Computational Structures Technology
Chapter 5 M. Jirásek+ and B. Patzák*
+Laboratory of Structural and Continuum Mechanics, Swiss Federal Institute of Technology, Lausanne, Switzerland *Department of Structural Mechanics, Faculty of Civil Engineering, Czech Technical University, Prague, Czech Republic Keywords: nonlocal continuum, localization, quasibrittle failure, damage, softening, adaptivity, extended finite elements. In many structures subjected to extreme loading conditions, the initially smooth distribution of strain changes into a highly localized one. Typically, the strain increments are concentrated in narrow zones while the major part of the structure experiences unloading. Such strain localization can be caused by geometrical effects (e.g., necking of metallic bars) or by material instabilities (e.g., microcracking, frictional slip, or nonassociated plastic flow). In the present paper we concentrate on the latter case. To keep the presentation simple, we consider only the static response in the small-strain range. The initiation, growth, interaction and coalescence of microdefects lead on the macroscopic scale to a gradual degradation of the effective mechanical properties such as stiffness or strength. Continuum-based description of these phenomena requires constitutive models with softening, i.e., with decreasing stress under increasing strain. It is well known that stress-strain laws with softening used in the context of a standard Boltzmann continuum cause the loss of ellipticity of the governing differential equations and lead to ill-posed boundary value problems with physically meaningless solutions. Numerically obtained results suffer by a pathological sensitivity to the finite element discretization, and the total energy consumed by the failure process is grossly underestimated and sometimes even tends to zero as the computational grid is refined. These deficiencies can be overcome by regularized formulations that enrich either the kinematic description or the constitutive equations and serve as localization limiters, i.e., enforce a certain finite size of the zone of localized inelastic strain, independent of the finite element mesh or other numerical discretization. Advanced regularization methods introduce an additional material parameter -- the characteristic length, which is related to the size and spacing of major inhomogeneities and controls the width of the numerically resolved fracture process zone. Typically, regularization is achieved by a suitable generalization of the standard continuum theory. Generalized continua in the broad sense can be classified according to the following criteria:
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