CONSTITUTIVE BEHAVIOR OF GRANITIC ROCK AT THE BRITTLE-DUCTILE TRANSITION Josie Nevitt, David Pollard, and Jessica Warren Department of Geological and Environmental Sciences, Stanford University, Stanford, CA 94305 e-mail: [email protected]
fracture to ductile flow with increasing depth in the Abstract earth’s crust. This so-called “brittle-ductile transition” has important implications for a variety of geological Although significant geological and geophysical and geophysical phenomena, including the mechanics processes, including earthquake nucleation and of earthquake rupture nucleation and propagation propagation, occur in the brittle-ductile transition, (Hobbs et al., 1986; Li and Rice, 1987; Scholz, 1988; earth scientists struggle to identify appropriate Tse and Rice, 1986). Despite the importance of constitutive laws for brittle-ductile deformation. This deformation within this lithospheric interval, significant paper investigates outcrops from the Bear Creek field gaps exist in our ability to accurately characterize area that record deformation at approximately 4-15 km brittle-ductile deformation. Such gaps are strongly depth and 400-500ºC. We focus on the Seven Gables related to the uncertainty in choosing the constitutive outcrop, which contains a ~10cm thick aplite dike that law(s) that govern(s) the rheology of the brittle-ductile is displaced ~45 cm through a contractional step transition. between two sub-parallel left-lateral faults. Stretching Rheology describes the response of a material to an and rotation of the aplite dike, in addition to local applied force and is defined through a set of foliation development in the granodiorite, provides an constitutive equations. In addition to continuum excellent measure of the strain within the step. We use mechanics parameters (e.g., stress, strain), constitutive the geometry of this well-constrained outcrop to motivate the geometry and boundary conditions of a finite element model of the deformation. The model is then used to test the ability of six constitutive laws (von Mises elastic-plasticity, Drucker-Prager elastic- plasticity, Drucker-Prager Cap elastic-plasticity, power-law creep, hyperbolic-sine creep, and viscoplasticity) to reproduce the deformation features observed in outcrop. The results indicate that the constitutive behavior is likely frictionless and has a yield criterion that depends on the hydrostatic stress. In addition, the flow rule should not contain a volumetric strain component, as this results in volume loss in the modeled aplite dike. Of the constitutive laws tested, the viscoplasticity law most accurately depicts the outcrop deformation. This result should motivate future laboratory research into the creep properties of granitic rock. The results of this study help to eliminate constitutive laws (e.g.., frictional plasticity) that could potentially describe brittle-ductile deformation, which leads to a better understanding of the rheology of the continental crust in the brittle-ductile transition.
Keywords: Figure 1. Strength profile constructed using brittle-ductile transition, constitutive law, rheology, Byerlee’s Law (red) and a dislocation creep flow strike-slip fault, ductile fabric, finite element analysis, law for quartzite (blue) published by Hirth et al. water weakening (2001). The dashed purple line indicates that the strength of the crust within the brittle-ductile transition is likely overestimated by the Introduction intersection between Byerlee’s Law and the flow The failure mode of rock transitions from brittle law.
Stanford Rock Fracture Project Vol. 23, 2012 G-1 equations also include material parameters and thus are strength using the Coulomb criterion for frictional specific to the material of interest. Most modeling sliding: efforts assume an over-simplified constitutive law with (1) = + a surprising degree of success. For example, the upper where is the shear stress, is the normal stress, is brittle section of the crust is commonly assumed to be a the coefficient of friction, and is the frictional homogenous, isotropic, linear elastic material (Cooke and Pollard, 1997; Fiore et al., 2007; Pollard and cohesive strength. Byerlee (1978) empirically determined that for , and Segall, 1987). In contrast, the lower continental crust is < 200 = 0.85 commonly modeled using a flow law for dislocation , while for , and = 0 > 200 = 0.6 creep in quartz (Behr and Platt, 2011; Brace and . = 60 Kohlstedt, 1980; Ranalli and Murphy, 1987) . In these In contrast, failure under higher pressure and examples, the models focus on capturing only the most temperature conditions in the ductile lower crust occurs characteristic features of the deformation (e.g., fault slip through a variety of viscous mechanisms, including the in an elastic model, steady-state flow in a creep model). migration of dislocations and the solid-state bulk However, a mathematical theory “meets its limits of diffusion of vacancies (Ji and Xia, 2002; Karato, 2008; applicability where a disregarded influence becomes Poirier, 1985). Seismic anisotropy observed in the important” (Flugge, 1967). It seems that continental mantle (e.g., Silver, 1996) and direct observations of crust located in the brittle-ductile transition is subject to crystallographic preferred orientations in rocks from the many important influences, and it remains unclear as to lower crust suggest that dislocation creep is the what constitutive law or combination of constitutive dominant deformation mechanism in these regions. The laws provides the best fit. Weertman equation (Weertman, 1978) describes This paper provides a new perspective on the brittle- dislocation creep through a power law relation between ductile rheology of granitic rock by investigating a strain rate, , and the differential stress, : naturally deformed contractional fault step in (2) granodiorite that developed at a depth of 4-15 km and = exp − temperature of 400-500°C. We combine field where is the pre-exponential factor, is the stress observations with mechanics-based numerical modeling exponent, is the apparent activation energy, is the to test the ability of several constitutive laws to ideal gas constant, and is the absolute temperature (in reproduce the deformation observed in outcrop. Kelvin). Because the models use mechanical properties for Together, Equations (1) and (2) can be used to granite and aplite published in the rock mechanics define a strength profile through the lithosphere (e.g., literature, they also test the validity of applying Brace and Kohlstedt, 1980, Kohlstedt et al., 1995). experimentally-determined properties to studies of Figure 1 gives an example of a strength diagram naturally deformed rock. This paper explores the constructed with a commonly cited dislocation creep constitutive properties representative of the continuum flow law for quartzite published by Hirth et al. (2001). deformation under brittle-ductile conditions, and does The intersection between Byerlee’s Law and the Hirth not explicitly take into account the underlying micro- et al. (2001) flow law forms a sharp peak and suggests a mechanisms. maximum differential stress of ~180 MPa in the lithosphere. The transition from brittle to ductile Lithospheric strength and the brittle- behavior, however, is likely a more gradual process, ductile transition and the true strength through the brittle-ductile The brittle-ductile transition is frequently discussed transition may lie closer to the purple line dashed in in terms of the strength of the lithosphere (e.g., Figure 1. An improved understanding of the constitutive laws governing deformation in the brittle- Kohlstedt et al., 1995). The term “strength” is used ductile transition will allow for more accurate strength throughout the crustal deformation literature, often calculations in this complicated interval of the without an explicit definition for its physical meaning in the context of rock mechanics. Broadly, strength is lithosphere. defined as the maximum stress a material can experience without failure. The definition of strength, Constitutive behavior of granitic rocks therefore, is specific to the mode of failure (i.e., brittle Large-scale models often idealize the continental or ductile). Because failure in the upper brittle crust is crust as granitic in composition and this has motivated generally attributed to frictional sliding on pre-existing abundant research on the deformation of granitic rocks. planes of weakness (e.g., faults), many papers (Brace Granite is composed of a variety of minerals, generally and Kohlstedt, 1980; Kohlstedt et al., 1995) define the including feldspars, quartz, micas, hornblende, and sphene. Each of these minerals is defined by a distinct
Stanford Rock Fracture Project Vol. 23, 2012 G-2 yield strength and characterized by a specific requires solid state transport through the crystalline deformational behavior under a certain set of conditions lattice or along crystal defects (Kronenberg et al., (Tullis, 1990; Tullis and Yund, 1977, 1980). A 1990). This proposed mechanism for water penetration complicated relationship between the strengths, volume along microcracks and mobile dislocations suggests that proportions, and geometric arrangement of the hydrolytic weakening is most prevalent under brittle- constituent minerals determines the overall strength of ductile conditions (Kronenberg et al., 1990). the polymineralic rock (DellAngelo and Tullis, 1996). Because they are the most abundant constituent Bear Creek field area minerals, quartz and feldspars play important roles in The Bear Creek field area (Figure 2) is located in controlling the behavior of granite during deformation. the John Muir Wilderness of the central Sierra Nevada, Furthermore, the onset of crystal plasticity in quartz and CA and has garnered the attention of geologists over feldspar appear to control the upper and lower bounds, the past thirty years for its excellent exposure of respectively, of the brittle-ductile transition in structural features. While extensive work has focused continental crust (Scholz, 1988). on the brittle structures, including opening-mode joints, The shallow limit of the brittle-ductile transition is strike-slip faults and fault zones, the development of the defined by the onset of plasticity in quartz at 300°C ductile features, such as local foliations, remains largely (Scholz, 1988). Between 300-500°C, deformation of unresolved (Burgmann and Pollard, 1994; Bürgmann quartz generally produces bulging recrystallization and and Pollard, 1994; d'Alessio and Martel, 2005; Davies subgrain rotation recrystallization microstructures due and Pollard, 1986; Griffith et al., 2008; Griffith et al., to migration and build-up of dislocations (Hirth and 2009; Martel, 1990, 1999; Martel and Boger, 1998; Tullis, 1992; Passchier and Trouw, 2006; Stipp et al., Martel and Pollard, 1989; Martel et al., 1988; Pachell et 2002). Deformation at high deviatoric (i.e., flow) stress generally results in grain size reduction (e.g., Stipp et al., 2002). In addition, quartz can recrystallize into highly connected "ribbon structures" that greatly lowers the strength of the aggregate (DellAngelo and Tullis, 1996). In contrast, feldspars do not deform plastically until 450°C, which defines the deepest limit of the brittle-ductile transition in continental crust (Passchier and Trouw, 2006). Thus, brittle-ductile microstructures in granitic rock are generally characterized by recrystallization and grain size reduction in quartz and microcracking and cataclasis in feldspars (Bürgmann and Pollard, 1994). Temperature constraints for the brittle-ductile transition, however, are extremely sensitive to the presence of water. Tullis and Yund (1980) found that at high confining pressure, adding trace amounts of water decreases the strength of granite and lowers the transition from brittle to plastic behavior by 150-200°C in both quartz and feldspars. Thus, granite deformed under wet conditions and at high confining pressure will exhibit “high temperature” deformation microstructures (Tullis and Yund, 1980). Tullis and Yund (1980) observed that during laboratory experiments, water weakening results in inhomogeneous deformation by concentrating strain in Figure 2. Geologic map of the Bear Creek field area, grains adjacent to fractures. Kronenberg et al. (1990) located in the Mount Abbot quadrangle, central built upon this observation by investigating fractured Sierra Nevada, CA (modified from Lockwood and granitic rock naturally deformed under brittle-ductile Lydon, 1975). Kmr – Mono Recesses quartz conditions. They proposed that the mechanism for monzanite; Kle – Lake Edison granodiorite; Kl – water weakening is related to the ability of fluids to Lamarck granodiorite; Kj – granitic rocks of uncertain affinities; J Tr – Metavolcanic rocks; Tt – migrate along dilatant microcacks surrounding Olivine trachybasalt; Q – alluvium. The Bear Creek fractures. Fluids transported along the microcracks Meadow, Kip Camp, and the Seven Gables outcrop likely promote crack growth as well as crack healing are marked in their locations. and continued fluid penetration into the microcracks
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Figure 3. Map view of three stages of development of the joints and faults in Bear Creek during the late Cretaceous; (a) joints develop soon after crystallization of the Lake Edison granodiorite; (b) reorientation of the principal stresses causes joints to reactivate as left-lateral strike-slip faults; (c) Secondary structures, including splay cracks and ductile fabric, develop in extensional and contractional fault steps, respectively. Modified from Burgmann and Pollard (1994). al., 2003; Segall and Pollard, 1980, 1983a, b). lateral strike-slip faults (Segall and Pollard, 1983b). The field area is comprised of several northwest The occurrence of pseudotachylyte veins suggests that trending plutons of late Cretaceous age, each with a some of the faults experienced dynamic slip events steeply dipping foliation that trends approximately (Griffith et al., 2008). The orientation of most northwest. This paper focuses on structures located in compressive stress, σ3, during this time of faulting has the Lake Edison granodiorite, a medium- to fine- been determined by bisecting the angle of intersecting grained biotite-hornblende granodiorite with a left-lateral and right-lateral faults located near the Bear crystallization age of 88 ±1 Ma (Tobisch et al., 1995). Creek Meadow fault, shown in Figure 4. Such Aplite dikes, composed of ~70% feldspar and ~30% relationships indicate that the most compressive stress quartz, along with volcanic xenoliths are scattered trends approximately 224°. throughout the Lake Edison granodiorite. The depth of emplacement of these plutons is broadly constrained to 4-15 km according to the amphibole geobarometry pressure estimates of 100-400 MPa (Ague and Brimhall, 1988; Griffith et al., 2009). Because joints and faults developed shortly after pluton crystallization, between 85-79 Ma (Segall et al., 1990), the depth of emplacement approximately corresponds to the depth of deformation. Previous studies have given estimates for the temperature of deformation ranging from 300- 350°C (Burgmann and Pollard, 1992) to >500°C (Pennacchioni and Zucchi, in review). The variability in temperature estimates may be related to the locations of those studies relative to the Mono Creek pluton that intruded at a younger age (see Figure 2). The structural history of Bear Creek is generally well-established (see Figure 3), although many of the details remain a source of contention (Pennacchioni and Zucchi, in review). Following crystallization, a set of cooling joints developed that strike predominantly ENE and contain quartz, chlorite and epidote mineral fill (Martel et al., 1988; Segall and Pollard, 1983a). Joints attain lengths that range from ~50 cm to tens of meters and apertures that reach up to a few centimeters in Figure 4. The orientation of the maximum width. During the next ~5 million years, a change in the compressive stress, σ3, was determined by stress regime caused the joints to reactivate as left- bisecting the angle between intersecting left-lateral and right-lateral faults. Outcrops shown in (a) and (b) are located near the Bear Creek Meadow.
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Figure 5. Four examples of contractional fault steps. In each case, the fault tips deflect outward from the step center and the region within the step contains locally foliated granodiorite.
The faults frequently cross-cut aplite dikes and
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Figure 6. Field evidence for water weakening; (a) alteration halos (marked by pens laid out on the outcrop) occur as the granodiorite is bleached near areas of opening along a fault trace; (b) exposed joint and fault (pictured here) surfaces have a distinct green appearance as they are coated by hydrothermal minerals, including quartz, epidote and chlorite. The faults frequently cross-cut aplite dikes and volcanic pressure conditions (Tullis and Yund, 1980). Evidence xenoliths that record displacements ranging from that water was present during deformation in Bear millimeters to 2 m along the fault traces. The length Creek includes alteration halos, where hydrothermal dimension of the faults is generally inherited from that fluids bleached the wall rock (Figure 6a), in addition to of the predecessor joints. In cases where the step-to- fracture fill consisting of hydrothermal minerals, width ratio is small between en echelon fault tips, including chlorite, epidote and quartz (Figure 6b). however, fault interaction resulted in stress In addition to contractional fault steps, ductile concentrations and the development of asymmetric features accompany many structures throughout the secondary features (Bürgmann and Pollard, 1992, 1994; Bear Creek field area. For example, although the faults Segall and Pollard, 1980, 1983b). sharply offset aplite dikes and xenoliths at the outcrop The nature of these secondary features depends on scale, mineral fill within the faults often display the step geometry (i.e. left-stepping or right-stepping) mylonitic fabrics at the microscopic scale (Segall and and the resulting state of the stress concentration (i.e. Simpson, 1986). In addition, mylonitic shear zones tension or compression, respectively). Left-stepping commonly localize on aplite dikes oriented geometries result in tensile stress states and include approximately E-W (Christiansen and Pollard, 1997), extensional secondary features, such as splay cracks. shown in Figure 7a. Figure 7b gives another example On the other hand, compression within right-stepping of ductile deformation in the Bear Creek area, which is geometries is often associated with locally strong the occurrence of right-lateral monoclinal kink bands ductile fabrics. A ductile fabric occurs where the that inelastically warp closely spaced left-lateral faults granodiorite has undergone crystal plastic deformation, (Davies and Pollard, 1986; Martel, 1999). The size of resulting in an elongation and alignment of mineral these kink bands ranges from the meter-scale in outcrop grains. Segall and Pollard (1983b) refer to such left- to the quadrangle scale (Pachell et al., 2003). stepping and right-stepping discontinuities as extensional and contractional steps, respectively. Seven Gables outcrop Contractional steps (see Figure 5) containing ductile Although the field area contains numerous outcrops fabrics adjacent to strike-slip faults provide beautiful that feature adjacent brittle and ductile structures, this examples of "brittle-ductile" structures. study focuses on the Seven Gables outcrop, located on Based on elastic models of fault steps in Bear the south bank of the East Fork Creek near Seven Creek, Bürgmann and Pollard (1994) concluded that the Gables Mountain. This outcrop contains ~15 individual local ductile fabrics within contractional fault steps fault segments that vary in length from approximately result from a dependence of the rheology on the mean 1-10 m. Many of the right-stepping discontinuities stress distribution. Furthermore, the transition to ductile between left-lateral fault terminations contain ductile deformation in contractional steps may be related to fabric. In one case, however, a contractional step offsets water weakening, which is only activated at high an aplite dike and the deformed aplite dike within the
Stanford Rock Fracture Project Vol. 23, 2012 G-6 most compressive regional stress. The aplite dike is offset 45 cm through the right step between the two left-lateral faults. Within the step, the aplite dike has been stretched to 3 times its initial length and rotated 36° counter-clockwise. The apparent width of the aplite dike at the surface decreases from 10 cm outside the step to 2 cm inside the step. Although this apparent change in width suggests that the volume of the dike decreases within the step, the three-dimensionality of the outcrop indicates otherwise. The dip of the aplite dike increases from 25°outside the step to 61° within the step. This significant rotation of the aplite dike means that volume was approximately conserved during deformation within the step region. In addition to the rotation and stretching of the aplite dike, the development of a ductile fabric contributes to inelastic deformation within the step. The weak regional foliation trends ~310° and can be seen near core sample 1 in Figure 8. With decreasing distance to the step center, the fabric trend rotates ~45° into the orientation of the stretched aplite and becomes much stronger as the dark phases, predominantly biotite, join together in elongate aggregates. This "disappearing dike" structure, in which an aplite dike is displaced and undergoes stretching within a contractional step, occurs in numerous locations throughout the field area. The scale of these structures ranges from the decimeter to tens of meters, with the Figure 7. Brittle-ductile deformation takes place in larger examples occurring near the contact with the the form of several different types of structures; (a) ductile shear zone that nucleated on a small aplite Mono Creek Granite (Kmr in Figure 2). Three dike; (b) kink bands are defined by inelastically examples of disappearing dikes are given in Figure 10. warped left-lateral faults that are closely spaced In our analysis of these examples, we have found that and often cross-cut aplite dikes. the volume of the aplite dike is always approximately conserved within the step, thus the contribution of diffusive mass transfer to the overall deformation is step provides an excellent graphic measure of the total likely negligible. In addition, we note that the fault tips strain. This contractional step, along with the relevant consistently deflect outward from the step center, measurements detailed in the following paragraph, is presumably to accommodate granodiorite moving into shown in Figure 8. the step. A final key characteristic of these structures is Because the outcrop is glacially polished, we used that the transition from the foliated granodiorite to well-exposed nearby faults (Figure 9a) and core unfoliated granodiorite occurs sharply across fault samples (Figure 9b) to determine the 3 dimensional traces. This observation is highlighted in Figure 11. orientations of the structures in the Seven Gables A conceptual, kinematic model for the evolution of outcrop. The contractional step is defined by two sub- the Seven Gables disappearing dike is presented in parallel left-lateral faults that trend 259° and dip 76°, Figure 12. Initially, the aplite dike has a uniform such that there is a 35° angle between the faults and the thickness and is cross-cut by two fractures that overlap
Table 1: Characterization of the Seven Gables outcrop
Dike Dike Constant dike Dike Step Dike Fabric thinning 1 stretch 2 volume? Rotation widening 3 displacement in step (°CCW) (cm) only? Seven Gables 0.4 3.0 Yes 36 1.8 42 Yes 1Dike thinning: final width/initial width 2Dike stretch: final length/initial length 3Step widening: final step width/initial step width
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Figure 8. (a) photograph looking down at the Seven Gables outcrop (a sub-horizontal surface); (b) important measurements made on the outcrop. by ~10 cm and have a step width of ~10 cm. As the by
Stanford Rock Fracture Project Vol. 23, 2012 G-8 intensifies and the fault tips deflect outward to accommodate the material moving into the step region. Because of the well-constrained geometry and strain distribution, we use the Seven Gables disappearing dike to motivate the geometry and boundary conditions of a mechanical model. Table 1 summarizes the important characteristics of the Seven Gables disappearing dike that we use to evaluate the model results.
Constraints on temperature Because there has been recent debate regarding the temperature of deformation in Bear Creek, we use microstructural observations from sample 1 (Figure 13)
Figure 9. In order to determine the three- dimensional orientation of the structures in the glacially polished Seven Gables outcrop, we considered the following: (a) a well-exposed fault surface approximately 30 m in height gives an idea about the vertical extent of the Seven Gables faults; (b) core samples (Sample 6 shown here) along the faults and aplite-granodiorite contacts indicate the dip of the faults and dikes and reveal the Figure 10. Additional examples of dikes stretched subhorizontal rake of slickenlines. and rotated through contractional steps. (a) and (b) are located in Kip Camp, while (c) is located approximately 2 km east of the Seven Gables ~10 cm and have a step width of ~10 cm. As the faults outcrop, near the Kle-Kmr contact. The slip, the step region becomes a site of contraction and occurrence of this type of structure across the left-lateral shear. This initiates the counterclockwise field area suggests that it is a characteristic rotation of the aplite dike and foliation within the step. structure of the deformation, independent of With increasing deformation, the local foliation regional variability in temperature.
Stanford Rock Fracture Project Vol. 23, 2012 G-9 to put constraints on the temperature of deformation for the Seven Gables outcrop. Quartz grains (~130 µm-1.3 mm in diameter) show clear evidence for both bulge and subgrain rotation recrystallization (Figure 13a). Feldspar microstructures include both brittle indicators, namely microcracks (Figure 13c), and evidence for plastic deformation, such as a bulging texture along the grain boundaries (Figure 13b). In addition, feldspars exhibit subtly bent twinning, flame perthite (Figure 13d), and myrmekite development (Figure 13e). Together, these quartz and feldspar microstructures are consistent within the temperature range of 400-500°C (Passchier and Trouw, 2006). An additional constraint on the temperature of deformation comes from the apparent growth of biotite into the foliation plane (Figure 13f). This indicates that biotite was a stable phase during deformation and suggests a temperature of 400-450°C (Amato et al., 2002). Thus, the microstructures record a temperature of deformation in the range 400-500°C. Because water weakening has been observed in shear zones in the Bear Creek field area (Kronenberg et al., 1990), this temperature estimate may be 150-200°C too high (Tullis and Yund, 1980). The microstructures observed in the Seven Gables sample are similar to microstructures found in wet Westerly granite deformed experimentally at a strain rate of 10 -6 s-1 and temperature 500-700°C (Tullis and Figure 11. The transition from locally foliated to undeformed granodiorite occurs sharply across Yund, 1980). In that study, cross-cutting relationships fault planes; (a) Close-up photograph of the were used to determine that grain-scale fractures western-most fault tip in the Seven Gables preceded the plastic deformation in the feldspars and outcrop. The area below the fault is strongly that the water enabled the mode of deformation to foliated in the direction of the stretched dike. change progressively from microfracturing to Immediately above the fault, the granodiorite is dislocation glide and climb with increasing strain characterized by the undeformed regional within a certain temperature range. This mechanism of foliation. (b) foliation contrast across a fault near deformation may explain the brittle and ductile Hilgard. The section of the photograph above the microstructures observed in the feldspars in the Seven fault has a strong mylonitic fabric, while the granodiorite below the fault remains undeformed. Gables sample. to put constraints on the temperature of deformation for
Figure 12. Conceptual model for the development of the disappearing dike structure. See text for discussion.
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Figure 13. Microstructural observations from sample 1 in the Seven Gables outctop were used to constrain the temperature of deformation to 400-500°C; (a) bulging (arrows) and subgrain rotation recrystallization (boxed) in quartz (Qz); (b) bulging texture (arrows) along a grain contact between plagioclase (Pl) and potassium feldspar (K); (c) microfracture (arrows) in plagioclase grain filled with quartz; (d) flame perthite (boxed) in plagioclase; (e) myrmekite (boxed) growth in plagioclase; (f) tapering (arrows) of biotite (Bi) grain in the direction of the regional foliation, S1.
Possible constitutive laws Stanford Rock Fracture Project Vol. 23, 2012 G-11 Possible constitutive laws frictional properties and a dependence on the hydrostatic stress. The Drucker-Prager yield criterion is In general, deformation can be broken into elastic defined as (recoverable) and inelastic (irreversible) components. (4) Inelastic deformation results in permanent strain and is = + a complicated phenomenon potentially described by a where and are material parameters related to the large number of constitutive laws. In this paper, we angle of friction and cohesion, respectively. In stress consider only a handful of the possible constitutive laws space, this yield surface takes the shape of a cone, such and have focused on end-member scenarios to best that the diameter of the yield surface increases with identify the mechanical sensitivities of brittle-ductile increasing hydrostatic stress (Figure 15a). deformation. Although Drucker-Prager elastic-plasticity takes the hydrostatic stress into account, the material is restricted Plasticity from yielding during pure hydrostatic loading. In Figure Plasticity describes the behavior of a material that 15a, one can imagine the stress state increasing along experiences permanent deformation when subjected to a the hydrostatic axis to infinity without causing the stress that exceeds a critical value, called the yield material to yield. This type of behavior is unrealistic for stress (Fung, 1965). Material behavior is often deemed many materials, particularly porous rocks. To correct elastic-plastic if it includes a component of both elastic for this, a cap can be added to the Drucker-Prager yield and plastic deformation. This is illustrated in a surface (Figure 15b). The cap is often referred to as the phenomenological model shown in Figure 14, in which volumetric yield surface, as volumetric strain occurs the elastic-plastic model contains both elastic and when the stress state intersects the cap (Desai and plastic elements. The constitutive laws for plasticity Siriwardane, 1984). have two main objectives: to define the yield criteria as An elastoplastic constitutive law makes it possible a three-dimensional surface in principal stress space, for a model to capture the brittle (elastic) to ductile and to define the flow rule once yielding has occurred (Fung, 1965). Although numerous formulations exist for elastic-plasticity, this paper focuses on three: von Mises perfect plasticity, Drucker-Prager frictional plasticity, and Drucker-Prager Cap plasticity. An associated flow law, in which the direction of flow is perpendicular to the yield surface, is assumed for each of these models. The von Mises yield criterion and flow rule define the simplest plastic material (von Mises, 1913). Yielding occurs when the second invariant of the deviatoric stress tensor, , reaches a critical value (Fung, 1965): (3) = The von Mises yield criterion can be represented as a smooth, cylindrical surface in stress space (Figure 15a). The yield criterion is independent of the first invariant of the Cauchy stress tensor, also called the hydrostatic stress. This is illustrated by the constant diameter of the cylinder with increasing hydrostatic stress in Figure15a. Materials described by a criterion independent of the hydrostatic stress are called frictionless materials (Desai and Siriwardane, 1984). Although frictionless plasticity accurately Figure 14. Phenomenological models for elastic- represents the behavior of some materials, primarily plastic, elastic-viscous, and elastic-viscoplastic metals, the strength of many geologic materials is constitutive laws. Elastic behavior is represented as a spring characterized by the Young’s modulus, strongly dependent on the hydrostatic stress. For E. Plastic behavior is represented as a sliding example, the strength of soils and porous rock increases element characterized by the yield stress, σy. with hydrostatic stress and has frictional characteristics. Viscous behavior is represented as a dashpot Several formulations, including Drucker-Prager elastic- characterized by the pre-exponential factor and the plasticity (Drucker and Prager, 1952) incorporate stress exponent of the creep flow law.
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Figure 15. Depictions of the yield surfaces used for the plasticity models: (a) von Mises yield surface fits as a right-cylinder within the cone-shaped Drucker-Prager yield surface in principal stress space; (b) in Drucker-Prager Cap plasticity, a cap is placed over the end of the Drucker-Prager shear yield surface in p-q space.
(plastic) transition with increasing confining pressure. The low-stress creep regime is characterized by the Although elastic-plastic constitutive laws have been diffusion of vacancies either through the crystal lattice tailored for either metals or granular materials, Nova (Nabarro-Herring creep; Nabarro, 1948; Herring, 1952) (1986) suggests that the basic structure of elastic- or along grain boundaries (Coble creep; Coble, 1963). plasticity could be used as a foundation for modeling In diffusion creep, materials behave as Newtonian brittle-ductile deformation in low-porosity, crystalline fluids, in which the strain rate is linearly related to the rocks, such as granite. differential stress ( =1). Furthermore, the strain rate is sensitive to the grain size ( = 2-3). Creep At moderate differential stress, creep occurs primarily by the movement of dislocations. This The long-term steady-state flow of rock throughout mechanism is independent of grain size ( ), but most of the earth can be characterized by a creep = 0 very sensitive to the value of the differential stress: rheology (Ji and Xia, 2002). Creep is a time-dependent (6) deformation that results when a material is subjected to = exp − stress over a long period of time. Unlike plasticity, This flow law is referred to as the Weertman creep deformation may occur at any stress and is not equation in which the strain rate of creep is controlled characterized by a yield criterion. Many variables by dislocation climb. Although the stress exponent, , influence creep deformation in rocks, including the is often quoted to range between 2 and 5 (e.g., Nicolas temperature, confining pressure, presence of water, and and Poirier, 1976; Poirier, 1985), experimental results chemical environment (Ji and Xia, 2002). The following power-law constitutive equation is generally indicate that may vary from 1.5 to 11 (e.g., Kirby, used to describe creep behavior (Poirier, 1985): 1983; Kirby and McCormick, 1984). (5) In areas of high stress, such as near a fracture tip, = exp − the stress-dependence of the creep rate exceeds what where is the steady-state strain rate, is the pre- would be predicted by the power law constitutive equation. This phenomenon is referred to as “power law exponential factor, is the differential flow stress, is breakdown” (PLB) (Sherby et al., 1954; Tsenn and the stress exponent, is the apparent activation energy, Carter, 1987). The cause of PLB is not fully is the ideal gas constant, is the absolute understood, but may be related to the generation of temperature (K), is the grain size and is the grain excess vacancies and an increased contribution from size exponent. Importantly, this constitutive law is pipe diffusion to the overall deformation (Sherby and appropriate only in the case of steady-state creep when Burke, 1968; Sherby et al., 1975; Tsenn and Carter, either the differential flow stress or the strain rate is 1987). Rather than a power-law relationship, the strain constant. The steady state creep rate is a strong function rate has an exponential dependence on the differential of the differential stress, resulting in three different stress (Samanta, 1971; Sherby et al., 1954): stress-dependent creep regimes (Sherby and Burke, (7) 1968). = exp − exp ( )
Stanford Rock Fracture Project Vol. 23, 2012 G-13 where and are constants and is the activation Finite element model energy for creep in the PLB regime. The commercial finite element software Efforts have been made to derive empirical Abaqus/CAE is used to numerically test the ability of equations that can simultaneously describe both the six different constitutive laws to accurately depict the moderate and high stress regimes. A commonly cited brittle-ductile deformation observed in the Seven equation is the power law hyperbolic-sine law Gables outcrop. We use forward modeling from the (Garofalo, 1965; Sellars and Mctegart, 1966) [Eq 8]: undeformed state, which is based on the kinematic (8) model given in Figure 12. The fault length and step = (sinh ) exp − dimensions are true to the outcrop measurements. where is the uniaxial equivalent strain rate, is the Faults assume a Coulomb friction law with a coefficient equivalent deviatoric stress , , are user defined of friction, µ=0.4. The boundary conditions are material parameters. It follows that moderate stress employed in two steps: (1) an isotropic pressure of 250 values (such that < 0.8) result in Equation (8) MPa to simulate the lithostatic load at 10 km depth; and reducing to the power law equation, Equation 6, with (2) horizontal contraction of 3% to induce left-lateral = . For high stress values (such that > 1.2) slip on the faults. The model geometry and boundary Equation (8) reduces to the exponential form, Equation conditions are illustrated in Figure 16. (7), with = /2 and = (Tsenn and Carter, While maintaining the same initial model geometry 1987). and boundary conditions, the model is run six times with different constitutive laws defining the rheology of Viscoplasticity the granodiorite and aplite. The material properties used Viscoplasticity essentially considers plastic to define the constitutive laws (see Table 2) are taken deformation as a function of time (Cristescu, 1989; from the published rock mechanics literature when Desai and Zhang, 1987). Although experimental studies available, and are estimated when the data is lacking. on the viscoplastic properties of granite are rare, The elastic moduli are held constant in each of the Maranini and Yamaguchi (2001) developed a models. viscoplastic model that accurately predicts the deformation of granite in a series of triaxial creep tests. Results Although the maximum confining pressure in these Modeling results for the most contractional experiments was limited to 40 MPa, the model could principal inelastic strain, are given in Figure 17. potentially be extrapolated to conditions more Calculations and observations made for each model are representative of the mid-crust (Maranini and given in Table 3 and are briefly discussed in this Yamaguchi, 2001). Viscoplasticity can be conceptually section. The results given in Table 3 can be compared visualized as an elastic-plastic model in parallel with an to Table 1, which summarizes the important features in elastic-viscous model, as shown in Figure 13. In this the Seven Gables outcrop. The distribution of paper, the elastic-plastic component is defined by von contractional inelastic strain (either plastic or creep) is Mises elastic-plasticity while the elastic-viscous used as an indication of where ductile fabrics are component is defined by a power-law creep flow law. expected to develop. Note that the asymmetric strain
Figure 15. Model geometry and boundary conditions, which are employed in two steps.
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Table 2: Modeling parameters
Elastic (GPa) Source aplite 60 0.2 Leeder and Pérez-Arlucea (2006) granodiorite 74.8 0.2735 Morrow and Lockner (2006)
(GPa) Source Von Mises 1.1 Estimate
(MPa) (°) Source Ducker-Prager 76 51 Morrow and Lockner (2006)
(GPa) Source Cap 1 0 0 2.85 Estimate
-n (MPa /s) (kJ/mol) Source Power-law creep 2 x10 -4 1.9 141 Hansen and Carter (1983)
-n (MPa /s) (kJ/mol) Source Hyp.-sine creep 3 x10 -2 2 x10 -10 1.9 141 Estimate
(GPa) (MPa -n/s) (kJ/mol) Source Viscoplasticity 1.1 2 x10 -4 1.9 141 Estimate
distribution in the model results is due to the difference dike achieves little thinning (0.9) with no stretch or in fault lengths, in addition to the oblique angle of the rotation. In addition, the step width does not grow aplite dike. during deformation and the dike is displaced only ~8 cm across the step. The step region does not accumulate Von Mises elastic-plasticity any plastic strain (note that the red contour represents 0.0 contractional plastic strain), so the dike thinning This model is fairly successful in achieving represents elastic deformation only. Extensional plastic significant thinning (0.6), stretching (1.4) and rotation strain in this model concentrates in the near-tip regions (16˚) of the aplite dike. The aplite dike is displaced ~18 outside the fault step, which is not consistent with cm through the step and the step width increases to 1.3 outcrop observations. times its initial width. Furthermore, the fault tips gently deflect outward from the fault step, which is consistent with the observations in the outcrop, shown in Figure 8. Drucker-Prager Cap elastic-plasticity The model results indicate that the greatest In this model, the dike experiences considerable contractional plastic strain occurs within the aplite thinning (0.5), rotation (7˚), and displacement (~13 adjacent to each fault and gradually decreases outward cm), but little stretch (1.1). As a result, the area of the from the step center. This distribution of contractional dike decreases by ~50% which violates the field plastic strain is similar to that of the ductile fabric in the observations of volume conservation in the deformed Seven Gables outcrop; however, there are some dikes. The distribution of contractional plastic strain is important discrepancies. In addition to plastic strain fairly successful, in that it is limited to within the step localizing within the fault step, it also occurs in the region with sharp transitions across the faults. On the near-tip regions outside of the model fault step. This is other hand, the modeled distribution of contractional inconsistent with field observations, which demonstrate plastic strain is more localized than the fabric a sharp transition between foliated and unfoliated distribution observed in outcrop. In addition, the granodiorite across fault surfaces (Figure 11). maximum magnitude of the contractional plastic strain maximum magnitude of the contractional plastic strain Drucker-Prager elastic-plasticity (1.032) is far greater than that calculated in the von Mises (0.66) and Drucker-Prager (0.0) models. This model is the least successful in reproducing the deformation observed in the Seven Gables outcrop. The
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Figure 17. Model results for most contractional principal inelastic strain. (a) von Mises elastic-plasticity; (b) Drucker-Prager elastic-plasticity; (c) Drucker-Prager Cap elastic-plasticity; (d) Power-law creep; (e) Hyperbolic-sine creep; (f) Viscoplasticity.
Power-law and hyperbolic-sine creep Stanford Rock Fracture Project Vol. 23, 2012 G-16 Table 1: Summary of results
Dike Dike Constant dike Dike Step Dike Fabric thinning 1 stretch 2 volume? Rotation widening 3 displacement in step (°CCW) (cm) only? Von Mises 0.6 1.4 Yes 16 1.3 18 No Drucker-Prager 0.9 1 No 0 1 8 No Cap 0.5 1.1 No 7 1 13 Yes Power-law creep 0.9 1.1 Yes 0 1.1 8 No Hyp.-sine creep 0.9 1.1 Yes 0 1.1 8 No Viscoplasticity 0.5 2 Yes 14 1.7 22 No 1Dike thinning: final width/initial width 2Dike stretch: final length/initial length 3Step widening: final step width/initial step width
Clearly, the Drucker-Prager elastic-plastic Power-law and hyperbolic-sine creep constitutive law is not capable of depicting the outcrop Because these models produced nearly identical deformation. This is likely due to the frictional nature results, they are discussed together in this section. For of the constitutive law and the hardening of the yield both creep models, the aplite dike achieves little criterion with increasing hydrostatic stress. The step thinning (0.9), stretch (1.1), displacement (~8 cm), and region is an area of increased hydrostatic stress relative no rotation of the aplite dike. The most contractional to its surroundings (Bürgmann and Pollard, 1994). principal creep strain is located within the dike adjacent Therefore, yielding within the step necessitates a higher to each of the faults. The hyperbolic-sine creep model deviatoric stress in the Drucker-Prager model than in has a greater magnitude of the most contractional the models governed by different constitutive laws. We principal creep strain (1.6e-01), compared to the power- therefore conclude that a constitutive law that law model (1.5e-01). The creep strain decreases moving accurately describes brittle-ductile rheology is likely to outward from the step and covers a larger area than be frictionless. what is observed in the previous models. In addition, The von Mises model demonstrates that a near-tip regions outside of the fault step accumulated frictionless, perfect plasticity model provides a fairly creep strain, which is inconsistent with field accurate representation of the deformation within the observations. step. However, the von Mises model also results in plastic deformation occurring outside the step region, Viscoplasticity which contradicts the field observations. Plasticity in the non-step region is likely related to the exclusive This model is the most successful in reproducing the dependence of the von Mises yield criterion on . outcrop deformation. The dike experiences significant Elastic analyses of contractional fault steps demonstrate thinning (0.5), stretch (2), rotation (14˚), and that the maximum shear stress distribution is symmetric displacement (~22 cm). In the final geometry, the step about the fault tips (Bürgmann and Pollard, 1994). This width has increased to 1.7 times its initial width and the allows to reach the critical value outside the step, fault tips deflect gently away from the step center. The resulting in plastic deformation. In order to limit the distribution of the most contractional plastic strain is plastic deformation to within the step region, the yield similar to what was observed in the von Mises model. criterion should include a dependence on the The maximum magnitude of contractional plastic strain hydrostatic stress. Because there is a difference in sign (0.83) is less than that reached in the cap model (1.032) of the hydrostatic stress inside and outside the step, the and occurs within the aplite dike adjacent to each of the yield criterion could potentially restrict the plastic faults. The magnitude of this strain decreases gradually deformation within the step. moving outward from the step center. Contractional The Drucker-Prager Cap model provides an plastic strain also occurs in the near tip regions outside example of a yield criterion that depends on the the step. hydrostatic stress. Although this model does an excellent job of limiting the plastic deformation to Discussion within the step region, the non-conservation of area of Although none of the models were completely the aplite dike suggests that the plastic flow rule is successful in reproducing the deformation observed in incorrect for the style of deformation observed in the outcrop, there is still much we can learn about brittle- field. Yielding along the cap surface is designed to ductile deformation from the relative success of each emulate compaction through volumetric plastic strain. constitutive law. Although this is appropriate for other geologic
Stanford Rock Fracture Project Vol. 23, 2012 G-17 materials (e.g., soils, high-porosity rocks), aplite dikes is a series of numerical models that test the ability of in Bear Creek do not strain volumetrically within six different constitutive laws to accurately depict the contractional fault steps. Nevertheless, this model observed deformation. The main conclusions from the demonstrates that a dependence of the yield criterion on results of the modeling are as follows: the hydrostatic stress is critical to limiting plastic strain 1. Frictionless elastic-plasticity (e.g., von Mises) is to the step region. This pressure-sensitive yielding may more appropriate than frictional elastic- be related to water weakening, which occurs only at plasticity (e.g., Drucker-Prager). high confining pressure (Tullis and Yund, 1980). 2. A dependence of the plastic yield criterion on Compared to the von Mises model, the creep the hydrostatic stress is necessary to localize models are relatively unsuccessful in reproducing the plastic deformation to within contractional fault outcrop deformation. As in the von Mises and Drucker- steps. Prager models, the lack of dependence on the 3. The plastic flow law should not contain a hydrostatic stress likely contributes to the inaccuracy of volumetric strain component as this would the final creep strain distribution. In addition, the creep represent volume loss within contractional material parameters determined experimentally may be steps. invalid or out-of-date. 4. The viscoplastic constitutive law is most It is difficult to perform laboratory experiments on successful in reproducing the deformation polymineralic rocks at high temperature-pressure within the step, but is not totally accurate in its conditions for a number of reasons, including mineral prediction of the distribution of ductile fabrics. reactions and partial melting (Ji and Xia, 2002). The final geometry of the structure in the Seven Compared with monomineralic flow laws available, the Gables outcrop is difficult to reproduce numerically and parameters determined by Hansen and Carter (1983) are the finite element model results are quite different for quite different. For example, Hirth et al. (2001) each constitutive law tested. This indicates that this methodology has the potential to eventually identify a determined that for quartzite, = 10 and = 4. In addition, Hirth et al. (2001) incorporate a term for the constitutive law that properly depicts the observed water fugacity, which is not explicitly included in the deformation. In addition to this study of brittle-ductile Hansen and Carter (1983) flow law. The difference in deformation, the workflow used in this project could have applications in other endeavors, as well. For the stress exponent between the wet granite ( = 1.9) example, one could investigate the mechanical and the quartzite ( = 4) has a significant impact on the resulting strain rate. It is therefore critical to use properties of shale by using field or well-bore accurate material parameters and this should motivate observations to motivate the geometry and boundary further laboratory research into the creep properties of conditions of a finite element model and then polymineralic rocks. experiment with the constitutive law and material The viscoplastic model appears to have been the properties. most successful, in terms of achieving the greatest Finally, there is a need for more experimental data thinning, stretch, and rotation of aplite within the fault on the inelastic properties of polymineralic rocks. Using step while conserving area. However, limited laboratory accurate parameters is critical to attaining accurate research has explored the viscoplastic properties of model results. Not only is there a need for a better granitic rocks. Such information could provide great understanding of plastic and viscous deformation of insight into the rheology of the crust under brittle- granite, but of aplite, as well. It seems likely that the ductile conditions. It is important to note that the contrast in material properties between these different viscoplastic properties depend wholly on the plasticity lithologies could be key to numerically representing the and creep laws implemented. As those constitutive laws observed deformation. and relevant material properties evolve in future modeling, the success of the viscoplastic model will References change as well. Ague, J.J., and Brimhall, G.H., 1988, Magmatic Arc Asymmetry and Distribution of Anomalous Conclusions Plutonic Belts in the Batholiths of California - This paper has discussed structures located in the Effects of Assimilation, Crustal Thickness, and Depth of Crystallization: Geological Society of Bear Creek field area that likely developed under America Bulletin, v. 100, p. 912-927. brittle-ductile conditions (4-15 km depth and 400- Amato, J.M., Miller, E.L., and Hannula, K.A., 2002, 500°C). We focus specifically on the Seven Gables Orthogocal flow directions in extending continental outcrop, which includes an aplite dike that is rotated crust: An example from the Kigluaik gneiss dome, and stretched though a contractional step between two Seward Peninsula, Alaska, in Miller, E.L., Grantz, left-lateral faults. Complementing the field observations A., and Klemperer, S., eds., GSA Special Paper, Volume 360, p. 133-146.
Stanford Rock Fracture Project Vol. 23, 2012 G-18 Behr, W.M., and Platt, J.P., 2011, A naturally constrained International Journal for Numerical and Analytical stress profile through the middle crust in an Methods in Geomechanics, v. 11, p. 603-620. extensional terrane: Earth and Planetary Science Drucker, D.C., and Prager, W., 1952, Soil mechanics and Letters, v. 303, p. 181-192. plastic analysis or limit design: Quarterly of Brace, W.F., and Kohlstedt, D.L., 1980, Limits on Applied Mathematics, v. 10, p. 157-165. Lithospheric Stress Imposed by Laboratory Fiore, P.E., Pollard, D.D., Currin, W.R., and Miner, D.M., Experiments: Journal of Geophysical Research, v. 2007, Mechanical and stratigraphic constraints on 85, p. 6248-6252. the evolution of faulting at Elk Hills, California: Burgmann, R., and Pollard, D.D., 1992, Influence of the State AAPG Bulletin, v. 91, p. 321-341. of Stress on the Brittle-Ductile Transition in Flugge, W., 1967, Viscoelasticity: London, Blaisdell. 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Stanford Rock Fracture Project Vol. 23, 2012 G-20 Tsenn, M.C., and Carter, N.L., 1987, Upper Limits of Power Law Creep of Rocks: Tectonophysics, v. 136, p. 1- 26. Tullis, J., 1990, Experimental studies of deformation mechanisms and microstructures in quartzo- feldspathic rocks, in Barber, D.J., and Meredith, P.G., eds., Deformation processes in minerals, ceramics and rocks: London, Unwin Hyman Press, p. 190-227. Tullis, J., and Yund, R.A., 1977, Experimental Deformation of Dry Westerly Granite: Journal of Geophysical Research, v. 82, p. 5705-5718. Tullis, J., and Yund, R.A., 1980, Hydrolytic Weakening of Experimentally Deformed Westerly Granite and Hale Albite Rock: Journal of Structural Geology, v. 2, p. 439-451. von Mises, R., 1913, Mechanik der festen Körper im plastisch deformablen Zustand: Göttin. Nachr. Math. Phys. Kl., v. 1913, p. 582-92. Weertman, J., 1978, Creep Laws for Mantle of Earth: Philosophical Transactions of the Royal Society of London Series a-Mathematical Physical and Engineering Sciences, v. 288, p. 9-26.
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