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, ʖ- , and the differential stress, : naturally deformed contractional fault step in (2) - ) ͋ granodiorite that developed at a depth of 4-15 km and ʖ = ̻ exp ƴ− Ƹ 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