PAGEOPH, Vol. 138, No. 4 (1992) /92/ / Birkh/iuser Verlag, Basel. The Mechanisms of Finite Brittle Strain

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1 PAGEOPH, Vol. 138, No. 4 (1992) /92/ / Birkh/iuser Verlag, Basel The Mechanisms of Finite Brittle Strain Geoffrey C. P. KING 1 and Charles G. SAMMIS 2 Abstract--The mechanical processes that lead from first fracture in an undeformed rock mass to the fault gouge observed in a highly sheared fault zone are outlined. Tensile fracture, dilation, rotation, the collapse of beams and filling of voids are the basic mechanical elements. Repeated many times, over a wide range of scales, they accommodate finite strain and create the complex fabrics observed in highly deformed rocks. Defects that nucleate tensile cracks in the earth are both spatially clustered and occur on a wide range of scales. This inhomogeneity is responsible for features that distinguish deformation of rocks from deformation of laboratory samples. As deformation proceeds, failure at one scale leads to failure at another scale in a process of evolving damage. Abrupt catastrophic failure never extends indefinitely throughout the earth as it does in rock samples. The mechanics of the interactions between scales are investigated. Approximate expressions are modified from engineering damage mechanics for this purpose and their validity is demonstrated by detailed numerical modeling of critical examples. The damage that results as deformation proceeds extends over a range of scales and is consistent with the observed fractal nature of fault systems, joints and fault gouge. The theory for the mechanical evolution of fractal fault gouge which is based on the mechanical interaction of grains of different sizes is discussed. It is shown that the damage mechanics description and the granular deformation mechanism are alternative descriptions of the same process. They differ mainly in their usefulness in describing different stages of damage evolution. Field examples of features described in the geological literature as faults, joints, fault gouges, megabreccias and melanges are shown to be plausibly explained by the mechanical processes described. Key words: Fractal, rock deformation, damage, faults, joints, breccia, melange. Introduction The accommodation of large strains by brittle deformation is unique to the deformation of the earth's crust. Unlike engineering structures, total catastrophic failure does not occur. Engineering fracture mechanics successfully predicts total failure but does not describe the progressive damage associated with the strain evolution of the crust of the earth, nor does it predict the geological structures and rock fabrics associated with brittle deformation. 1 Institut de Physique du Globe de Strasbourg, Universit6 Louis Pasteur, 5, rue Ren6 Descartes, Strasbourg, Cedex, France. 2 Department of Geological Sciences, University of Sourhern California, Los Angeles, CA , U.S.A.

2 612 Geoffrey C. P. King and Charles G. Sammis PAGEOPH, There are two reasons for the difference between the behavior of the earth and an engineering structure or rock sample. The first is that confining pressure prevents the earth from developing large-scale tensile structures. The earth cannot fly apart and always fails in shear; no single flaw is fatal to the integrity of the planet. Second, in engineering materials the distribution of flaws can be assumed to be uniform (e.g. SAMMIS and ASHBY, 1986; ASHBY and SAMMIS, 1990). Uniformity implies that the shear instability in one region is necessarily associated with incipient failure in the rest of the sample. Macroscopic fracture always extends to the sample boundaries. Although cracks in the earth grow from stress concentrations, the distribution of defects is far from homogeneous and macroscopic failure does not extend to the boundaries. An instability which would produce failure in a finite specimen with a uniform distribution of defects simply produces brittle behavior at a different scale-length within the crust. Examing how behavior at one scale influences behavior at another is the basic topic of this paper. Although failure in shear is the dominant mode of deformation of the crust of the earth, the basic mode of failure involves a change in volume. Brittle deformation shares this basic feature with ductile deformation. Under ductile conditions voids at an atomic scale move in such a way as to accommodate the deformation. If local defects do not exist prior to deformation (either as a result of lattice imperfections or thermal energy), then they are brough into existence by the motion itself. Although the effect is slight, a ductile material must change volume locally to change shape. The volume change is usually positive, but decreases of volume associated with phase changes have been shown to be associated with shear deformation (KIRBY et al., 1991). Brittle deformation exploits local volume increases in the form of open cracks to permit deformation. In common with creep both pre-existing and deformation induced voids are exploited. However, unlike creep, the voids involved and the way in which they interact occur on a wide range of scales. The deformation processes considered here, or the rock fabrics that result, have all been observed in the laboratory or the field. The new focus is on the implications for scale interactions. If the observer is prepared to repeatedly readjust the scale at which deformation is being viewed, the same simple processes can be seen to repeat themselves as deformation proceeds and the total number of fractures, and damage over a range of scales, increases. Increasing damage is a characteristic of progressive finite brittle deformation. With a few trivial exceptions, no matter how much a brittle body has been deformed, new fracture always accompanies a further change in shape. Thus the fracture system in a much deformed body is very complicated. We show the stages that an initially undeformed system goes through to approach such a multiply fractured condition. The evolution is associated with the repeated growth and interaction of tensile fractures followed by the rotation, bending and collapse of beams. The process always creates fractures at a series of scales smaller than that

3 Vol. 138, 1992 Mechanisms of Finite Brittle Strain 613 of the initial tensile cracking and sometimes at larger scales. The result is a fracture system that tends towards a fractal geometry. It has been shown that the observed fracture and particle size distribution of highly deformed rocks is fractal. We discuss the evolution of damage in highly deformed rocks which already have a fractal geometry. This can be described in a statistical way using the same concepts of tensile fracture, rotation and collapse of beams used to describe the initiation processes, furthermore the fractal geometry is maintained by the process. An Overview of the Evolution of Damage from Finite Strain Strain and Stress Conditions under Finite Deformation The earth subjects rocks to boundary conditions which are much more varied than can be readily reproduced in the laboratory (KINO, 1983). This is illustrated in Figure 1. At a large scale, upper and lower boundaries are defined in terms of displacement. At the lower boundary normal and tangential displacements are fixed and at the upper boundary a constant normal displacement rate is imposed and tangential displacements are zero. Lines indicate the flow pattern if the material was a fluid. As finite strain deformation proceeds, the stress state within a piece of material is not determined by the remote boundary conditions alone, but also depends on its strain history. It is evident that small subregions such as A and B experience very different local stress conditions, and that a piece of the material carried along by the flow pattern can experience different conditions as it moves. It is clear however, that the simple shear conditions of B are most common and we will consider boundary conditions like B in the rest of this paper. Although a brittle solid may not flow in the simple fashion illustrated at the local scale, finite brittle deformation must exhibit similar features on a large scale if it is to accommodate the imposed regional boundary conditions. Some of the geometrical implications of finite brittle strain being relieved on shear planes have been discussed by KING (1983). Here we examine a region subject to simple shear such as B in the figure and illustrate how shear planes or zones evolve from pre-existing defects or from earlier fractures. The Distribution of Starter Defects in the Crust of the Earth The region in Figure 1 is asssumed to have an initial distribution of defects which are clustered in some fashion and have a range of sizes. For convenience we shall assume the clustering to be fractal although strictly fractal behavior is not esssential. We simply require that clustering should appear on whatever scale we view the region, and that the numbers of small defects should increase as their size

4 614 Geoffrey C. P. King and Charles G. Sammis PAGEOPH, Finite displacement T T ~ V V Figure 1 Boundary conditions experienced by samples of rock in a larger rock mass subject to finite strain. In the upper figure, the lower boundary is regarded as fixed and the upper boundary moves. Any component of motion perpendicular to the boundary will cause complex "flow" paths. The example shown is for simple normal compression. As deformation proceeds samples in environments such as B are more common than those in environments such as A. Throughout this paper we examine finite strain under simple shear displacement boundary conditions that approximate B. Defects in the material are indicated by small dots that are clustered. These may result form earlier episodes of deformation. diminishes. This regular scaling has an upper limit; individual defects and clusters of defects greater than a certain size do not occur. The distribution of sizes and clustering is important since regular arrays of identical defects produce failure of the entire system at the same stress. No matter how distant boundaries may be, fracture will always reach them; a situation which may be regarded as pathological. Similarly, if there is no upper limit of size, a large defect or cluster of defects will always exist which is similar in dimensions to the region being considered, no matter how large that region may be. Thus the region will always be dominated by the behavior of that defect and by its interaction with the boundaries of the region. The supposition of multiscale clustering of defects with a range of sizes permits an analysis of interaction processes and plausibly describes the likely defect distribution in the brittle crust. It is also plausible that the upper size of voids is limited by confining pressure~

5 Vol. 138, 1992 Mechanisms of Finite Brittle Strain 615 Because we assume an approximately fractal distribution, the geometries of the fractures that result will also be fractal. However, none of the processes that we discuss remove fractal geometries and some create them. Thus an episode of deformation always leaves behind a fractal distribution of defects to be exploited by the next episode. Although the structures that we describe will never come into existence in a material that is defect free or contains uniform, identical defects, once the material reaches fractal conditions, the deformation processes that we describe are indefinitely sustainable. Stages in the Evolution Of Multiply Fractured Rock Before discussing phases of deformation quantitatively, we shall describe the phases of deformation that lead from a cluster of pre-existing flaws to a multifractured material like fault gouge. The first step is the nucleation of tensile cracks at the largest and most suitably oriented starter flaws within the cluster (Figure 2a). Stress concentration at the starter flaws drives the tensile cracks which propagate in a plane which contains the greatest principal stress aj. In this case, al is at 45 ~ to the orientation of the remote applied displacement vectors. Since this process is associated with opening of tensile cracks, net dilation occurs. In an extended medium this implies elastic compression in the region around the cluster. Dilation associated with a cluster thus acts to Suppress crack formation in its immediate neighborhood. In Figure 2b the solid vector shows the displacement at a distance and the shaded vector the relative displacement across the array of developing cracks. We show these as having equal horizontal components. The difference is accommodated in the elastic compression referred to earlier. As finite strain proceeds this elastic component becomes progressively less important. As the tensile cracks grow, the aspect ratio (length divided by width) of the slabs of rock between them increases. Finite displacements at the boundaries result in a bending moment on the slabs. SAMMIS and ASHBY (1986) showed that bending is also enhanced by asymmetries due to the starter defects which make the slabs irregular. They also showed that the contribution of the starter flaws diminishes as the cracks extend. The effects of beam bending increasingly dominate the crackgrowth producing further dilation as indicated in Figure 2c. In Figure 2d the aspect ratio of the slabs of rock between cracks has increased to the point where they have become unstable. Buckling occurs with the formation of a transverse "second generation" of tension cracks, and additional dilation. Compressive stresses now concentrate where the beams have split resulting in smaller starter defects becoming locally active. Smaller arrays of tensile cracks form readily (Figure 2e) since they are close to the free surface associated with the earlier tensile fissure. Where earlier processes have produced a net dilation, buckling failure of the slabs now produces a slight collapse. The smaller beams formed at the corners become unstable and also fail perpendicular to their length. The resulting

6 616 Geoffrey C. P. King and Charles G. Sammis PAGEOPH, (',, II C i X/// d Figure 2 A schematic evolution that can lead from a few defects to a multifractured material like fault gouge. Initially four defects are shown shedding tensile fractures (a). Displacement boundary conditions applied at a distance are shown by opposing arrows above and below each figure. As the tensile fractures create beams, motion close to the beams has an outward component. This is shown by angled arrows within the figures (a-d). When the beams start to fail similar arrows show inward motion (e, f). Further deformation involves repeated episodes of such expansion and contraction associated with dilation and collapse of the region being fragmented. Except for (a) where the initial stress conditions are shown, vector diagrams to the right of each figure indicate the cumulative displacement distant from the zone (horizontal arrows) and the corresponding displacements close to the zone (angled arrows). The zone cycles through episodes of expansion as voids are created and contractions as they are filled.

7 Vol. 138, 1992 Mechanisms of Finite Brittle Strain 617 particles move to fill the void space associated with the earlier tensile cracking at the larger scale. At the same time the material contracts along the length of the original beams. We can summarize the basic processes as follows: 1) the creation of void space, 2) the rotation of beams to accommodate shear, and 3) the collapse of those beams to refill the void space. The process can now repeat (Figures 2g and 2h) with the dilatational growth of new tensile cracks, rotation of the rock slabs, followed by buckling, longitudinal splitting, and collapse. It is evident for two reasons that fragment sizes and the pattern of fractures will become more irregular as deformation proceeds. First, the starter flaw distribution is not regular and second each cycle of deformation causes particles to rotate. The different rotation rates for different particles will create a jumble of orientations. Although we cannot detail the mechanics of this phase of deformation, we can describe the processes once deformation has proceeded much further. SAMMIS et al. (1987) show that similar rotation and fracture processes occur in fault gouge. Mechanical Processes Involved in Finite Brittle Strain Within the framework just described we can now examine each step in more detail. In each case we provide examples of the process from the field or from the laboratory. Initiation of Tensile Failure Figure 3 shows the process of initiation of tension cracks from the starter flaws within a cluster. Although many types of starter flaws are possible, the spectrum is bracketed by the two end members indicated in the figure. Low aspect-ratio (spherical) pores or inclusions and high aspect ratio (elliptical) cracks can both initiate tension cracks in a compressive stress field. As illustrated by the graph in Figure 3, the behavior of these defects is surprisingly similar (SAMMIS and ASHBY, 1986; ASHBY and HALLAM, 1986; NEMAT-NASSER and HORII, 1982). The graphs show the scaled deviatoric stress required to extend the tensile cracks to a scaled length L = 1/a from a flaw of size 2a. Crack interaction has been neglected in this figure. Its effect is to make the two cases even more similar at longer crack lengths. In general, the growth of these cracks is stable; an increase in deviatoric stress is required for each increment of crack growth. However, when the length of the wing cracks exceeds the size of the starter flaws, crack growth may become unstable, particularly for low values of the mean stress al + a3,~/k~c where the smallest principal stress, a3, becomes tensile. The curves in Figure 3 are universal in the sense that they have been scaled to include the effects of the size of the starter flaw.

8 618 Geoffrey C. P. King and Charles G. Sammis PAGEOPH, Stress Exaggerated displacement Tensile cracking mechanism ~I00,,. 9 9 sr j 9 SI+S3= 10 O S 1+S3= 1 cracks... ~d~= n o~. r/3! i i Scaled crack extension L=_.t a Figure 3 The initiation of tensile failure from starter defects. The upper part of the figure shows a set of tensile fractures shed from larger defects (not shown). The stress conditions created by the displacement boundary conditions applied at a distance from the region are shown and an exaggerated view of the types of defects that could be involved. End members are a circular flaw and a shear defect at an appropriate angle. The lower part of the figure shows the geometric idealization of the defects used to develop approximate expressions for their behavior. The graphs in the lower right of the figure indicate the behavior of the lwo types of defects under varying stress conditions. Larger starter flaws thus nucleate tensile cracks at a lower value of the applied stress 05 and also require a smaller applied stress to produce the same scaled crack growth L. Under conditions of no confining stress a3, the stress required for crack growth scales as,,~, exactly as for tensile loading. Subject to confining pressure the effect is partially but never completely suppressed. It is rarely possible to see this stage of evolution in rocks, the amount of strain required to produce out-of-plane tensile crack growth is only 10-4 and nearly all rocks are more highly deformed. However, examples can be found in the Navaho sandstone of Zion Canyon at the edge of the little deformed Colorado Plateau. Figure 4a shows a face of a tensile fissure about 1 meter in diameter with the starter

9 Vol. 138, 1992 Mechanisms of Finite Brittle Strain 619 flaw indicated. Striations on the fracture surface leave little doubt that the starter flaw is correctly identified. Figure 4b shows a much larger rock face in which several subcircular features can be seen where a slab of rock has detached to leave the rear face of a tensile crack. The arches that remain provide a primary tourist attraction and in Zion National Park examples ranging in size from centimeters to tens of Figure 4(a)

10 620 Geoffrey C. P. King and Charles G. Sammis PAGEOPH, Figure 4(b) Figure 4 Initial stages of deformation in the Navaho sandstone of Zion Canyon, Utah. Collapse of the canyon walls has revealed numerous disk-shaped traces of tensile fractures. In a few cases (a) the initiation defect can be identified. The features range in scale from tens of centimeters to tens of meters (b). meters can be admired. Except in a few cases it is not possible to identify the starter flaw that initiated the crack. The field examples shown are very similar to fractures that can be created under laboratory conditions. A photograph of a similar crack emanating from a deliberately emplaced defect in an epoxy resin cylinder loaded in a triaxial test rig is shown in SAMMIS and ASHBY (1986). The Buckling and Transverse Fracture of Slabs Figure 5 shows the stage at which slabbing has reached the point where buckling occurs. If the array of slabs is assumed to buckle in unison, then each may be assumed to have one fixed end and one end which is free to move laterally as shown in the figure. For these end conditions, a slab of length h and width w and depth

11 Vol. 138, 1992 Mechanisms of Finite Brittle Strain 621 Beam bending and brenkln~ mechanism Stress 151 //// Figure 5 Beam bending and breaking. The upper part of the figure shows the original tensile fractures linked by shorter perpendicular tensile fractures. The stress conditions are also shown. The central part of the figure shows the geometry used to develop approximate expressions for the beam buckling phase of failure. The right central figure shows where regions of tension and compression will form in the buckling beam and the lower figure stlows the tensile fractures which will develop. b will buckle under the critical force rc2e (bw3"~ Fcr,t.- (1) which corresponds to a critical stress of zc2e (w~ 2 o-, = ~.- \~-//. (2) This buckling stress is independent of the confining stress a3, but h is not.

12 622 Geoffrey C. P. King and Charles G. Sammis PAGEOPH, Spalling and Collapse Stress cyi Collapse and Void filling Figure 6 Spalling and collapse of the broken beams. The upper part of the figure shows the smaller scale fracture in parts of the broken beams. The stress conditions are also shown. The lower part of the figure shows further detail. The next phase of deformation is shown in Figure 6. Compressive stresses are concentrated at the corners of blocks formed by the earlier stages and it is evident that exactly the same tension crack initiation followed by beam instability and failure will occur at this smaller scale. These phases of deformation are more readily seen in rock exposures. An example is shown from a granite rock face in Boulder Canyon, Colorado (Figure 7). Fractures in the photograph are meters in length. In this example, conditions have been such that transverse fractures have just started to form. Although it is not generally possible to determine the time relations between fractures in such a system, a notable exception is provided by the work of BARTON and HSIEH (1989), BARTON (1990). By mapping jointed pavements at Yucca Mountain, Nevada over a range of fracture lengths from centimeters to tens of meters, they have determined the order of fracture creation. The order of creation and location of the joints that they describe is in agreement with our model. A more advanced stage of deformation is shown in Figure 8 (fractures are millimeters in length) where the remnants of slabbing can be seen in part of the section and a gouge-like structure is seen in other parts.

13 Vol. 138, 1992 Mechanisms of Finite Brittle Strain 623 Figure 7 Long tensile fractures with shorter perpendicular cracks in a rock face in Boulder Canyon, Colorado. Deformation in an Established Fault Gouge Although we cannot detail all the processes that are required to establish a gouge it is possible to describe the way in which an established gouge will continue to deform. To see how finite shear strain is accommodated consider Figure 9. The

14 624 Geoffrey C. P. King and Charles G. Sammis PAGEOPH, Figure 8 Deformed rock from faulting associated with the Castle Cliff Detachment in Utah. The sample is 20 cm in the longest dimension. Some parts of the sample have not passed far beyond the slabbing phase while others have a fragmented gouge-like structure. upper row of figures a, b, c correspond to a series of different magnifications of gouge. The largest particles are A and A' with overall directions of motion indicated by arrows, with A' taken as fixed. The displacements are labelled as products of strain and sample dimensions, since they should be regarded as scaled to the size of the figure. Between particles A and A' are three solid lines and six dashed lines that indicate idealized trajectories of compressive stress. Between particles A and A' are two smaller particles B and B'. These are shown in expanded view in b and have the same scaled displacements as for A and A' but at this scale we take B' as fixed. In this "blow-up," part of one of the solid stress trajectories and parts of two of the dashed ones become solid lines and six more dashed trajectories are shown. Two particles C and C' lie between B and B' in b and are reproduced at a larger scale in c. It is clear that the relations described between pictures a and b also exist between b and c and the system can be extended indefinitely to greater or lesser scales than those shown. At any instant the mechanical behavior of the gouge can be approximated by the beam models d, e and f shown below. The beams may be thought to follow the lines of the stress trajectories shown by solid lines in the figures immediately above

15 Vol. 138, 1992 Mechanisms of Finite Brittle Strain 625 o) /~C22,1%Xb(~22 Xoel2 b) I--~ Xb CI2,1~ Xc s c) L-~Xce,2 T C.-'~ ~ ~ FIXED N=.i FIXED,t~Xo e22, /i,.xb CZ2,, AXc E;22,t).!L-~xoe,2 e~. L_~xt,r " ~ l~xce,2 FIXED XO FIXED FIXED FIXED Figure 9 Network of ligands at different scales. The upper figures demonstrate how each ligand is composed of a hierarchy of ligands at smaller scales. The lower figures show the ligands idealized as simple beams to demonstrate how simple shear places beams in longitudinal compression. o them. Each beam should be thought to be comprised of many smaller beams. Thus e shows a detail of d, and f shows a detail of e. The relative displacements are the same as those for the upper figures. As motion proceeds A moves relative to A' and the beams rotate. This reduces the contact between adjacent beams. The net effect of this motion is to marginally increase the compressional stress down the beam but dramatically reduce it in a perpendicular direction. The amount that the longitudinal stress increases depends on the value of e22. If the deformation is completely constrained this will be zero and longitudinal stress will rise more rapidly for a given e12 than if e22 is not zero. This expresses the intuitive observation that any attempt to move the beam system will lead to stresses tending to increase volume. This may be regarded as the origin of pressure dependence in fault gouge deformation. As a consequence of motion, the set of particles that comprise the beam are placed in uniaxial compression and will fail when the appropriate stress difference is reached. As soon as failure occurs, the e22 can return to zero and the increment of el2 becomes a permanent deformation. The rearrangement of particles causes the

16 626 Geoffrey C. P. King and Charles G. Sammis PAGEOPH, y Figure 10 Stress conditions experienced by a particle in a fault gouge. stress trajectories to change and follow a new course through the gouge. A new system of beams is thus formed that has to fail before the next increment of strain can be accommodated. The mode of tensile failure of particles is shown in Figure 10. Particles that are of similar size are loaded at two points in the fashion shown in Figure 10a and fail by axial splitting (this geometry forms the basis of the Brazilian strength test, JAEGER and COOK, 1979). If a particle of a given size is surrounded by smaller particles as in Figure 10b, the load it experiences is distributed and it is less likely to fail. Similarly a small particle surrounded by larger particles as in Figure 10c is less likely to fail in tension. It is evident that the condition that all particles are similarly "cushioned" requires that they are all surrounded by smaller particles. SAMMIS et al. (1987) and STEACY and SAMMIS (1991) point out that this is satisfied by a fractal distribution of gouge fragments with a fractal dimension of about Figure 11 shows a random fractal of approximately this dimension created by an automaton that breaks into four equal parts "particles" of the same size that share a side or a vertex. An example of an established gouge with a measured fractal dimension of 2.6 is shown in Figure 12 which can be compared with the less advanced deformation in Figure 8. Both may be compared with the theoretical gouge produced by the automaton. The examples of gouge shown have particles and hence fractures on the scale of centimeters or less. However, megabreccias (the geological term for gouges which contain large lumps) such as those exposed around Death Valley, California have blocks tens of meters in size and there is reason to suppose that the melanges often but not exclusively associated with ophiolites (e.g., MCCALL, 1972) can be regarded as "mega fault gouges" with blocks reaching kilometers in size. Interaction of Clusters." Localization or DeIocalization So far we have considered the processes associated with a single cluster of defects. The figures indicate distant displacement boundary conditions above and

17 Vol. 138, 1992 Mechanisms of Finite Brittle Strain 627 Figure 11 Fractal figure produced by an automaton that breaks into four equal parts "particles" of the same size that share a side or a vertex. below the defects but do not consider what happens to the left and right. The clusters of columns in Figure 2 are not free to buckle and fracture in the way shown unless failure occurs outside the region shown. We shall first consider how this occurs qualitatively and then examine some quantitative models. Two kinds of evolution are proposed. The first involves scaling down and localization. The second requires scaling up before localization can occur. Scaling Down and the Creation of a Shear Zone In Figure 13 clusters of initial failure are sufficiently close that they interact strongly. Stress concentrations in the gaps between them initiate small starter flaws and a zone of shear forms that can continue to evolve. Shear is localized within a planar zone and clusters that initially started to develo p throughout the region become inactive when the zone forms. Scaling up and the Creation of a Shear Zone of Larger Dimensions Unlike Figure 13 the clusters in Figure 14a are not sufficiently close to exploit the smaller defects between them. Instead they initiate tension cracks that propa-

18 628 Geoffrey C. P. King and Charles G. Sammis PAGEOPH, Figure 12 Optical photomosaic of a section of fault gouge. The gouge was prepared for sectioning by vacuum impregnation with epoxy resin (SAMMIS et al., 1986). A scale is provided by noting that the shorter sides of the photograph represent a length of 7 mm for the original sample.

19 Vol. 138, 1992 Mechanisms of Finite Brittle Strain 629 a) Localization and formation of a shear zone b) 2C *-'2b-* c) i Condition for formation of a shear zone: c>b/2 Shear zone Figure 13 Localization and the formation of a shear zone. The superdefects formed from clusters of original defects (a) are sufficiently close to interact (b) and form a shear zone (c). The right part of the figure shows the approximation to the real geometry used to calculate the conditions under which interaction Occurs. gate along the directions of maximum principal stress (Figure 14b). The behavior of the clusters of defects is similar to that described earlier for simple defects. We shall refer to the clusters as "super defects." In Figure 14c we look at a larger region that contains a number of superdefects. These are also clustered. It is evident that if these larger scale clusters are sufficiently close (Figure 14d), the interaction processes shown in Figure 13 will occur and a shear zone will form (Figure 14). If the original clusters were the same size as those shown in Figure 13a, the shear zone finally formed in Figure 14e will be much wider than that in Figure 13c. Before a shear zone could form, the less dense clusters had to link to form clusters at a larger scale. There is no reason that only one scale jump should occur. Further scale jumps may be needed before the slabs created by the process have the required length to thickness ratio for buckling to occur.

20 630 Geoffrey C. P. King and Charles G. Sammis PAGEOPH, SCALING UP BY CRACK EXTENSION If the condition to form a shear zone is not satisfied at one scale, a zone can form at a larger scale a) q ::.::: '' ::.: i::! Scaling up by crack growth h) 9 Shear zone cannot form c)! i! ii' Formation of a shear zone at the larger scale d) e) 9 :... 9 ~: " i{ir :-: c>- b 2 Shear zone can form Figure 14 Scaling up by crack extension. The superdefects (a) are insufficiently close to form shear zones and consequently shed tensile fractures (b). The figure on the fight shows the geometric approximation used to calculate the conditions under which this will occur. When a sufficient number of superdefects shed tensile fractures in this way a superdefect is created at a larger scale (c). These larger superdefects (d) can interact to form a larger scale shear zone (e). The figure on the right shows the geometric approximation used to calculate the conditions under which this will occur. The Formation and Interaction of Superdefects Simple quantitative models for the behavior of superdefects shown in Figures 13 and 14 can be developed. Consider first the out-of-plane propagation tensile cracks from a superdefect as illustrated in Figure 14. Even if the buckling condition in

21 Vol. 138, 1992 Mechanisms of Finite Brittle Strain 631 equation (2) is satisfied, the slabs will not buckle unless the nucleation condition for tensile fractures is also satisfied. If we treat the superdefect as a crack of length 2a and take the effective coefficient of friction to be zero during the collapse of the slabs, then the nucleation condition for an out-of-plane shear crack at the next scale is (ASHBY and HALLAM, 1986; ASHBY and SAMMIS, 1990) O-1 = O-3 + (3),#s Both this initiation condition and the buckling condition (2) must be satisfied before the superdefect can activate tensile cracking at the next scale. These conditions are shown graphically in Figures 15a and 15b. In these figures the horizontal axis is o- 1 and the vertical axis is the right-hand side of the buckling equation (2). The dashed line at 45 ~ to the axes thus marks the onset of buckling. The superdefect nucleation equation (3) is plotted as the vertical broken line in these figures. The heavy solid line with arrows indicates the path of a developing superdefect in this space. As the maximum remote principal stress O-1 increases, the cracks grow, and h increases causing (~2E/12)(w/h)2 to decrease as indicated. Two scenarios for superdefect formation are shown in Figure 15. In Figure 15a the buckling instability (equation (2)) is satisfied at a smaller value of O-1 than the nucleation condition (equation (3)). In this case, the system must develop well into the single column buckling field before the nucleation condition is satisfied (as indicated by the star) and the superdefect can nucleate a tensile crack at the next scale. In Figure 15b the superdefect is potentially unstable before the slabs can buckle. In this case nucleation cannot occur until the buckling equation is satisfied as indicated by the star. If the superdefects are sufficiently close together, smaller flaws between them may be activated to produce a through-going shear zone. The conditions under which this will occur can also be modeled approximately. Consider two coplanar superdefects each of length 2a which are separated by a distance 2b as in Figure 16. Assume that a smaller defect (or superdefect) of length 2c is centered between the two larger superdefects as shown in the figure. We now wish to compare the nucleation condition for the smaller defect in the local field of the two larger superdefects with the condition for the out-of-plane nucleation of tensile cracks from the two superdefects discussed above. If remote stresses a~ and a3 are applied as indicated in the figure, then the principal stresses at the midpoint between the two superdefects can be found by superimposing the well-known equations for the stress concentration in the vicinity of a crack (PARIS and SIEH, 1964). They are O'ma x ~-~- N~ ~ 0"1 U O'min = N~ /a O-3- Vo (4)

22 632 Geoffrey C. P. King and Charles G. Sammis PAGEOPH, Figure 15 Alternative conditions for superdefect instability. In (a) the buckling instability occurs before the condition for the nucleation condition for shedding tensile fractures. In (b) the nucleation condition is reached and before the buckling instability occurs. Assuming for simplicity that o" 3 = 0 and applying the nucleation equation to both the large superdefects in the remote stress field, and to the smaller flaws in the local field (4), the condition that the smaller flaws will nucleate first is

23 Vol. 138, 1992 Mechanisms of Finite Brittle Strain 633 Figure 16 Geometric approximation used to calculate the conditions for the creation of a shear zone. which, using equation (4) simplified to c > b/2. (6) This says that the formation of a shear zone requires that the superdefects be so close together that the next smallest defects are more than half the dimension of the gaps between them. These approximations can be illustrated with calculations of the deformation around sets of cracks. In Figure 17 an array is shown tending to shed tensile fissures when the fracture toughness of the material is reached. The size of the black lobes associated with the edges of the fissures is a measure of the stress intensity factor generated at those points by the geometry of the fractures and the applied boundary conditions. In Figure 17b tensile fissures have been allowed to extend and the stress intensity factor associated with their ends has increased. In Figure 17c an array of half the size as that in Figure 17a but under the same boundary conditions is shown. The stress intensity factors are half that for the larger array.

24 634 Geoffrey C. P. King and Charles G. Sammis PAGEOPH, Figure 17 Figure 18

25 Vol. 138, 1992 Mechanisms of Finite Brittle Strain 635 In Figure 18a two arrays are shown which will also tend to shed fissures. If an array of half the dimensions is introduced between the two arrays (Figure 18b) the stress intensity factor associated with the smaller array is as great as for the larger arrays. In isolation the smaller arrays would have stress intensity factors of half the amplitude. Thus when placed between the larger arrays the small array has the same tendency to "grow" as the larger arrays. If two further arrays half as small again are added the same process repeats (Figure 18c). These smaller arrays are again associated with high stress intensity factors. It is evident that the process can cascade downwards to utilize defects of smaller sizes and permit a through-going shear zone to form. In the figures light shading is associated with compression. In Figures 18a and 18b compression is localized around the defect clusters but in Figure 18c it can be seen to spread in a zone parallel to the forming shear zone. This compression will tend to suppress any defects that had started to grow in these regions. It is commonly observed in rock sample experiments that acoustic emission indicates the localization of damage towards the evolving shear zone and the suppression of damage away from it (LOCKN~ et al., 1990). Figure 17 Tensile strains associated with arrays of tensile cracks used to approximate superdefects. In each figure simple shear displacement boundary conditions are applied at a distance I00 times greater than the dimensions of the region shown. No confining pressure is applied. An array of 7 tensile cracks (a) has lobes of tensile strain (dark shading) associated with the outer tensile cracks. The stress intensity is proportional to the size of these lobes. In (b) the outer cracks have been allowed to extend and are associated with larger lobes of tensile strain. This is in agreement with the graphs in Figure 3 that indicate that long cracks are stable at lower stresses than shorter ones. Consequently, once initiated, tensile fractures from a superdefect can reach very large dimensions when the confining pressure is low. The tensile lobes associated with a similar array of cracks of half the size (c) are commensurately reduced. In the absence of interction between superdefects, the largest will shed tensile fractures. The figures are calculated using the methods of CROUCH and STARFIELD (1983). Each crack is approximated by 11 subelements in (a) and (c). In (b) the number of elements is increased to 21 for the longer cracks. Experiments with different numbers of subelements show that the results are numerically stable when more than 5 and more than 11 subelements are used for short and long cracks, respectively. Figure 18 The interaction between superdefects to form a shear zone. Two superdefects of similar size separated by their own longest dimension do not interact significantly (a). A superdefect of half the size midway between them (b) however produces tensile lobes of the same dimensions as the larger superdefects. Two further superdefects a quarter of the size of the large superdefects placed in the remaining gaps (c) also produce tensile lobes of the same dimensions as the larger superdefects. Tensile damage can occur at small scales and a shear zone can form. The geometry is similar to that described earlier for the condition permitting the formation of shear zones. For those calculations the superdefects were approximated by ellipsoids. In (c) displacements have become larger than in (a) or (b) resulting in the formation of a zone of compression (light shading) around the forming shear zone. This will suppress the growth of features outside the evolving zone. See the caption of Figure 17 for information about the calculation procedure.

26 636 Geoffrey C. P. King and Charles G. Sammis PAGEOPH, Figure 19 Superdefects that do not interact. The largest superdefects have a separation 50% greater than in Figure 19. The separation between the smaller defects is similarly increased. The largest tensile lobes are associated with the largest defects (a). When these extend (b) the size of the lobes on the smaller defects reduces. For this geometry the largest superdefects will shed tensile fractures. See the caption of Figure 17 for information about the calculation procedure. The separation of the arrays cannot be much greater than that shown if a shear zone is to form. In Figure 19a the separation has been increased by 25% compared to Figure 18c. The maximum stress intensities are now associated with the arrays of the largest cracks and there is no tendency to exploit smaller scale defects. In Figure 19b the larger arrays have been allowed to shed cracks and it can be seen that the stress concentrations associated with the smaller arrays diminishes dramatically as their tendency to grow is suppressed.

27 Vol. t38, 1992 Mechanisms of Finite Brittle Strain 637 The Nature of Superdefects The labeling of the collective behavior of a cluster of defects as a "superdefect" provides a convenient method of illustrating the way in which scales can interact. However, these features need not be as clearly defined or grouped as our figures suggest and their exact behavior will depend on the nature of the initial clustering. We make no attempt to characterize clustering in this paper and consequently do not expand further on this topic here. Superdefects differ from simple defects in one very important respect. Because they are composed of collections of linear fractures aligned parallel to the direction of maximum compressive stress they will not cause any local rotation of stress axes. Thus any tensile extension of their member cracks will always occur in the plane of those cracks and will not be a curved tensile crack shed by an inclined Griffith flaw (ASHBY and HALLAM, 1986). It is observed that rock joints are remarkably planar in form, except where obvious material heterogeneity intervenes, and this broad characteristic of crustal fractures is consistent with the mechanisms that we propose. It is also noteworthy that superdefects must dilate in order to shed cracks. This dilation is additional to the dilation associated with new cracking. Pressure acting to suppress this dilation will act in similar but not identical fashion to friction imagined to act on an inclined shear plane (friction does not require dilation). It is plausible to propose however, that frictional behavior is in reality the combined effect of arrays of dilating centers and that, at a small enough scale, it is the inherent tensile fracture toughness of native material that provides friction with its critical value. The Relation between Tensile Fracture Models and Granular Gouge Models Describing the deformation of solids in terms of the interaction of particles or grains seems very different from the shedding of tensile fractures from defects. The difference is less than might appear, as can be appreciated from Figure 20. Figure 20a shows two large particles of similar size in contact. This is the condition in which they are vulnerable to axial splitting because of the stress localization where they touch (see Figure 10a). The condition of two particles in this geometry implies that two regions of smaller particles lie on each side. The existence of many smaller particles is associated with a lot of small-scale damage and hence the regions will act as super defects (Figure 20b). They are sufficiently separate however, that the linkage condition to form a shear zone is not satisfied and hence they will shed out-of-plane tension cracks. These are the cracks that split the larger particles. In contrast, Figure 20c shows the larger particles separated with smaller particles in between. This is equivalent to smaller defects lying between the superdefects and consequently a shear zone can form (Figure 20d) and the larger particles will survive unbroken.

28 638 Geoffrey C. P. King and Charles G. Sammis PAGEOPH, Figure 20 The relation between the fracture methanical description and the fractal particle description of rock deformation. Two large particles are shown in contact (a) with regions of smaller particles on each side. The particle description predicts that the high stresses associated with contact will cause tensile fractures to break the large particles. A superdefect approximation to (a) is shown in (b) where superdefects represent the regions of smaller particles and the large particles and their close area of contact is represented by an absence of significant pre-existing defects. Under these conditions the superdefects will shed tensile fractures into regions equivalent to the large grains in (a). The large particles in (c) are shown separated by smaller particles. This can be approximated (d) by superdefects of various sizes that are close together. The conditions for the formation of a shear zone are satisfied. The description of a gouge as a granular material with a fractal geometry implies a fractal distribution of interfaces and a fractal distribution of interface sizes. Thus such a geometry has the fractal distribution of defects required by the tensile failure mechanisms described earlier. Conclusions Earilier descriptions of the deformation of rock have concentrated on explaining the formation of a shear zone in laboratory experiemnts. RUDNICKI and RICE (1975) for example modified the expressions for shear in soils (RICE, 1973) to damage in rock. However, these are continuum models based on lumping failure

29 Vol. 138, 1992 Mechanisms of Finite Brittle Strain 639 conditions into constitutive equations. Localization results from a strain softening rheology. Micromechanical behavior associated with brittle deformation and shear localization has been recognized for many years (e.g., HALLBAUER et al., 1973; TAPPONIER and BRACE, 1976; WONG, 1982; SCHOLZ, 1990) and implies that continuum models are of limited value since they are only loosely related to the underlying micromechanical processes. HORII and NEMAT-NASSER (1985) have examined the micromechanics of shear localization by studying the failure of uniform arrays of identical Griffith flaws. Their model succesfully predicts the dependence of the failure angle in triaxial experiments on the confining pressure, but does not explain the evolution of damage and associated rock fabrics created by finite brittle deformation of the crust. In this paper we show that apparently complex failure in the earth involves the interaction between simple processes that occur on a wide range of scales. Such multiscale processes provide plausible explanations of many features associated with the deformation of rock observed both in the laboratory and in the field. An important aspect of brittle deformation is its fractal nature. The elements of multiscale deformation which we explore both exploit and contribute to the fractal structure. Two alternative approaches are used to explain the evolution of damage. The first considers the extension and interaction of tensile fissures shed from inhomogeneously distributed starter flaws and is an extension of the damage mechanics of ASHBY and SAMMIS (1990) and the array analysis of HORII and NEMAT-NASSER (1985). The second regards the crust as behaving in a granular fashion (SAMMIS et al., 1987; BIEGEL et al., 1989; SAMM~S, 1990). Deformation is associated with stress concentrations caused by grain interactions. Both descriptions are clearly appropriate under certain circumstances. Early stages of brittle deformation are best described using a fracture mechanics approach, while multiply fractured materials in well developed shear zones are best described using the granular mechanics approach. Both approaches, however, invoke underlying processes of tensile fracture, dilation, rotation and collapse and we show that they can be reduced to alternative descriptions of the same phenomona. REFERENCES ASHBY, M. F., and HALLAM, S. D. (1986), The Failure of Brittle Solids Containing Small Cracks under Compressive Stress States, Acta Metall. 34, ASHBY, M. F., and SAMMIS, C. G. (1990), The Damage Mechanics of Brittle Solids in Compression, Pure and Appl. Geophys. 133, BARTON, C. C., and HSlEH, P. A., Physical and Hydrologic-Flow Properties of Fractures, Am. Geophys. Union Guidebook T 385, BARTON, C. C., Fractal analysis of the scaling and spatial clustering of fractures. In Fractals and their Use in the Earth Sciences (eds. Barton, C. C., and LaPointe, P. R.) (GSA Memoir 1992) in press.

30 640 Geoffrey C. P. King and Charles G. Sammis PAGEOPH, BIEGEL, R. L., SAMMIS, C. G., and DIETER1CH, J. H. (1989), The Frictional Properties of a Simulated Gouge Having a Fractal Particle Distribution, J. Struct. Geol. i1, CROUCH, S. L., and STARFIELD, A. M., Boundary Element Methods in Solid Mechanics (Allen Unwin, London 1983). HALLBAUER, D. K., WAGNER, H., and COOK, N. G. W. (1973), Some Observations Concerning the Microscopic and Mechanical Behaviour of Quartzite Sepcimens in Stiff, Triaxial Compression Tests, Int. J. Rock Mech. Min. Sci. 10, HORII, H., and NEMAT-NASSER, S. (1985), Compression-induced Microcrack Growth in Brittle Solids: Axial Splitting and Shear Failure, J. Geophys. Res. 90, JAEGER, J. C., and COOK, N. G. W., Fundamentals of Rock Mechanics (Chapman and Hall, 1979). KING, G. C. P. (1983), The Accommodation of Large Strains in the Upper Lithosphere of the Earth and Other Solids by Self-similar Fault Systems: The Geometrical Origin of b-value, Pure and Appl. Geophys. 121, KIRBY, S. H., DURHAM, W. B., and STERN, L. A. (1991), Mantle Phase Changes and Deep-earthquake Faulting in Subducting Lithosphere, Science 252, LOCKNER, D. A., BYERLEE, J., KUKSENKO, V., PONOMAREV, A., and SIDORIN, A. (1990), Observations of Quasi-static Fault Growth from Acoustic Emissions, LOS, Trans. Am. Geophys. Union 71, LOCKNER, D. A., The Modeling of Brittle Failure and Comparisons to Laboratory Experiments. Ph.D. Thesis (M.I.T., February 1990). MCCALL, G. J. H. (ed.), Ophiolitic and Related Mblanges. Benchmark Papers in Geology~66 (Hutchinson Ross Publishing Company, Stroudsburg, Pennsylvania 1972). NEMAT-NASSER, S., and HORII, H. (1982), Compression-induced Non-planar Crack Extension with Application to Splitting, Exfoliation, and Rockburst, J. Geophys. Res. 87, PARIS, P. C., and SIEH, G. C. (1964), Stress Analysis of Cracks, Fracture Toughness Testing and its Applications, ASTM Spec. Tech. Publ. 381, RICE, J. R., The initiation and growth of shear bands. In Plasticity in Soil Mechanics (ed. Palmer, A. C.), Proc. Symp. on Role of Plasticity in Soil Mechanics, Cambridge, Sept, 1973 (Cambridge University Engineering Dept., Cambridge, England 1973) pp RUDNICKI, J. W., and RICE, J. R. (1975), Conditions for the Localization of Deformation in Pressure-sensitive Dilatant Materials, J. Mech. Phys. Solids 23, SAMMIS, C. G., Fractal fragmentation in a granular crust. In Fractals in Geology (ed. Barton, C.) (GSA Memoir 1992) in press. SAMMIS, C. G., and ASHBY, M. F. (1986), The Failure of Brittle Porous Solids under Compressive Stress States, Acta Metall. 34, SAMMIS, C. G., and BIEGEL, R. L. (1989), Fractals, Fault-gouge, and Friction, Pure and Appl. Geophys. 131, SAMMIS, C. G., KING, G., and BIEGEL, R. L. (1987), The Kinematics of Gouge Deformation, Pure and Appl. Geophys. 125, SCHOLZ, C. H., The Mechanics of Earthquakes and Faulting (Cambridge University Press, New York 1990). STEACV, S. J., and SAMMIS, C. J. (1991), An Automaton for Fractal Patterns of Fragmentation, Nature 353, TAPPON1ER, P., and BRACE, W. F. (1976), Development of Stress-induced Microcracks in Westerly Granite, Int. J. Rock Mech. Min. Sci. 13, WONG, T. F. (1982), Shear Fracture Energy of Westerly Granite from Post-failure Behaviour, J. Geophys. Res. 87, (Accepted December I, 1991)