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:

1. Generalized kinematic relations (and the dual equilibrium equations).
   1. Continua with microstructure, e.g., Cosserat-type continua or strain-gradient theories.
   2. Continua with nonlocal strain, e.g., nonlocal elasticity.
2. Generalized constitutive equations.
   1. Material models with gradients of internal variables (in some cases also with gradients of thermodynamic forces).
   2. Material models with nonlocal internal variables (in some cases also with nonlocal thermodynamic forces).

The paper focuses on the second class of models, with enhancements on the level of the constitutive equations. Their advantage is that the kinematic and equilibrium equations remain standard, and the notions of stress and strain keep their usual meaning. Particular attention is paid to integral-type nonlocal formulations of continuum damage mechanics, which replace a suitably chosen local quantity by its nonlocal counterpart, obtained as a weighted average over a certain finite neighborhood (domain of influence) of each material point. The size of this domain and the shape of the nonlocal weight function determine the characteristic length imposed by the model, which should be related to the material microstructure. The nonlocal quantity is then inserted into the original constitutive equation. Besides acting as a localization limiter, the nonlocal approach can substantially reduce the mesh-induced directional bias and improve convergence of the incremental-iterative solution strategies. First of all, it is shown that not all enhancements of the constitutive equations by spatial averages of certain variables act as efficient and sound localization limiters. In damage mechanics, certain nonlocal formulations lead to a special type of stress locking and cannot provide a clean description of the complete material degradation process up to the final failure. The next issue to be addressed is the computational efficiency of nonlocal simulations. Numerical examples illustrate the performance of Newton-Raphson iterations with a consistently derived "nonlocal" tangent stiffness matrix. Finally, attention turns to techniques that allow an accurate resolution of highly localized strain profiles. One possible approach is based on adaptive refinement of the finite element mesh. As an alternative, the quality of the numerical approximation can be improved by adding special enrichments of the standard shape functions in the spirit of the partition-of-unity method.

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