The inelastic stress increments in a microcracking material are equal to the stresses that the load increment would produce on the cracks if they were temporarily “frozen" (or “glued”), i. e., prevented from opening and growing. The nonlocality arises from two sources: (1) crack interactions, which means that application of the pressure on the crack surfaces that corresponds to the “unfreezing" (or “ungluing”) of one crack produces stresses on all the other frozen cracks; and (2) averaging of the stresses due to unfreezing over the crack surface, which is needed because crack interactions depend primarily on the stress average over the crack surface (or the stress resultant) rather than the macroscopic stress corresponding to the microcrack center. The crack interactions (source 1) can be solved by Kachanov’s (1987a) simplified version of the superposition method, in which only the average crack pressures are considered.
The resulting nonlocal continuum model involves two spatial integrals. One integral, which corresponds to source (1) and has been absent from previous nonlocal models, is long-range and has a weight function whose spatial integral is 0; it represents interactions with remote cracks and is based on the long-range
Figure 13.3.9 Localization in a bar predicted by nonlocal model based on crack interactions: (a) normalized strain profiles along a bar; the profiles are symmetric with respect to the origin, and an exponential softening law was used; (b) load-displacement diagrams for various bar lengths L. (From Jirasck and Baz. ant 1994.)
asymptotic form of the stress field caused by pressurizing one crack while all the other cracks are frozen. Another integral, corresponding to source (2), is short-range, involves a weight function whose spatial integral is 1, and represents spatial averaging of the local inelastic stresses over a domain whose diameter is roughly equal to the spacing of major microcracks (which is roughly equal to the spacing of large aggregates in concrete).
As an approach to continuum smoothing when the macroscopic field is nonuniform, one may seek a continuum field equation whose possible discrete approximation coincides with the matrix equation governing a system of interacting microcracks.
The long-range asymptotic weight function of the nonlocal integral representing crack interactions (source 1) has a separated form which is calculated as the remote stress field of a crack in infinite body. It decays with distance r from the crack as r~2 in two dimensions and v "3 in three dimensions. This long – range decay is much weaker than assumed in previous nonlocal models. In consequence, the long-range integral diverges when the damage growth in an infinite body is assumed to be uniform. This means that only the localized growth of damage zones can be modeled.
In contrast to the previous nonlocal formulations, the weight function (crack influence function) in the long-range integral is a tensor and is not axisymmetric (isotropic). Rather, it depends on the polar or spherical angle (i. e., is anisotropic), exhibiting sectors of shielding and amplification. The weight function is defined statistically and can be obtained by evaluating a certain averaging integral in which the integrand is the stress field of one pressurized crack in the given structure.
When an iterative solution of crack interactions according to the Gauss-Seidcl iterative method is considered, the long-range nonlocal integral based on the crack influence function yields the nonlocal inelastic stress increments explicitly. This explicit form is suitable for iterative solutions of the loading steps in nonlinear finite element programs. The nonlocal inelastic stress increments represent a solution of a tensorial Fredholm integral equation in space, to which the iterations converge.
The constitutive law, in this new formulation, is strictly local. This is a major advantage. It eliminates difficulties with formulating the unloading criterion and the continuity condition, experienced in the previous nonlocal models in which nonlocal inelastic stresses or strains have been part of the constitutive relation.