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_Journal of Structural Geology 33 (2011) 220-243_ _Microfabrics as energy minimisers: Rotation recrystallisation as an example_ _Alison Ord a,*, Bruce Hobbs a,b_ _a School of Earth and Environment, The University of Western Australia, 35 Stirling Highway, Crawley, Perth, Western Australia 6009, Australia b CSIRO Earth Science and Resource Engineering, PO Box 1130, Bentley, Western Australia 6120, Australia_ _Article info_ _Article history: Received 14 May 2010; Received in revised form 18 October 2010; Accepted 1 November 2010; Available online 24 November 2010_ _Keywords: Fractals Energy minimisation Microstructure Microfabric Rotation recrystallisation Deformation lamellae_ _Abstract_ _Microfabrics are discussed as features that minimise Helmholtz energy in a system undergoing deformation and metamorphism. The energy minimisation process leads to inhomogeneous deformations and hence the formation of microfabrics at a number of scales. This process together with the requirement for compatibility both with the imposed deformation and local gradients in deformation means the microfabric must refine on smaller and smaller scales in a self-similar manner leading to fractal geometries. Nine independent levels of refinement are necessary to match the nine independent components of a general imposed deformation gradient. The process of rotation recrystallisation is proposed as one example of self-similar refinement so that crystallographic preferred orientations (CPO) associated with rotation recrystallisation are presented as fractals whose fractal dimensions reflect the conditions of deformation. Compatibility also has implications for the formation of other microstructures such as non-rational deformation lamellae. The evolution and orientations of microfabrics that minimise energy are related to the history of the imposed deformation gradient (as was originally proposed by Sander) and not the strain tensor as is commonly assumed. As examples, possible models for CPO development in deformed quartz aggregates by rotation recrystallisation and the development of deformation lamellae in deformed quartz grains are presented._ _? 2010 Elsevier Ltd. All rights reserved._ _1. Introduction_ _The hallmark of deformed metamorphic rocks is heterogeneity in the distribution of deformation. At the electron microscope scale this heterogeneity is expressed in the distribution of dislocations (McLaren et al., 1970; White, 1973; Twiss, 1976) defining lamellar structures and subgrain boundaries. At the grain scale one observes deformation and B?hm lamellae that can correspond to rational or non-rational crystallographic planes (Fairbairn, 1949; Twiss, 1974, 1976), and subgrain boundaries of various rational and nonrational orientations (Trepied et al., 1980). At the optical microscope scale, one sees domains of grains of like orientation (Sander, 1970; Hobbs, 1966; Pauli et al., 1996; Heilbronner and Tullis, 2006) although the orientations of grains within these domains can vary considerably if the complete crystallography is established (Halfpenny et al., 2006; Stipp and Kunze, 2008; Jiang et al., 2000; Bestmann and Prior, 2003) and porphyroblasts, lineations and foliations defined by variations in mineralogical composition and micro-folds. At larger scales, chevron folds, rounded folds, mineral lineations and so on are ubiquitous. Heterogeneity is developed at all scales and has been emphasised in a vast number of papers over the past 100 years (for example, Sander, 1911, 1970; Turner and Weiss, 1963; Hobbs, 1966; Oertel, 1983; Cobbold and Gapais, 1986; Gapais and Cobbold, 1987; Pauli et al., 1996; Menegon et al., 2008; Lloyd et al., 2010). The aim of this paper is to present a basis for such heterogeneity based on the formation of domainal structures as minimisers of Helmholtz energy. In addition there is a requirement that the heterogeneous deformation must be compatible from one part of the aggregate to another and also with the imposed deformation (as defined by the instantaneous deformation gradient). We show that this means fractal geometries can develop._ _The observation in deformed rocks of ubiquitous unstable deformation modes as represented by inhomogeneous deformations contrasts with parallel developments in continuum mechanics that concentrated on the stability of deformation and hence the development of homogeneous deformations. This stability of deformation arises from the Drucker postulate (Houlsby and Puzrin, 2006, p. 31) that the Helmholtz energy of deformation is convex. We will worry later in this paper what this statement actually means. In the early 1970’s and onwards (Ericksen, 1975) people who were concerned with martensitic transformations realised that convexity of the Helmholtz energy did not explain the heterogeneity of deformation that they observed and they developed various approaches to the subject based on non-convexity of the Helmholtz energy but within a framework of finite non-linear elasticity (Ericksen, 1975; Ball and James, 1987; Truskinovsky and Zanzotto, 1996). The fact that this theoretical base was involved with elastic deformations seemed to remove it from any application to plastic deformations or to the development of fractures. This approach has recently been extended to both the plastic deformation of materials and fracture development with the realisation that heterogeneity of deformation commonly observed in such materials can be approached from the same point of view as was developed for finite non-linear elasticity (for plastic deformation: Ortiz and Repetto, 1999; Carstensen et al., 2002; Miehe et al., 2004; for fracture: Francfort and Marigo, 1998; Choksi et al., 1999; Del Piero and Truskinovsky, 2001). The application of these concepts to the formation of microfabrics in deformed rocks forms the subject of this paper. The aim of this paper is to outline the general theory of heterogeneous deformation based on the above principles and then to present two possible examples namely deformation lamellae in quartz and quartz CPO development by rotation recrystallisation._ _The deformation and metamorphism of rocks involves the development of fabrics at various scales. The term fabric is used in the sense of Turner and Weiss (1963) to mean the internal ordering of both geometrical and physical spatial data in a deformed rock; the term microfabric refers to the micro-scale. Vernon (2004) divides the term microfabric into microstructure comprising the spatial arrangements and relationships of microfabric elements such as grains, grain boundaries and foliation planes, and preferred orientation referring to the spatial orientations of microfabric elements including grains, domain boundaries and of crystallographic features. This paper is involved only with fabrics developed at the micro-scale and so involves length scales ranging from a few microns up to about a metre. The principles discussed here are applicable at all scales but it simplifies the discussion to concentrate on the micro-scale where for geological strain-rates the deformation can be considered isothermal (Hobbs et al., 2010, Hobbs et al., in press). As such the structures we are talking about are subgrain and grain boundaries and their geometry, the distribution of various mineral phases within the rock, the size distributions of grains and subgrains and their shapes and crystallographic preferred orientations (CPO). The distribution of mineral phases involves a huge range of microstructures including the geometries of schistosity, gneissosity, mineral lineations, porphyroblasts and various forms of metamorphic differentiation. Our concern in this paper is to develop an integrated approach to microfabric development that explains the various relations and geometries we observe in deformed metamorphic rocks and provides a framework for relating such geometries to the conditions under which the microfabrics formed._ _In some instances, microstructures have been described as fractal. Examples are sutured grain boundaries associated with recrystallisation (Kruhl and Nega, 1996), subgrain size (Streitenberger et al., 1995; H?hner et al., 1998) and grain shape (Mamtani, 2010). The fractal characteristics of these microstructures have been used to indicate temperature and strain-rate (Kruhl and Nega, 1996; Takahashi et al., 1998; Mamtani, 2010; Mamtani and Greiling, 2010). One question we seek to answer in this paper is: Why do fractal geometries exist and what is the control on the fractal dimension that is responsible for temperature and strain-rate dependence? The answer lies in the development of microstructures as minimisers of the Helmholtz energy of the system together with the necessity for compatibility with the imposed and local deformation gradients in order to minimise long range stresses. A second question is: Are there other common microfabrics that have a fractal geometry and what is the significance of this fractal geometry? One answer here concerns CPO development, especially by rotation recrystallisation. We show that such CPO patterns are fractal in nature and suggest that the fractal dimension is a measure of a range of parameters including amount of strain, temperature and strain-rate._ _In order to develop this approach it turns out that the important feature controlling the development at a particular instant of all of these microfabrics is the deformation, as measured by the deformation gradient and not the strain as is the current dominant paradigm. This means that there is a class of structures in deformed rocks that, integrated over the deformation history, is related to the kine_ Ключевые слова: laminate, geological, rational, complete array, homogeneous deformation, fractal geometry, maxwell construction, acta, dissipative material, interface, increasing hardening, miehe, spatial arrangement, journal, subgrain boundary, mechanics, geometrical softening, jef relation, invisibility criterion, re?nement, domainal structure, fractal analysis, independent component, normal, geometry, plane, continuum mechanic, identity matrix, curve, metamorphic differentiation, microfabrics, direction, tullis, helmholtz, develop, domain, rotation recrystallisation, microfabric, volume proportion, analysis, maxwell stress, fractal microstructures, stress, heilbronner, solids, mathematical treatment, boundary, fractal, imposed deformation, imposed strain, sm m, chemical, dimension, structural, single crystal, single slip, applied, physical, non-convex, cantor dust, bulk deformation, approach, basal, prior halfpenny, deformation gradient, ball, journal structural, york, compatibility, gibbs energy, geological society, homogeneous, journal structural geology, plastic, individual domain, non-rational, rhomb, gradient, hobbs journal, deformed, recrystallisation, negative rhomb, quartz, springer, slip direction, grain, single plane, hai, extension direction, burgers vector, geometrical, tectonophysics, hansen, number, strain, scalar product, geophysical, scale, fractal dimension, form, repetto, specic volume, subgrain rotation, basal slip, grain boundary, mineral reaction, fabric, appendix, individual grain, latent hardening, ha, invariant surface, sequential laminate, strain tensor, adjacent domain, structure, lister, rock, white, plasticity, hobbs, result, internal node, crystal slip, shear, ortiz repetto, plastic deformation, developed, maintain compatibility, chemical potential, dyadic product, reaction, node, taylor-bishop-hill approach, paterson, compression axis, plastic strain, jef, deformation proceeds, ortiz, microstructures, condition, inhomogeneous deformation, stressestrain, granular, del piero, order, dissipation function, dynamic recrystallization, identical a-axis, cpo, slip, cpo predicted, structural geology, stressestrain curve, volume fraction, patchy slip, synthetic quartz, orientation, surface energy, length scale, self-similar renement, material, imposed, model, average deformation, minimise, hardening, geology, shear band, precise measurement, crystal, rice, shearing direction, principal stretch, degenerate, sander, volume, microstructure, deformation, direction cosine, slip plane, materials science, ericksen, press, point, vector, physics, deformation lamella, dislocation, single, subgrain, mineral lineation, thermodynamics, mechanism, helmholtz energy, energy, history, development, discussion, relation, general deformation, softening, independent, dissipative deformation, paper, crystallographic, silhavy, rotation, fractal character, schmid, s m, surface, basal plane, ortoleva, process, convex, etchecopar, work, internal energy, progressive deformation, crystal plasticity, recrystallised grain, coarse microstructure, parallel, american, subgrain structure