viscoplastic modeling of texture development in quartzite

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 94, NO. B12, PAGES 17,895-17,906, DECEMBER 10, 1989

Viscoplastic Modeling of Texture Development in Quartzite

H.-R.WENK

Department of Geology and Geophysics, University of California, Berkeley

G. CANOVA AND A. MOLINARI

Laboratoire de Physique et M•canique des Materiaux, Facult• des Sciences, Ile du Saulcy, Metz, France

U. F. KOCKS

Center for Materials Science, Los Alamos National Laboratory, Los Alamos, New Mexico

Polycrystal plasticity theory has been successfully used to simulate development of preferred orientation in rocks. In particular, G. Lister and coworkers have done a comprehensive study, applying the Taylor theory to quartzite. In these calculations it was assumed that deformation is homogeneous; that is, all grains deform at the same rate, that critical resolved shear stresses (CRSS) of slip systems remain constant, and that deformation is rigid-plastic; that is, dislocations move only when the CRSS has been reached and then with indeterminate velocity. We have been investigating the influence of work hardening and of strain rate dependence on texture development. In particular, modification of the viscoplastic Taylor theory suggests that textures are greatly dependent on the rate sensitivity of flow stress when this stress exponent is small, such as in quartz where it is near 3. The influence of work hardening is less critical in the case of quartz. Results from Taylor simulations are also compared with those from a self-consistent theory. The latter sacrifices local strain continuity for better stress equilibrium. In the self-consistent scheme, grains which are favorably oriented for slip are allowed to deform at a faster rate. Texture patterns obtained with the two theories are moderately different. In self-consistent deformation a c axis maximum in the intermediate strain direction (¾) is generated which is absent in Taylor deformation but is a common feature of many natural quartz fabrics. Grains associated with this maximum are most strongly deformed. Deformation modeling with more realistic boundary conditions adds complexities but appears to be necessary in the case of anisotropic and rate sensitive rocks.

INTRODUCTION

Of all minerals, quartz has received the most attention with respect to deformation studies. This is partly due to its importance in controlling the rheology of large por- tions of the continental crust, but it is also due to the interesting variety of texture types observed at different conditions of metamorphism. Literally thousands of fabric diagrams have been measured, beginning with the first fabric diagram [Schmidt, 1925] to the recent orientation distribution function (ODF) studies of Schmid and Casey [1986]. In interpreting textures, structural geologists have relied primarily on intuition. A favored notion based on Sander's [1950] concept of "movement picture" has been that the crystallographic slip plane aligns itself normal to the principal compressive stress (Figure l a). This is a very hard orientation because no shear stress is active on the slip plane. Another contention [e.g., Schmidt, 1927] has been that the slip plane aligns itself within the macroscopic shear plane (Figure lb). This is a soft orientation in which the shear stress is a maximum. Such intuitive models do not take account of the micros-

copic physical processes which occur during mechanical deformation of a polycrystal and have therefore often been misleading. Among the different attempts to interpret preferred orientation of quartz, the application of the Taylor theory has provided the most significant progress,

Copyright 1989 by the American Geophysical Union.

Paper number 89JB01506. 0148-00227/89/89JB-01506-$05.00

and the landmark studies of Lister and his coworkers

[e.g., Lister et al., 1978; Lister and Paterson, 1979; Lister and Hobbs, 1980; Lister and Williams, 1979] have illustrated a method for understanding the development of textures in terms of mechanisms acting in crystals and of the macroscopic deformation history.

Theories of plastic deformation relate stress, strain, and/or strain rate to each other and predict a material response assuming that certain mechanisms are active. The most significant deformation mechanism in rocks at medium to high temperature is slip. It is often accom- panied by syntectonic recrystallization. Whereas texture development during recrystallization is poorly under- stood, even in metals [Doherty et al., 1988], there are comprehensive theories for deformation by slip. The model of Sachs [1928] assumes that in each grain of a polycrystal that single system becomes and remains active on which the critical resolved shear stress (CRSS) is first reached [Schmid, 1928]. This leads to incompatibilities at grain boundaries causing overlaps and holes. Since this is not observed, Taylor [1938] suggested assuming homogeneous deformation (Figure 2a). In order to produce an arbitrary shape change, activation of up to five independent slip systems is necessary. The big advantage of this theory is that during deformation the polycrystal remains coherent. A disadvantage is that grains which are favorably oriented for slip ("soft") and grains which are unfavorably oriented for slip ("hard") have to deform at the same rate. Slip begins when the stress state on all five systems has reached a critical value and the flow stress is therefore higher than in the Sachs theory which is

17,895

17,896 WENK ET AL.: TEXTURE DEVELOPMENT IN QUARTZITE

COMPRESSION SIMPLE SHEAR

(a) (b)

Fig. 1. Intuitive interpretation of textures. (a) In compression the slip plane is oriented normal to the compression direction. (b) In simple shear the slip plane rotates into the macroscopic shear plane.

best illustrated in the single-crystal yield surface (SCYS) (Figure 3). The yield stress is controlled by the hardest of the five systems.

In recent years, metallurgists have generalized the Tay- lor theory in several respects. Strain continuity can be "relaxed" within a Taylor framework [Honnejf and Meck- ing, 1978; Kocks and Canova, 1981; Van Houtte, 1982] (for quartz see also Ord [1988]). So-called "self- consistent" theories sacrifice some of the local strain continuity for a better stress equilibrium [Molinari et al., 1987]. In minerals there is also often a much stronger strain rate dependence of the flow stress than there is in metals. Stress exponents range between 3 and 9 in minerals as compared to between 35 and 400 for metals deformed at low temperatures. This can be approached with a viscoplastic model [e.g., Canova and Kocks, 1984] which is applicable both to the Taylor and to the self- consistent theory. Finally, the critical shear stress is not a material constant but changes as the microstructure evolves. Work hardening due to dislocation interactions is most significant at low temperature.

We would like to explore effects of these generalizations in the case of quartz and see how much conclusions such as those of Lister and Hobbs [1980] may need to be modified if more realistic assumptions are made. The discussion of quartz textures may also be of interest to metallurgists. Effects of stress disequilibrium, of strain rate dependence, and of hardening are much more severe in quartz than in cubic metals, but slip mechanisms in quartz are quite similar to those in hcp metals, and conclusions about trends (not about specific textures) may also apply to those materials.

In the first section of this paper we specify parameters, boundary conditions, and properties of the theories. A

S2

Fig. 3. Two-dimensional sketch of the single-crystal yield surface in stress space illustrating differences between Sachs (S), Taylor (T), and self-consistent (SC) assumptions. Two slip systems, S• and S2 are shown. At the vertex, Taylor deformation covers the strain range &. Sachs deformation is initiated when the CRSS, •c is reached. Self-consistent deformation is intermediate.

second section presents results for plane strain deformation, particularly the distinction between pure shear (coaxial, plane strain) and simple shear (noncoaxial). The rate dependence will emerge as a crucial factor and will be discussed and compared with results of earlier investi- gations.

THEORETICAL CONSIDERATIONS

Slip Systems

We assume that strain is accommodated predominantly by slip. As input for our models, we need to know all potentially active slip systems defined by slip plane, slip direction and critical resolved shear stress (CRSS). For simplicity and for lack of adequate experimental data we assume a hexagonal slip system geometry of symmetry 6/mmm for quartz rather than the true symmetry 32. Reasons for this simplification (e.g., enantiomorphism, distinction of positive and negative rhombs and of Dau- phin• twinning) have been discussed by Takeshita and Wenk [ 1988].

We have made calculations for three model quartzites, labeled a, /•, and 3' (Table 1). Model a corresponds closely to Lister and Hobbs [1980] A and Takeshita and Wenk [1988] LT, model/• corresponds to B and HT, and model 3' is exactly the same as the Lister and Hobbs [1980] model C. Stress units are such that the CRSS for basal slip is 1. In a, basal slip dominates, which is representative of low temperature. Prismatic slip is excluded from the Bishop-Hill yield surface. In /•,

TAYLOR SELF-CONSISTENT

a b

Fig. 2. (a) Schematic representation of homogeneous deformation as required in Taylor theory and (b) heterogeneous deformation if the influence of the slip system orientations is taken into account.

TABLE 1. Slip Systems, Number of Systems in Each Family, and Critical Shear Stress Ratios Relative to Basal Slip

Slip System n a /• -•

( 0001 ) <2110> basal 3 1 1 1 { 1010 ) < 1210> prismatic 3 6 0.4 1 ( 1011 ) < 1210> pyramidal 6 3 3 c• ( 1011 ) <2113> pyramidal 12 6 6 c• (2111 )<1123> pyramidal 6 c• c• 2.5 ( 2111 ) < 1213> pyramidal 6 c• c• 3

Hardening per 5 5 5 unit strain

WENK ET AL.: TEXTURE DEVELOPMENT IN QUARTZITE 17,897

4-

2 I

H = 5 •J-0.8

,,' • •' •-0.4

ß :..-c. ................................. -0.2

1

I o 0.5 1 .o

SHEAR STRAIN •:

Fig. 4. Isotropic hardening model illustrating CRSS (r) (solid and dashed lines) and CRSS ratio (rl/v2) (dotted line) for two slip systems as a function of %

prismatic slip is preferred, representative of high temperature. Rhomb <a> slip is inactive. In % base and prism are equally easy. In addition, steep pyramids are active. Keep in mind that we are working with hypothetical models to test different influences. There is still consider-

able uncertainty about actual slip systems in quartz and their CRSS at particular conditions, and experimental data are incomplete.

Hardening We used two models to study the influence of micros-

tructure. In the first model we kept the CRSS for all slip systems at their respective initial values during the whole deformation path, which is what all Taylor calculations on geological materials have assumed so far. In a second model, microscopic hardening is represented through a

linear law such that all initial values of CRSS ratios •o/rc • are adjusted according to the shear .•s' on each slip system

The hardening coetficient Hs•, is also normalized to the CRSS of system 1 (basal slip) and is here assumed to be the same for all slip systems (isotropic hardening) and with a value of 5.0 times the CRSS value of system 1. It corresponds to hardening observed in experiments at intermediate temperature. Figure 4 illustrates hardening for two slip systems. Whereas absolute differences between CRSS remain the same, the ratio changes and the two systems become more similar with increasing strain.

Strain Rate Sensitivity There is considerable literature on the creep behavior

of quartz [e.g., Linker and Kirby, 1981; Heard and Carter, 1968]. In these references a low stress exponent n, rang- ing between 2.5 and 3.8, was determined. So far, all texture calculations on minerals have assumed that the

exponent was infinity, in accordance with the strict Taylor-Bishop-Hill condition. In physical terms this implies that dislocation movements do not occur until the CRSS is reached, at which point immediate movement occurs. This is approached in metals, but in minerals

with high Peierls stress there is a viscous drag for dislocations which is most significant at fast strain rates. Also, at intermediate temperatures, lattice diffusion and climb become active and lead to a velocity dependence. A phenomenological expression which is often used to describe the strain rate sensitivity during dislocation- controlled deformation is a power law which is applicable at least in the vicinity of some reference stress r0 coupled to some reference strain rate •0.

(2)

where -• is the shear strain rate and r is the effective critical shear stress for a stress exponent n. This function is

shown in Figure 5. If n = 1, the rate is proportional to stress as in a Newtonian liquid. If n = c•, no deforma-

tion occurs until r•/ro reaches a critical value. This value cannot be exceeded. In between these extremes, the viscoplastic range, the function is more complicated; that is, at slow strain rates, deformation occurs before the critical shear stress is reached, and at fast strain rates the reference stress can be exceeded. We will only consider stresses below the reference stress.

This can also be viewed in terms of the SCYS. For

rigid-plastic behavior the yield surthce is a polyhedron in five dimensions, each hyperface representing a slip system [see, e.g., Hobbs, 1985]. Figure 6 shows some two- dimensional sections through the SCYS of model a quartz. At a high stress exponent (n = 99), the vertices are so sharp that the appearance is that of a polyhedron. For n = 9, vertices are considerably rounded. Metallur- gists have introduced the viscoplastic condition to elim- inate the slip system ambiguity at vertices in cubic metals [Canova and Kocks, 1984]. In low symmetry minerals this ambiguity (i.e., more than five slip systems associated with a vertex) is generally of no concern, but the same approach can be used to model deformation at realistic stress exponents. Lowering n to 3, the yield surface attains a smooth curvature, not too different from a cir-

•0

2.0

o

o 1.o

Fig. 5. Viscoplastic assumption giving the relationship between normalized CRSS (r/to) and strain rate ('•/'•0) for different stress exponents n.


(•13

(•12

(•23

(•12

(•23

(•13

Fig. 6. Two-dimensional sections through single-crystal yield surface of model • quartz for stress exponents n = 3, 9, 99. Stress coordinates a13-a12, a23-a12, a23-a13 are indicated (compare Hobbs[1985, Figure 4]).

cle. We will illustrate the influence of the stress exponent (strain rate sensitivity) on texture development. With a power law model, essentially all slip systems are active but some only to a very small degree.

Taylor Versus Viscoplastic SelJ:Consistent Theory

Two polycrystal plasticity theories will be used to simulate texture development in model quartzites. The Taylor [1938] theory assumes homogeneous strain and requires activation of up to five slip systems in each crystal to provide enough degrees of freedom to accommodate an arbitrary shape change [see, e.g., Van Houtte and Wagner, 1985]. All crystals deform at the same rate and therefore have the same shape at each step of the deformation path (Figure l a). This leads to stress disequilibrium at grain boundaries which is assumed to be rectified by elastic strains.

For small elastic deformation, self-consistent theories were introduced to take into account the interaction of

each grain with its surroundings, satisfying both stress equilibrium and strain compatibility [e.g., Budiansky and Wu, 1962; Hill, 1965; KrOner, 1961]. Recently, Molinari et al. [1987] have introduced a viscoplastic "self- consistent" theory for large plastic deformation. In the following discussion we refer to this model which neglects elastic deformation but uses a similar formalism as

KrOner [1961]. Each crystal is assumed to be embedded

in a homogeneous anisotropic matrix consisting of the average over all crystals. The grain response is affected by both the compatibility and the equilibrium with its environment. A local interaction equation is solved to give the grain stresses which induce activity on slip systems providing lattice rotations. The macroscopic stress is determined after each deformation increment and

adjusted to fulfill compatibility. In this scheme, compatibility is enforced only on the macroscopic scale, but there is incompatibility between individual grains, with soft grains deforming faster than hard grains. In the self- consistent approach, deviations in stress are only about half as large as in the Taylor theory. Whereas in the Tay- lor theory each grain can be analyzed individually, in the self-consistent model, each grain deforms according to the orientation distribution of all grains, and the number of grains influences the behavior.

The input parameters for calculations with both theories are identical. A set of starting orientations is required. We chose 200 random orientations which we prefer over regular orientations because they give greater uniformity. Slip systems need to be specified by slip plane, slip direction, CRSS, and hardening coet•cient. The stress exponent is required. The deformation path is described by an imposed velocity gradient L (also called displacement gradient or distortion tensor per unit time). We show calculations for plane strain in pure shear

and simple shear

0.05 0 0 0 -0.05 0 0 0 0

(3)

o -o.1 o ] L = 0 0 0 (4) 0 0 0

Such deformation increments provide the same Tresca "finite" strain for pure and simple shear. The viscoplastic yield condition is obtained through an interactive procedure in both the Taylor and self-consistent models [Canova et al., 1988]. Results of the texture simulations after applying distortion tensors (3) and (4) 20 times are shown as c axis pole figures displaying individual orientations. They are plotted in equal-area projection. The position of axes 1 (right), 2 (top), and 3 (center) are indicated on Figures 7-9.

TEXTURE RESULTS

Pure Shear

Figure 7 displays c axis diagrams of the 200 starting orientations for the three models a, fl, and % We use c axis fabric diagrams rather than three-dimensional orientation distributions because geologists are more familiar with them and such diagrams are sut•cient to illustrate most of the points which we wish to make. We are aware that such a representation is incomplete. The symbol size is proportional to the Taylor factor. Crystals with a large Taylor factor are in orientations which are unfavorable for slip ("hard"), those with a small Taylor factor are favorably oriented ("soft"). For pure shear in all three models orientations with c axes close to the principal compression (axis 2) and extension (axis 1) direction are


Pure shear

-IX • X x • &• • x x xx•

•--•• 2.5 - 9.1

Simple shear

Fig. 7. A c axis diagram of 200 starting orientations in model a, •, and % Symbol sizes are proportional to Taylor factor (effective stress) for pure shear deformation (top) and simple shear deformation (bottom). The range in values for the three models is indicated. Axis 1 in pure shear is extension, axis 2 is compression.

hard. The range is large; for example, for model • from 0.7 to 9.0 stress units (r for basal slip - 1.0), illustrating that plastic properties of the model quartzites are very anisotropic. For models • and % orientations close to axis 3 are soft. For simple shear the hardest orientations have c axes 45 ø to the shear plane normal (axis 2) and the shear direction (axis 1), respectively. This 45 ø rotation for Taylor factors for pure and simple shear, respectively, is expected, inasmuch as Taylor factors do not depend on rotations but only on strain.

Figure 8 shows c axis diagrams of texture predictions after 20 distortion increments (a Tresca strain of 1.0) according to the full constraint Taylor theory on the left and using the self-consistent model to the right. Taylor calculations in the first column are for a stress exponent n - 99 which is close to a rigid assumption; the second column for n - 9, and the third for n - 3, which is closest to the actual value for quartz. Predictions without hardening and with hardening are alternating. At all conditions, strong textures develop. The c axis diagrams for n - 99 without hardening (first column) are virtually identical to those of Lister and Hobbs [1980]. The effect of microstructure hardening is relatively minor, but the influence df strain rate sensitivity is considerable. With decreasing stress exponents, textures become more diffuse; for example, submaxima in models a and • merge

into a single concentration parallel to the compression direction. In models a and •, increasing n has a similar effect as adding hardening. Only in model -• where pyramidal slip systems are active does the pattern remain complex with cross girdles.

The right two columns in Figure 8 show results of self- consistent calculations for n = 31 and n = 3. The

numerical procedure did not allow us to go to a higher stress exponent. In this case the influence of hardening is more significant. The patterns resemble overall those of Taylor calculations, particularly for n = 3, but there are slight differences, most notably in models • and -• in the concentration in the intermediate strain direction (axis 3 in center of diagrams), which is absent in the Taylor predictions. Since grains have freedom to deform at different rates there is a spread of grain shapes (indicated by symbol size), which is correlated with orientation. There is a striking concentration of strongly deformed grains with c axes in the intermediate direction in models • and % These grains are deformed more than twice as much as the least deformed grains.

Simple Shear Whereas pure shear pole figures (Figure 8) have overall

orthorhombic symmetry, simple shear pole figures are monoclinic (Figure 9). In Figure 9 we indicate the shear plane and the principal axis of finite strain which is


TAYLOR SELF CONSISTENT

31 3

Fig. 8. The c axis fabric diagrams after 20 deformation steps in pure shear simulated with the Taylor and the self-consistent theory for model a, •, and •, quartz; n is the stress exponent, H are simulations with fnicrostructure hardening. Equal-area projection. Symbol sizes proportional to strain.

inclined at an angle 0 (tan 20 = 2/•). The symmetry is not only expressed in reference to the distribution with respect to the macroscopic shear plane and shear direction but also to the distribution itself. In general (but there are exceptions!), texture development in simple shear is less pronounced than in pure shear for the same Tresca "finite" strain. In Taylor predictions (Figure 9,

left), we notice again the strong influence of the stress exponent, particularly drastic in model % Self-consistent calculations (Figure 9 right) demonstrate heterogeneous deformation. In model •, grains whose c axes are in the shear plane and normal to the shear direction (axis 3) undergo the greatest shape changes, while some other grains barely deform at all.


TAYLOR SELF CONSISTENT

n=99 9 3 31 3 2 •.

Fig. 9. The c axis fabric diagrams after 20 deformation steps in simple shear simulated with the Taylor and the self-consistent theory for model a,/•, and -• quartz; n is the stress exponent, H are simulations with microstructure hardening. Shear plane and principal axis of finite strain are indicated. Equal-area projection. Symbol sizes proportional to strain.

DISCUSSION

Introduction of new parameters and deformation conditions has added variety to geological texture predictions. If we use assumptions which are in better accord with the physical properties of the material, texture patterns are significantly different from simulations reported

in the literature, and some caution is advised for applying theoretical simulations to interpret the history of natural geological deformation. There have been several applica- tions of Lister and Hobbs' [1980] results [e.g., Lister and Dornsiepen, 1982; Lister and Snoke, 1984; Price, 1985; Ord, 1988] which went as far as comparing minor features


14-

12-

10-

8

6-

4-

2-

0

0

Tgg

SC31 _

_ _

-

I I I i

0.2 0.4 0.6 0.8

SHEAR STRAIN e

topology of the single-crystal yield surface, which therefore needs to be reevaluated after each strain increment.

The importance of hard systems increases with progressive deformation, and the plastic anisotropy decreases. In this sense, hardening has a similar effect as decreasing the stress exponent n, but this is not always expressed in the texture. For Taylor models in particular, strain is accommodated rather uniformly by many systems (Figure 12) because of the compatibility requirement and because changing CRSS within topologic macrodomains has no significant effect on texture [see Takeshita et al., 1987]. This is different in the self-consistent case.

At n - 3 textures for Taylor and viscoplastic self- consistent conditions are rather similar and differences

are less severe than we described for halite, with n = 9 [Wenk et al., 1989], and are similar to those for olivine with n = 5 [Takeshita et al., 1989]. Whereas at high n self-consistent deformation in model a requires on the average only 1.5 slip systems (which is close to Sachs), this number increases to 9-10 for n - 3, and this is close to Taylor. The greatest difference between the two theories is for rigid material which does not harden. Here the self-consistent model only activates the softest systems to considerable strain when deviations in stress become high enough to activate the next harder systems (Figure 12). Without hardening the soft system (basal slip in model a) is mainly active. With hardening, its impor-

Fig. 10. Average number of active slip systems for model a quartzite as a function of strain e. Pure shear: solid lines; simple shear: dashed lines. T, Taylor; SC, self-consistent.

in patterns. Much of these minor features change if we use a realistic rate sensitivity. In fact, we have been particularly impressed (and surprised) by the strong influence of rate sensitivity on texture. The reason for this is the large increase of active slip systems as illustrated for model a in Figure 10. In viscoplastic deformation modeling, mathematically all systems are active, but some contribute only a very small amount to the total shear. In each grain we count those slip systems as significant which contribute at least 5% of the total shear and calculate an average number of significant slip systems by averaging over all grains and weighting each by the effective strain rate. Whereas around five slip systems are active for rigid Taylor conditions, at n - 3 the number increases to 14 because systems which are outside the flow surface also contribute to deformation. Strain is more uniformly distributed which has the effect that the flow stress is lowered from seven to four stress units (Fig- ure 11). The influence of rate is less significant for minerals with a larger stress exponent such as calcite [e.g., Wenk et al., 1987; Takeshita et al., 1987] and halite [e.g., Wenk et al., 1989], and the classical Taylor theory should be more applicable in these systems.

The influence of microstructural hardening is smaller than we anticipated. An effect of hardening is that soft and hard systems become relatively more similar with increasing deformation, as shown by the rl/r2 plot in Fig- ure 4. The change of this ratio can lead to changes in

w

1

/ /

/ T3

2

I

0

SC31

0!2 0!4 0!6 0!8 SHEAR STRAIN

Fig. 11. Effective stress-strain curves for model a quartzite. Pure shear: solid lines; simple shear: dashed lines. T, Taylor; SC, self-consistent.


w 80

--- 60

O >- 40

O < 20

SC

(0001) <1210>

{1011} <1210>

..... {1011} <1123>

,.""SC .....""

0.2 0.4 0.6 1.0

SHEAR STRAIN

Fig. 12. Activity of slip systems in pure shear deformation of model a quartzite. (0001)<a> slip: solid lines; (1011)<a> slip: dashed lines; (1011)<a +c> slip: dotted lines. Heavy lines are for slip with hardening. SC is for self-consistent.

rance decreases, strain is distributed over more systems, and we approach Taylor. Whereas rate sensitivity and microstructural hardening increase the number of slip systems, "self-consistency" reduces it.

A quantitative comparison of our simulations with natural quartz textures is premature and would have to rely on full orientation distributions. But it is interesting to note that some of our c axis diagrams (e.g., Figure 13a, model •, self-consistent, n = 3) compare very well with the most common fabric types, the crossed girdle, such as that represented in the first published fabric diagram [Schmidt, 1925] (Figure 13b). Note particularly the concentration in the Y fabric direction (our axis 3) which could not be obtained with Taylor simulations and is the most distinct feature of natural medium-to-high temperature quartz fabrics. Even more revealing is a comparison with the study of Bouchez [1977], who correlated grain deformation and orientation. He reported for an amphibolite facies quartzite strongly deformed grains in the fabric Y direction (normal to linearion and foliation normal, Figure 13d) and much more weakly deformed grains along the crossed girdles (Figure 13c), and we find indeed c axes of the most strongly deformed grains (large sym- bols) are in the fabric Y direction (Figure 13a). The self-consistent model has the advantage of allowing heterogeneous deformation.

Also natural simple shear textures [e.g., Law et al., 1989] (Figure 14b) have very similar c axis distributions as simulations at n - 3 (Figure 14a). Self-consistent models • and 3' with a secondary c axis concentration normal to the shear direction (axis 3) show the best agreement. This also is consistent with experimentally pro- duced shear textures [Dell'Angelo, 1985].

At this stage the self-consistent model may give too much freedom, and we expect a new model which main- tains compatibility within a grain cluster to be more applicable. We also consider this study of quartz as a first step in treating polyphase assemblages composed of hard and soft crystals which can be treated with this theory.

An important result is the difference between pure shear and simple shear deformation. Geologists (with the exception of Lister and coworkers) have often interpreted textures intuitively, assuming that an active slip plane is oriented in the macroscopic shear plane (Figure lb) and an active slip direction in the macroscopic shear direction. Concentrations in the texture were associated with

"ideal orientations" and slip systems [e.g., Etchecopar, 1977; Schmid and Casey, 1986]. However, plasticity theory and experiments on rocks and metals show that the situation is much more complicated. For example, in torsion experiments of fcc metals there is no concentration of orientations with (111) in the shear plane and [110] parallel to the shear direction, although at some conditions, (111) may form a maximum normal to the shear plane and at other conditions, a (110) maximum parallel to the shear direction is observed [Canova et al., 1984]. We should keep in mind that in plastic deformation, several slip systems are active simultaneously. The Taylor requirement of five active slip systems may be high, but the self-consistent theory with 1.5-3 active systems also predicts textures for quartz in which the prom- inent slip plane is not aligned in the macroscopic shear plane.

Another contention is that textures can be interpreted in terms of finite strain coordinates, i.e., that a simple shear texture is simply a rotated pure shear texture which could be superficially assumed by comparing n = 99 textures for models a and 3' (but not •!) in Figures 8 and 9. Although both pure and simple shear strain increment tensors can be brought to coincidence by a geometrical rotation, the corresponding velocity gradients can not. This means that instantaneous stress (including the Tay- lor factor) and active slip systems can be deduced for both tests by simple symmetry arguments but the orientation changes are different. To investigate this, it is useful to look at rotation trajectories of individual orientations, displayed in Figure 15 for 20 grains in model 3'. Squares are the initial orientations, crosses are orientations at 10% strain increments. The symbol size expresses the Taylor factor (effective stress). In the case of pure shear, grains rotate quickly into stable orientations. These orientations are in this case rather hard, contradicting another intuitive conclusion that stable orientations correspond to "easy glide." The trajectories for simple shear are totally different. Rotation paths are much longer and only go in one sense. There are no stable orientations, and texture maxima correspond to regions where grains rotate more slowly. They are sometimes hard and sometimes soft orientations. The long rotation paths and the lack of stable orientations account for the slower texture buildup in simple shear. The texture represents a dynamic situation. As the stress exponent decreases, rotations become more uniform and textures become weaker which has also

been observed in metals [Toth et al., 1988].


a

I

I I ß

/ ß ß

c

ß

ß ß

ß ß

ß ß

ß ß

ß

ß ß

I I

b

l

Fig. 13. (a) Self-consistent model • pure shear simulation compared with the first published quartz fabric diagram [Schmid, 1925] with a typical crossed girdle. Mugl, Austria. (b) Foliation normal (s) and linearion (1) are indicated, and amphibolite facies quartzite [from Bouchez, 1977]. (c) Grains with an aspect ratio <4. (d) Grains with an aspect ratio >_ 8.

b

ee

Fig. 14. (a) Self-consistent model • simple shear simulation compared with (b) a fabric diagram of quartzite from NW Scotland, deformed in simple shear [Law et al., 1989].


n - 99 9 3

?r /

// /

/

Fig. 15. Rotation trajectories for 20 orientations in Taylor simulations for different stress exponents n for model 'v quartzite, no microstructural hardening. Top diagrams are for pure shear, bottom diagrams for simple shear. Strain increment is 0.1 units, 40 deformation steps. Initial orientation: square. Symbol size is effective stress. Equal-area projection.

We noted above that the most commonly observed textures in deformed metamorphic quartzites are crossed girdles of c axes in the foliation plane and normal to the lineation. In high-grade mylonites this maximum becomes very strong. For a realistic strain rate sensitivity we can only get crossed girdles for conditions where steep pyramidal slip systems are active (model 'r); also, they are only present in pure shear deformation. Models a and •, which have so far been preferred [e.g., Price, 1985] may not be applicable in many cases, and a search for new slip systems in naturally deformed quartzites is desirable. We are encouraged that the viscoplastic self-consistent model predicts textures with strong concentrations of c axes in the intermediate strain direction which was not possible to produce with the Taylor theory.

The c axis concentration in Y of predictions is not dominating as it is in most natural mylonite fabrics. This could be due to recrystallization favoring nucleation in the most strongly deformed grains; these new grains in Y then dominate the fabric, consuming the less deformed grains by boundary migration.

We find self-consistent pure shear textures in best agreement with many natural quartz fabrics and wonder if the role of simple shear in texture development of metamorphic tectonites has not been overemphasized in recent years, even though simple shear appears to be ener- getically favored over pure shear (Figure 11). Clearly, it

would be desirable to compare predictions with experiments at low deformation symmetry; these are unfortunately not available for quartz.

Acknowledgments. We are grateful to S. Schmid and J. Tullis for constructive reviews which improved the manuscript. H.R.W. acknowledges support from the A. V. Humboldt Founda- tion during a sabbatical leave at T. U. Hamburg-Harburg, from NSF through grant EAR-8709378, and from IGPP Los Alamos.

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H.-R. Wenk, Department of Geology and Geophysics, Univer- sity of California at Berkeley, Berkeley, CA 94720.

(Received February 9, 1989; revised July 6, 1989;

accepted July 14, 1989.)

viscoplastic modeling of texture development in quartzite

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