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_Journal of Structural Geology 32 (2010) 997-1008_ Contents lists available at ScienceDirect Journal of Structural Geology journal homepage: www.elsevier.com/locate/jsg Drawing parallels: Modelling geological phenomena using constraint satisfaction R. Edmunds*, B.J. Hicks, G. Mullineux Innovative Design and Manufacturing Research Centre, Department of Mechanical Engineering, University of Bath, Bath BA2 7AY, UK Article info Article history: Received 10 December 2009 Received in revised form 31 May 2010 Accepted 16 June 2010 Available online 30 June 2010 Keywords: Parallel folding Kink banding Friction Energy methods Constraints Optimization Abstract This paper gives insight into the transition between different folding-types seen in nature. Using constraint satisfaction and optimization to study least energy solutions of an elastic, frictional model for concentric parallel folding, kink band waveshapes resulting from the same model are discovered. Simplifying the concentric parallel folding model down to a two-layer formulation, and assuming the geometry of the whole layered material is governed by this, the behaviour of the central interface is represented using a number of points whose displacement is constrained. With a linear foundation, the full large-deflection energy formulation reaches a point where the whole system is locked up after only two folds, matching experimental evidence. This is overcome by adding a nonlinearity to the foundation, where the sequential destabilization and restabilization of experimental load-deflection plots is observed and the wave-profiles agree with naturally occurring geological phenomena. Increasing the nonlinearity in the foundation and the magnitude of overburden pressure, the phenomenology of the concentric folding model can be altered to one that is more kink band-like in structure. Thus a “trigger” is found, relating two prevalent folding patterns which are generally considered to be at opposite ends of the spectrum of geometries. © 2010 Elsevier Ltd. All rights reserved. 1. Introduction This paper builds upon previous work (Hunt et al., 2006), which presented a rigorous analysis of serial concentric parallel folding. Fig. 1 shows a series of folds, where each fold has been instigated in sequence. Such behaviour is called serial (Blay et al., 1977), sequential (Peletier, 2001) or cellular (Hunt et al., 2000a; Hunt, 2006) buckling. Whilst many types of folds are observed in the field, the exact ordering or formation of the buckles is often not apparent. The process of serial folding has long been recognized by geologists as the most common phenomenon in the folding of rocks, with field observations (Price, 1970, 1975) and analogue experiments (Cobbold, 1975; Blay et al., 1977) supporting this. However, this type of behaviour is markedly different to the synchronous wave-trains where all of the folds occur uniformly throughout the material that are predicted by Biot’s viscous models (Biot, 1961, 1963, 1964). In particular, the wavelength resulting from sequential buckling does not correspond to the dominant wavelength, the one that amplifies most rapidly in the spontaneous formulation (Budd et al., 2001). With folding occurring at many different levels of the Earth’s crust, the rheology of the rocks during the folding process is often unknown. However, fold amplification is achieved by non-elastic (e.g. viscous or plastic) behaviour. In order to explore these processes, researchers have investigated a variety of theoretical approaches to aid the understanding of the phenomenology and the governing mechanisms. In particular, sophisticated analytic and numerical models have evolved as a result of the studies of both single and multilayer buckling of various rheological combinations by Biot and Ramberg (Biot, 1965; Ramberg, 1961; Ramberg and Ström?rd, 1971). Whilst Biot and Ramberg both recognized that elastic effects are important in the early stages of the folding process, they put the viscous effects as the dominant deformation type controlling the folding process. Opposing this idea, Johnson explored elastic-plastic deformation further, proving that behaviour similar to that observed in the Biot and Ramberg models could also develop in these models, although again only for synchronous folding (Johnson, 1977). This paper is complementary to these works, as it studies non-synchronous folds and shows that elastic buckling is of geological importance. Parallel folds in particular are usually found in the younger, upper parts of an orogenic belt which supports the use of elastic theory (de Sitter, 1964). When an elastic multilayer comprising stiff material, embedded in a soft matrix, is loaded axially, the layers slip at the interfaces (Donath and Parker, 1964) and deform into the softer surrounding medium. The multilayer bends about the centres of curvature such that a regular periodic concentric buckle pattern is created (Fig. 1). Correspondingly, when the multilayer and foundation are of a similar competency and subjected to loading along the length, the layers are unable to move into the matrix and sections of the layers rotate across the width of the multilayer, forming straight limbs and sharp corners. This leads to kink banding or box folding (Price and Cosgrove, 1990) (Fig. 2). As both concentric folding and kink banding phenomena only occur under high overburden pressure (Hobbs et al., 1976), the importance of the layering becomes paramount as the interfaces provide the natural slip planes necessary for the system to adopt these modes (Hobbs et al., 1976; Price and Cosgrove, 1990). Central to understanding this behaviour is the need to consider the frictional properties along the interfacial planes. These considerations have led to a series of papers that investigated multilayer slippage under large overburden pressures using elastic, frictional models. The formation of individual kink bands was initially considered by Wadee et al. (2004) and the propagation of this system to a series of bands was explored by Wadee and Edmunds (2005). The model was also extended to investigate fibrous materials (Edmunds and Wadee, 2005). Limiting themselves to two layers and the formation of the initial buckle, Budd et al. (2003), assuming that the folding seen in nature corresponds to a minimal energy solution which penalizes voids, formulated a potential energy model for concentric parallel folding where deformation is by flexural buckling. This formulation includes an energy contribution due to the slip at the layer interface. By extending the model to a multilayer of n layers, Edmunds et al. (2006) were able to successfully compare the solutions of the formulation to the first instability of a set of experiments using layers of paper in foam. Using the small-deflection two-layer formulation, a primitive form of serial buckling—following the transition from a single fold to a second fold (it was computationally too intensive to go beyond two folds)—was shown by Hunt et al. (2006), who restricted the waveshapes to cubic B-splines and added a restabilizing nonlinear component to the foundation. Of course, this imposed the waveshape upon the solution, rather than allowing a natural one to emerge from the formulation. The shortcomings of using B-splines in this way are covered in a recent paper (Edmunds et al., submitted for publication), which uses constraint-based techniques to successfully follow the loaded-deflection paths and fold evolution of the small-deflection energy formulation over a number of humps. However, an additional consideration of the B-spline analysis is that it is difficult to apply the methodology to the full, large-deflection problem. The purpose of this paper is therefore to deal with explicitly the large-deflection formulation and the limitations of the B-spline approach. Constraint-based modelling is concerned with what is to be achieved rather than how it is to be achieved. It is particularly useful in the early stages of problem solving where precise information is not available. Often in these stages exact knowledge of the solution is not possible, but rather a sense of the limitations placed upon the system is more apparent. The intersection of the limitations is then the feasible solution space. It is possible to explore the solution alternatives by creating a set of criteria that must be satisfied by the system—i.e., the constraints—finding a configuration that gives the smallest perturbation from these goals. To resolve the constraint set and thus find these configurations, a constraint-based modelling environment (Mullineux, 2001) has been created which uses a number of optimization codes. This methodology has been used effectively for many of the general engineering design applications that are found in practice, including products, machines and technical systems (Hicks et al., 2006; Mullineux et al., 2005). The constraint-based modeller in particular has been used to explore several engineering domains: - Design synthesis and analysis of mechanisms (Mullineux et al., submitted for publication); - Design analysis and optimization of machines (Hicks et al., 2001); - Investigation of machine-material and machine-product interaction (Mullineux et al., in press); - Evaluation of processing equipment to handle product variation (Matthews et al., 2006); - Modelling and understanding of human motion (Mitchell et al., 2007). 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