Abstract
Geologic structures are repetitive in a quasi—periodic or erratic manner. The geometries of these structures are also manifestations of the mechanical behaviour of a deforming rock mass. Can we therefore obtain information on the dynamics of a complex geologic system from an examination of the geometry of geologic structures? We examine this question from the point of view of non—linear dynamics.
We investigate the variability in space of the velocity of growth of crenulations in a model rock mass which is undergoing a (numerical) simple shearing deformation. That is, we follow the distribution and evolution of the velocity in space, rather than in time. We investigate the thesis that the behaviour of this one variable reflects the presence of all other variables participating in the dynamics and, by use of increasing multiples of a fixed space lag, discretize the system, and unfold the system’s dynamics into a multidimensional phase space. The trajectories within this phase space of the system converge to a subspace which is the geometrical attractor for the system. We infer from this that our deforming model rock can be described by a set of deterministic laws. The dimension of this attractor is about 2.5; that is, the system may be completely represented by a fractal attractor. Further, this fractal attractor embeds in a phase space of three so that at least three variables must be considered in the description of the underlying dynamics. These are the variables involved in the three independent differential equations of the numerical model: the stress equations of motion, the yield criterion and the flow rule. We conclude that geological systems may be successfully modelled on the basis of such a system of equations, and analysed using the concepts of fractal geometry.
This new application of nonlinear dynamics to the spatially erratic structures of deformed rocks results in an improved understanding of rock deformation behaviour and in an improved prediction of the distributions of structurally—controlled phenomena.
‘... and I doubt that we are founding a new science, but at least we are having fun. ’ Nonlinear dynamics, chaos and mechanics. P. Holmes, 1990
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Ord, A. (1994). The Fractal Geometry of Patterned Structures in Numerical Models of Rock Deformation. In: Kruhl, J.H. (eds) Fractals and Dynamic Systems in Geoscience. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-07304-9_11
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DOI: https://doi.org/10.1007/978-3-662-07304-9_11
Publisher Name: Springer, Berlin, Heidelberg
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