A review of the mechanics of heterogeneous materials and their implications for relationships between kinematics and dynamics in contients

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1 University of Montana, Missoula From the SelectedWorks of Rebecca Bendick 2013 A review of the mechanics of heterogeneous materials and their implications for relationships between kinematics and dynamics in contients Rebecca Bendick, University of Montana - Missoula Available at:

2 TECTONICS, VOL. 32, , doi: /tect.20058, 2013 A review of heterogeneous materials and their implications for relationships between kinematics and dynamics in continents R. Bendick 1 and L. Flesch 2 Received 7 September 2012; revised 22 May 2013; accepted 19 June 2013; published 2 August [1] The variation of mechanical properties as a function of depth in the lithosphere controls the relationship between surface deformation and whole-lithosphere deformation. Where mechanical competence (elastic strength or viscous stiffness) is a continuous function with depth, surface deformation can be used to constrain either force balance or rheological parameters for the lithosphere. However, where competence is vertically discontinuous, the surface deformation and deformation at depth may have a range of relationships, including complete independence. Therefore, both inversions and forward models designed to test either rheology or stress state are highly sensitive to the a priori choice of a mechanical model. Citation: Bendick, R., and L. Flesch (2013), A review of heterogeneous materials and their implications for relationships between kinematics and dynamics in continents, Tectonics, 32, , doi: /tect Introduction [2] Initial efforts to develop a comprehensive theory of plate tectonics, whether qualitative [Wegener, 1915] or more quantitative [McKenzie and Parker, 1967; Morgan, 1968; LePichon, 1968], were restricted to a description of the pattern of relative motion of parts of the Earth s lithosphere, or kinematics. Further deciphering why plates move and deform entails identifying the set of forces and torques that drive or excite displacement and deformation as well as quantifying their relative importance. From the very earliest attempts to list and compare possible contributors [e.g., Solomon and Sleep, 1974; Forsyth and Uyeda, 1975; Richter and McKenzie, 1978; Cloetingh and Wortel, 1986; Fu and Huang, 1990] to more sophisticated modern numerical simulations [e.g., Hager and O Connell, 1981; Billen and Gurnis, 2001; Steinberger et al., 2001; Conrad and Lithgow- Bertelloni, 2002; Gerya, 2011], evaluations of model success often rely heavily on comparisons of predicted plate motions with observed plate motions. This kind of test requires an assumption that accessible observations of plate motions and internal plate strains are representative of the complete displacements of lithospheric volumes. Because most kinematic observations are made at the surface, the entire field of tectonic dynamics therefore embeds an assumption that surface observations of plate motion plus boundary and intraplate strain are indicative of the response of the whole lithosphere. 1 Department of Geosciences, University of Montana, Missoula, Montana, USA. 2 Department of Earth and Atmospheric Sciences, Purdue University, West Lafayette, Indiana, USA. Corresponding author: R. Bendick, Department of Geosciences, University of Montana, 32 Campus Dr. 1296, Missoula, MT , USA. (Bendick@mso.umt.edu) American Geophysical Union. All Rights Reserved /13/ /tect [3] However, this question of whether surface deformation is indeed the same as whole-lithosphere deformation, hence expresses sufficient information about dynamic state, has become increasingly contentious with the advent of new data sets recording tectonic displacements in continental settings. When only data about displacements of oceanic lithosphere were available, such as the orientation and pattern of magnetic stripes on the seafloor [Vine and Matthews, 1963], the strike and slip rate on transform faults [McKenzie and Parker, 1967; Morgan, 1968], and the geometry of tectonic boundaries from structural mapping and seismic epicenters, only the kinematics and dynamics of oceanic lithosphere were relevant to the theory. Rock mechanics results [e.g., Brace and Kohlstedt, 1980] for oceanic lithosphere indicated that the likely yield strength envelope, or competence function, for oceanic materials is continuous, so that oceanic plates could reasonably be approximated as an elastic beam or sheet with a finite effective thickness. Hence, using these data resulted in reasonable estimates of the magnitude of forces and torques on plates, suggesting that the implied assumption of homogeneous vertical deformation within the lithosphere was at least reasonable, although bounds on dynamic state from velocity calculations were very weak [e.g., Solomon and Sleep, 1974; Cloetingh and Wortel, 1986]. However, new types of observations of surface kinematics, especially strain calculations from seismic moment release, GPS, radar interferometry and correlation, optical image correlation, and both radial and azimuthal seismic anisotropy provide vast improvements to spatial resolution of displacement, velocity, and strain fields both at the surface and at depth. More complete observations of finite strain from the geological record, such as fault displacements, structural indicators, and geochronology, also provide improved spatial and extended temporal resolution of deformation. These data sets are all preferentially collected in continental lithosphere. They, along with indirect estimates of the competence of continental lithosphere, such as from gravity admittance and coherence [e.g., Burov, 2011], 2-D 980

3 Figure 1. Classes of deformation of vertically heterogeneous materials based on competence contrasts between layers. Cartoons of the dominant style of deformation for characteristic arrangements of layers. Type 1 materials have continuous competence functions and can be approximated as homogeneous monolayers. Type 2 materials have discontinuous competence, with more over less competent layers. They can be approximated in the limit as quasi-rigid blocks. Type 3 materials have less over more competent layers and can be approximated by gravity flows. Type 4 materials are complex multilayered composites and have a variety of emergent deformation patterns. and 3-D coupled thermomechanical forward modeling [e.g., Lavier and Steckler, 1997; Beaumont et al., 2004], and rock mechanics studies for continental materials [Ord and Hobbs, 1989; Kohlstedtetal., 1995], indicate that continental rheology is more complicated than oceanic and that an assumption of coherent deformation throughout the continental lithosphere may not always be appropriate, with profound implications for the study of continental dynamics. [4] Direct applications of methods developed for tectonic dynamics of the oceanic lithosphere to continental settings clearly illuminates the resulting discrepancies. For example, although slab pull and slab suction appear sufficient to account for the motion of oceanic plates [Conrad and Lithgow-Bertelloni, 2002], additional forces such as strong mantle drags must often be added to match continental settings [Steinberger et al., 2001; Liu and Bird, 2002; Ghosh and Holt, 2012]. In other locations, other ad hoc contributions to the dynamic equations of equilibrium are required, often including a large role for body forces related to gravitational potential gradients. However, these more complex force and torque balances can only be evaluated if the observational constraints accurately represent the dynamics of the whole lithosphere. That is, dynamics can be inferred from kinematic observations only if those observations are correlated through the whole lithosphere. Furthermore, dynamics of the combined lithosphere and mantle can be inferred only if density variations in the mantle are transmitted through to the surface through density and strength variations within the lithosphere [Burov, 2010]. [5] This work explores the general validity of the assumption that surface kinematics are related to whole-lithosphere kinematics by cataloging and comparing a range of numerical, analog, and analytical solutions for end-member mechanical configurations (Figure 1) along with observations of tectonic deformation in specific terrestrial settings. The goal of this effort is to explore whether predominantly surface observations of tectonic deformation can be used to study tectonic dynamics in continental settings. We emphasize limiting cases and general solutions (often 2-D) as a means of efficiently exploring and delimiting a vast solution space; newer 3-D mechanically heterogeneous numerical simulations and stability analyses [e.g., Medvedev and Podladchikov, 1999b; Lechmann et al., 2011] illustrate that more complicated solutions are still related to the bounding cases. We also emphasize general computational efforts over compilations of observations from specific continental settings. These limiting solutions are described according to a general mechanical classification in the following section (Figure 2), with a subsequent suggestion for further classification of deformation modes. We then consider specific examples from regions of current tectonic activity (Figure 3). We expand on Flesch and Bendick [2012] both by providing a broad compilation of results from geophysics and soft materials physics for each of the end-member mechanical configurations, by revising our proposed classification system to better reflect the physical attributes of lithosphere, and by providing more detailed analyses of specific terrestrial tectonic settings in the context of our classification. The former effort is intended to facilitate selection and implementation of models for tectonic problems, the latter to illustrate the relationship between such models and real systems. 2. Numerical, Analytic, and Empirical Analyses [6] The problem of strain or velocity continuity in a layered medium is not restricted to a tectonic context. Rather, it is a more general question of classical mechanics and condensed matter physics. Computational results are classified 981

4 many different specific formulations for these more complicated materials One Homogeneous Layer or Sheet [7] Models involving the deformation of a single sheet or layer of constant competence are the most common in the literature. Derivations for the deformation of such layers entail only conditions placed on monolayers, but they can also be thought of as special cases of layered media where the competence difference between adjacent layers is so great that they do not interact. For Earth, the monolayer approximation, where used, assumes the latter condition through a stress-free Figure 2. Examples from the geophysical literature of numerical and analytic solutions for each material type. either according to boundary conditions (whether the state of stress is extensional, compressional, or shearing) or according to the vertical arrangement of materials with variable rheologies. We take the latter strategy and organize notable computational results according to the initial variation of mechanical properties in the vertical (or radial) direction. Very few general solutions have been generated for laterally heterogeneous materials, although these are certainly important in real settings. Because these cases are highly nonlinear and highly sensitive to local effects, we consider them outside of the scope of this paper. Where appropriate, we point out differences that arise in extensional versus compressional settings, but many of the published solutions are exclusively for the compressional case. Throughout this summary, we will use the term competence to interchangeably include strength for elastic materials or viscosity for viscous materials. Competence parameters, elastic moduli for elastic materials and viscosity for Newtonian fluids, are directly interchangeable in linear approximations through the Stokes-Rayleigh analogy (Table 1). This convention allows us to discuss the vertical (or radial) distribution of competence rather than exclusively strength or stiffness, thus allows us to compare and contrast solutions with elastic, viscous, viscoelastic, or viscoplastic constitutive relations, even though there are Figure 3. Compiled surface velocity fields illustrated by published GPS (blue vectors and mapped fault data (red lines). Regions are paired with their most likely mechanical approximation. Tibet observations are from Gan et al. [2007]; Basin and Range from Plate Boundary Observatory (www. earthscope.org) and Tom Herring (personal communication, 2012); and East Africa from Stamps et al. [2008] and Kogan et al. [2012]. 982

5 Table 1. The Stokes-Rayleigh Analogy Between the Mathematical Theories for the Bulk Equilibrium of Hookean Linear Elastic Solids and the Bulk Creeping Flow of Newtonian Viscous Liquids Static Elasticity Displacement u Strain γ Shear modulus G Creeping Flow Velocity v Strain rate γ Shear viscosity μ basal boundary for the lithosphere. The original formulation of the plate tectonic approximation assumes that the lithosphere is divided into a small number of adjacent sheets with infinite competence over an asthenosphere with finite competence. These rigid sheets therefore serve as perfect stress guides and are translated and rotated by a set of forces and torques [McKenzie and Parker, 1967; Morgan, 1968]. [8] Modification of this model for finite but large (compared to the asthenosphere) lithospheric competence allows out of plane deformations through buckling and bending [e.g., Biot, 1961a; Turcotte and Schubert, 2002]. Consideration of curved shells introduces additional terms to the buckling analysis of O(w/R) due to geometric stiffening where w is the buckling deflection and R is the radius of curvature of the sheet [Landau and Lifshitz, 1986; Howell, 1996; Mahadevan et al., 2010]. Buckling and bending monolayers have similar behavior over more than 9 orders of magnitude in length scale, from the lithosphere to monolayers of surfactant polymers [Milner et al., 1989]. For both flat and curved sheets of any size, deformation and velocity have the same form through the entire thickness of the monolayer. [9] Homogeneous sheets may alternatively be addressed using the thin viscous sheet (TVS) approximation [England and McKenzie, 1982], allowing for pure shear [Medvedev and Podladchikov, 1999a]. These solutions, like plate models, also entail an assumption that velocity does not vary vertically, analogous to the continuity of deflection in the elastic cases, thus a reduction of the solutions for viscous flow from three to two dimensions. Therefore, solutions for TVS problems give velocities as horizontal but not vertical position. They provide information about thickening if volume is conserved. Bird and Piper [1980] first described such a model for the Earth, but with a nonlinear constitutive law intended to incorporate brittle failure of the upper crust. A more generic solution including inertial terms and allowing buckles or sags in the sheet is provided by Howell [1996] and with a geometric stiffening term by Ribe [2002]. At large strains, the symmetry of buckling in both elastic and viscous sheets breaks into highly localized folds and creases [Lobkovsky and Witten, 1997; Diamant et al., 2001; Pocivavsek et al., 2008]. [10] Even in cases that violate the small deflection approximation [Turcotte and Schubert, 2002] in bending, both homogeneous elastic bodies and homogeneous viscous sheet solutions enforce continuity by definition such that displacement and velocity have the same form through the entire thickness of the beam or sheet. As a direct consequence, any measurements of displacement or velocity taken at any point in the sheet give the values of displacement or velocity for the whole sheet. Practically, this means that surface observations of displacement or velocity represent the displacement or velocity of the whole sheet, the critical condition required to invert surface deformation for dynamic state. These conditions are met when the lithosphere is coupled, so where there are no mechanical discontinuities in the vertical direction [Ellis, 1996]. Garthwaite and Houseman [2011] show that the continuity condition is met for any such sheet regardless of the boundary conditions More Over Less Competent [11] Next most common in the geophysical literature is the case of more competent materials over less competent, almost always as an elastic lid over a viscous half space. These models are intended to represent the amply documented existence of seismicity and faults in the uppermost crust; they are also sometimes used to investigate buckling instabilities in competent layers, although we defer this discussion to the multilayer section below. They present the widest range of qualitative results for velocity correlation through the entire system, ranging from a strong relationship between surface velocities and those at depth [Bourne et al., 1998] to no relationship at all [Li and Rice, 1987; Savage, 2000]. The interaction between layers is highly dependent on the nature and strength of coupling across the interface separating the more and less competent materials [Mancktelow and Abbassi, 1991; Zatman, 2000; Schueller et al., 2009], which may also be expressed through the absolute competence contrast. Smaller competence contrasts imply greater coupling. [12] At the highly (vertically) coupled extreme, models where the dominant force acting on discrete blocks of elastic upper crust is basal traction, the long-term velocity field at the surface calculated from finite strain (fault slip rates) equals the average velocity field in the viscous substrate [Bourne et al., 1998]. Any instantaneous observations of surface velocity also include elastic effects related to the frictional interaction from block to block in the more competent layer; these must be negligible compared to the basal tractions. The critical condition in this formulation appears to not only be the high level of basal coupling, but the presence of discontinuities in the uppermost material, making its integrated competence far less than the inherent competence of the material. These zero competence dislocations effectively reduce the competence of the upper layer to approach that of the substrate, making this solution asymptotically approach the homogeneous cases discussed above. [13] When, in contrast, the integrated competence of the upper layer is much greater than that of layers below, either because dislocations are not included in the upper layer or because they retain finite competence, velocities in the two modeled layers are unrelated [Zatman, 2000]. When the elastic lid has a free-slip basal boundary condition, no traction passes through the horizontal interface between layers, and again the solution approaches that for a monolayer, where now the layer is only the strong lid, not the composite of stiff lid with softer substrate. [14] An intermediate case with a single dislocation is given by Savage [2000]. This solution is more like the intact elastic lid analyzed by Zatman [2000] because stresses in the elastic lid are also determined by the boundary conditions on the sides of the model volume. There are no regions of the model space surrounded laterally by free-slip boundaries, as the internal blocks are in Bourne et al. [1998], so the driving mechanism to provide elastic lid stresses is always a constrained edge velocity rather than basal tractions. 983

6 [15] Results for materials at smaller scales reinforce the conclusion that deformation in a stiff lid can be excited by basal tractions only when there is coupling across the interface. In particular, thin metal films over soft polymers buckle in response to volumetric strain in the polymer [Bowden et al., 1998] and do so in regular sinusoidal structures such that the integrated horizontal finite strain at the surface is equal to that in the substrate near but below the interface. As long as the interface between the elastic film and the viscous substrate has a no-slip condition, so that the tractions and velocities are continuous across the interface, bending in the film and flow in the fluid interact to determine both the scaling and the strain rate of the resulting deformation, with the bending cost of the elastic deformation and the rate of viscous flow trading off to form the characteristic buckle wavelength [Huang and Suo, 2002; Mahadevan et al., 2010]. This is also true for engineered bilayer solids where, again, the buckling pattern is determined by the tradeoff between bending of the stiffer layer and stretching of the weaker layer [Concha et al., 2007]. [16] Therefore, for the generic scenario of greater over lesser competence, the competence contrast across the horizontal interface separating layers is the critical parameter. In separating this class of models from the set of monolayer models, we are assuming that some interaction of some kind occurs between layers, although it may be very limited and asymptotically approaches the monolayer problem as the competence contrast becomes very large or very small Less Over More Competent [17] The case of a less competent material flowing over a more competent base is classified as a gravity current (gravitationally induced flow) when gravitational potential energy excites the flow. Huppert [1982] provides a general similarity solution for flow on a rigid base where continuity of either tractions or velocities between the current and the base is precluded. Observations of velocity, displacement, or stress in the less competent material are independent of those in the more competent one and therefore cannot be used to infer the state of the base. [18] Copley and McKenzie [2007] consider the gravity current problem with either the stress-free lower boundary used by Huppert or a rigid deformable lower boundary. The rigid deformable condition allows for continuity of normal stresses across the interface, but precludes continuity of horizontal velocity across the same interface. Therefore, the whole velocity field is not continuous between the upper and lower layers, and information about deformation at the surface does not give a complete description of deformation at depth, but flows in both layers are related through the vertical stress continuity, such that displacements at the surface excite flows in the base and vice versa through gravitational potential gradients. The rigid deformable condition requires that the competence contrast across the interface is sufficiently small that the base cannot support normal stresses applied by the gravity current, such as the weight of the material in the current [Bendick and Flesch, 2007; Bendick et al., 2008]. [19] As in the more over less competent case, the relationship between deformation above and below the mechanical interface is determined by the magnitude of the competence contrast. When the materials have a smaller competence contrast, the displacement and velocity fields are related across the interface; when they have a larger competence contrast such that the interface approaches a free slip or stress-free boundary, displacement and velocity are independent in the two layers and solutions again approach the monolayer formulation, with only information about the deformation of the uppermost layer expressed at the surface Multiple Layers [20] Several geophysical models of continental crust with an elastic lid, a weak viscous lower crust, and a more viscous mantle lithosphere have been constructed [e.g., Li and Rice, 1987; Hetland and Hager, 2004; Burov, 2010] following from results in rock mechanics [e.g., Goetze and Evans, 1979; Burov and Diament, 1992; Kohlstedt et al., 1995]. In these cases, the competence contrast between the upper crust and mantle lithosphere is less important than the competence contrasts between the upper and lower crust and the lower crust and mantle lithosphere. If the lower crust is much less competent than either of the other layers, then there is no continuity of either velocity or stress through a column of lithosphere and the secular velocities of the surface and the mantle lithosphere are independent and, for a very weak lower crust, also independent of the lower crustal layer. The weak lower crust dissipates any stress of the layers above and below. Hetland and Hager [2004] point out, however, that transient velocity coupling may occur across the weak lower crust but this relation again depends on the magnitude and nature of the competence contrast, with strain rate-dependent rheology in the lower crust allowing for such coupling at short timescales. Ord and Hobbs [1989] suggest an even more complicated layered structure based on likely distributions of minerals in continental crust, which could be interpreted to suggest that any arbitrary arrangement of layers of varying competence may occur in this setting. [21] Sandbox multilayer models [Brun, 2002] similarly reproduce a range of responses from localized surface failure to quasi-continuous deformation depending on strain rates, the degree of coupling between crustal and mantle lithosphere strong layer analogs, and the relative competence of the crustal and lithospheric strong layer analogs [Haq and Davis, 2008]. The continuous, homogeneous deformation end-member is associated with strong vertical coupling or less competent mantle lithosphere; strong strain localization is associated with weak vertical coupling and more competent mantle lithosphere. This pattern suggests that the limiting cases for the sandbox configurations approach the more over less competent and less over more competent end-members discussed in the previous subsections. [22] Multilayered systems differ most from simpler laminates in their buckling response. In general, both folding and buckling instabilities are a consequence of abrupt and large contrasts in mechanical competence across layers [Cloetingh et al., 1999] because buckling minimizes work in more competent layers relative to homogenous shortening in less competent layers. The general stability analysis for folding in multilayered systems has a rich literature [e.g., Biot, 1961a, 1961b; Ramberg, 1961; Cobbold et al., 1971]. However, these solutions may sometimes not apply to geological problems because of the strong influence of initial perturbation amplitude and wavelength on fold stability [Mancktelow, 1999]. 984

7 [23] As in simpler laminates, coupling between layers of varying competence acts as a critical parameter in the stability analysis, such that bonding of the matrix-layer interface appears to have a much greater effect on the growth rate curve than theoretically predicted, with these effects especially significant at low viscosity contrast (<100) [Mancktelow and Abbassi, 1991]. Furthermore, when the viscosity contrast is small, the consequent stiffening of the laminated system results in shorter wavelength, more periodic folds than expected from the standard stability analysis [Mancktelow and Abbassi, 1991]. [24] More generalized studies of buckling and folding in multilayer systems confirm this relation. Schmalholz et al. [2002] identify folding modes that depend on a dimensionless combination of the Argand number (differently defined from that in England and McKenzie, but still a ratio of the boundary and body forces) and the effective viscosity contrast. Velocity is insensitive to depth only when the ratio of the length scale of deformation to the thickness of the deforming sheet satisfies the relation ðl=hþ 1= n 1, where L is the length of the sheet, H is its thickness, and n is the rheological exponent. Therefore, the relationship of surface deformation to that at depth is linear only at such scales. Where strain is localized more than this limit, buckling or folding contributes as much to deformation as sheet thinning and thickening, so surface observations are not simply related to deformation in nonsurface layers. Zuber [1987] provides stability analysis for a layered model of buckling on Venus under both extension and compression. When the effective viscosity contrast between layers is large, the development of dominant wavelength folding breaks down, so the form of folding in such layers is very unstable and often aperiodic [Mancktelow, 1999; Schmalholz and Podladchikov, 1999] precluding any estimate of layer deformation from general boundary conditions. For large deflections of layered material, even if the layers are mechanically connected, the distribution of strain and the subsequent patterns of deformation vary strongly from layer to layer. At large strains, the form of surface deformation is entirely different from that at depth, and from the integrated deformation of the composite multilayer package [Schmalholz et al., 2001]. Cloetingh et al. [1999] point out that folding of multilayered laminates is generally scale invariant, so these results can be applied at the scale of the whole lithosphere or to the crust alone. [25] Relatively new numerical formulations attempt to directly address the possibility of multiple deformation modes in multilayered bodies, rather than through stability analysis. For example, Medvedev and Podladchikov [1999a] extend the thin viscous sheet formulation to allow pure shear (the standard TVS model), simple shear (channel flow) and folding simultaneously. Results for a range of boundary conditions show that as higher-order terms in the force-balance formulation become important, the velocity, stress, and strain vary more with depth. Therefore, as vertical mechanical heterogeneity increases, the relationship between surface observations of deformation and deformation at depth becomes more complicated [Medvedev and Podladchikov, 1999b]. [26] In a full 3-D multilayer mechanical model consisting of alternating more and less competent layers, deformation consists of contributions from buckling, thickness changes, and lateral flow [Lechmann et al., 2011] as in the extended viscous sheet formulation. Solutions presented for horizontal velocity fields are most like equivalent solutions for thin viscous sheets when the buckling contribution is small [Lechmann et al., 2011], hence when competence differences from layer to layer are small, when the finite strain is small, when the viscosity is Newtonian, and when the Argand number is large. When these conditions are met, the velocity fields are similar in each layer, and therefore have little depth dependence. Therefore, the surface velocity observations are related to the velocities at depth. When competence contrasts are instead large, viscosities are nonlinear, or finite strains are large, competent layers buckle and less competent layers deform by internal flow plus thickness changes, such that the velocity field for each layer differs substantially. [27] Simpler numerical simulations specifically designed to investigate the role of competence contrasts in a layered viscous volume show that velocities at depth diverge from surface velocities either when an intermediate (lower crustal) layer is much less competent than layers above and below or when upper and lower layers themselves have very different competence and are separated by a low competence layer [Flesch and Bendick, 2012] (Figure 1). In that paper, sensitivity was evaluated using a 3-D multilayer finite element model where the relative competence of layers was systematically varied to represent common vertical rheological profiles for continental lithosphere under constant boundary velocity conditions. [28] A separate family of numerical simulations of the lithosphere without explicit internal layering consists of coupled thermomechanical formulations. In these cases, internal competence contrasts emerge as a consequence of the temperature and strain distribution in the model volume [e.g., Buck and Toksoz, 1983; Kusznir and Park, 1987; Willett et al., 1993; Hirth and Kohlstedt, 1996; Beaumont et al., 2004; Rey et al., 2009]. For cases where the effective viscosity obeys an Arrhenius-type law including a term with the form exp Q nrt where Q is an activation energy, n is a rheological exponent (and n = 1 is the Newtonian case), R is the ideal gas constant, and T is the temperature, regions in the model volume become less competent at high temperature and more competent at lower temperature. When erosion or other advective terms are included, these systems are often stable, reaching an equilibrium morphology and thermal state once the tectonic inflow of material matches the erosional outflow [e.g., Willett and Brandon, 2004]. In these cases, the deformation field within the orogen is internally coherent, or coupled everywhere, but is decoupled by the thermal evolution from material below, such as in a subducting slab or indenter, and from material laterally adjacent in a fixed backstop. In coupled thermomechanical models also including strain dependent terms, typically of the form ε fðþ n where the strain rate, ε, or sometimes a strain invariant such asi 2 is raised to some function of the rheological exponent, the model volume typically evolves to a vertically decoupled state [e.g., Beaumont et al., 2004] as a consequence of the positive feedback between progressive strain and decreasing effective viscosity. Although the development of these models does not require explicitly specifying competence contrasts in the crust or lithosphere, once they emerge, they can be classified in the same way as the other models cataloged in this paper. Specifically, coupled thermomechanical models where large 985

8 (though not fully discontinuous) competence contrasts exist display strong vertical gradients in deformation and velocity and approach the explicitly multilayered case. Therefore, information about the deformation or velocity of one layer cannot be used to constrain the deformation or velocity of a different layer. Where the competence contrasts are limited, such as by negative feedbacks between temperature and advection of mass out of the model, deformation coherence persists. [29] In summary, the theoretical development of continuum mechanics over a wide range of length scales demonstrates that a critical parameter determining the relationship of surface deformation to deformation at depth is the magnitude of vertical competence contrasts within the material volume. Where competence varies little, mechanical coherence is maintained throughout the evolution of deformation, so surface observations are illustrative of the whole lithosphere. Where competence varies much, surface observations are unrelated to deformation at depth and the response of the integrated system cannot be inferred from them. The role of competence contrast is similar in systems with two to an infinite number of layers. For monolayers, the concept of competence contrast does not apply, and deformation is by definition coherent throughout the medium. 3. Classification and Comparison [30] Because competence contrast is a critical parameter in determining the relationship of surface deformation to deformation at depth in the lithosphere, it is also a critical parameter in determining the relationship of the surface deformation to the dynamic state of the lithosphere. Specifically, the magnitude, the depth, and the sign of abrupt changes in mechanical competence all change the amount of information about integrated lithospheric deformation, hence the balance of forces in the lithosphere, accessible in the surface kinematic field. [31] We therefore classify plausible mechanical profiles for continental lithosphere based on the magnitude and sign of competence contrasts, in keeping with the general classification of mechanics solutions in the previous section. This classification should serve the joint purposes of suggesting appropriate end-member formulations for forward modeling of surface kinematics and indicating the extent to which dynamic state can at all be inferred from surface observations (Figure 1) Type 1 (Single Layer) [32] Type 1 lithospheric settings are those where the competence profile with depth is continuous (Figure 1). That is, there are no abrupt changes in competence with depth in the lithosphere. Oceanic lithosphere is typically assumed to have this characteristic, supported by rock mechanics [Brace and Kohlstedt, 1980] with competence increasing linearly to a maximum under the brittle failure regime in the crust and then decreasing nonlinearly with increasing temperature and depth in the mantle lithosphere. Materials with a single competence maximum and continuous competence can be represented by a single layer with equivalent effective competence. In elastic approximations, this competence is represented by the effective elastic thickness, Te, or the flexural rigidity, D. In viscous approximations, this competence is represented by an effective Newtonian viscosity or power law rheology. In either case, analytic solutions exist which efficiently describe the mechanics of bending, buckling, stretching, translation, and rotation for equivalent sheets, even when they have nonplanar geometry, all from the catalog of solutions for monolayers (Figure 2). These analytic solutions also give the strain or velocity throughout the entire layer, and in all cases, the integrated strain or velocity is closely related to or equal to the surface strain or velocity. Therefore, for type 1 settings, surface observations of deformation are representative of deformation throughout the lithosphere. Either surface observations or integrated observations of deformation include responses to all boundary and body forces, so can be used to either invert for the dynamic state of the lithosphere if the effective constitutive law is known or for the constitutive law if the dynamic state is known. [33] Some lithospheric settings have discontinuous but approximately symmetrical competence profiles with depth. These cases typically consist of upper crust and mantle lithosphere with similar maximum viscosities or elastic moduli separated by less competent lower crust. The magnitude of the competence contrast determines whether an equivalent monolayer can be substituted for the multilayer composite. Specifically, if the weaker lower crustal layer is still competent enough to couple vertical stresses then the surface deformation is a harmonic of the individual layers displacements [Cerda and Mahadevan, 2003]. The TVS assumption that deformation is not depth dependent mostly holds because the upper crustal and mantle lithosphere layers have the same set of imposed forces and the same maximum viscosity [Lechmann et al., 2011] so both have very similar deformation responses. However, if the lower crustal layer is so weak that it mechanically decouples the lithosphere, the surface deformation is instead a linear sum of the individual layers displacements. Furthermore, in the decoupled case, stresses due to topography are not transferred from the crust to the mantle lithosphere [Bendick and Flesch, 2007], so the mantle lithosphere responds mostly to boundary forces and contributes little to the surface deformation and only at very long wavelengths. [34] Inversion or forward models using monolayer approximations in these cases can still produce good fits to observations of surface deformation, as long as the effective material parameters are adjusted. This condition then differs from the simply continuous case, because the effective viscosity or flexural rigidity of the model sheet is not representative of the actual mechanical properties of any of the materials involved, and may not have a physical meaning. Therefore, the choice of a multilayer formulation or an equivalent monolayer should be made based on the process to be targeted Type 2 (More Over Less Competent) [35] Type 2 lithosphere consists of materials with discontinuous and asymmetrical competence (Figure 1). In this case, the surface displacement field is a highly nonlinear combination of the deformation of different parts of the lithosphere, themselves dissimilar. Deformation through the lithosphere is strongly depth dependent, so surface displacements do not have a clear relationship to displacements at depth. The surface velocity field for the case of a stronger upper crust is dominated by the boundary forces. The most 986

9 common approach in the geologic literature for this case is block modeling, in which nearly rigid domains (based on some threshold velocity residual) are defined by finding sets of Euler poles that best describe observed surface velocities, usually from GPS geodesy (Figure 2). In some cases [e.g., Thatcher, 2007; Loveless and Meade, 2011] the boundaries of the rigid blocks are specified a priori using constraints from regional fault mapping. In other cases [e.g., Payne et al., 2012] block boundaries are estimated by inversion from the velocity field, either by finding a set of blocks that satisfies the residual tolerance or by mapping gradients in the velocity field Type 3 (Less Over More Competent) [36] As in type 2, type 3 lithosphere consists of materials with discontinuous and asymmetrical competence and deformation through the lithosphere is strongly depth dependent, so surface displacements do not have a clear relationship to displacements at depth. The surface velocity field for weak upper layer is dominated by buoyancy forces even though both boundary and buoyancy forces are important to the model dynamics, therefore this case is typically modeled as a gravity current or gravitational collapse in the geologic literature [e.g., Copley and McKenzie, 2007; Copley et al., 2011] (Figure 2). It has most often been applied to the Tibetan Plateau and western North America [Sonder and Jones, 1999] Type 4 (Multilayer) [37] An arbitrary arrangement of layers with varying competence has long been used at smaller length scales to describe buckling and folding of crustal layers or rock units, especially in sedimentary sequences. Because of the scale invariance of buckling, the same principles apply to a crustal composite with a complicated geologic history, or, indeed, to the whole continental lithosphere. In these cases, the displacement field of each layer approaches independence both from neighboring units and from the average displacement of the entire composite (Figure 2). Although numerical solutions exist for the full multilayer solution [e.g., Lechmann et al., 2011], they are very sensitive to the choices of mechanical properties for each layer, and the continuity conditions imposed internal to the model volume. Therefore, applications to real geologic settings, such as active intracontinental tectonics in Asia or North America, are highly nonunique. Independent constraints on important parameters, such as effective viscosities, temperatures, or lateral continuity [e.g., Yao et al., 2008; Yao et al., 2010] are difficult to acquire, if not impossible, but entirely determine the relationships of interest between dynamics and kinematic observations. Unless both the kinematics and dynamic state are known a priori, the kinematic observations cannot be used to invert for mechanical parameters. Therefore, although multilayer formulations are of considerable theoretical interest for continental tectonics, their practical utility in specific settings is hampered by severe underdetermination. [38] This classification differs from that presented in Flesch and Bendick [2012], but better parallels the categories of general mechanics solutions available in the literature. We therefore find it more useful for identifying appropriate matches between terrestrial tectonic settings and mechanics solutions, as in the following section. 4. Observations of Continental Tectonic Settings [39] The development of deformation analysis in specific tectonic settings on Earth mirrors the theoretical development, in that the choice of model conditions has a strong influence on the interpretation of regional dynamics. In particular, a surface velocity field can be fit to the same level of statistical significance by a wide range of models with fundamentally different mechanics and dynamics [e.g., Flesch and Bendick, 2007]. Therefore, the most important contributions to tectonic reconstructions, especially reconstructions of dynamics, are typically the a priori assumptions about competence contrasts in the lithosphere. Often, the nonuniqueness of the problem and the sensitivity of the interpretation to these assumptions are not addressed in individual manuscripts, leading to multiple, mutually exclusive dynamic interpretations of identical observational data sets. [40] Three regions of distributed tectonically-excited deformation are considered the type cases for continental tectonics, and have the largest amount of surface deformation data available: Tibet for continental convergence, the North American Basin and Range for wide rifting or transtensional kinematics, and East Africa for continental extension. We illustrate the task of choosing appropriate mechanical models, and the implications thereof, in the following examples Tibet [41] The surface deformation field for the Tibetan Plateau is relatively well constrained by GPS, although the observations are not equally distributed in space [Gan et al., 2007] (Figure 3). Additional constraints for the more remote parts of the plateau are available from seismic focal mechanism and moment tensor analyses [Carey-Gailhardis and Mercier, 1987; Holt et al., 1991; Mitra et al., 2005], Quaternary fault slip rates [e.g., Murphy et al., 2000; Lacassin et al., 2004, Cowgill, 2007], and remote sensing, especially radar interferometry [Wright et al., 2004]. The very wide range of interpretations for Tibetan dynamics illustrates the profound influence of mechanical assumptions. Specifically, the same surface velocity field can be fit with a thin viscous sheet (type 1) [England and Houseman, 1986; Flesch et al., 2001], elastic blocks (type 2) [Thatcher, 2007], an elastic lid over a viscous half space (type 2) [Ryder et al, 2007; Loveless and Meade, 2011], a gravity current over a more competent base (type 3) [Copley and McKenzie, 2007; Copley et al., 2011], and a variety of weak lower crust approximations (type 4) [Royden, 1996; Bendick et al., 2008; Lechmann et al., 2011]. For each of these solutions, the inferred fundamental dynamic state differs, reiterating the conclusion from theoretical analyses of deformation in vertically heterogeneous materials that the surface observations depend both on the dynamic state of the system and on its mechanical properties, and that if neither is known, surface deformation cannot be used to uniquely identify the dynamic state. As a corollary, the wide range of similarly successful models suggests that if the study target is not the full dynamic state of the lithosphere in a particular region, then different simplifying assumptions may be appropriate depending on the parameters of interest. [42] In detail, homogeneous sheet based analyses typically find contributions of similar magnitude from tectonic boundary forces and gravitational body forces [England and 987

10 Houseman, 1986; Flesch et al., 2001]. Elastic block [Tapponnier et al., 2001], elastic over viscous [Thatcher, 2007; Meade, 2007], or viscoelastic over viscous [Loveless and Meade, 2011] kinematic models, which are all variants of the type 2 formulation, imply that tectonic boundary forces dominate the regional dynamics. They also imply that GPS in continents records primarily or exclusively elastic deformation within the context of the seismic cycle, and topography is not an important contributor to the state of stress. In contrast, Copley and McKenzie [2007] and Copley et al. [2011] include long wavelength topography data with the surface velocity field in gravity current models (type 3) to reach the opposite conclusion: that gravitational body forces dominate the regional dynamics. Finally, multilayer models [Royden, 1996; Clark and Royden, 2000; Beaumont et al., 2004; Bendick and Flesch, 2007; Bendick et al., 2008; Lechmann et al., 2011] conclude that the surface deformation field primarily provides information about the upper crust at short wavelengths and limited information about the lower crust at longer wavelengths. Little or no information about mantle lithosphere deformation is expressed at the surface [Ellis, 1996]. The dynamics include contributions from both boundary and body forces, with the former mostly exciting displacement of the competent upper crust and the latter exciting flow in the lower crust. [43] In summary, surface observations of deformation can be matched by versions of the entire theoretical range of mechanical configurations for continental lithosphere. Differences in the mechanical assumptions for each type lead to mutually exclusive interpretations of the dynamic state of the collision zone, from boundary force dominated to body force dominated. These conflicts emerge not from the observations themselves, but from assumptions about the variation of deformation with depth in the lithosphere, and the consequent contributions to the surface field of different deformation mechanisms. [44] In Tibet, independent constraints from seismic anisotropy of the mantle lithosphere are available to constrain the vertical coherence of deformation. The use of these data is sensitive to the interpretation of the anisotropic signal [Lechmann et al., 2011], but for reasonable relationships between fast direction orientation and strain tensor orientation [McNamara et al., 1994; Hirn et al., 1995; Huang et al., 2000; Flesch et al., 2005; Wang et al., 2008], deformation in the mantle lithosphere appears to match the surface deformation, at least at long wavelengths. This coherence requires transmission of at least vertical stresses [Bendick and Flesch, 2007] and suggests that the deformation field in Tibet is at least quasi-continuous with depth, so that the surface field is indicative of integrated deformation throughout the lithosphere [e.g., Bendick and Flesch, 2007], although this conclusion remains contentious Basin and Range [45] The North American Basin and Range has abundant geodetic observations (Figure 3) typically modeled using more over less competent formulations [Dixon et al., 1995; McClusky et al., 2001; Hammond and Thatcher, 2004; Hammond et al., 2011]. In this case, the mechanical assumptions are supported by independent seismic results suggest a very thin or missing mantle lithosphere over a weak asthenosphere [Li et al., 2007]. In this case blocks or microplates with length scales less than the lithospheric thickness dominate the surface deformation field and mainly express the contribution of tectonic boundary forces. In the Basin and Range, the surface strain is associated with individual normal faults and upper crustal blocks rotating in response to Pacific shear [Hammond and Thatcher, 2004;Hammond and Thatcher, 2007; Hammond et al., 2011]. Thin viscous treatment of the regions has demonstrated that GPE variations produced by the elevated Basin and Range regions provides enough stress to drive deformation [Jones et al., 1998; Sonder and Jones, 1999] however the modeled deformation is nearly 90 off from the observed extension direction [Flesch et al., 2007], and this highlights the importance of stresses associated with Pacific-North American relative plate motion to fit the observed GPS and geologic data. Humphreys and Coblentz [2007] also argued that tractions due to large-scale convection were not coupled to surface motions. Additionally, combining thin viscous sheet methodology with fault strength modeling showed that below 20 km depth there is very little contribution to the depth integrated deviatoric stresses from GPE in the Basin and Range [Klein et al., 2009]. Using shear-wave splitting data as an approximation for deformation in the mantle in the Basin and Range shows no correlation to the surface velocities or geologic structure but instead reveals the direction of the convecting mantle below [Silver and Holt, 2002; Zandt and Humphreys, 2008]. All the above studies show that none of the dynamic contributions from either the sublithospheric mantle or the mantle lithosphere are transferred to the surface kinematics in usable form, and the kinematics cannot be used to infer those dynamics. It is interesting to note that estimates of vertically averaged effective viscosity in the Basin and Range [Flesch et al., 2000] are smaller than those in Tibet [Flesch et al., 2001] where the estimated effective viscosity reflects the integrated viscosity over the whole lithosphere indicating that the competence of the mantle maybe weak and does not contribute to the integrated strength. [46] However, intermediate to short wavelength topography, but not the surface instantaneous velocity field, is best reproduced using a multilayer model with a less competent lower crust over either a rigid deformable or rigid base, hence a slightly to much more competent mantle lithosphere [McKenzie et al., 2000]. This model suggests a role for gravitational potential gradients in promoting flows in the less competent layer, but only at length scales comparable to the width of typical basins and ranges. As with all multilayer models, contributions to the surface deformation field from layers at depth in this approximation are scale dependent and difficult to separate. Like the more over less competent approximations, the deformation through the lithosphere cannot be inferred from surface signals, so neither can the dynamic state. [47] To summarize, as in the theoretical development of deformation in vertically heterogeneous materials, the interpretation of surface kinematics in terms of dynamic state is utterly dependent on assumptions about the mechanical properties. In particular, a full range of possibilities from exclusively boundary force driven to exclusively body force driven can fit surface observations if the vertical distribution of competence in the continental lithosphere is adjusted. 988