

Activity of slip systems determines not only texture devel- opment but also the mechanical properties of the polycrys- tal. Again we have to bear in mind that our model only pre- dicts behavior for deformation by slip and does not account

8344 X, VENK ET AL.: MODELLING PI•STIC DEFORMATION OF PERIDOT1TE

100

8o

6o

4o

2o

0.0

COMPRESSION

............... ...(100)[001]

(001)[100]

ß i ß i ß , ß 14 ß 0.1 0.2 0.3 .0. 0.5

lOO

• 40

o

:_• 20

o

o.o

PURE SHEAR

mmmm•mmm m ...... (100)[001]

(001)[100]

(010)[100]

(010)[001] ß I ' I ' I ' I '

0.1 0.2 0.3 0.4 0.5

lOO

80

6o

40

2o

SIMPLE SHEAR

mmmmmmmmmmmm•m -~~..(.1.0 0) [001 ]

(001)[100]

(010)[100]

(010)[001] 0 ' I ' ' I ' I ' 0.0 0.1 012 0.3 0.4 0.5

Equivalent Strain

Fig. 9. Slip system activity (in percent) as a function of equivalent strain for olivine and enstafite polycrystals, predicted from the n site viscoplasfie self-consistent model. Dashed lines are for en- statite.

for diffusion-related processes which are significant in exper- imental and natural deformation of peridotire [e.g., Borch and Green, 1989]. Figure 10 plots effective stress-strain curves. Changes in polycrystal strength with deformation in these calculations are solely due to changes in crystal orientation and not to other factors such as microstructural hardening. In all deformation modes the effective stress increases and

for olivine roughly doubles at 50% strain. The increase for pure enstatite is more drastic and, for reasons outlined above, we do not consider this behavior realistic. For peridotitc the behavior is similar to that of olivine, but stresses are higher due to the higher plastic strength of enstafite. The stress in- creases if, on average, crystals become less favorably ori- ented for further slip (a decrease in Sehmid factor) or if, with

600

COMPRESSION

4OO

200

0.0

mmmmmmmmmmmmmmlmmlmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmlmmmmmmlmmmmlmmmm

' I ' I ' I ' I '

0.1 0.2 0.3 0.4 0.5

600

400

2O0

0

0.0

PURE SHEAR

ß 01 ' I ' I ' I ' .1 0.2 0.3 0.4 0.5

600

SIMPLE SHEAR

400

200 • ,,

,,,,,,,,,,

0 ' I ' I ' I ' I ' 0.0 0.• 0.2 0.3 0.4 0.5

Equivalent Strain

Fig. 10. Effective stress versus Von •ses eq•vflent stm• for de- fo•afion • •mpression, pure sh•r •d s•ple sh•n Solid •es are for pure oHv•e, dott• Hnes am for a 70% oHv•e-30% enstafi• m•tum, •d dash• •es am for pure enstafite. U•ts am MPa for • effective stra• rate of 0.5 •es •e stra• rate at w•ch critical shear s•esses we• dete•ed (Table 1).

WENK ET AL.: MODELLING PLASTIC DEFORMATION OF PERIDOTrrE 8345

texture development, harder slip systems become activated. This case holds for both olivine and enstatite.

An important aspect of the self-consistent scheme is that individual grains are allowed to deform differently, and it is useful to investigate the grain shape evolution. This hetero- geneous deformation of grains should not be confused with the homogeneous deformation of the microstructure dicussed earlier. The latter is only used to define the topology of neighbors. In the pole figures (Figures 3-7) we have plotted orientations with different symbol sizes. The symbol size was chosen proportional to • which is a bulk measure for the grain plasticity and determined as 4-•/2 times the inte- gral over the Von Mises effective strain rate. The average grain shape parameter • is a crude measure of the total de- formation of a grain. The wide range in symbol sizes illus- trates that some grains barely deformed, whereas others changed their shape a lot. More information is provided by an analysis of individual grain shapes (length of ellipsoid axes, a>b>c) which are represented in Figure 11 for compres- sion and pure shear as log a/b versus log b/c plots ("Flinn diagram" [Flinn, 1962]). The distance from the origin in- dicates the degree of deformation. The self-consistent vis- coplastic model predicts a wide spread of grain shapes. It is most pronounced for compression which, for pure olivine, has a bimodal distribution. In the olivine-enstatite mixture,

enstatite deforms less than olivine. For pure enstatite the distribution is asymmetric with many grains not deforming at all. In addition, Flinn diagrams give information about the relative shape. In all eight diagrams there is a concen- tration along the diagonal, representing plane strain. This is expected for pure shear, but it occurs surprisingly for compression as well where one might expect originally spherical grains to attain a saucer shape (b = c) and plot along the abscissa. However, on closer view it is clear that olivine and particularly enstatite cannot, with the available slip systems, deform to such a shape. Enstatite crystals, with a single slip system, can only deform in plane strain, as is illustrated in the Flinn diagram for the olivine-enstatite mix- ture. For pure enstatite, deviations from plane strain are only introduced at higher deformation when the model activates the artificial strong slip system. The macroscopic shape of the polycystal (indicated by the solid circle in the Flinn dia- grams) is obtained through superposition of ellipsoidal grains with different orientation relative to the compression axis. This expression of grain heterogeneity is called "curling" and was originaly described for fcc metals deformed in compression [Hosford, 1964]. Later, it was also observed in calcite [Wenk et al., 1986]. Enstatite is an excellent ex- ample of predicted curling due to slip system geometry.

DISCUSSION

We have applied the viscoplastic self-consistent theory to model the evolution of preferred orientation in a system which is composed of two phases, plastically softer olivine and harder enstatite. Both minerals have fewer than five in-

dependent slip systems, and deformation has to be heteroge- neous. In compression as well as in plane strain, strong preferred orientation develops producing plastic and elastic anisotropy which is significant for modelling convection in the upper mantle [e.g., Christensen, 1987] or in interpreting seismic observations such as low-velocity layers [e.g., Fuchs, 1975, 1983].

The texture results look reasonable. In olivine aggregates deformed experimentally in a dislocation creep regime [010] axes align with the compression direction [Nicolas et al., 1973] as they do in our simulated textures (Figure 4). The predicted texture (Figure 4) agrees with their medium strain texture (Figure 1D of Nicolas et al.). At larger strain (Figure 1E of Nicolas et al.), Nicolas et al. observe a stronger (001) girdle normal to the compression axis and particularly a de- pletion of (001) poles parallel to the compression axis. We consider such details beyond the present resolution. On the one hand, this experiment was quite heterogeneous; on the other hand, there are still minor deficiencies with the self- consistent model. Also there are uncertainties with critical

shear stresses, hardening, and effects of recovery. For ex- ample, making the slip system (001)[100] twice as strong as (010)[100] reduces its relative activity and produces a texture which is in closer agreement with that of Nicolas et al. [1973] (Figure 12).

In experimentally deformed enstatite, [100] aligns with the compression direction [Carter et al., 1972]; however, in these experiments the material was partially recrystallized. Unfortunately, data on experimental deformation of peridotite are scarce, but there is a wealth of information on naturally deformed rocks. Mercier [1985] described olivine and enstatite fabrics from peridotites of the Bay of Islands ophiolite complex which show a good resemblance to the simple shear simulations (Figures 13a and 13b) even though there are admittedly always uncertainties in identifying shear plane and foliation in natural settings. Peridotire fabrics from Alpine mylonites (Alpe Arami [Mackel, 1969]) differ in that olivine [001] axes are normal to the schistosity; en- statite [100] is normal to the schistosity and [001] is paral- lel to the linearion as in our pure shear simulations. On the other hand, in a lherzolite from the Ivrea zone [Skrotzki et al., 1990] the olivine agrees with our pure shear textures, whereas [001] axes of enstatite lie preferentially in the folia- tion plane and perpendicular to the lineation. However, in this material internal deformation of enstatite is weak, and the texture was attributed to anisotropic grain shape. Olivine fabrics from basal sections of several ophiolite complexes are distinctly asymmetric relative to a macroscopic planar structure [Boudier et al., 1982]. If this planar feature is interpreted to represent the shear plane, the asymmetry indicates simple shear and can be used to determine the sense of shear. It is important to note that different simulations [e.g., Takeshita et al., 1990; Ribe and Yu, this issue], and our new calculations, but not those of Etchecopar and Vasseur [1987], predict for simple shear tex- ture maxima for (010) in the case of olivine and (100) in the case of enstatite which are asymmetric to the macroscopic shear plane and displaced against the sense of shear. Their position does not change appreciably with increasing strain.

Our simulations, which include neighbor interaction, pre- dict little influence of 30% enstatite on the olivine texture.

The textures of pure olivine and pure enstatite polycrystals are similar to those of olivine and enstatite in peridotitc. We have commented on the fact that the olivine texture in

peridotire is stronger than in pure olivine rock due to distri- bution of the same total strain over the smaller olivine vol-

me. We have recently generalized the self-consistent model to allow for intragranular heterogeneity by dividing each grain into cells. With such a scheme, addition of stiff platelets to a quartz aggregate reduces preferred orientation of

8346 WF2qK ET AL.' MODELLINO PI•STIC DEFORMATION OF PERIDOTITE

COMPRESSION

0.8

o.6 100% Olivine

0.4

o.o i • • • i • • • i • • • i • • • I 0.0 0.2 0.4 0.6 0.8

PURE SHEAR

0.8

0.6

:****

0.0 0.2 0.4 0.6 0.8

70% Olivine

_

0.0 0.2 0.4 0.6 0.8

** , ,•**•*.,.• *** ** .,

ß

0.0 0.2 0.4 0.6 0.8

0.0

0

30% Enstatite

0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8

0.8 0.8 -- _

_

_

o.6 100% Enstatite o.6 -

0.4 0.4 --

0.2 0.2 --

0.0 T• • • I • •"" I • • • I • • • I 0.0 0.0 0.2 0.4 0.6 0.8

log b/c

**,•*•, , ß

, •* * ß

0.0 0.2 0.4 0.6 0.8

Fig. 11. Flinn diagrams displaying the grain shapes of individual grains after 50% deformation in compression and pure shear for pure olivine, a 70% olivine-30% enstatite mixture, and pure enstatite as predicted with the self-consistent viscoplastic n site model. A solid circle in the pure enstatite diagrams indicates the macroscopic deformation. For convention of strain ellipsoid axes a )b)c.

WF2qK ET AL.: MODELLING PLASTIC DEFORMATION OF PERiDOTITE 8347

(100) (010) (001)

XX x•< x

X= •" X X X•

x X x •

Fig. 12. (100), (010), and (001) pole figures after 50% equivalent strain in axial compression for a pure olivine aggregate. Critical resolved shear stress for (010)[100] slip is 18 MPa, for (001)[100] slip 36 MPa, and (010)[001] slip 45 MPa. Compare with Figure 4. Equal-area projection, 200 grains total.

a

b

o o] o px o] o p x [OOOopx

Fig. 13. Pole figures from a naturally deformed peridotitc. Bay of Islands ophiolite complex, Newfoundland [Mercier, 1985]: (a) olivine; (b) enstatite. The schistosity plane and the lineation L are indicated. Fig. 14. (010) pole figures after 40% equivalent strain in pure shear for pure olivine and a 70% olivine-30% enstatite mixture. Results from a uniform grain model (left) are compared with those from a cell structure model (right) which relies on a self-consistent scheme in which 200 grains have been divided into 4096 cells and which allows for heterogeneous intragranular deformation [Canova et al., 1991]. Equal area projection.

quartz by spreading orientations of individual cells [Canova et al., 1991]. In the case of 70% olivine-30% enstatite this model with 4096 cells instead of 200 uniform grains does not change the pattern appreciably (Figure 14). Pure olivine displays a stronger preferred orientation than the mixture; however, there is a general smoothing which is in accord with the observed weaker strength of textures in experiments than in polycrystal plasticity predictions. It also conforms with microstructures which document considerable grain bending [e.g., Nicolas et al., 1973, Plate lB]. In contrast to quartz, the difference in plastic strength between olivine and enstatite is not large. In fact, the single allowed system has a lower critical resolved shear stress than olivine (Tablel).

The grain shape analysis (Figure 11) illustrates that enstatite deforms, albeit more slowly. Clearly, the interaction aspect needs to be investigated further with different systems to evaluate the importance of volume fractions, grain shapes, and relative strength of the components.

We find that in all deformation modes the yield strength increases moderately during texture development. Such a be- havior is typical of materials with few slip systems when crystals rotate into orientations which are less favorably ori- ented for further slip. This is true for compression (orientations with (010) normal to the compression axis have no resolved shear stress), pure shear, and simple shear (where no "easy slip" orientation is predicted). The textural hardening is particularly severe for simple shear because of restrictions imposed by the shear sense. From our model it would appear that for shear strains •, larger than 1.0, other mechanisms than slip need to become activated, particularly grain boundary sliding. The texture hardening described here has not been predicted in relaxed Taylor calculations [Takeshita et al., 1990] in which normal strain components were arbitrarily omitted. It is also different from experi- ments at high temperature where a more steady state flow be- havior can be attributed to diffusional processes during creep

8348 WENK ET AL.: MODELLING PLASTIC DEFORMATION OF PERIDOTITE

uniform grains cell structure 100% Olivine 70% Olivine 100% Olivine 70% Olivine

30% Enstatite 30% Enstatite

Fig. 14. (010) pole figures after 40% equivalent strain in pure shear for pure olivine and a 70% olivine-30% enstatite mixture. Results from a uniform grain model (left) are compared with those from a cell structure model (right) which relies on a serf-consistent scheme

in which 200 grains have been divided into 4096 cells and which allows for heterogeneous intragranular deformation [Canova et al., 1991 ]. Equal area projection.

[e.g., Chopra and Paterson 1981; Borch and Green 1989]. In our model, enstatite strengthens the aggregate which is dif- ferent from Hitchings et al.'s [1989] experimental observa- tions. Perhaps grain boundary processes were active in theix experiments and were rate determining. The yield stress analysis of Takeshita et al. [1990] which assumes that nor- mal strains are accommodated by climb and predicts that for simple shear the material neither hardens nor softens may be more applicable.

The new self-consistent simulations for olivine predict sim- ilar textures as the models of Parks and Ahzi [1990], Ribe and Yu [this issue], and Takeshita et al. [1990], all of whom have used different theories or procedures, and they differ from predictions by Etchecopar and Vasseur [1987]. The Ribe and Yu model has some similarities to a relaxed Taylor model. It differs from Taylor (and the application to olivine by Takeshita et al. [1990]) in that it does not rely on the maximum work principle to select slip systems but instead minimizes incompatibility. It is not clear how this model can be generalized to systems with more than five indepen- dent slip systems. It is actually surprising that the textures simulated with the Ribe and Yu approach are so similar. We attribute this to the special case of olivine: with only a few slip systems, it is insensitive to slip system selection cri- teria. In detail there are some differences. Ribe and Yu

predict for the same critical resolved shear stresses chosen by Takeshita et al. [1990] a compression texture with a pronounced (001) maximum in the shortening direction which is absent in the predictions of Takeshita et al. [ 1990] and Parks and Ahzi [1990]. Of course, if different models predict the same textures, it does not follow that they are equally applicable, and alternatively, the best agreement between experiments and a theory does not prove that the theory is correct. Polycrystal plasticity in the slip regime is governed by a balance of stress equilibrium and strain compatibility, and it is unclear how this can be addressed with a purely kinematic model. We maintain that the actual physical processes during deformation of a polycrystal by dislocation glide are best accounted for in the version of the viscoplastic self-consistent model with intragranular heterogeneity [Canova et al., 1991] even though the model is not perfect. The anisotropic viscoplastic self-consistent model as described here is the only one which takes the stress interaction between neighboring grains into account.

We have illustrated the development of preferred orientation in peridotitc during ductile flow for different histories. Dur- ing convection in the mantle, deformation in simple shear, pure shear, and compression applies in different sectors, but on the whole the history in a convection cell may be quite complicated. Textures develop, become erased, or are super- posed. With polycrystal plasticity theory there is no diffi- culty modelling a complex strain history. It should be noted that the development of preferred orientation and re- sulting anisotropy is a function of the whole strain history and not of overall finite strain [e.g., Takeshita et al., 1989].

Acknowledgments. Constructive reviews by A. Etchecopar, N. Ribe, and J. Tullis were helpful in improving the manuscript and we are thankful to numerous discussions with G. Johnson. We grate- fully acknowledge financial support through NSF grant EAR-87 09378 and from IGPP, Los Alamos. H.R.W. is appreciative for the hospitality shown at LPMM Metz on the occasion of two visits in 1989 and 1990.

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K. Bennett and H.-R. Wenk, Department of Geology and Geo- physics, University of California, Berkeley, CA 94720.

G. Canova and A. Molinari, Laboratoire de Physique et Mtcanique des Mattriaux, Facult6 des Sciences, 57045 Metz-Cedex, France.

(Received June 1, 1990; revised January 2, 1991;

accepted January 10, 1991.)