show that analyses of geological structures using kinematics alone provide, at
best, ... the development of geological structures and therefore they argue for a ...
Kinematics vs. Mechanics in Understanding Rock Deformation Raymond C. Fletchera, David D. Pollardb a
Dept. of Geological Sciences, University of Colorado, Boulder, CO 80309-0399, USA
b
Dept. of Geological and Environmental Sciences, Stanford University, Stanford, CA 94305-2115, USA
Abstract The products and processes of rock deformation may be understood using a complete mechanics, of which kinematics is an integral part. We describe what is meant by a complete mechanics and show that analyses of geological structures using kinematics alone provide, at best, limited insight. To make these points we refer to general concepts from dimensional analysis and the equations of motion, and we describe particular results from models of pluton expansion, oblique plate convergence, ductile shear zones, and spreading nappes. We conclude that kinematics is necessary, but not sufficient to achieve understanding. ________________________________________________________________________ 1. Introduction Tikoff and Wojtal (1999) take the point of view that displacements or velocities control the development of geological structures and therefore they argue for a kinematic approach to structural analysis. On the other hand Fletcher and Pollard (1999) take the point of view that kinematics is not sufficient: a complete mechanics is required to understand the processes and products of rock deformation. These two papers, as well as others in the 20th Anniversary Issue of this journal, motivated a debate at the annual meeting of the Geological Society of America on the usage of kinematics and mechanics in understanding rock deformation. The debate teams were composed of the authors of these two papers. This paper addresses the relative merits of the approaches advocated in the debate using examples drawn from Tikoff and Wojtal (1999), their correlative papers, and the antecedent papers authored by others. A complete mechanics includes a complete sub-set, or “N equations in N unknowns”, of the laws of conservation of mass, momentum and energy, of relations describing kinematics, and of the constitutive relations describing material behavior. Note, in particular, that the kinematic
2 vector quantities of displacement, u, velocity, v, and acceleration, a, and the associated deformation gradient tensor (strain and rotation) and velocity gradient tensor (rate of deformation and vorticity) are found throughout these equations, but are by no means the only physical quantities therein. We concede that tectonic processes and their products will not be completely described by mechanical models, but advocate the position that the mechanics must be complete. Thus, for prescribed initial and boundary conditions, a forward model is generated that may produce likenesses of some of the observed geological structures and fabrics, to some satisfactory degree of approximation. If the forward model fails to produce satisfactory likenesses we learn that one or more of the postulates is inappropriate and must be excluded or modified. One cannot, however, exclude or modify the fundamental laws upon which the mechanical model is based. The general method we advocate is the construction of a sequence of quantitative models, graduated in their degree of detail and successively providing an improved understanding. Tikoff & Wojtal (1999) suggest that: “There are two distinct ways of viewing the development of geological structures”. We propose a single distinct way: one applies the extensive conceptual framework of the physics and chemistry of the processes involved, most centrally rock deformation, to both suggest observations and to analyze, visualize, and interpret them. In this way, a coherent and self-consistent, if idealized, and non-unique, re-construction of the development of a geological structure may be achieved. To demonstrate how the methodologies differ we review a number of kinematic models and ask three questions: (1) Does the model produce a valid and insightful result? (2) Does it contain error? (3) Can the use of a complete mechanics substantially correct or advance the result? 2. Dimensional Analysis The distinction between a complete mechanics and kinematics alone is, perhaps, most simply appreciated by considering a dimensional argument (Hubbert, 1937; Brodkey & Hershey, 1988, ch. 8). Here we refer symbolically to the fundamental dimensions of length [L], mass [M], time [T] and temperature [Θ], and point out that these compose a complete set for the mechanical analysis of structures. Deformation is quantified in terms of the initial and final positions of the particles of a body, so it involves quantities with dimensions of length [L]. The description of deformation, also involves dimensionless quantities (e.g. strain and rotation [LL-1 = L0]).
3 Physical quantities of a kinematic analysis of flow, such as velocity [LT-1] and acceleration [LT2
], have dimensions that are composed of length and time raised to various powers. Time itself
might be eliminated using a monotonic quantity such as a principal strain or displacement as a parameter of advance. In contrast, a complete mechanics is based upon the conservation laws of mass, momentum and energy which are formulated using physical quantities that include mass and temperature in their dimensions in addition to length and time. Some of these quantities are, for example, mass density [ML-3], stress [ML-1T-2], viscosity [ML-1T-2] and heat capacity per unit mass [L2T-2Θ-1]. None of these quantities can be derived from the kinematic quantities alone. Based upon dimensional analysis we assert that one cannot formulate a description of a process of deformation that ignores mass and temperature, yet necessarily satisfies the conservation laws. Put another way, no purely kinematic description of deformation can be certified as obeying these laws of nature. In the context of a complete mechanics one begins with these fundamental laws and derives the kinematic fields for the structure under investigation from the solution to a particular initial and/or boundary value problem. Thus, the complete mechanics makes full use of kinematics, but pre-determines that all of the derived kinematic quantities, and their spatial and temporal variations, are consistent with the relevant laws of nature. The aim of kinematic modeling, as we understand it, is to investigate the path of deformation. Since geological information on this is available (Elliott, 1972), we do not deny that such modeling may be useful. On the other hand, kinematic modeling alone does not lead to an understanding of the mechanics of deformation. For example, statements such as: “This kinematic analysis also provides a quantitative tool to understand the mechanics of oblique convergence: i.e. the fundamental difference between kinematic partitioning and mechanical decoupling” (Tikoff & Teyssier 1994, p. 1575) violate the basic principles of dimensional analysis. Based upon these principles kinematic modeling must rigorously exclude conclusions about mechanical quantities, mechanisms, or processes that involve mass or temperature. Application to the pluton space problem Limiting ones view to kinematics places constraints upon the questions that can be asked, legitimately, in the course of a structural analysis. Consider the pluton space problem addressed by Tikoff, de Saint Blanquat, and Teyssier (1999). These authors consider a part of the Mono
4 Creek pluton where its eastern contact locally bulges into the Round Valley pluton and apparently deflects the metamorphic host rocks about 7 km to the east (see their Fig. 1). “We have shown that accounting for the role of translation explains that the observed strain does not record the total emplacement strain. Rather, the emplacement strain is primarily a result of translation, itself caused by push of magma." This statement is inconsistent with kinematic principles because strain results from displacement gradients and translation, by definition, involves no such gradients. None-the-less, the authors conclude: …"this approach does lead directly to two important questions: How is the translation accommodated? And how large is the magma pressure?" Consider the second question in light of the dimensions of pressure [ML-1T-2]. Having only geometric and kinematic quantities in hand, there is no procedure to estimate pressure, because the requisite equation must be dimensionally homogeneous yet mass is not included in the dimensions of the geometric and kinematic quantities that were investigated. Missing from the kinematic analysis are the constitutive properties, such as viscosity [ML-1T-1]. The authors acknowledge the necessity for including material properties when they suggest “translation was certainly facilitated by shells of gradually increasing rheological strength that surrounded the pluton” (Tikoff et al. 1999). They conclude that “the exact source of the magma pressure and its evolution through time remain exciting questions in the study of granite intrusions.” We agree, but a methodology limited to kinematics cannot address these questions. The ‘shell’ model used to address the pluton space problem (Tikoff et al. 1999) is virtually equivalent to Ramsay’s (1989) kinematic model for the expansion of emplaced, partially solidified magma and wall rock by addition of new magma. Ramsay uses this model to rationalize the distribution of finite strain recorded in deformed xenoliths in the composite Chindamora Batholith. The only distinction is that the ‘shell’ model is for cylindrical geometry, whereas Ramsay uses spherical geometry. The success of Ramsay’s model could be viewed as supporting the author’s kinematic analysis, but it was not cited. As an example to compare with these kinematic models we offer the following simple, but not completely satisfactory, two-dimensional mechanical model of steady-state intrusion from a vertical dike (Fig. 1). This model is developed using the methods described by Johnson and Fletcher (1994, p.81), so the details are omitted. Intrusion takes place from a vertical dike into an unbounded homogeneous viscous medium, and material input is prescribed by two
5 boundary conditions along the base of the model: the normal velocity is parabolic across the dike and otherwise zero; the horizontal velocity is zero. For mathematical convenience the model is spatially periodic and the dike dimension, X, is a modest fraction of the spacing, L; in the case shown, X/L = 0.2. The resulting flow in the viscous half-space is “steady,” although the temporal history of magma input may be arbitrary. After one dimensionless time unit, the form of the ‘intrusion’ and the kinematics of an initially square grid are shown in Figure 1. The particle initially at the origin has moved out to the far edge of the intrusion, but one would not speak of the motion, or the underlying mechanism that resulted in it, as ‘translation’. Indeed, the distorted grid illustrates a heterogeneous deformation outside the ‘intrusion’ and the velocity field within is not homogeneous. In this mechanical model the boundary conditions were expressed in terms of prescribed velocity components at the surface of the half-space. One might suppose, then, that the motion within the half-space could be spoken of as ‘velocity controlled,’ a concept introduced by Tikoff and Wojtal (1999), but this is inappropriate. First, the process within the half-space is governed by the relevant field equations, including the constitutive relations. Second, although boundary conditions on velocity are stipulated for the surface of the half-space, a further condition is the vanishing of motion and associated stresses far from the boundary. To evaluate some of the mechanical parameters of this model suppose the viscosity of the country rock and emplaced intrusion is η = 1020 Pa-s, and the cylindrical intrusion has been emplaced in 100,000 years and is 5 km in radius. This gives a required maximum input velocity of (15π/8)cm/a ≅ 6 cm/a, and the maximum pressure at the center of the dike is 94 MPa. Thus the mechanical model leads directly to an estimate for pressure. However, the large value suggests that a modification to the model is in order. The viscosity might be lowered substantially, for example, on the supposition that the most strongly deformed region is one in which intrusion has led to greater temperature. While considerable refinement of this simple model would undoubtedly enhance comparisons to the bulge in the Mono Creek pluton, it is adequate to demonstrate the dimensional necessity to look beyond kinematics for answers to questions about space problems and magma pressures. 3. The Independent and Dependent Variables of Deformation
6 In the abstract of their paper Tikoff and Wojtal (1999) state: "… taking velocities and displacements as independent variables in deformation and stresses as dependent variables requires fewer assumptions and is more consistent with the observed geology." In contrast, we suggest that deformation is both understood by and constrained by the conservation laws, the kinematics relations, and the constitutive relations of a complete mechanics. Ignoring mass and energy conservation, as in the cited paper, one may identify the independent and dependent variables of deformation in the context of momentum conservation. For any volume element, fixed with respect to an inertial reference frame, and located within a deforming continuous rockmass, conservation of linear and angular momentum are prescribed as follows (Bird et al. 1960, p. 76-78, Fung 1969, p. 199-202):
∂ ∂ ∂ ∂ (ρv x ) = − (ρv xv x ) − (ρv x vy )− (ρv xv z ) ∂t ∂x ∂y ∂z ∂σ ∂σ ∂σ + xx + yx + zx + ρ gx ∂x ∂y ∂z ∂ ∂ ∂ ∂ ρ vy )= − (ρ vy v x )− (ρ vy vy )− (ρ vy vz ) ( ∂t ∂x ∂y ∂z ∂σ xy ∂σ yy ∂σ zy + + + + ρ gy ∂x ∂y ∂z ∂ (ρvz ) = − ∂ (ρv zv x ) − ∂ (ρvzv y )− ∂ (ρv zvz ) ∂t ∂x ∂y ∂z ∂σ yz ∂σ zz ∂σ + xz + + + ρgz ∂x ∂y ∂z σ xy = σ yx , σ yz = σ zy , σ zx = σ xz
(1a)
(1b)
(1c)
(1c)
The dependent variables in these equations of motion are the mass density, ρ, the components of the velocity vector, vi, the components of the stress tensor, σij, and the components of the acceleration of gravity vector, gi. The independent variables are the three spatial coordinates (x, y, z) and time, t. In general, mass density, velocity, stress, and gravitational acceleration are
functions of all three spatial coordinates and time. The deformation governed by these equations may be both heterogeneous and non-steady. Many tectonic processes are confined to rock masses for which the density and gravitational acceleration may justifiably be considered homogeneous in space and constant in time. Under these conditions we conclude from (1) that velocity and stress are the dependent
7 variables, while the spatial coordinates and time are the independent variables. Together, these equations describe the motion of a deforming rock mass regardless of length and time scales. Furthermore, although they do play an important role in more specialized versions of the equations of motion, the constitutive relations are not invoked to derive (1)—the fundamental principle of momentum conservation applies to any rock mass regardless of its constitutive properties. In this conceptual framework one is not required to choose between velocity and stress as the independent variable. Can the stress components be eliminated from (1) in favor of the velocity components, and does this reduce the study of deformation to kinematics? The answer is yes to the first question, but no to the second. For example, consider the simplest and most commonly applied constitutive law for rocks that flow: a homogeneous, isotropic and incompressible viscous fluid. This material behavior is defined in terms of the fluid pressure, p, the viscosity, µ, and the components of the rate of deformation using (Fung, 1969, p. 213-215): ∂v ∂v ∂v x , σ xy = µ x + y = σ yx ∂y ∂x ∂x ∂v ∂v ∂v σ yy = − p + 2µ y , σ yz = µ y + y = σ zy ∂z ∂y ∂y ∂v ∂v ∂v σ zz = − p + 2µ z , σ zx = µ z + x = σ xz ∂x ∂z ∂z
σ xx = − p + 2µ
(2a,b) (2c,d) (2e,f)
The equations of motion for this material are (Bird et al. 1960, p. 80): Dv x ∂p ∂ 2 v x ∂ 2v x ∂ 2 vx ρ =− +µ + + 2 + ρgx ∂x 2 ∂y 2 Dt ∂x ∂z 2 2 ∂ v ∂ v ∂ 2 v Dv y ∂p y y y ρ =− +µ 2 + 2 + 2 + ρgy ∂x Dt ∂y ∂y ∂z Dv ∂p ∂ 2v ∂ 2v ∂ 2v ρ z = − + µ 2z + 2z + 2z + ρgz ∂x Dt ∂z ∂y ∂z
(3a) (3b) (3c)
The stress components are eliminated from (1), leaving the Navier-Stokes equations, but constants mass density and viscosity, and the dependent variables pressure, acceleration of gravity and velocity remain. Eliminating stress in favor of velocity does not reduce the study of deformation to kinematics.
8 Investigating just the kinematics of structures is advocated because: “Kinematics is an accurate description of a physical system, with many fewer assumptions than an interpretation derived from an inferred stress distribution. We do not need to assume forces or rheology (constitutive relations), both of which are required by dynamic models and are poorly known for naturally deformed rock" (Tikoff and Wojtal 1999). The argument that kinematics involves many fewer assumptions sounds attractive, but consider the following (rarely stated) assumptions that follow from (1) – (3) and are integral to kinematic analyses: 1) Mass is irrelevant; 2) Conservation of mass may be violated; 3) Conservation of linear and angular momentum may be violated; and 4) The strength and constitutive properties of rock are irrelevant. The utility of a purely kinematic analysis might be viewed in a different light were these assumptions acknowledged and assessed. 4. Homogeneous Deformation and Flow
To illustrate the kinematic methodology Tikoff and Wojtal (1999) conceptualize and then analyze structural problems in terms of homogeneous deformation and homogeneous flow (see their Figs. 1 and 4). For these conditions the components of the strain tensor are uniform (Means 1976, p. 199) and the components of the rate of deformation tensor (Fung 1969, p. 120) are not functions of the spatial coordinates (x, y, and z), so the partial derivatives of these components (∂/∂x, ∂/∂y, ∂/∂z) all are zero. If one’s view is restricted to a homogeneous deformation or flow, then x, y, and z are essentially irrelevant and much of the machinery of a complete continuum mechanics can be ignored. While such a position has the attractive consequence of making the tools of structural analysis simpler both conceptually and mathematically (this can be thought of as a ‘pre-Calculus’ version of structural analysis), in our opinion the limitations outweigh the advantages. The most obvious limitation is that homogeneous deformation can, at best, only describe a restricted sub-volume of a developing structure. Because straight and continuous material lines remain straight and continuous within this sub-volume, homogeneous deformation cannot, for example, describe a volume of rock that includes a fold or straddles a fault (Ramsay, 1967, p. 54; Means 1976, p. 132).
9 Application to ductile shear zones and oblique plate convergence Homogeneous deformation may be applied to a sub-volume within a structure—a common occurrence being the interior of ductile shear zones according to Tikoff and Wojtal (1999). The rationale for studying such a sub-volume could be the determination of the local strain or rate of deformation components, and this certainly has merit. However, there must be a spatial variation of deformation across any shear zone (at least near the boundaries) and along the zone (at least near the tip line) as the deformation changes from substantial to negligible. Documenting these spatial variations in deformation near the boundaries and tip line could be accomplished in a piecemeal fashion by sampling and analyzing homogeneous sub-volumes, if suitable strain markers are present. However, understanding the spatial variations, we argue, requires one to abandon the context of a homogeneous deformation and grapple explicitly with the spatial derivatives of the strain or rate of deformation components as treated in a boundary or initial value problem using a complete mechanics. A motivation for such an effort is, for example, the possibility of addressing questions about how shear zones grow in width and length (Simpson, 1983; Segall & Simpson, 1986; Bürgmann & Pollard, 1994; Christiansen & Pollard, 1997). Tikoff and Teyssier (1994) use homogeneous deformation to model crustal-scale zones of oblique plate convergence, notably the Great Sumatran and the San Andreas fault systems. The kinematic model (see their Fig. 1) is composed of pure shear perpendicular to the fault zone and simple shear parallel to the fault zone in varying proportions. To address the possibility of uniform slip on a discrete fault through the middle of the zone they treat each half as homogeneously and similarly deformed, but allow one half to translate relative to the other along the fault (see their Fig. 7). Rigid plates bound the fault zone and move with prescribed convergent velocities (‘boundary conditions’), while deformation within the zone is ‘partitioned’ among pure shear, simple shear, and slip on the discrete fault to perfectly accommodate the relative motion of the plates. Slip reduces the amount of simple shear in the adjoining parts of the fault zone and material within the zone extrudes upward, maintaining a constant volume. The authors “assume that initial faulting in a homogeneous block is controlled by the orientation of the instantaneous strain responding to the initial movement of material points.” Thus, for example, strike-slip faults form initially in a zone dominated by simple shear, whereas
10 later “thrust faults striking near-parallel to the orogen must form to accommodate finite rather than instantaneous strain.” For a zone dominated by pure shear “thrust faults form first, in response to the applied instantaneous strain. … strike-slip faults form in response to finite strain … .” We note that these relationships between strain and faulting are distinct from, and not easily reconciled with a vast literature that has accumulated over the past fifty years and is summarized in textbooks and monographs (e.g. Anderson, 1951; Griggs & Handin, 1960; Price, 1966; Jaeger and Cook, 1969; Johnson, 1970; Hubbert, 1972; Paterson, 1978; Turcotte & Schubert, 1982; Atkinson, 1987; Scholz, 1990; Engelder, 1993; Middleton & Wilcox, 1994). In the physical picture established to date, shear fracturing and faulting are related to the mechanical properties of rock (strength, friction, elastic moduli, etc.) and criteria for fault formation and slip are written in terms of the stress state and not the strain state. According to Tikoff and Teyssier (1994) their new relationships between strain state and faulting are “corroborated by physical experiments” and they mention, among others, the clay cake models of Withjack & Jamison (1986). Analogue (scaled) model experiments such as these have an important role to play in structural geology, but they do not corroborate the new theory of faulting that would, if correct, overturn half a century of laboratory and theoretical research. Among other inconsistencies we mention: 1) Withjack and Jamison do not describe a sequence of faulting that is similar to the new theory; 2) they do find a reasonable correlation between fault trends in the clay cake and trends predicted by Anderson’s theory of faulting, based upon the state of stress; 3) the analogue experiments are for transtension not transpression, yet rocks behave quite differently in tension and compression; and 4) rock strength depends upon factors not included in these experiments such as confining pressure, temperature, and strain rate. Upon formation of the first fault in the zone of oblique convergence the deformation becomes, by definition, heterogeneous. This should obviate the use of a homogeneous strain analysis to account for both the initial and secondary faults. However, the new theory is applied to the Great Sumatran and the San Andreas fault systems (Tikoff and Teyssier 1994) where the axial trends of young folds are interpreted as perpendicular to the axis of ‘minimum instantaneous strain’ (see their Figs. 9 and 11). Then, the angular relation between fold axes and the fault zone boundary is used to estimate how much of the relative motion between the rigid blocks bounding the zone is taken up by slip on discrete faults (see their Fig. 10). In this way the
11 deformation is ‘partitioned’. However, the presence of these active folds, as well as their nonuniform distribution, is additional evidence that the deformation is heterogeneous. It is an acceptable practice to idealize a theoretical model by invoking certain postulates about the geometry, material properties, and boundary conditions. Three goals of this procedure are: 1) to make the problem tractable from a mathematical point of view; 2) to make the solution interpretable in terms of its simplicity; and 3) to retain the first order effects that one wishes to understand, explain, or predict. Tikoff and Teyssier (1994) have successfully achieved the first two goals in that the mathematics of homogeneous deformation is understood and easily interpreted. However, in our opinion, they have made significant compromises in terms of the last goal. The first order structures within these zones are the strike-slip and thrust faults and the folds, yet none of these structures could form in a rock mass that is strictly constrained to homogeneous deformation. The kinematic model for ‘displacement-field partitioning in transpressional orogens’ is introduced by asserting (Tikoff and Teyssier 1994): “This phenomenon of partitioning has been approached in two distinct ways, either by assuming that stress is partitioned (e.g. Zoback et al. 1987, Rice 1992, Zoback & Healy 1992) or that strain is partitioned (e.g. Oldow 1990, Molnar 1992).” We note that the investigations of Zoback et al. and Rice are not limited to homogeneous deformation within the fault zone and rigid behavior outside the zone, nor do these authors make any explicit assumptions about ‘stress partitioning’. In the investigations of Rice (1992), for example, mechanical equilibrium (a consequence of momentum conservation) is used to demonstrate that certain components of the stress tensor (those acting on planes parallel to the fault zone) are essentially the same within and near the fault zone. The constitutive relations and the condition of strain rate continuity near the fault zone boundary determine the other components of stress. Using these fundamental constraints on both stress and strain rate, Rice (1992) concludes: “… if the mature fault zone is weak … the equations of continuum mechanics generally preclude the possibility that all components of stress are the same within and outside the zone.” He then evaluates pore pressure differences as a possible explanation for a weak fault zone using the machinery of a complete continuum mechanics. In contrast, ‘partitioning’ of deformation among slip, simple shear, and pure shear within a homogeneously deforming fault zone is an explicit assumption of the kinematic model, and not a result derived from fundamental relationships, such as mechanical equilibrium or strain rate
12 continuity. Therefore, at best, no more is forthcoming from the kinematic model than the proportions of the three kinematic quantities, and these proportions are based on homogeneity and ad hoc assumptions about the angular relationships between fold axes and the fault zone boundary. On these grounds, we suggest that the efficacy of the kinematic model is overstated when Tikoff and Teyssier (1994) conclude that: “An exact relationship between angle of plate convergence, instantaneous strain, and finite strain is calculated, providing a predictive tool to interpret the type and orientation of geological structures in zones of oblique convergence.” Our point here is not to claim that Zoback et al. (1987) or Rice (1992) solved the outstanding problems of oblique plate convergence. Indeed, these researchers and their colleagues have continued to bring new insights, model results, and data to bear on problems of the San Andreas and Sumatran fault zones (e.g. Zoback & Beroza, 1993; Segall & Rice, 1995; Townend and Zoback, 2000; McCaffrey et al., 2000; Faulkner and Rutter, 2001). Instead, we emphasize the methodological differences between the kinematic and the complete mechanical approaches, and the consequent differences in level of substantiation. Tikoff and Teyssier (1994) would seem to agree: “no rheological assumptions were made in our model about the San Andreas or the Great Sumatran faults. Therefore, this model does not distinguish whether these faults are, in any sense, ‘weak’.” On the other hand, they contradict themselves: “It is unlikely that the San Andreas fault is rheologically ‘weaker’ than the Great Sumatran fault.” Such a statement is not a valid conclusion from a kinematic model. 4. Steady and Non-Steady Homogeneous Deformation and Minimum Strain Paths
We surmise that the position taken in Tikoff and Wojtal (1999) is based on and reflected in the models they have used and their experience in applying them. A model that plays a central role in their researches consists of a tabular region with one or two finite dimensions and a third, along-strike dimension of indefinite length. In this region, a homogeneous deformation takes place and the path of deformation is accounted for. A relationship is established between the deformation within the zone and the motion of external rock masses bounding it on its parallel sides. The model and its applications have appeared in a substantial body of previous work. Some of this work pays close attention to field data, especially to the variation of deformation within a rock mass. Coward and Kim (1981) apparently first eliminated time as an explicit
13 variable from results given by Ramberg (1975). Neither in that paper nor in the authors’ papers does this procedure lead to any valid extension, simplification, or generalization of Ramberg’s results. The only such generalization possible for the case of constant ratios of the components of the velocity gradient tensor, follows a procedure in part suggested by Sanderson (1982). The attempt to establish an element of causality based on the purely kinematic concept of a ‘minimum strain path’ by Fossen and Tikoff (1997) is without foundation. Its application to a spreading-gliding nappe leads to results at odds with those of complete mechanical modeling and laboratory model experiments. ‘Minimum strain path’ and ‘minimum work path’ models are remarkably close in terms of their kinematics. The ‘uniqueness’ claimed for the former is shown to be the result of requiring a maximum displacement for one particle, conveniently but otherwise arbitrarily chosen, in the deforming rock mass. ‘The unified deformation matrix’ Ramberg (1975) derived the model used here for the deformation gradient tensor of a region of indefinite shape and size that undergoes homogeneous deformation with constant velocity gradient tensor. Earlier derivations are given in the continuum mechanics literature. McKenzie and Jackson (1983) independently arrived at identical results. Their method seems different, because they consider the more general possibility of inhomogeneity, but they do not carry out any further development along these lines. Their computation involves integration of the rates of change of components of the deformation gradient tensor. The deformation is obtained by integration of equations describing the rate of change in the position of particles in a homogeneous deformation dx/dt = vx = Dxxx + (Dxy – ωz)y + (Dxz + ωy)z
(4a)
dy/dt = vy = (Dxy + ωz)x + Dyyy + (Dyz - ωx)z
(4b)
dz/dt = vz = (Dxz - ωy)x + (Dyz + ωx)y + Dzzz
(4c)
with initial conditions x(0) = X,
y(0) = Y,
z(0) = Z
(5)
where X, Y, and Z are the initial positions of any particle in the body. The constant coefficients in (4) are the components of the velocity gradient tensor, L, with Lxx = Dxx, Lxy = Dxy – ωz,….; Dxx,… , are the components of the rate of deformation tensor; and ωx,…are the components of the vorticity vector. The solution has the form
14 x = FxxX + FxyY + FxzZ
(6a)
y = FxyX + FyyY + FyzZ
(6b)
x = FzxX + FzyY + FzzZ
(6c)
where F is the deformation gradient tensor. This notation conforms to that in Malvern (1969). In most papers discussed, many of the components in (4) or (6) are zero, so that the results take a much simpler form. For example, the equations dx/dt = Dxxx + 2Dxyy
(7a)
Dyyy
(7b)
dy/dt = dz/dt =
Dzzz
(7c)
are used for transpression/transtension (McKenzie and Jackson, 1983; Sanderson and Marchini, 1984; Fossen and Tikoff, 1993), with x parallel to the zone, y normal to it, and z vertical. The plane flow case with Dzz = 0 and y taken to be vertical is used in an application to a spreadinggliding nappe (Sanderson, 1982; Fossen and Tikoff, 1997). Coward and Kim (1981, p. 291), in an application to strain within a thrust sheet, treat the case Dyy = 0. The solution to (7) taken from Ramberg (1975) and used by Coward and Kim (1981, equation 16) may be written: x = exp(Dxxt)X + (2Dxy/Dxx)[exp(Dxxt) – 1]Y
(8a)
y=
(8b)
Y
They are the first, of the papers we have examined, to introduce the following modifications: √λ = exp(Dxxt) and 2Dxy/Dxx = (2Dxyt)/(Dxxt) = γ/ln(√λ), so (8a) becomes: x = √λX + [γ/ln(√λ)](√λ – 1)Y
(9)
The key issue is the value of recasting Ramberg’s (1975) result in terms of the variable γ. As is made clear in Fossen and Tikoff (1993), this corresponds to no discrete factorization of the strain. Beyond showing that the result yields a plot comparable to those for their factorizations, Coward and Kim (1981) make no further use of it. They suggest that γ = γ•t “is the finite shear strain.” This is incorrect in the sense of a shear strain that can be extracted from the final deformation by a process of factorization – except in the limit that the deformation is simple shear. This quantity is merely a dummy variable which substitutes for time, t. Their ratio γ•/ε•, equivalent to 2Dxy/Dxx in our notation, is a model constant and might simply have been designated as such.
15 A generalization of Ramberg’s result, starting with the equations (4) or (7), is possible if all ratios of these quantities are constant so one can recast the time-dependence such that integration with constant coefficients may be performed. This amounts to replacing time, t, with another time-like variable. While the variable selected would be convenient to think about if monotonically increasing, there is no mathematical necessity for this. Suppose that Dxx(t) is reasonably well behaved and positive, as for a spreading nappe, and replace the differential dt with: dτ = Dxx(t)dt
(10)
In a different context, Sanderson (1982, p. 217) suggested using ε• (Dxx) as a time-like parameter. All “constant” coefficients in (4) or (7) are then normalized by dividing through by Dxx(t), to become truly constant in the variable τ. Applying (10) to (7) with Dzz = 0 and Dyy = Dxx, we obtain: dx/dτ = x + 2αy
(11a)
dy/dτ =
(11b)
-y
where α = Dxy(t)/Dxx(t)
(12)
Integration of (11) yields: τ
τ
τ
x = e X + 2α(e – e- )Y = FxxX + FxyY y=
τ
e- Y =
FyyY
(13a) (13b)
For example, consider an “episode of extension’ of duration t* in which the rate of extension starts at zero, goes to a maximum at t = t*/2, and falls to zero at t = t* Dxx(t) = (4Dmax/t*)(t – t2/t*), 0 ≤ t ≤ t*
(14a)
τ(t) = (2Dmax/t*)t2[1 – 2/3(t/t*)]
(14b)
Then Suppose we directly solve the equation dy/dt = -Dxx(t)y
(15a)
Substituting (14a) into (15a), and integrating, we obtain y = exp{-(2Dmax/t*)t2[1 – 2/3(t/t*)]}Y This result may also be obtained by direct substitution for t in the second equation in (13).
(15b)
16 We recast the solution (13) in terms of quantities equivalent to those of Coward and Kim (1981), Merle (1986), and Tikoff and Fossen (1993). The coefficient of Y in the expression for x is: τ
τ
Fxy = 2α(e – e- ) = 2α(k – 1/k) = γ/[2ln(k)](k – 1/k) = (γ/τ)(k – 1/k)
(16)
Since γ = 2Dxyt, this can only be correct if τ = Dxxt, or if the velocity gradient tensor is rigorously constant in time. That is, while Ramberg’s results may be generalized in the manner indicated, by using the variable τ and assuming constant ratios of the components of the velocity gradient tensor, the use of the dummy variable γ= 2Dxyt, where Dxy is, of necessity, constant in t, does nothing to generalize or simplify the results. It merely “hides” the variable t. Tikoff and Fossen (1993) follow Ramberg (1975) and factor the deformation gradient tensor into a product of discrete deformation increments. In the limit of an infinite product, the continuous solution of Ramberg (e.g., (3b)) is recovered, a result already understood from the well-known properties of exponential functions. For example, et can be written as the product of n terms of the form et/n. The same ‘hiding’ of t and exp(Dxxt) used by Coward and Kim (1981) occurs in obtaining the authors’ equation 6, which is not only “similar to the one derived by Merle (1986)” but identical to it. Both here and in Merle (1986), the symbol γ is used in equations in which it does represent a simple shear, and, again, for the dummy symbol γ•t. The authors remark (p.269): “If the deformation matrices of pure and simple shear are written as “ …[these matrices, in the second of which the symbol γ enters] …we want a single deformation matrix D of the form …[in which the off-diagonal term is written as f(γ,k)]…” It is natural, again, to conclude from this that the authors intend that both γ’s are the same quantity. This notational problem is for the most part cleared up in Fossen and Tikoff (1993) who introduce variables γp,s and γs,p. Tikoff and Fossen (1993, equation 12) give for the vorticity number: Wk = cos{arctan[2ln(k)/γ]} = |Dxy/Dxx|/{1 + (Dxy/Dxx)2]1/2 = α/(1 + α2)1/2
(17)
The second expression, in our notation, is for the same special case, Dyy = -Dxx, that they consider. The author’s statement that: “Wk is the same for all increments of deformation, as well
17 as the final deformation state.” is made clear in the second expression of (17), since the ratios of components of the velocity gradient tensor are constant. A question suggested, but not entirely answered, by our reading of these papers may deserve further thought by structural geologists. It has two parts. First, is a decomposition of the deformation, such as those carried out by Coward and Kim (1981), useful in interpreting natural data? Second, ought one to think in the framework, suggested by Tikoff and Fossen (1993) and Fossen and Tikoff (1993) in which the deformation is decomposed, or represented, in terms of pure shears and simple shears? Our answer to the second questions is that it is not useful to think in terms of this decomposition in the general case. Their formalism is based on the appeal that quantities such as γ (and γxy, γyz, γxz) are conducive to understanding deformation. Because these quantities are dummy variables used to hide time, we find no value in their use or in the misleading notion that they refer to quantities that may somehow be readily visualized in the general case. The representations, as we have shown, are only valid in the case that the velocity gradient tensor is strictly constant in the variable t, in which case they are identical to those given by Ramberg (1975). It should be clear, in fact, that any generalization must be sought before a set of equations is solved, not afterwards. The information contained in the strains, which equation 8 of Fossen and Tikoff (1993) is postulated to model, may be most directly represented in terms of contours of the vorticity number, Wk, or of α, in the space of the orientation of the principal axis of extension to the x-axis τ
and, say, √λ1 (Figure 2a). Contours of the time-like parameter, s = e are also plotted. In Figure τ
2b, the alternative quantity γp,s = α(e2 – 1) is contoured, again with contours of s. These results may be compared with figure 2 of Fossen and Tikoff (1993); only the portion of the plot up to their contour “k=1” is given in Figure2. This comparison demonstrates that the following statements are without foundation (Tikoff and Fossen, 1993, p. 268): “Ramberg (1975a, b) and several subsequent workers have approached this problem in a purely continuum mechanics framework…. Conversely, this work uses an incremental mathematical approach to find the exact solutions and puts the results in time-independent solutions. ……Although threedimensional deformation matrices are given by Ramberg (1975a, b) and McKenzie and Jackson (1983), these use instantaneous strain rates and are less general than the matrix derived in this article.”
18 Minimum work paths and minimum strain paths The minimum strain path hypothesis of Fossen and Tikoff (1997) attributes a causal role to a purely kinematic criterion. It is of interest to trace the partial history of this idea. Since the authors compare their minimum strain path with ‘Nadai’s minimum work path.” (Nadai, 1963), the idea for a minimum strain path might have come from this source. More recent references to the minimum work path are found in the structural geology literature. Wojtal (1986, p. 351, figure 10) uses an illustration, close to the one presented in Nadai (1963), without further mention of it. Sanderson and Marchini (1984, p. 452) remark: “In general we cannot predict deformation paths with any degree of certainty, although the use of incremental strain indicators (Elliott, 1972) may constrain the choice. If stresses and material properties of the rock remain constant, then we might predict constant strain increments, but clearly these situations are unlikely to apply to natural deformation. Arguments based on the minimum work principle (Nadai, 1963) seem equally impractical since they require the rock to prejudge its final strain state and be able to compute the required strains.” Fossen and Tikoff (1997) similarly remark (p. 990): “With a steady-state deformation, as considered above, a rock would have to prejudge its final state to choose the correct Wk to maximize its offset – an unlikely event except for very clever rocks.” They do not cite Sanderson and Marchini (1984). The question of ‘predicting a deformation path’ and the authors’ suggestion that a ‘steady-state’ path is unlikely, as raised by Sanderson and Marchini (1984), are central to Fossen and Tikoff’s effort. We re-derive their results and obtain others to better grasp their argument. The deformation and the associated velocity gradient tensor are homogeneous. In the ‘steadystate’ flow, the ratios of the components of the velocity gradient tensor are constant; in the ‘nonsteady’ flow, they are not. The deformation is two dimensional and in the (x,y)-plane such that material elements lying along the x-axis remain there. For a steady-state flow, the coordinates of a particle in the body with the initial coordinates X and Y at any time t are: x = sX + α(s – 1/s)Y
(18a)
y = Y/s
(18b)
where s = exp(Dxxt) and α = Dxy/Dxx. Here, s is a monotonic time-like variable, computed for a truly constant velocity gradient tensor, and α describes the nature of the flow between the limits of pure shear, α → 0, and simple shear, α = ∞. The variable s is the same as the authors’ variable k. The vorticity is ωz = ω = -Dxy. Instead of α, the authors use the kinematic vorticity
19 number, Wk, which, in the present case, is given by (17). Only the case of an incompressible material is considered. The authors attach a particular meaning to the horizontal offset, d, of a particle initially at the normalized position X = 1, Y = 1, d = s + α(s – 1/s) – 1
(19)
and the principal stretch, λ1. The ‘minimum strain path’ gives the minimum value of λ1 for a given offset, d. Relations between λ1 and d for pure shear (α = 0) and for simple shear (α → ∞) are, respectively: λ1 = (1 + d)2
(20a)
λ1 = [1 + d2/2 – d(1 + d2/4)1/2]-1
(20b)
The authors use the ratio λ1/λ2, which is equal to (λ1)2 for an incompressible material in plane flow; we use λ1 alone. An example of a steady-state minimum strain path for d = 2 is given in the authors’ figure 2, and it may be noted that the long axis of the strain ellipse lies close to or along the diagonal of the deformed initially square element. If we guess that this is precisely the case for the ‘minimum strain path’, we may derive relations between α or Wk, λ1, and d that correspond to this condition. We find λ1 = (α + 1)/(α – 1)
(21a)
s = [(α2 + 1)/(α2 – 1)]1/2
(21b)
The value of α or Wk for a given d that corresponds to this steady-state strain path is found by substituting for s in (19), giving a relation between d and α alone, and then Wk may be computed: Wk = 2λ1/[λ12 + 1]
(21a)
d = √(2λ1) λ1/(λ12 + 1)1/2
(21b)
These results allow us to plot λ1 versus d for pure shear, simple shear, and the “guess” (Figure 3). These curves appear to match the curves plotted in figure 3 of Fossen and Tikoff (1997). The “guess” gives a lower strain for a given offset. A numerical search for the ‘minimum strain path’ for steady flow produces data points that fall nearly on the “guess” (Figure 3a). In fact, the numerical search produces a slightly smaller quadratic elongation so, the guess is wrong. Results for Wk versus offset, d, are also
20 obtained from both procedures (Figure 3b). These are quite different for small offset, but converge for large offset. The authors then consider a non-steady flow in the course of which Wk is varied to give the minimum strain increment for a given increment in offset. Because this procedure sets a condition to be satisfied along the entire path of deformation, it may be more plausibly viewed as modeling a “process” followed in a natural deformation, and they view it as such. We do not agree, because an essential element of causality – i.e., identification of what determines that the deforming rock will follow such a path - is either lacking or unstated. The authors use a numerical procedure to obtain the non-steady-state minimum strain path. Fitting of our results for Wk(d) to a fourth order polynomial, we obtain the relation Wk ≅ 0.9002 – 0.0633d – 0.0655d2 + 0.0139d3 – 0.0008d4 This bears a close relationship to that of the authors’ (p. 992), if typographical errors in sign and decimal position are accounted for. To base an important concept, here the ‘minimum strain path’, on results of a numerical search leaves something to be desired, since it is difficult to gain concrete insight. Since the steady-state minimum strain path was remarkably close to the path of the material line through the initial position (1, 1), we guess that a path close to the nonsteady-state minimum strain path would be that in which the maximum rate of extension always lay along the diagonal of the parallelogram into which the initial square was deformed. To conform to the conditions imposed on the deformation, the parallelogram is simultaneously rotated to maintain contact of its edge with the x-axis. The velocity components for the guessed non-steady-state minimum strain path, referred to axes x’ and y’ along and normal to the diagonal are vx’ = dx’/dt = Dxx’x’ – ωy’
(22a)
vy’ = dy’/dt =
(22b)
- Dxx’y’
Here, the rate of extension along the diagonal may be taken as constant, but the vorticity, ω, will be a function of the orientation of the diagonal to the x-axis, denoted θ. The condition determining ω is that the vertical velocity of particles along the x-axis be zero. Since dy/dt = (dx’/dt)sinθ + (dy’/dt)cosθ = [Dxx’sin(2θ) + ω]x – Dxx’cos(2θ)y
(23)
ω = -Dxx’sin(2θ) and Wk = |ω|/Dxx’ = sin(2θ)
(24)
we must have
21 The rate of change of θ is dθ/dt = ω = -Dxx’sin(2θ)
(25a)
with initial value θ(0) = π/4. Integration yields tan(θ) = exp(-2Dxx’t)
(25b)
The offset is the projection of the diagonal of the parallelogram onto the x-axis less unity, d = -1 + √2exp(-2Dxx’t)/[1 + exp(-4Dxx’t)]1/2
(26)
Wk = 2exp(-2Dxx’t)/[1 + exp(-4Dxx’t)]
(27)
From (25b) and (26) The quadratic elongation is λ1 = exp(2Dxx’t)
(28)
The quadratic elongation, and the kinematic vorticity number are plotted versus offset in Figures 4a and b for the minimum strain path [1] and the “guess” [2]. While Wk = 0.894 at zero offset for both steady state and non-steady state paths, the value for this guess is also Wk = 1. The proportion of simple shear is higher at first, but then drops below that for the minimum strain path. Fossen and Tikoff (1997) correctly claim that the non-steady-state ‘minimum strain path’ does not correspond to the minimum work path of Nadai (1963, p. 96 – 105). Nadai remarks (p. 96): “For this particular strain path the principal axes of finite, plane strain will coincide with those of the incremental strain tensor and consequently also with the instantaneous axes of principal stress in every point of the path.” Coincidence of the principal axes of the incremental strain tensor, or the rate-of-deformation tensor, and those of stress depends on the material’s rheological behavior. It occurs for a material that was initially isotropic and remained so during deformation. Extension to anisotropic materials is mentioned in Chung and Richmond (1992). Nadai shows that such paths are minimum work paths for the case of an isotropic perfectly plastic solid. The path of our “guess” for non-steady state minimum strain path satisfies the kinematic condition for a minimum work path of this type. The further requirement goes beyond kinematics, since work is a mechanical concept. In the case of our “guess”, the material line that remains throughout the principal extension direction and the instantaneous direction of the maximum rate of extension is the diagonal of the initial square, at 45o to the x-axis. Other “minimum work paths” may be obtained by selecting a different orientation for this material line. For example the curve [3] in Figures
22 4(a) and (b) correspond to the case of a line at initial orientation θ0 = 31.7o. The initial value of Wk for this line is the minimum strain value 0.894. Although the initial line orientation is quite different, the displacement – strain variations for both are close to each other. The pure shear curve (Wk = 0) is also a “minimum work path.” For paths of this type, relations for d and Wk in terms of quadratic elongation may be readily derived. The essential relationship is that between the orientation of the principal axis and its initial value, as a function of strain. This is tanθ = tanθ0/λ1
(29a)
The final relations obtained are Wk = 2λ1tanθ0/[λ12 + tan2θ0]
(29b)
d = √λ1[M + (1/M)(1 – 1/tanθ0)]
(29c)
M = λ1[(1 + tan2θ0)/( λ12 + tan2θ0)]1/2
(29d)
and where The additional restriction on these paths, as required in Fossen and Tikoff’s specification, is that they are confined to sub-simple shear. The ‘minimum work’ path giving the closest approximation to the non-steady-state minimum strain path is that for the case θ0 = 45o. Because of the kinematic condition that the principle axes of the rate of deformation tensor coincide with the direction of maximum extension throughout the deformation, a path of this type corresponds to the coaxial rotational deformation of Lister and Williams (1983) or the spinning coaxial deformation of Means et al. (1980). The internal or shear-related vorticity is zero. Since the offset-quadratic elongation relation for the case θ0 =45o is close to that for the non-steady minimum strain path, we ask whether this path is also one of small internal vorticity? We concur with Lister and Williams (1983, p. 6) that “Shear induced vorticity …is of fundamental importance in affecting the development of fabrics and microstructures.” This is relevant to the remarks of Fossen and Tikoff (1997), who appear to view this ‘minimum strain path’ as likely to arise in a natural deformation in a shear zone, and who then discuss its relevance to fabric development in a shear zone. These authors did not compute the shear induced vorticity for the minimum strain path, although they suggest that “a dramatic change in Wk during deformation” is an important factor. On the other hand, most structural geologists who study shear zones would likely be skeptical of
23 a model approximating a coaxial spinning deformation! Decomposition of the vorticity into internal and external parts for the non-steady-state minimum strain path shows that the internal vorticity represents at most 0.55 that of the total, and has effectively vanished at a modest offset d = 3. An offset of 3 is small for a shear zone. Fossen and Tikoff (1997, p. 995) offer a comparison of their ‘minimum strain path’ with the minimum work path whose major thrust is to emphasize a separation in the nature of the two. They remark: “…it should be made clear that the minimum strain path differs significantly from Nadai’s minimum work path….” As we have shown, a path satisfying the kinematic criterion for a minimum work path lies close to their minimum strain path and produces an offset versus quadratic elongation relation that is virtually indistinguishable from that for it. Zero or strongly reduced internal vorticity characterizes both. Thus, similarity, rather than dissimilarity, is supported. Since the kinematic criterion for a minimum work path – at least for the constitutive relations treated by Nadai – is that it be a coaxial spinning path, it is clear that any finite deformation may be achieved, as the authors suggest. They cite an example (see also Wojtal, 1986, figure 10) in which the deformation path involves, quite obviously, Wk > 1, but seem not to appreciate that deformations involving only sub-simple shearing (Wk ≤ 1) may also be achieved this way. Fossen and Tikoff’s ‘non-steady-state minimum strain path is indeed uniquely determined, whereas any final state of strain may be achieved by the coaxial spinning deformation required for the minimum work path. However, this uniqueness is tied to the peculiar condition that the displacement of a special particle at the upper surface of the layer along a line drawn at 45° to an arbitrary position at which the displacement is zero be the parameter of advance with respect to which the strain is to be minimized. Choice of other particles on this surface, relative to a fixed origin, leads to quite different variations of vorticity number with offset. The criterion does not correspond to a global condition on the deforming body –i.e., a condition involving all of it – as does a minimum work path. We conclude that the non-steady state minimum strain path cannot be used to characterize even the homogeneous deformation of a body. Hence, it cannot be used to establish a criterion for the path of deformation. Fossen and Tikoff suggest that “non-steady-state deformations are potentially important for geometric reasons” with the minimum strain path as one example. We do not agree. We
24 would agree that it is of the greatest importance to consider velocity fields, and fields of the velocity gradient tensor, that may vary both in time and space. Such fields are those most often encountered in natural processes. To further document this point, we consider application of the kinematic model under discussion and simple mechanical models to a gravitationally driven spreading-gliding nappe. Application to a spreading-gliding thrust nappe Fossen and Tikoff (1997, p. 993) comment on this application as follows: “Potentially the best example of deformation that is not limited by strain compatibility restrictions is the gravitationally driven movement of ice sheets over bedrock. The bedrock-ice interface can be treated as a detachment, above which the ice is free to undergo any type of deformation. Many authors consider this situation analogous to gliding and spreading nappes in the upper to middle crust.” Later (p. 995), these authors remark: “The minimum strain path may be applicable to gravitationally induced movement, and other flows which are not strongly limited by strain compatibility requirements.” They draw a spreading nappe (Figure 4d) in which a layer above a rigid basement undergoes homogeneous sub-simple shear corresponding to the minimum strain path for the offset shown (d = 2). An upper layer undergoes horizontal extension in pure shear. Displacement is continuous at the interface, but the vertical derivative of the horizontal displacement, uniform in each layer, undergoes a jump. Slip occurs between the lower layer and rigid basement. Excluding the role of a ‘minimum strain path,’ the authors’ model, like that of Sanderson (1982, Figure 3), is crudely consistent with observation: shear deformation is concentrated near the base of a structure of this type. This is primarily, we point out, because the shear stress component, σxz, increases from zero roughly linearly with depth. Thus, near surface deformation must consist of nearly layer-parallel shortening or extension. The presence of a detachment surface is likewise to be expected. However, the ‘minimum strain path’ cannot ‘explain’ any of these features. In contrast, by means of a complete mechanics (in this case lubrication theory) one may obtain valid solutions for the gravitational spreading of a sheet of material. This method is based on the assumption that the form of the body is sufficiently slender, as when its thickness, h, is much less than its breadth, L, or h/L
