A Dynamical System Theory of Large Deformations and Patterns in Non-Cohesive Solids
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A DYNAMICAL SYSTEM THEORY OF LARGE DEFORMATIONS AND PATTERNS IN NON-COHESIVE SOLIDS Pierre EVESQUE* and Didier SORNETITE** *Laboratoire de M6canique : Sols, Structures et Mat~riaux, CNRS URA 850, Ecole Centrale de Paris, 92295 Chatenay-Malabry Cedex, France *Laboratoire de Physique de la Matire Condense, URA CNRS 190, Universitd de NiceSophia Antipolis, Parc Valrose, 06034 NICE Cedex, France ABSTRACT We propose a dynamical system theory of triaxial-test deformationS and localization bifurcation in brittle media. We apply it to predict that localization may occur in a packing looser than "critical" and that the general localization shape is a spiral staircase in axisymmetric 3-D cells. These two facts have recently been confirmed experimentally. This theory provides a framework for understanding the development of complex deformation patterns from the mechanics of localization and rupture. INTRODUCTION The mechanics of large deformations offers an intringuing and novel realm of research from the point of view of the physical sciences, in between the classical fields of solid mechanics (usually involving small deformations) and fluid mechanics (involving very large displacements associated with flows). Large deformations appear for instance in non-cohesive brittle materials, such as granular media of various kinds, and constitute one of the basic problem of soil mechanics. They are also omnipresent in rocks associated with plate tectonic deformations occuring in the earth crust and lead to complex structures such as mountain ranges and fault (crack) patterns that can be observed at the earth surface, over a very large range of scales [1]. The standard way to gain access to the physics of the deformation process in non-cohesive solids is through so-called triaxial test experiments [2,3], in which a cylindrical (rock or granular medium) specimen is subjected to an axial compression defining the maximum principal stress component a, and to a lateral confining pressure p=cY3 , while being kept at a fixed temperature T. Typical stress-strain curves (q=al-0 3 as a function of deformation el along the axial compression a 1 ) are shown on fig.1 [2,3]. In the regimes of relatively small deformations studied during a typical triaxial test, the deformation is essentially homogeneous over the sample, except at the point of instability where q is maximum[3], at which a localized shear band appears. The interest of triaxial test is to allow to predict the behavior of larger inhomogeneous samples by using finite element calculation. However the mechanics of large deformations (as in plate tectonics) leads to complex structures (mountains, faults) over a very large range of scales[ 1] for which classical computation methods are not' applicable. It is the purpose of this paper to suggest a novel framework to address this problem. In this letter, we restate triaxial results in terms of the general theory of dynamical systems and use it to outline a theory of large deformations in non-cohesive solids. This program is difficult, due to
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