# Nonlocal Continuum Modeling of Damage Localization

In concrete and other quasibrittlc materials, fracture develops as a result of localization of distributed damage due to microcracking. Indiscrete fracture models which have been discussed in previous chapters, the damage due to distributed cracking is lumped into a line, but this is not sufficiently realistic for all applications. The width and microcracking density distribution at the fracture front may vary depending on structure size, shape, and type of loading.

Such behavior can be captured only by continuum damage models. However, such models cannot be implemented in the sense of the classical, local continuum, i. e., a continuum in which the stress at a point depends only on the strain at the same point. Rather, one must adopt the more general concept of a nonlocal continuum, defined as a continuum in which the stress at a point depends also on the strains in the neighborhood of that point or some type of average strain of the neighborhood. The reasons for introducing the nonlocal concept arc both mathematical and physical:

1. The mathematical reason is that, as we discussed in Chapter 8 (Bazant 1986c), a local strain softening continuum exhibits spurious damage localization instabilities, in which all damage is localized into a zone of measure zero. This leads to spurious mesh sensitivity. The energy that is consumed by cracking damage during structural failure depends on the mesh size and tends to zero as the mesh size is refined to zero. The reason is that the energy dissipation, as described by the local stress-strain relation per unit volume, and thus also the total dissipation, converges to zero if all damage is localized into a band of single element width as the element size tends to zero. Such spurious localization on a set of measure zero is prevented by the nonlocal concept.

2. The physical reason is that microcracks interact (Bazant 1994b; Bazant and Jirasek 1994a, b)). The formation or growth of one microcrack either promotes or inhibits the formation or growth of adjacent microcracks. Continuum smearing of such interactions inevitably leads to some kind of a nonlocal continuum. The interaction of microcracks is the physical reason why a continuum model for distributed strain softening damage ought to be nonlocal. A secondary physical reason is that a crack has a macroscopically nonnegligible dimension, causing the crack growth to depend on the macroscopic stress field in a zone larger than the crack (Bazant 1987c, 1991b).

Spatial averaging integrals and interaction integrals are not the only way to describe a nonlocal con­tinuum. If the strain field in the neighborhood of a point is expanded into a Taylor series, the strains in the neighboring points arc approximately characterized by the spatial partial derivatives (gradients) of the strain tensor at the given point. Thus, the nonlocal continuum may alternatively be defined as a continuum in which the stress at a point depends not only on the strain at that point but also on the successive gradients of the strain tensor at that point (Bazant 1984b, Triantafyllidis and Aifantis 1986). This approach may be regarded as a generalization of Cosserat’s couple stress continuum or Eringen’s micropolar elasticity. Cosserat’s continuum was considered as an alternative approach to achieve reg­ularization of-the strain-localization problem in softening materials, but they have been superseded by fully nonlocal or high-gradient models, and will not be presented here. The interested reader may refer, among others, to the works by de Borst, R. (1990) Vardoulakis (1989), de Borst (1991), de Borst and Sluys (1991), and Dietschc and Wiliam (1992).

In Chapter 8 we have already seen the crudest but simplest type of nonlocal approach—the crack band model, in which the dependence of stress on the average deformation of a certain representative volume

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Figure 13.1.1 Spalial averaging, (a) Profiles of micro strain and average strain along a segment centered at point x in the center of a representative volume, (b) Sketch of the representative volume centered at x. (c) Representative volume near the surface of the body, (d) Uniform vs. smoothly decaying averaging functions. (Adapted from Bazant 1990c.) of the material is enforced by prescribing the minimum crack band width, coinciding with a minimum element size (the reason is that, in a constant strain finite element, the strain approximates the average strain of the material within the element area). The first part of this chapter will describe the nonlocal models based on the idea of averaging, approached in a phenomenological manner. The second part of this chapter will present a recent development in which the nonlocal concept is derived from micromechanical analysis of crack interactions. The mathematical aspects of localization instabilities and bifurcations will not be discussed in detail and the reader is referred to the book by Bazant and Ccdolin (1991, Ch. 13).