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Wave propagation, localization and dispersion in a gradient-dependent medium

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  • Physics

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PII: 0020-7683(93)90010-5 hf. J. Solids Sfrucrures Vol. 30, No. 9, pp. 1153-l 171, 1993 GUZO-7683/93 $6.00+ .30 Printed in Great Britain 0 1993 Pergamon Press Ltd WAVE PROPAGATION, LOCALIZATION AND DISPERSION IN A GRADIENT-DEPENDENT MEDIUM L. J. SLUYS Delft University of Technology, Department of Civil Engineering, P.O. Box 5048, 2600 GA Delft, Netherlands R. DE BORST Delft University of Technology, Department of Civil Engineering/TN0 Building and Construction Research, P.O. Box 5048, 2600 GA Delft, Netherlands and H.-B. M~HLHAUS CSIRO Division of Geomechanics, P.O. Box 54, Mt. Waverley, Victoria 3149, Australia (Received 15 June 1992; in revised form 18 October 1992) Abstract-A continuum model that incorporates a dependence upon the Laplacian of the inelastic strain is used to regularize the initial value problem that results from the introduction of strain softening or non-associated flow. It is shown that the introduction of this gradient dependence preserves well-posedness of the initial value problem and that wave propagation in the enhanced continuum is dispersive. An analysis of the dispersive wave propagation reveals the existence of an internal length scale. Numerical analyses of one-dimensional and two-dimensional problems confirm that this internal length scale sets the localization zone and show that the results are insensitive to the fineness of the discretization and to the direction of the grid lines. This holds true with respect to the strain profiles, the energy dissipation and the extent of wave reflection. 1. INTRODUCTION Micro-structural phenomena such as micro-cracking and void or pore nucleation and growth, which occur in a localization zone, cause discontinuous deformation processes which cannot be described with classical continuum models. Therefore, various kinds of modifications and generalizations of standard continuum plasticity have been proposed to avoid a spurious solution for the localization zone and an excess

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