Field Equation for Nonlocal Continuum

Now comes the most difficult step. We need to determine the nonlocal macroscopic field equation which represents the continuum counterpart of (13.3.8). The homogenization theories as known are inapplicable, because they apply only to macroscopically uniform fields while the nonuniformity of the macroscopic field is the most important aspect tor handling localization problems. The following simple concept has been proposed (Bazant 1994b):

The continuum field equation we seek is the equation whose discrete approximation can be written in the form of the matrix crack interaction relation (13.3.8).

This concept leads to the following field equation for the continuum approximation of microcrack interactions (Bazant 1994b):

ДSw{x) – [ Л(х:£)ДSil)(£)dV(£) = AS0)(x) (13.3.9)


Indeed, an approximation of (he integral by a sum over the continuum variable values at the crack centers yields (13.3.8). Here we denoted Л(хм, £„) = £(Л^„)/1Д = crack influence function, Vc is a constant that may be interpreted roughly as the material volume per crack, and £ is a statistical averaging operator which yields the average (moving average) over a certain appropriate neighborhood of point x or £. Such statistical averaging is implied in the macro-continuum smoothing and is inevitable because, in a random crack array, the characteristics of the individual cracks must be expected to exhibit enormous random scatter.

It must be admitted that the sum in (13.3.8) is an unorthodox approximation of the integral from (13.3.9) because the values of the continuum variable are not sampled at certain predetermined points such as the chosen mesh nodes but are distributed at random, that is, at the microcrack centers. Another point to note is that (13.3.8) is only one of various possible discrete approximations of (13.3.9). Since this approximation is not unique, the uniqueness of (13.3.9) as a continuum approximation is not proven. Therefore, acceptability of (13.3.9) will also depend on computational experience (which has so far been favorable; see Ozbolt and Bazant 1996).

When (13.3.9) is approximated by finite elements, it is again converted to a matrix form similar to (13.3.8). However, the subscripts for the sum then runs over the integration points of the finite elements. This means the crack pressures (or openings) that are translated into the inelastic stress increments are only sampled at these integration points, in the sense of their density, instead of being represented individually as in (13.3.8). Obviously, such a sampling can preserve only the long-range interactions of the cracks and the averaging. The individual short-range crack interactions will be lost, but they are so random and vast in number that aspiring to represent them in any detail would be futile.

For macroscopic continuum smearing, the averaging operator — over the crack length now needs reinterpretation. Because of the randomness of the microcrack distribution, the macro-continuum variable at point x should represent the spatial average of the effects of all the possible microcrack realizations within a neighborhood of point x whose size is roughly equal to the spacing t of the dominant microcracks (which is, in concrete, approximately determined by the spacing of the largest aggregates, which is in turn proportional to the maximum aggregate size); hence,

АБЩх) – J AS^(£)cx(x,£)dV(£) (13.3.10)

The weight function a(x,£) is analogous to that in Eq. (13.1.1). It should vanish everywhere outside the domain of a diameter roughly equal to L For computational reasons, it seems preferable that a have a smooth bell shape. Because of randomness of the microcrack distribution, function a(x,£) may be considered as rotationally symmetric (i. e., same in all directions, or isotropic).

Strictly speaking, the macroscopic averaging domain could be a line segment in the direction of the dominant microcrack (that is, normal to ASl”(x)), or an elongated, roughly elliptical domain. However, averaging only along a line segment seems insufficient for preventing damage from localizing into a line, in the case of a homogeneous uniaxial tension field, and it would also be at variance with the energy release argument for nonlocality of damage presented in Bazant (1987c, 1991b).

Equation (13.3.9) represents a Fredholm integral equation (i. e., an integral equation of the second kind with a square-integrablc kernel) for the unknown Дб’О(х), which corresponds in Fig. 13.3.2 to the segment 35. The inelastic strain increment tensors ДS^)(x) on the right-hand side, which correspond in Fig. 13.3.2 to the segment 32, are calculated from the strain increments using the given local constitutive law (for example, the microplane model, continuum damage theory, plastic-fracturing theory, or plasticity with yield limit degradation).