Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
J. metamorphic Geol., 2006 doi:10.1111/j.1525-1314.2006.00644.x The effects of porphyroblast growth on the effective viscosity of metapelitic rocks: implications for the strength of the middle crust W.G. GROOME, S.E. JOHNSON AND P.O. KOONS Department of Earth Sciences, University of Maine, Orono, ME 04469, USA ([email protected]) ABSTRACT Numerical models are used to examine the effects of porphyroblast growth on the rheology of compositionally layered rocks (metapelites and metapsammites) and by extension the middle crust during prograde metamorphism. As porphyroblast abundance increases during prograde metamorphism, metapelitic layers will strengthen relative to porphyroblast-free metapelitic units, and potentially relative to quartzofeldspathic metapsammitic units. As metapelitic layers become stronger, the integrated strength of compositionally layered successions increases, potentially causing large volumes of midcrustal rock to strengthen, altering the strain-rate distribution in the middle crust and affecting the geodynamic evolution of an orogenic belt. The growth of effectively rigid porphyroblasts creates strength heterogeneities in the layer undergoing porphyroblast growth, which leads to complex strain-rate distributions within the layer. At the orogen scale, the strengthening of large crustal volumes (on the order of thousands of cubic kilometres) changes the strain-rate distribution, which may change exhumation rates of high-grade metamorphic rocks, the geothermal structure and the topography of the orogen. The presence of a strong zone in the middle crust causes strain-rate partitioning around the zone, suppressed uplift rates within and above the zone and leads to the development of a basin on the surface. Key words: metamorphic strengthening; geodynamics; numerical modelling; porphyroblasts; rheology; strain-rate partitioning. INTRODUCTION The rheological structure of the lithosphere is a firstorder control on the distribution of strain within an orogen, which can in turn affect processes such as the localization of metamorphism (e.g. Kerrich et al., 1977; Rubie, 1983; Brodie & Rutter, 1985; Koons et al., 1987; Frueh-Green, 1994; Baxter & DePaolo, 2004), the exhumation of deep crustal rocks (e.g. Platt, 1986; Koons, 1987; Beaumont et al., 2001; Zeitler et al., 2001; Jamieson et al., 2002; Koons et al., 2002) and the topographic evolution of an orogen (e.g. Koons, 1989, 1995; Williams et al., 1994; Carminati & Siletto, 1997). Much of our knowledge concerning the rheology of earth materials comes from laboratory experiments conducted on monomineralic aggregates (natural and synthetic) under pressure, temperature and strain rate conditions not generally expected in natural settings (e.g. Tullis & Yund, 1977; Arzi, 1978; Paquet et al., 1981; Ji & Zhao, 1993; Shea & Kronenberg, 1993; Farver & Yund, 1999; Renner et al., 2000; Hirth et al., 2001; Rosenberg, 2001; Stunitz & Tullis, 2001; Mecklenburgh & Rutter, 2003; Tenthorey & Cox, 2003). In this paper, rock ÔstrengthÕ is used to characterize the resistance to deformation of a rock, which can be described in several ways, including the yield stress (r1 ) r3) at failure for plastic deformation, _ for viscous deformation the effective viscosity (r=e) (either Newtonian or non-Newtonian), or the elastic 2006 Blackwell Publishing Ltd modulus (r/e) for elastic deformation. Broadly, strength can be viewed as the resistance to deformation of a rock unit, and, unless otherwise stated, this broad definition of strength is used. Understanding the strength of polymineralic rocks is problematic and has been approached in four main ways: (1) analytical treatments viewing rocks as composites of individual minerals with known strengths, such that the bulk strength of the rock is a function of the volume fraction and relative strengths of different phases (e.g. Burg & Wilson, 1987; Jordan, 1988; Handy, 1990; Tullis et al., 1991; Ji & Zhao, 1993; Treagus, 2002; Ji, 2004); (2) experiments using analogue materials such as normcamphor (e.g. Bons & Cox, 1994; Bons & Urai, 1994) and silicone putty (e.g. Treagus & Sokoutis, 1992), and two-phase mixtures of weak minerals such as halite and calcite (e.g. Jordan, 1987; Ji et al., 2001); (3) numerical experiments studying the effects of different volume fractions and relative strengths of constituent materials (e.g. Tullis et al., 1991; Bons & Cox, 1994; Treagus, 2002; Johnson et al., 2004); and (4) field measurements to constrain the relative strengths of different rock types deformed under natural conditions (e.g. Smith, 1977; Lisle et al., 1983; Kanagawa, 1993; Treagus, 1999; Treagus & Treagus, 2002; Kenis et al., 2005; Groome & Johnson, 2006). During orogenesis, parts of the middle crust will periodically strengthen and weaken during 1 2 W.G. GROOME ET AL. metamorphism (e.g. Beach, 1980; Rubie, 1983; Brodie & Rutter, 1987; Koons et al., 1987; De Bresser et al., 2001; Handy et al., 2001). The strength of a rock during metamorphism will change if: (1) reactants have a different strength than the products, which alters the ductile strength of the rock (e.g. Arzi, 1978; Beach, 1980; Poirier, 1982; Brodie & Rutter, 1987; Handy et al., 2001); (2) product minerals are fine-grained enough for the deformation mechanism to change, decreasing differential stress or increasing strain rate, which alters the ductile strength of the rock (e.g. Brodie & Rutter, 1987; Koons et al., 1987; De Bresser et al., 2001; Stunitz & Tullis, 2001); and (3) fluid pressure changes, which alters the brittle strength of the rock (e.g. Etheridge et al., 1983; Sibson, 2003). Weakening or strengthening can occur if product phases have different strengths than reactant phases. An example of this process occurs during partial melting, when relatively weak partial melt is produced, leading to an overall weakening of the rock (e.g. Arzi, 1978). A solid-state example of weakening by the replacement of relatively strong minerals by relatively weak minerals occurs during the serpentinization of mafic rocks, when relatively strong olivine grains are replaced by relatively weak serpentine grains (Barnes et al., 2004). Conversely, strengthening should be expected during the prograde growth of garnet, which is relatively strong (e.g. Wang & Ji, 1999), in metapelitic rocks because the relative volume fraction of strong garnet increases. Weakening can occur if reaction products are finer grained than the reactants because the dominant deformation regime in the zones of reaction may switch from intracrystalline creep mechanisms to grain size-dependent creep mechanisms, which generally accommodate higher strain rates (e.g. Passchier & Trouw, 1996). However, a situation may develop in which the grain size is maintained near the transitional grain size between intracrystalline and intercrystalline creep mechanisms (e.g. De Bresser et al., 2001). During continued reaction, grain size may increase, shifting the dominant deformation mechanism into the intracrystalline creep regime, causing strengthening (e.g. Brodie & Rutter, 1987; Koons et al., 1987; De Bresser et al., 2001). Fluids liberated during dehydration metamorphism may remain trapped in the rock at grain boundaries, causing weakening, and possibly brittle fracture, as the pore pressure increases (e.g. Hubbert & Rubey, 1959; Sibson, 2003). This is probably a transient process, because as the pore pressure increases and the rock weakens, deformation will partition into the weakened zone, which will serve to liberate the fluids, thus strengthening the rock again until fluid pressure can build up to weaken the rock (e.g. Etheridge et al., 1983; Selverstone et al., 1991; Tobisch et al., 1991; Handy et al., 2001; Sibson, 2003). Fluid-enhanced weakening may also occur during the retrograde metamorphism of relatively anhydrous rocks if the fluids catalyse reactions that produce relatively weak minerals (e.g. Rubie, 1983; Brodie & Rutter, 1987; Koons et al., 1987; Selverstone et al., 1991; FruehGreen, 1994; Moecher & Wintsch, 1994; Barnes et al., 2004). In this paper, we explore the effects of porphyroblast growth in layered sequences on the effective viscosity structure and strain-rate partitioning at the grain and bed scale, and address the strengthening effects of porphyroblast growth on lithosphere-scale strength profiles and the geodynamics of orogens. THEORETICAL BACKGROUND The strength of polyphase aggregates at given pressure and temperature conditions is generally considered to be controlled by the volume fraction, distribution and relative strength of the constituent phases (e.g. Jordan, 1988; Handy, 1990; Tullis et al., 1991; Ji & Zhao, 1993; Takeda, 1998; Ji & Xia, 2002; Treagus, 2002; Takeda & Obata, 2003; Ji, 2004; Johnson et al., 2004). Two theoretical bounds, relating the elastic strength of an aggregate to the volume fraction and relative strengths of the constitutive phases, are generally considered to define the maximum and minimum strength of an aggregate (e.g. Handy, 1990; Tullis et al., 1991; Ji & Zhao, 1993; Takeda, 1998; Ji & Xia, 2002; Ji, 2004). The Voigt bound assumes that all grains within an aggregate will experience the same strain, and that the distribution of differential stresses will differ from grain to grain such that strong grains will accumulate higher differential stresses to achieve the same strain as weak grains, which deform at lower differential stress (e.g. Handy, 1990; Tullis et al., 1991; Ji & Zhao, 1993; Ji & Xia, 2002): ra ¼ /s rs þ /w rw ; ð1Þ where r is the differential stress in the aggregate (a), strong phase (s) and weak phase (w), and / is the volume fraction of a given phase. The Reuss bound assumes that all grains within the aggregate experience the same differential stress and that strain in individual grains will be different depending on the bulk modulus of each grain (e.g. Handy, 1990; Tullis et al., 1991; Ji & Zhao, 1993; Ji & Xia, 2002): /s /w 1 þ : ð2Þ ra ¼ rs rw Strictly, the Voigt and Reuss Bounds are defined for homogeneous elastic materials deformed at low elastic strain; however, the two bounds can be reformulated for homogeneous viscous materials by assuming that the bulk modulus (elastic stress/elastic strain) is analogous to viscosity (flow stress/viscous strain rate) (e.g. Tullis et al., 1991; Ji & Zhao, 1993; Ji & Xia, 2002). In this paper, the following Voigt and Reuss formulations are used for viscous materials (Fig. 1): 2006 Blackwell Publishing Ltd Aggregate viscosity ) VB 1( 75 0. 50 5 0. 5 0 2 0. –0 .2 5 –0 .5 Aggregate viscosity 5 .7 –0 1 0 –1 0.25 0.5 0.75 Volume fraction of strong phase 5 5 .2 –0 5 –0. –0.75 B) (R 1.0 0 5 0.2 0.5 1 (V 10 0.7 5 100 B) EFFECTIVE VISCOSITY OF PORPHYROBLASTIC ROCKS 3 –1 (RB) 1 0 0.25 Volume fraction of strong phase 0.5 Fig. 1. (a) Plot of the Voigt (VB) (Eq. 3a) and Reuss (RB) (Eq. 3b) bounds and the generalized mixing rule of Ji (2004) (Eq. 4) for an arbitrary two-phase mixture with two orders of magnitude difference in viscosity between the weak and strong phase. Dashed lines show the effects of different J values in Eq. 4. (b) Close-up of the light grey box on panel a. Note that the rate of viscosity increase with increasing volume fraction of the strong phase is a function of the J exponent. ga ¼ / s gs þ / w gw ; ga ¼ /s /w þ gs gw ð3aÞ 1 ; ð3bÞ where g is the effective viscosity: r g¼ : ð3cÞ e_ Ji (2004) provided a generalized mixing rule that describes the behaviour of two phase mixtures: ( " #)1=J Ms J 1 ; ð4Þ Ma ¼ Mw 1 þ Vs Mw where M is a specific mechanical property (e.g. viscosity) of the composite (a), weak (w) or strong (s) phase, J is a constant dependent on the specific microstructure, and V is the volume fraction of the strong grain (Fig. 1). The parameter J can be varied to account for specific microstructural features (e.g. grain shape, connectivity and continuity) in the aggregate. For instance, J 0 approximates the geometrical mean (similar to the formulation of Tullis et al. (1991), J ¼ ±0.5 approximates aggregates with spherical grains, and J ¼ ±0.25 approximates aggregates with grains of arbitrary shape (Ji, 2004). The sign of J describes whether the aggregate consists of strong grains in a weak matrix (J > 0) or weak grains in a strong matrix (J < 0). The Voigt and Reuss limits can be described with J ¼ 1 and J ¼ )1, respectively. In natural systems, the strength of a polyphase aggregate is considered to lie between the Voigt and Reuss bounds, with the Reuss bound being an approximation of strength for a material with strong grains embedded in a weak matrix and the Voigt bound being an approximation of strength for a material with weak grains embedded in a strong matrix (e.g. Arzi, 1978; Handy, 1990; Ji & Zhao, 1993; Ji 2006 Blackwell Publishing Ltd et al., 2001; Treagus, 2002). However, divergence from the theoretical bounds is observed in experiments, particularly if the volume fraction of the embedded phase approaches c. 40% (e.g. Arzi, 1978; Jordan, 1987, 1988; Tullis et al., 1991; Ji & Zhao, 1993; Treagus, 2002; Takeda & Obata, 2003), at which point the strength–volume fraction trend varies between the two theoretical bounds. In a system where volume fraction of the strong inclusion phase increases with time, the transition from Reuss- to Voigt-type behaviour corresponds to the point at which the system becomes an interconnected network of strong inclusions, and the aggregate strength unavoidably trends towards the strength of the strong phase (e.g. Ji & Xia, 2002). Finally, the Voigt and Reuss bounds only describe the theoretical strength evolution trends of homogeneous aggregates in which the embedded phase is evenly distributed through the aggregate volume. Other formulations proposed for describing the strength of two-phase aggregates consider the effects of clast shape (e.g. Jordan, 1988; Handy, 1990; Tullis et al., 1991; Treagus, 2002), power law constitutive relationships for the phases (e.g. Tullis et al., 1991; Ji & Zhao, 1993; Ji et al., 2001), the presence of a truly rigid phase (e.g. Duva, 1984; Yoon & Chen, 1990; Ravichandron & Seetharaman, 1993; Treagus, 2002) and the effects of layer anisotropy and orientation (e.g. Treagus, 1993; Ji et al., 2001). The strength–volume fraction relationships of these formulations all lie between the end-member Voigt and Reuss bounds, indicating that these limits are good descriptions for the maximum and minimum strengths of two-phase aggregates, respectively. Based on theoretical considerations, the following generalizations can be made about the strength of polyphase rocks: 1. the strength of a two-phase aggregate will lie between the end-member strengths of the constituent phases; 4 W.G. GROOME ET AL. 2. a two-phase aggregate will become stronger if the volume fraction of the strong phase increases; 3. the shape of the strong or weak phase will affect the strength of an aggregate, with square inclusions resulting in stronger aggregates than circular inclusions (e.g. Tullis et al., 1991; Treagus, 2002); 4. the presence of a truly rigid phase will strengthen an aggregate more than the presence of a strong deformable phase (e.g. Duva, 1984); 5. layered successions compressed perpendicular to layering will be stronger than homogeneously mixed two-phase aggregates (e.g. Ji & Xia, 2002); and 6. layered successions deformed in layer-parallel simple shear will be weaker than successions deformed in layer-perpendicular simple shear (e.g. Treagus, 1993). In nature, phyllosilicate-rich pelitic layers in turbidite couplets are considered to be weak because they generally record higher finite strains (e.g. Treagus, 1988, 1993, 1999 and references therein). Based on foliation refraction measurements from sedimentary and low-grade metasedimentary rocks, Treagus (1999) estimated that psammitic units have between two and 10 times higher effective viscosities than pelitic units and Kenis et al. (2005) estimated that pelitic units had effective viscosities approximately two to five times lower than psammitic units based on mullion shape measurements. Experimental deformation of mica-rich rocks suggests that mica is quite weak when sheared parallel to {001}, relative to quartz at low temperatures (e.g. Shea & Kronenberg, 1993; Tullis & Wenk, 1994), and numerical models (e.g. Johnson et al., 2004) indicated that rocks experience significant weakening when mica grains become interconnected to form a foliation. These observations suggest that the relative weakness of low-grade, phyllosilicate-rich, pelitic rock is due at least in part to the weakness of mica. In general, prograde metamorphism of pelitic rocks results in a decrease in the volume fraction of weak hydrous minerals, such as clays, and a corresponding increase in relatively strong minerals, such as garnet (e.g. Bucher & Frey, 1994). In many cases, the volume fraction of relatively strong porphyroblastic minerals (i.e. staurolite, garnet, andalusite, kyanite) can be quite high (20–30%), which should significantly alter the strength of these rocks (see Eq. 3a,b). If the volume fraction of effectively rigid porphyroblasts increases sufficiently, pelitic layers should become stronger than interlayered psammitic layers, which typically have inappropriate bulk compositions for the growth of porphyroblastic phases. NATURAL EXAMPLE The White Mountains region of the New England Appalachians (Fig. 2) preserves field evidence for the strengthening of metapelitic layers relative to metapsammitic layers in amphibolite facies metaturbidites. During prograde low-pressure, high-temperature metamorphism, large (up to 15 cm long) andalusite porphyroblasts grew in the metapelitic layers (Fig. 3a), causing these layers to become strong relative to interlayered, porphyroblast-free metapsammitic layers during a period of isoclinal, km-scale nappe folding. Based on field relationships and porphyroblast-matrix microstructure relationships, it appears that andalusite growth was prior to or synchronous with the early stages of nappe folding (e.g. Eusden et al., 1996; Groome & Johnson, 2006). The foliation developed during the nappe-stage folding anastamoses around andalusite porphyroblasts, and pressure shadows are ubiquitously developed along the margins of andalusite porphyroblasts (Fig. 3b,c), supporting the interpretation that these large porphyroblasts were present during the early deformation (e.g. Johnson, 1999; Vernon, 2004). Additionally, andalusite porphyroblasts locally define a mineral lineation that lies in the plane of the fold axis (e.g. Eusden et al., 1996; Rodda, 2005). Later regional metamorphism led to the pseudomorphing of andalusite porphyroblasts to aggregates of blocky and fibrolitic sillimanite, muscovite and staurolite (e.g. Wall, 1988; Allen, 1992; Eusden et al., 1996; Groome & Johnson, 2006). However, andalusite porphyroblasts do not appear to have undergone intracrystalline deformation prior to being pseudomorphed, suggesting that they behaved as essentially rigid objects during nappe-stage folding. The relative strengths of metapelitic and metapsammitic layers were determined using the foliation refraction technique of Treagus (1999) (Groome & Johnson, 2006). To use this technique, it is assumed that the difference in finite shear strain recorded in two rock types is a function of the effective viscosities of the two layers, such that the unit with higher effective viscosity will record lower finite shear strain (e.g. Treagus, 1983, 1988, 1999). Using geometric arguments, Treagus (1999) proposed that the ratio of bedding–foliation angles for two layers (hA/hB) is a function of the effective viscosity contrast between the layers (gA/gB) via: cA tan wA tan hB gB ¼ ¼ ; cB tan wB tan hA gA ð5Þ where c is the shear strain, w is the shear angle and subscripts A and B refer to two rock layers with contrasting viscosity (Fig. 3d–f). Using foliation refraction angles, Groome & Johnson (2006) estimated that the porphyroblast-rich metapelitic layers in the amphibolite facies metaturbidites of the White Mountains region had effective viscosities approximately two to three times greater than the interlayered metapsammitic layers. In unmetamorphosed pelite–psammite successions, the psammitic layers are expected to have higher effective viscosities because they have a higher volume fraction 2006 Blackwell Publishing Ltd Pe ab od yR . US Rte 16 EFFECTIVE VISCOSITY OF PORPHYROBLASTIC ROCKS 5 Peabody Granite anite d Gr Mig Front kfor Bic 3 St ut St O Ou t Sil In Fro nt Mig St Out Sil In 1 R. 2 hic North t rus Th 1 = Staurolite zone (St + Gt + Bt + Qz + Ms) 2 = Lower sillimanite zone (Sil + St + Bt + Gt + Qz + Ms) 3 = Upper sillimanite zone (Sil + Bt + Gt + Qz + Ms) 3 3 rp mo Peak metamorphic zones St Ou t West Peabody AR MW eta Extent of pseudomorphed andalusitebearing metaturbidites Two-mica granite 2 3 Legend Undivided migmatite st-m Po Mig Fron Mt Washington t O ut St Out St 3 t ron F Mig 2 3 2.5 km W60 W65 W70 W75 N50 PQ White Mountains Region NB ME ON NY N45 NS VT NH MA CT RI PN 200 km NJ of relatively strong quartz and feldspar as well as a generally coarser grain size. Using foliation refraction angles, Treagus (1999) estimated that unmetamorphosed (or low grade) metapsammites had effective viscosities two to 10 times greater than metapelites, and Kenis et al. (2005) estimated that porphyroblastfree metapelitic units were up to five times weaker than interlayered metapsammitic units based on mullion-shape characteristics. In the White Mountains region, metapelitic units contain up to c. 25–30% andalusite porphyroblasts, and Groome & Johnson (2006) hypothesized that the high volume fraction of effectively rigid porphyroblasts led to the metapelitic units having higher effective viscosities than metapsammitic units. The numerical modelling presented below was designed to test the hypothesis that prograde metamorphism resulting in the growth of effectively rigid porphyroblasts preferentially in 2006 Blackwell Publishing Ltd Fig. 2. Location map and simplified metamorphic assemblage map for the study area in the eastern White Mountains region of New Hampshire. NY, New York State; VT, Vermont; NH, New Hampshire; ME, Maine; NB, New Brunswick; PQ, Quebec; NS, Nova Scotia; MA, Massachusetts; CT, Connecticut; RI, Rhode Island; NJ, New Jersey; PN, Pennsylvania; MWAR, Mount Washington Auto Road. metapelitic bulk compositions could lead to a strength reversal relative to unmetamorphosed pelite–psammite successions. GRAIN- AND BED-SCALE NUMERICAL EXPERIMENTS Grain- and bed-scale models were constructed to explore the effects of porphyroblast growth on the evolving strength of metapelitic layers, the evolving strength contrast between porphyroblast-rich metapelitic layers and interbedded metapsammitic layers, and the changing strain-rate partitioning within the metapelitic layer as porphyroblast abundance increases. A linear viscous, plane strain, finite element formulation is used to investigate the problems outlined. The model domain consists of a three-layer system analagous to a single metaturbidite couplet (Fig. 4). 6 W.G. GROOME ET AL. 5 cm (a) (b) And And 5 mm (d) (c) S1 St Pel Psa Pel And S4 Pseud P-Sha dow 1m 5 mm (f) (e) ηA ψA el θP θA S1 Psa sa Pel Pe l Ps a θP 1m ηB θB γA ψB γ B = tanψ A tan θ B η B ≈ = tanψ B tan θ A η A Fig. 3. (a) Photograph of a typical andalusite-rich (And) pelitic unit. (b) Photomicrograph of a partially pseudomorphed andalusite showing anastamozing S1 foliation and a synchronous pressure shadow, indicating that the porphyroblast was present during the formation of this porphyroblast. (c) Sketch of the photograph in panel b. (d) Sketch of an outcrop from near the summit of Mt Washington showing refracting foliation through a pelite (Pel) – psammite (Psa) layered succession. (e) Photograph of a refracting foliation through a psammite–pelite layered succession. The larger bedding-foliation angles in the pelitic layers indicate that they had higher effective viscosities during the development of this foliation. (f) Schematic diagram showing a refracting foliation with the angles used to estimate relative effective viscosities. Modelling environment Analytical expressions for the strength of aggregates are useful when dealing with a constant volume fraction of two or more phases. However, to explore the evolving strength of a rock undergoing metamorphic strengthening or weakening reactions during which the relative volume fractions of strong and weak phases changes, a numerical approach is more appropriate. Furthermore, numerical treatments allow us to investigate the effects of changing mineralogy on strain-rate partitioning during syn-deformational growth. Ideally, to numerically investigate the rheological effects of porphyroblast growth on both the 2006 Blackwell Publishing Ltd EFFECTIVE VISCOSITY OF PORPHYROBLASTIC ROCKS 7 Undeformed No porphyroblasts Deformed SSR 1.0 Psammite Pelite Psammite 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Log10viscosity Shear strain rate Fig. 4. Model results for the porphyroblast-free experiment with a viscosity contrast of 2:1 (gpsa:gpel). Far left panel: One-dimensional profile of the viscosity structure through the centre of the model. Note the log10 viscosity scale. Middle panels: Shear strain-rate contour maps and the model geometry in the undeformed (left) and deformed (right) states. Deformation was by dextral simple shear, as indicated by the shear arrows. Far right panel: One-dimensional profile of the shear strain rate distribution through the model. evolving strength and strain-rate partitioning within a given unit, we would use a formulation that: (1) allows for an arbitrary geometry of individual grains that can have irregular shapes and sizes; (2) allows the volume fractions of strong and weak grains to change with time; (3) stores information, such as effective viscosity, position and crystallographic orientation, for each grain; and (4) can couple the geometric evolution with a mechanical response to an imposed stress or displacement field. The ELLE microstructural modelling platform satisfies these requirements. ELLE is an open-source, Linux-based code that can be used to model the two-dimensional microstructural evolution of a system by executing a number of specific processes in sequence for small time steps (e.g. Jessell et al., 2001). Processes that have been modelled in the ELLE environment include Taylor–Bishop–Hill lattice rotations (e.g. Piazollo, 2000), grain growth via surface energy driving forces (e.g. Jessell et al., 2001), subgrain rotation recrystallization (e.g. Piazollo, 2000), grainboundary diffusion (e.g. Park et al., 2004), porphyroblast growth (this paper), and nucleation processes. Each process is a separate open-source code and can be modified to suit the user’s requirements, and a particular ELLE model is controlled by a central script that calls each sub-routine in sequence (e.g. Jessell et al., 2001). The microstructure of a system is simulated using a geometry consisting of several nodes that contain specific information such as viscosity, chemical concentration, lattice orientation and position. Individual grains are defined by a group of bounding nodes that can hold information such as mineralogy and viscosity. Porphyroblast growth is modelled using a routine that allows the surface area of selected grains to increase by a finite amount per time step in order to ultimately form large subcircular grains. The growth routine simulates porphyroblast growth from initially dispersed nuclei in the pelite matrix. In order to simplify our modelling and focus on the rheological effects of porphyroblast growth, local chemistry is not consi 2006 Blackwell Publishing Ltd Table 1. Parameters used in grain-scale numerical models. Grain g1 a g2 g3 Pelite Psammite Porphyroblast 1 2 100 1 4 100 1 6 100 a Dimensionless viscosity in Eq. 6. dered, nor do we prescribe specific porphyroblastforming reactions to control the growth of porphyroblasts. Conservation of volume is attained by allowing grains along the margins of the growing porphyroblast to decrease in size as the porphyroblast grows. Porphyroblast grains were assigned a normalized viscosity 100 times that of the matrix grains surrounding them, making the porphyroblasts effectively rigid during deformation (Table 1). In order to model the mechanical effects of porphyroblast growth, ELLE was coupled with the two-dimensional finite element code BASIL (Barr & Houseman, 1996), which solves for incompressible, plane-strain deformation using linear and non-linear viscous constitutive relationships: _ sij ¼ 2ge; ð6aÞ 1n g ¼ 0:5BE_ n ; ð6bÞ e_ ¼ @vy @vx þ ; @x @y ð6cÞ where sij is deviatoric stress, g is the effective viscosity, B is a material constant, E_ is the second invariant of the strain rate tensor, n is a stress exponent that can vary from 1 to 3, e_ is the shear strain rate and v is velocity. In the experiments presented below, a constant displacement rate is used on the upper and lower bounds of the model domain to solve for shear stress. In order to concentrate the modelling on the rheological effects of changing porphyroblast abundance, we have chosen to use a linear-viscous constitutive 8 W.G. GROOME ET AL. relationship (n ¼ 1) so that the effects of strain hardening can be ignored at this stage, though this may not provide a satisfactory description for the deformation behaviour of natural rocks in the middle crust. layer-parallel simple shear (e.g. Treagus, 1993), which allows us to asses the end-member strengthening effects of porphyroblast growth. Porphyroblast-free models Model results Two groups of models are presented here: (1) porphyroblast-free models and (2) porphyroblast growth models. All models consist of a three-layer succession with a model pelite layer between two model psammite layers (Fig. 4). The starting microstructure for the entire system is homogeneous, with most grains being subcircular, similar to a foam-textured quartzite. A simplified microstructure was used because we did not want to impose an arbitrary geometry on the model with mica, quartz and feldspar grains in varying proportions depending on layer bulk chemistry (i.e. quartz-feldspar-dominated psammite layers and micadominated pelite layers). The only difference between model pelite and model psammite layers is the bulk viscosity of that layer (Table 1). To establish the bulk viscosity of a given layer, all grains within that layer were assigned the same viscosity. This modelling assumption was made because we wanted to focus on the bulk strengthening effect of porphyroblast growth and decided to simplify the pelite layer rheology to a two-phase mixture – bulk matrix viscosity plus porphyroblasts. Furthermore, as porphyroblast growth in our models is not governed by chemical processes, we do not need to assign individual grains in the pelite layer to be mica, quartz or feldspar. The bulk viscosity contrast between model pelite and psammite layers ranged from 2 to 6 (gpsa:gpel) in the experiments reported here. In the porphyroblast model group, porphyroblast viscosity was two orders of magnitude greater than the pelite matrix, making the model porphyroblasts essentially rigid. In our natural example, microstructural evidence suggests that the andalusite porphyroblasts were effectively rigid during deformation, but as little is known about the flow properties of andalusite, we could not use effective viscosity estimates based on experimental literature to constrain the viscosity of our model porphyroblasts. Model porphyroblast shapes are ultimately subcircular; however, the andalusite porphyroblasts in our natural example are elongate. We chose to simplify the model porphyroblast shapes to be subcircular (i.e. garnet-like) so that we could focus our results on the strengthening effects of porphyroblast growth without consideration for grain shape. Previous numerical modelling assessing the role of clast shape suggests that elongate clasts may increase the strength of aggregates more if they are oriented at an angle to the flow field than if the clasts are oriented parallel to the flow field (e.g. Treagus, 2002). In all experiments, deformation is layer-parallel dextral simple shear. The layer-parallel simple shear boundary condition was chosen because layered rocks are considered to be weakest during Porphyroblast-free layered systems were deformed to a bulk shear strain of c. 1.0 to illustrate the strain partitioning behaviour expected in a porphyroblast-free layered succession during layer-parallel simple shear (Fig. 4). The strain partitioning in these models is controlled by the viscosity ratios between the model pelite and psammite layers. The model with the lowest viscosity contrast between pelite and psammite (gpsa:gpel ¼ 2) had the least amount of strain partitioning between the two layers. Furthermore, within each bed, strain rates are homogeneously distributed in these models (Fig. 4). These models serve as a reference frame to compare with the porphyroblast growth models presented below. Porphyroblast growth models Porphyroblast growth experiments were conducted to examine the evolving viscosity structure of our layered model during porphyroblast growth in the pelite layer. In these models, porphyroblast abundance varies from <1% to >70% of the model pelite layer (Fig. 5). Porphyroblast locations were handselected in the starting geometry to be widely dispersed in the model pelite layer, in order to simulate diffusion-controlled nucleation spacing (e.g. Porter & Easterling, 1992; Vernon, 2004). Model porphyroblast growth occurs by constant surface area expansion, and is not governed by any chemical processes because we only wanted to assess the mechanical effects of porphyroblast growth. In our natural example, porphyroblast modal abundance is on the order of 25–30%, as determined visually in the field and by photographic analysis. The bulk viscosity of the model pelite layer increases by more than an order of magnitude between 1% and 70% porphyroblast abundance (Figs 6 & 7). The evolving strength trend in these models lies close to the Reuss bounding limit, consistent with what would be predicted for aggregates consisting of a strong phase embedded in a weak matrix undergoing layer-parallel simple shear (e.g. Handy, 1990; Treagus, 1993, 2002; Ji et al., 2001). Of particular note is the nearly fourfold increase of pelite layer viscosity between 0 and 30% porphyroblast abundance. The increasing porphyroblast abundance in the pelitic layer also leads to changes in strain-rate partitioning, both within the pelitic bed and within the layered system as a whole (Fig. 7). Within the pelitic layer, the highest strain rates are in grains immediately adjacent to the growing porphyroblasts, with strain rates increasing in these porphyroblast-marginal grains as the adjacent porphyroblasts get larger. As the pelitic layers strengthen, 2006 Blackwell Publishing Ltd EFFECTIVE VISCOSITY OF PORPHYROBLASTIC ROCKS 9 Initial geometry 10% Porphyroblast Psammite Pelite Psammite 20% Porphyroblast 30% Porphyroblast Psammite Pelite Psammite 40% Porphyroblast 50% Porphyroblast Psammite Pelite Psammite 60% Porphyroblast 70% Porphyroblast Psammite Pelite Fig. 5. Initial geometries for the porphyroblast deformation experiments showing porphyroblast abundances within the pelitic beds ranging from 0 to 70%. Porphyroblasts are black. higher strain rates are recorded in the interlayered psammite (Fig. 7). The relative strength of porphyroblast-rich model pelite layers and psammite layers is strongly dependent on the relative viscosities of the two layers prior to 2006 Blackwell Publishing Ltd Psammite porphyroblast growth (Fig. 6). At low initial viscosity contrasts (gpsa:gpel ¼ 2), the pelitic layer becomes as viscous as the psammite layer at porphyroblast abundance of c. 11%, and is twice as viscous at a porphyroblast abundance of c. 30%. At an intermediate 10 W.G. GROOME ET AL. the increasing viscosity of the pelitic layers during porphyroblast growth causes an increase in the integrated viscosity of the layered succession. OROGEN-SCALE NUMERICAL EXPERIMENTS B) (V 50 0. 75 1 0 0. 25 0. 5 ata –0 .25 Normalized viscosity 100 ld nta –0 .5 _– 0. 75 e rim e xp E Figure b 1 0 –1 0.25 0.5 B) (R 0.75 1.0 0 ta al d a ent 9 1 (VB) 10 0.5 Porphyroblast abundance (areal fraction) erim 7 Exp 6 ηpel:ηpsa = 6.0 WM Modelling environment .5 5 ηpel:ηpsa = 4.0 4 –0 Normalized viscosity 8 3 ηpel:ηpsa = 2.0 2 1 0.0 Preliminary three-dimensional numerical models were constructed to explore the effects of metamorphic strengthening reactions in the middle crust on the strain-rate distribution and topography of a collisional orogen. The problem domain for this series of models is a 400 · 450 km crustal block with a thickness of 50 km (Fig. 8). The models provide information about the velocity and displacement fields, which are then used to calculate strain rates within the problem domain. The models consist of an upper crust described by a pressure-dependent plasticity constitutive relationship and a lower crust described by a temperature-dependent plasticity constitutive relationship constrained by empirical flow law data for wet quartzite (Carter & Tsenn, 1987). The models presented here are preliminary, but serve to illustrate the orogen-scale strain-rate distribution resulting from heterogeneously distributed metamorphic strengthening in the middle to lower crust, as well as the effects of these metamorphic reactions on the topography of an orogen, without considering the effects of erosion. B) –1 (R 0.1 0.2 0.3 0.4 0.5 0.6 Porphyroblast abundance (areal fraction) Fig. 6. (a) Normalized viscosity-porphyroblast volume fraction graph showing the Voigt and Reuss theoretical limits, calculated from the viscosity contrast between matrix and porphyroblast grains, and the trend of the experimental data. (b) Detail of the region outlined in panel a. The initial viscosity contrasts between pelitic and psammitic beds are indicated to compare the viscosity of the porphyroblast-rich layer with the psammite layer. WM indicates the estimated porphyroblast abundance in the study area. viscosity contrast (gpsa:gpel ¼ 4), the pelitic layer does not become as viscous as the psammitic layer until porphyroblast abundance exceeds 30%. Finally, at high viscosity contrasts (gpsa:gpel ¼ 6), the pelitic layer does not become as viscous as the psammitic layer until porphyroblast abundance exceeds 40%. At porphyroblast abundance >40%, the viscosity of the pelitic layer increases rapidly, which is consistent with predictions based on the theoretical Reuss bound. Regardless of the initial viscosity contrast between porphyroblast-free pelitic layers and psammitic layers, Three-dimensional orogen-scale models were developed using the numerical code FLAC3D, which was modified to accommodate large strains (e.g. Koons et al., 2002; Upton et al., 2003; Johnson et al., 2004). Materials in the models are represented by polyhedral elements within a three-dimensional grid using an explicit, time-marching solution and a form of dynamic relaxation. Each element responds according to a prescribed linear and non-linear stress–strain law in response to applied forces or kinematic boundary conditions. The pressure-dependent strength of the upper crust in our models is based on the Mohr– Coulomb constitutive relationship: rc ¼ rn tan / þ C; ð7Þ where rc is the critical shear stress at failure, rn is the normal stress, / is the internal angle of friction and C is cohesion. The models presented here have a 10-km thick upper crust, the strength of which is described by Eq. 7. The relatively thin Mohr–Coulomb upper crust was used to simulate the effects of an elevated geothermal gradient, which would decrease the depth to the brittle-ductile transition. The temperaturedependent strength of the middle and lower crust is based on the Von Mises failure criterion: rc ¼ K/ ; ð8Þ where K/ is the shear strength of the material. The temperature-dependent weakening of the ductile part of 2006 Blackwell Publishing Ltd EFFECTIVE VISCOSITY OF PORPHYROBLASTIC ROCKS 11 (a) Undeformed 10% Porphyroblasts Deformed SSR 1.0 Psammite Pelite Psammite 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 (b) 0.2 0.4 0.6 0.8 1.0 Shear strain rate Log10viscosity Undeformed 20% Porphyroblasts Deformed SSR 1.0 Psammite Pelite Psammite 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 (c) 0.2 0.4 0.6 0.8 1.0 Shear strain rate Log10viscosity Undeformed 30% Porphyroblasts Deformed SSR 1.0 Psammite Pelite Psammite 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 (d) 0.2 0.4 0.6 0.8 1.0 Shear strain rate Log10viscosity Undeformed 40% Porphyroblasts Deformed SSR 1.0 Psammite Pelite Psammite 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Log10viscosity 0.0 0.2 0.4 0.6 0.8 1.0 Shear strain rate Fig. 7. Model results for porphyroblast abundances of 10% (a), 20% (b), 30% (c) and 40% (d) with an initial viscosity contrast of 2:1 (gpsa:gpel). The panels are the same as in Fig. 5. Note the higher strain rates in the psammitic layer as porphyroblast abundance increases in the pelitic layer. the crust is characterized by decreasing the value of K/ with increasing depth in accordance with the flow stress for wet quartzite from Carter & Tsenn (1987). A perfectly plastic approximation of the power law flow law for wet quartzite is used because in our models there is 2006 Blackwell Publishing Ltd less than a threefold difference in strain rate from the reference strain rate of 10)14 used to approximate the flow stress for wet quartzite (Carter & Tsenn, 1987), so a perfectly plastic constitutive relationship is a reasonable approximation (e.g. see Chapple, 1978). 12 W.G. GROOME ET AL. 400 km Zone of metamorphic strengthening 15 km depth 50 km ∆σ BDT=10 km Elastic backstop Elastic lithospheric mantle 450 km 50 km ∆σ ∆σ Elastic lithospheric mantle Elastic backstop Zone of metamorphic strengthening In order to model the effects of a crustal volume that has undergone metamorphic strengthening, we assign a 40 · 100 · 5 km (X, Y, Z) crustal volume to have a Von Mises yield stress four times greater than the surrounding material, which is slightly greater than the amount of strengthening recorded in the bed-scale models at c. 30% porphyroblast. A fourfold increase in bulk strength (treated in this model as yield stress) would correspond to a fourfold difference in effective viscosity between porphyroblast-free pelitic units and psammitic units, such that 30% porphyroblast abundance would result in pelitic units being as viscous as psammitic units, and a fourfold increase in bulk effective viscosity of this region (Fig. 6). The X- and Z-dimensions of our rheological heterogeneity are intended to represent the size of the zone of metamorphic strengthening in eastern New Hamsphire and Western Maine during the early Acadian Orogeny, prior to overprinting by extensive migmatization (e.g. Guidotti, 1989; Solar & Brown, 1999). The 100-km Y-dimension is somewhat arbitrary and chosen to eliminate boundary effects when evaluating mechanical data in X-Z cross-sections. Model results Two models are presented here, a reference model with no strengthened zone and a model with a strengthened zone in the core of the orogen. We use a subducting elastic lithospheric mantle overlain by a 30-km thick deformable crustal section, simulating the effects of continental collision or the accretion of an exotic ter- Fig. 8. Schematic diagram of the orogenscale model domain. Top: Perspective view of the problem domain showing the rheological layering and applied boundary conditions. BDT ¼ Brittle-ductile transition. The dashed white line indicates the crosssection line. Bottom: Cross-section through the centre of the model showing the position of the zone of metamorphic strengthening (white box). rane. A constant velocity boundary condition is imposed along the base of the elastic slab and a fixed, farfield elastic backstop on the overriding plate, similar to models described in Koons et al. (2002). Internal velocities are not prescribed, but are allowed to develop in response to boundary kinematics and internal rheological definitions. Margin-parallel velocity in both models is fixed at zero, simulating orthogonal collision. A reference model is presented with a homogeneous 10-km-thick upper crust having a pressure-dependent yield strength underlain by a 20- to 40-km-thick plastic lower crust having a temperature-dependent yield strength is used for comparison with a model having a strengthened zone in the middle crust. The model topography after c. 10 km of convergence develops into a uniform, two-sided orogenic wedge, similar to those described in Koons (1990, 1995) and Willett et al. (1993) (Fig. 9). A cross-section of the velocity and vertical displacement fields for this model is shown in Fig. 10 along with a shear strain-rate map. Three zones of high shear strain-rate develop, which define shear zones along the crust-mantle boundary as well as orogen-bounding shear zones. The presence of a strengthened zone alters the topography of the orogenic wedge (Fig. 9) and changes the strain-rate distribution at depth (Fig. 10). The strengthened zone is intended to simulate the effects of a bulk strengthening of a particular stratigraphic succession, although in nature only certain layers within a compositionally layered succession will undergo metamorphic strengthening reactions. The size of the 2006 Blackwell Publishing Ltd EFFECTIVE VISCOSITY OF PORPHYROBLASTIC ROCKS 13 400 km size of the zone of metamorphic strengthening. However, the models presented here clearly show that metamorphic strengthening reactions can alter the geodynamic evolution of an orogen. DISCUSSION 400 km 450 km 450 km Fig. 9. Model topography for the reference (top) and metamorphic strengthened (bottom) models. The dashed line indicates the position of the cross-sections shown in Fig. 10. Contour interval ¼ 500 m, 10· vertical exaggeration. strengthened zone in our model is broadly constrained by the approximate extent of Devonian-aged andalusite-grade metamorphism in eastern New Hampshire and western Maine (e.g. Guidotti, 1989), although the actual size of the zone of strengthening in northern New England is unknown. The topography of this model is shown in Fig. 9, and a change in topographic slope is evident compared with the reference model, marking the location of the zone of metamorphic strengthening. The velocity and vertical displacement fields along with the shear strain-rate map for this model are shown in Fig. 10. Three high strain-rate zones are present in this model, defining a shear zone at the base of the crust as well as two orogen-bounding shear zones similar to the reference model, but the strong zone in the middle crust alters the location of high strain-rate zones within the orogenic wedge. Comparative diagrams showing the differences in vertical displacement and shear strain rate between the reference and strengthened models demonstrate the effects of a zone of metamorphic strengthening (Fig. 11). The presence of a strong zone in the middle crust leads to the development of zones of enhanced uplift outboard and inboard of the strengthened zone, which correspond to relatively high shear strain-rate zones at depth around the strengthened zone. As previously stated, these models are preliminary and are used to illustrate the possible orogen-scale effects of mid-crustal metamorphic strengthening; thus, we are presently unable to address the sensitivity of these results to geographical location, strength contrast and 2006 Blackwell Publishing Ltd It is clear from published analytical models and the numerical models presented here that the growth of effectively rigid porphyroblasts in metapelitic rocks will strengthen them relative to unmetamorphosed or porphyroblast-free pelitic rocks. The extent to which the porphyroblastic metapelitic rocks strengthen relative to interlayered rocks not undergoing porphyroblast-growth reactions is dependent on the final porphyroblast abundance in the layer and the initial relative strengths of the different bulk compositions. At low porphyroblast abundances, metapelitic layers are unlikely to become stronger than porphyroblastfree layers unless the initial viscosity contrast is small. Published estimates of viscosity contrasts between metapsammitic and porphyroblast-free metapelitic rocks are on the order of two to 10 (e.g. Treagus, 1999; Kenis et al., 2005). Based on the results presented here, in order for metapelitic layers to become stronger than interlayered metapsammitic units during porphyroblast growth, the initial effective viscosity contrast between the two units has to be at the low end of the published estimates (less than c. gpsa:gpel ¼ 4). Using our natural example from eastern New Hampshire, where the porphyroblastic metapelitic layers were approximately two to three times more viscous than the interlayered metapsammitic layers, we hypothesize that metapsammitic units have effective viscosities approximately two times greater than porphyroblast-free metapelitic units at amphibolite facies conditions. In our natural example, the metapelitic layers have c. 30% andalusite porphyroblasts by volume, which is in the mid-range of the porphyroblast abundance of our numerical models. In the numerical models, when porphyroblast abundance is c. 30%, the model pelite layer is approximately twice as viscous as the porphyroblast-free layer when the initial viscosity contrast (gpsa:gpel) is two, but the two layers have the same viscosity when the initial viscosity contrast (gpsa:gpel) is four. In the experiment with an initial viscosity ratio of six, the model pelite layer was still weaker than the model psammite layer at 30% porphyroblast abundance. The extent to which porphyroblast growth will strengthen the middle crust is dependent on the geographical extent of the metamorphic strengthening reaction. In the field example from eastern New Hampshire, andalusite-rich metapelitic rocks outcrop over a minimum area of c. 50 km2. The original extent of andalusite schist in the study are was probably much greater, perhaps as much as 100 km2 based on extent of correlative rocks in the study area, but overprinting Depth (km) Depth (km) Depth (km) 200 0 0 300 300 1 2 Shear strain rate (10–14/s) X-position (km) 250 0 00 +3 350 350 350 350 400 Elastic backstop 400 Elastic backstop 400 Elastic backstop 400 Elastic backstop 200 Shear strain rate 150 40 20 0 200 300 300 300 300 X-position (km) 250 X-position (km) 250 –1000 0 00 +1 X-position (km) 250 X-position (km) 250 Vertical displacement (m) 150 40 20 0 200 Velocity field 150 40 20 0 200 Lithospheric mantle 40 150 Ductile lower crust 20 +3000 Strengthened model Schematic Brittle upper crust 0 0 350 350 350 350 400 Elastic backstop 400 Elastic backstop 400 Elastic backstop 400 Elastic backstop Fig. 10. Cross-sectional profiles through the model domains showing the model schematic, velocity fields, contour maps of the vertical displacement and shear strain-rate maps. 150 40 300 X-position (km) 250 0 Shear strain rate 200 –1000 00 20 250 +1 0 150 40 20 0 300 X-position (km) 250 X-position (km) Vertical displacement (m) 200 Velocity field 150 40 20 0 200 Lithospheric mantle 40 150 Ductile lower crust 20 Depth (km) Depth (km) Depth (km) Reference model Schematic Brittle upper crust 0 00 +2 Depth (km) 0 00 +2 Depth (km) 0 14 W.G. GROOME ET AL. 2006 Blackwell Publishing Ltd EFFECTIVE VISCOSITY OF PORPHYROBLASTIC ROCKS 15 Difference in vertical displacement 100 300 Suppressed uplift up lift 100 –2 100 00 00 0 –100 10 200 Depth (km) up –1 200 0 20 0 0 0 10 200 100 20 0 0 40 300 200 400 X position (km) Strengthened zone Percent difference in shear strain rate Depth (km) 0 0 –20 20 –20 –40 +40+20 0 0 +20 +4 0 0 40 200 +20 –2 0 +20 +60 +40 +20 0 0 –20 0 metamorphism has obscured the ultimate extent of andalusite metamorphism. During Acadian deformation, this 100 km2 region would have behaved as a lozenge of relatively strong rock in the middle crust and would probably have caused strain to partition around it. This mid-crustal strain partitioning would have affected the geodynamic evolution of the developing orogen by: (1) potentially enhancing exhumation rates of deep crustal rocks during channel flow, which would alter the geothermal structure of the orogen (e.g. Koons, 1987; Beaumont et al., 2001; Zeitler et al., 2001; Jamieson et al., 2002; Koons et al., 2002, 2003); (2) altering the topographic expression above the zone of active metamorphism (e.g. Koons, 1995; Petrini et al., 2001; Husson & Sempere, 2003; Jackson et al., 2004), potentially leading to the development of intramontane sedimentary basins, similar to those described in the Himalaya attributed to lithospheric strength heterogeneities (e.g. England & Houseman, 1985; Neil & Houseman, 1997); and (3) shifting strain distribution, which may serve to catalyse metamorphic reactions in peripheral parts of the crust if they are in thermal disequilibrium (e.g. Brodie & Rutter, 1985; Rubie, 1986; Koons et al., 1987; Bell & Hayward, 1991; Bell et al., 2004). At the grain-scale, the growth of porphyroblasts leads to heterogeneities in strain-rate distribution within the layer experiencing porphyroblast growth 2006 Blackwell Publishing Ltd ed 0 ha En Fig. 11. Contour maps of the difference in vertical displacement (top) and shear strainrate between the reference model and the metamorphic strengthened model. Top: Perspective view through the middle of the model domain showing the difference in vertical displacement, calculated by subtracting the strengthened model displacement from the reference model. Contour interval ¼ 50 m. Note the development of zones of enhanced uplift around the periphery of the strengthened zone and a zone of suppressed uplift immediately above the strengthened zone. Bottom: Contour map of the percent difference of shear strain-rate between the strengthened model and the reference model, calculated by dividing the strain-rate of the strengthened model by the strain-rate in the reference model and subtracting 100. Contour interval is 20%. Positive values indicate zones where the shear strain-rate is higher in the strengthened model than the reference model. The position of the strengthened zone is indicated on both diagrams. nc d e nc ha lift En Yp osi tion (km ) 400 +40 +20 0 0 +20 +20 +40 300 400 X position (km) (Fig. 12). These heterogeneities lead to the development of strain-rate shadows and high strain-rate zones around the porphyroblast, which are oriented within the flow field. The formation of high strain-rate zones around porphyroblasts can enhance metamorphic reaction rates by: (1) increasing dislocation densities in quartz and feldspar, which adds strain energy to the reaction, lowering the energy barrier for reaction and increasing reaction rates (e.g. Brodie & Rutter, 1985; Bell & Hayward, 1991); (2) causing dilatent zones to develop along the grain boundaries, which would transiently increase the permeability of the rock, allowing fluids to either leave or enter the reaction site, thus increasing reaction rates (e.g. Etheridge et al., 1983; Brodie & Rutter, 1985; Frueh-Green, 1994; Graham et al., 1997; Farver & Yund, 1999; Zhang et al., 2000; Tenthorey & Cox, 2003); and (3) advecting reactant components to the site of reaction, which reduces the length scale for diffusive mass transfer (e.g. Koons et al., 1987). If a rock body is in disequilibrium with respect to ambient pressure–temperature conditions, the initial growth of porphyroblasts may lead to a feedback relationship between strain partitioning and metamorphic reaction. The development of high strain-rate zones around effectively rigid porphyroblasts in rocks in disequilibrium with respect to pressure and temperature could lead to the rapid advection of reactant 16 W.G. GROOME ET AL. Di l HSRZ Dilation along grain boundary HSRZ components to the site of reaction, which reduces the length scale for diffusional mass transfer and potentially increases the reaction rate (e.g. Koons et al., 1987). Furthermore, if high strain-rate gradients along the margins of porphyroblasts lead to the opening of grain boundaries, increasing permeability, the advection of fluids into or out of the reaction sites may catalyse reactions (e.g. Etheridge et al., 1983; Rubie, 1986; Graham et al., 1997). Thus, during porphyroblast growth, local heterogeneities may allow for rapid porphyroblast-producing reactions, which would in turn lead to rapid changes in rock strength, leading to shifting strain-rate distribution on a larger scale. ACKNOWLEDGEMENTS Funding for this project came from NSF Grant # EAR0207717 (to S.E.J. and P.O.K.), the Geological Society of America Grants in Aid Program (to W.G.G.) and the University of Maine (to W.G.G.). We thank M. Washburn for field assistance, D. Eusden for introducing us to the geology of the White Mountains region, M. Jessell, L. Evans and G. Houseman for help with ELLE and BASIL, and the Appalachian Mountain Club, the US Forest Service Androscoggin Ranger Station and the Mount Washington Auto Road for logistical assistance in the field. Constructive journal reviews by E. Baxter, S. Ji and D. Whitney improved this manuscript. Fig. 12. Shear strain rate contour map and model geometry for the 20% porphyroblast abundance experiment and detailed diagrams of select regions showing the high strain-rate zones around porphyroblasts (HSRZ) and zones of dilatation (Dil) between porphyroblasts. Also shown is a zone where grain boundary dilation would likely occur in a natural rock. REFERENCES Allen, T. T., 1992. Migmatite Systematics and Geology, Carter Dome-Wild River Region, White Mountains, New Hampshire. PhD Thesis, Dartmouth College, Hanover, NH, 249 pp. Arzi, A. A., 1978. Critical phenomena in the rheology of partially melted rocks. Tectonophysics, 44, 173–184. Barnes, J. D., Selverstone, J. & Sharp, Z. D., 2004. Interactions between serpentinite devolatization, metasomatism and strikeslip strain localization during deep-crustal shearing in the Eastern Alps. Journal of Metamorphic Geology, 22, 283–300. Barr, T. B. & Houseman, G. A., 1996. Deformation fields around a fault embedded in a non-linear ductile medium. Geophysical Journal International, 125, 473–490. Baxter, E. F. & DePaolo, D., 2004. Can metamorphic reactions proceed faster than bulk strain? Contributions to Mineralogy and Petrology, 146, 657–670. Beach, A., 1980. Retrogressive metamorphic processes in shear zones with special reference to the Lewison Complex. Journal of Structural Geology, 2, 257–263. Beaumont, C., Jamieson, R. A., Nguyen, M. H. & Lee, B., 2001. Himalayan tectonics explained by extrusion of a low-viscosity crustal channel coupled to focused surface denudation. Nature, 414, 738–742. Bell, T. H. & Hayward, N., 1991. Episodic metamorphic reactions during orogenesis: the control of deformation partitioning on reaction sites and reaction duration. Journal of Metamorphic Geology, 9, 619–640. Bell, T. H., Ham, A. P. & Kim, H. S., 2004. Partitioning of deformation along an orogen and its effects on porphyroblast growth during orogenesis. Journal of Structural Geology, 26, 825–845. Bons, P. D. & Cox, S. J. D., 1994. Analogue experiments and numerical modelling on the relation between microgeometry 2006 Blackwell Publishing Ltd EFFECTIVE VISCOSITY OF PORPHYROBLASTIC ROCKS 17 and flow properties of polyphase materials. Material Sciences and Engineering, A175, 237–245. Bons, P. D. & Urai, J. L., 1994. Experimental deformation of two-phase rock analogues. Materials Science and Engineering, A175, 221–229. Brodie, K. H. & Rutter, E. H., 1985. On the relationship between deformation and metamorphism, with special reference to the behaviour of basic rocks. In: Metamorphic Reactions: Kinetics, Textures and Deformation (eds Thompson, A. B. & Rubie, D. C.). Advances in Physical Geochemistry, 4, 138–179. Brodie, K. H. & Rutter, E. H., 1987. The role of transiently finegrained reaction products in syntectonic metamorphism: natural and experimental examples. Canadian Journal of Earth Sciences, 24, 556–564. Bucher, K. & Frey, M., 1994. Petrogenesis of Metamorphic Rocks. Springer-Verlag, Berlin, 318 pp. Burg, J. P. & Wilson, C. J. L., 1987. Deformation of two phase systems with contrasting rheologies. Tectonophysics, 135, 199– 205. Carminati, E. & Siletto, G. B., 1997. The effects of brittle-plastic transitions in basement-involved foreland belts: the Central Southern Alps case (N. Italy). Tectonophysics, 280, 107–123. Carter, N. L. & Tsenn, M. C., 1987. Flow properties of continental lithosphere. Tectonophysics, 136, 27–63. Chapple, W. M., 1978. Mechanics of thin-skinned fold-andthrust belts. Geological Society of America Bulletin, 89, 1189– 1198. De Bresser, J. H. P., Ter Heege, J. H. & Spiers, C. J., 2001. Grain size reduction by dynamic recrystallization: can it result in major rheological weakening? International Journal of Earth Sciences, 90, 28–45. Duva, J. M., 1984. A self-consistent analysis of the stiffening effect of rigid inclusions on a power-law material. Journal of Engineering Materials and Technologies, 106, 317–321. England, P. & Houseman, G., 1985. Role of lithospheric strength heterogeneities in the tectonics of Tibet and neighbouring regions. Nature, 315, 297–301. Etheridge, M. A., Wall, V. J. & Vernon, R. H., 1983. The role of the fluid phase during regional metamorphism and deformation. Journal of Metamorphic Geology, 1, 205–226. Eusden, J. D., Garesche, J. M., Johnson, A. H. et al., 1996. Stratigraphy and ductile structure of the Presidential Range, New Hampshire: tectonic implications for the Acadian Orogeny. Geological Society of America Bulletin, 108, 417–437. Farver, J. & Yund, R. A., 1999. Oxygen bulk diffusion measurements and TEM characterization of a natural mylonite: implications for fluid transport in mica-bearing rocks. Journal of Metamorphic Geology, 17, 669–683. Frueh-Green, G. L., 1994. Interdependence of deformation, fluid infiltration and reaction progress recorded in eclogitic metagranitoids (Seisia Zone, Western Alps). Journal of Metamorphic Geology, 12, 327–343. Graham, C. M., Skelton, A. D. L., Bickle, M. & Cole, C., 1997. Lithological, structural and deformation controls on fluid flow during regional metamorphism. In: Deformation-Enhanced Fluid Transport in the Earth’s Crust and Mantle (ed. Holness, M. B.), pp. 196–226. Chapman & Hall, London. Groome, W. G. & Johnson, S. E., 2006. Constraining the relative strengths of high-grade metamorphic rocks using foliation refraction angles: an example from the Northern New England Appalachians. Journal of Structural Geology, doi: 10.1016/j.jsg.2006.03.023. Guidotti, C. V., 1989. Metamorphism in Maine: an overview. In: Studies in Maine Geology; papers to commemorate the 150th anniversary of C.T. Jackson’s reports on the geology of Maine (eds Tucker, R. D. & Marvinney, R. G.), pp. 1–17. Maine Geological Survey, Agusta, ME. Handy, M. R., 1990. The solid state flow of polymineralic rocks. Journal of Geophysical Research, 95, 8647–8661. Handy, M. R., Mulch, A., Rosenau, M. & Rosenberg, C. L., 2001. The role of fault zones and melts as agents of weakening, hardening and differentiation of the continental crust: a syn 2006 Blackwell Publishing Ltd thesis. In: The Nature and Tectonic Significance of Fault Zone Weakening, Special Publications 186 (eds Holdsworth, R. E., Strachen, R. A., Magloughlin, J. F. & Knipe, R. J.), pp. 305– 332, Geological Society, London. Hirth, G., Teyssier, C. & Dunlap, W. J., 2001. An evaluation of quartzite flow laws based on comparisons between experimentally and naturally deformed rocks. International Journal of Earth Sciences, 90, 77–87. Hubbert, M. K. & Rubey, W. W., 1959. Role of fluid pressure in mechanics of overthrust faulting 1: mechanics of fluid-filled porous solids and its application to overthrust faulting. Geological Society of America Bulletin, 70, 115–166. Husson, L. & Sempere, T., 2003. Thickening the Altiplano crust by gravity-driven crustal channel flow. Geophysical Research Letters, 30, 4 pp. Jackson, J. A., Austrheim, H., McKenzie, D. & Priestley, K., 2004. Metastability, mechanical strength and the support of mountain belts. Geology, 32, 625–628. Jamieson, R. A., Beaumont, C., Nguyen, M. H. & Lee, B., 2002. Interaction of metamorphism, deformation and exhumation in large convergent orogens. Journal of Metamorphic Geology, 20, 9–24. Jessell, M. W., Bons, P., Evans, L., Barr, T. & Stuwe, K., 2001. Elle: the numerical simulation of metamorphic and deformation microstructures. Computers and Geosciences, 27, 17–30. Ji, S. & Xia, B., 2002. Rheology of Polyphase Earth Materials. Polytechnique Press International, Montreal, 235 pp. Ji, S. & Zhao, P., 1993. Flow laws of multiphase rocks calculated from experimental data on the constituent phases. Earth and Planetary Science Letters, 117, 181–187. Ji, S., Wang, Z. & Wirth, R., 2001. Bulk flow strength of forsterite-enstatite composites as a function of forsterite content. Tectonophysics, 341, 69–93. Ji, S., 2004. A generalized mixture rule for estimating the viscosity of solid-liquid suspensions and mechanical properties of polyphase rocks and composite materials. Journal of Geophysical Research, 109, doi: 10.1029/2004JB003124. Johnson, S. E., 1999. Porphyroblast microstructures: a review of current and future trends. American Mineralogist, 84, 1711– 1726. Johnson, S. E., Vernon, R. H. & Upton, P., 2004. Foliation development and progressive strain-rate partitioning in the crystallizing carapace of a tonalite pluton: microstructural evidence and numerical modelling. Journal of Structural Geology, 26, 1845–1865. Jordan, P., 1987. The deformational behaviour of bimineralic limestone-halite aggregates. Tectonophysics, 135, 185–197. Jordan, P., 1988. The rheology of polymineralic rocks: an approach. International Journal of Earth Sciences, 77, 285–294. Kanagawa, K., 1993. Competence contrasts in ductile deformation as illustrated from naturally-deformed chert-mudstone layers. Journal of Structural Geology, 15, 865–885. Kenis, I., Urai, J. L., van der Zee, W. Hilgers, C. & Sintubin, M., 2005. Rheology of fine-grained siliciclastic rocks in the middle crust: evidence from structural and numerical analysis. Earth and Planetary Science Letters, 203, 351–360. Kerrich, R., Fyfe, W. S., Gorman, B. E. & Allison, I., 1977. Local modification of rock chemistry by deformation. Contributions to Mineralogy and Petrology, 65, 183–190. Koons, P. O., 1987. Some thermal and mechanical consequences of rapid uplift: an example from the Southern Alps, New Zealand. Earth and Planetary Science Letters, 86, 307–319. Koons, P. O., 1989. The topographic evolution of collisional mountain belts: a numerical look at the Southern Alps, New Zealand. American Journal of Science, 289, 1041–1069. Koons, P. O., 1990. Two-sided orogen: collision and erosion from the sandbox to the Southern Alps, New Zealand. Geology, 18, 679–682. Koons, P. O., 1995. Modelling the topographic evolution of collisional belts. Annual Reviews of Earth and Planetary Sciences, 23, 375–408. 18 W.G. GROOME ET AL. Koons, P. O., Rubie, D. C. & Frueh-Green, G., 1987. The effects of disequilibrium and deformation on the mineralogical evolution of quartz diorite during metamorphism in the eclogite facies. Journal of Petrology, 28, 679–700. Koons, P. O., Zeitler, P. K., Chamberlain, C. P., Craw, D. & Meltzer, A. S., 2002. Mechanical links between erosion and metamorphism in Nanga Parbat, Pakistan Himalaya. American Journal of Science, 302, 749–773. Koons, P. O., Upton, P. & Terry, M. P., 2003. Three-dimensional mechanics of UHPM terrains and resultant P–T–t paths. European Mineralogical Union Notes in Mineralogy, 5, 415–441. Lisle, R. J., Rondeel, H. E., Doorn, D., Brugge, J. & Van de Gaag, P., 1983. Estimation of viscosity contrast and finite strain from deformed elliptical inclusions. Journal of Structural Geology, 5, 603–609. Mecklenburgh, J. & Rutter, E. H., 2003. On the rheology of partially molten synthetic granite. Journal of Structural Geology, 25, 1575–1585. Moecher, D. P. & Wintsch, R. P., 1994. Deformation-induced reconstitution and local resetting of mineral equilibria in polymetamorphic gneisses: tectonic and metamorphic implications. Journal of Metamorphic Geology, 12, 523–538. Neil, E. A. & Houseman, G. A., 1997. Geodynamics of the Tarim Basin and the Tian Shan in central Asia. Tectonics, 16, 571–584. Paquet, J., Francois, P. & Nedelec, A., 1981. Effect of partial melting on rock deformation: experimental and natural evidence on rocks of granitic compositions. Tectonophysics, 78, 545–565. Park, Y., Park, D., Evans, L. & Ree, J., 2004. An Elle-based 2-D model for cation exchange reaction between garnet and biotite. Journal of the Virtual Explorer, Electronic Edition, 15, paper 6. Passchier, C. W. & Trouw, R. A. J., 1996. Microtectonics. Springer-Verlag, Berlin, 289 pp. Petrini, K., Connolly, J. A. D. & Podladchikov, Y. Y., 2001. A coupled petrological-tectonic model for sedimentary basin evolution: the influence of metamorphic reactions on basin subsidence. Terra Nova, 13, 354–359. Piazollo, S. C., 2000. Shape Fabric Development During Progressive Deformation. PhD Thesis, Johannes Gutenberg Universitat, Mainz, 248 pp. Platt, J. P., 1986. Dynamics of orogenic wedges and the uplift of high-pressure metamorphic rocks. Geological Society of America Bulletin, 97, 1037–1053. Poirier, J. P., 1982. On transformation plasticity. Journal of Geophysical Research, 87, 6791–6797. Porter, D. A. & Easterling, K. E., 1992. Phase Transformations in Metals and Alloys, 2nd edn. Stanley Thornes (Publishers) Ltd, Chetham, UK, 514 pp. Ravichandron, K. S. & Seetharaman, V., 1993. Prediction of steady state creep behaviour of two-phase composites. Acta Metalurgica Material, 41, 3351–3361. Renner, J., Evans, B. & Hirth, G., 2000. On the rheologically critical melt fraction. Earth and Planetary Science Letters, 181, 585–594. Rodda, C. I., 2005. Early and Late Acadian Strain Partitioning: L1 Lineation and F4 Folding: Mt. Madison, New Hampshire. BS Honors Thesis, Bates College, Lewiston, ME, 167 pp. Rosenberg, C. L., 2001. Deformation of partially molten granite: a review and comparison of experimental and natural case studies. International Journal of Earth Sciences, 90, 60–76. Rubie, D. C., 1983. Reaction-enhanced ductility: the role of solid-solid univariant reactions in deformation of the crust and mantle. Tectonophysics, 96, 331–352. Rubie, D. C., 1986. The catalysis of mineral reactions by water and restrictions on the presence of aqueous fluid during metamorphism. Mineralogical Magazine, 50, 399–415. Selverstone, J., Morteani, G. & Staude, J.-M., 1991. Fluid channeling during ductile shearing: transformation of grano- diorite into aluminous schist in the Tauren Window, Eastern Alps. Journal of Metamorphic Geology, 9, 419–431. Shea, W. T. & Kronenberg, A. K., 1993. Strength and anisotropy of foliated rocks with varied mica contents. Journal of Structural Geology, 15, 1097–1121. Sibson, R. H., 2003. Brittle-failure controls on maximum sustainable overpressure in different tectonic regimes. American Association of Petroleum Geologists Bulletin, 87, 901–908. Smith, R. B., 1977. Formation of folds, boudinage and mullions in non-Newtonian material. Geological Society of America Bulletin, 88, 312–320. Solar, G. S. & Brown, M., 1999. The classic high-T, low-P metamorphism of west-central Maine: is it post-tectonic or syntectonic? Evidence from porphyroblast-matrix relations. Canadian Mineralogist, 37, 311–333. Stunitz, H. & Tullis, J., 2001. Weakening and strain localization produced by syn-deformational reaction of plagioclase. International Journal of Earth Sciences, 90, 136–148. Takeda, Y., 1998. Flow in rocks modeled as multiphase continua: application to polymineralic rocks. Journal of Structural Geology, 20, 1569–1578. Takeda, Y. & Obata, M., 2003. Some comments on the rheologically critical melt percentage. Journal of Structural Geology, 25, 813–818. Tenthorey, E. & Cox, S. F., 2003. Reaction-enhanced permeability during serpentine dehydration. Geology, 31, 921– 924. Tobisch, O. T., Barton, M. D., Vernon, R. H. & Paterson, S. R., 1991. Fluid-enhanced deformation: transformation of granitooids to banded mylonites, western Sierra Nevada, California, and southeastern Australia. Journal of Structural Geology, 13, 1137–1156. Treagus, S. H., 1983. A theory of finite strain variation through contrasting layers, and its bearing on cleavage refraction. Journal of Structural Geology, 5, 351–368. Treagus, S. H., 1988. Strain refraction in layered systems. Journal of Structural Geology, 10, 517–527. Treagus, S. H., 1993. Flow variations in power-law multilayers: implications for competency contrasts in rocks. Journal of Structural Geology, 15, 423–434. Treagus, S. H., 1999. Are viscosity ratios measurable from cleavage refraction? Journal of Structural Geology, 21, 895– 901. Treagus, S. H., 2002. Modelling the bulk viscosity of two-phase mixtures in terms of clast shape. Journal of Structural Geology, 24, 57–76. Treagus, S. H. & Sokoutis, D., 1992. Laboratory modelling of strain variation across rheological boundaries. Journal of Structural Geology, 14, 405–424. Treagus, S. H. & Treagus, J. E., 2002. Studies of strain and rheology of conglomerates. Journal of Structural Geology, 24, 1541–1567. Tullis, J. & Wenk, H.-R., 1994. Effect of muscovite on the strength and lattice preferred orientations of experimentally deformed quartz aggregates. Materials Science and Engineering, A175, 209–220. Tullis, J. & Yund, R. A., 1977. Experimental deformation of dry Westerly Granite. Journal of Geophysical Research, 82, 5705– 5718. Tullis, T. E., Horowitz, F. G. & Tullis, J., 1991. Flow laws of polyphase aggregates from end-member flow laws. Journal of Geophysical Research, 96, 8081–8096. Upton, P., Koons, P. O. & Eberhart-Phillips, D., 2003. Extension and partitioning in an oblique subduction zone, New Zealand: constraints from three-dimensional numerical modelling. Tectonics, 22, 14 pp. doi 10.1029/ 2002TC001431. Vernon, R. H., 2004. A Practical Guide to Rock Microstructures. Cambridge University Press, Cambridge, 594 pp. Wall, E. R., 1988. The Occurrence of Staurolite and its Implication for Polymetamorphism in the Mt. Washington Area, 2006 Blackwell Publishing Ltd EFFECTIVE VISCOSITY OF PORPHYROBLASTIC ROCKS 19 New Hampshire. MS Thesis, University of Maine, Orono, ME, 124 pp. Wang, Z. & Ji, S., 1999. Deformation of silicate garnets: brittleductile transition and its geological implications. Canadian Mineralogist, 37, 525–541. Willett, S., Beaumont, C. & Fullsack, P., 1993. Mechanical model for the tectonics of doubly vergent compressional orogens. Geology, 21, 371–374. Williams, C. A., Connors, C., Dahlen, F. A., Price, E. J. & Suppe, J., 1994. Effect of brittle-ductile transition on the topography of compressive mountain belts on Earth and Venus. Journal of Geophysical Research, 99, 19947–19974. Yoon, C. K. & Chen, I. W., 1990. Superplastic flow of two-phase ceramics containing rigid inclusions: zirconia-mullite compo- 2006 Blackwell Publishing Ltd sites. Journal of the American Ceramics Society, 73, 1555– 1565. Zeitler, P. K., Koons, P. O., Bishop, M. P. et al., 2001. Crustal reworking at Nanga Parbat, Pakistan: metamorphic consequences of thermal-mechanical coupling facilitated by erosion. Tectonics, 20, 712–728. Zhang, S., FitzGerald, J. D. & Cox, S. F., 2000. Reactionenhanced permeability during decarbonation of calcite + quartz ¼ wollastonite + carbon dioxide. Geology, 28, 911– 914. Received 3 October 2005; revision accepted 22 March 2006.