# Objectivity of Analysis

A physical theory must be objective, in the sense that the results of calculations made with it must not depend on subjective aspects such as the choice of coordinates, the choice of mesh, etc. If a theory is found

Figure 1.2.2 Illustration of lack of mesh-objectivity in classical smeared crack models (adapted from ACI Committee 446 1992).

to be unobjective, it must be rejected. There is no need to even compare it to experiments. Objectivity comes ahead of experimental verification.

A powerful, widely used approach to finite element analysis of concrete cracking is the concept of smeared cracking, introduced by Rashid (1968), which does not utilize fracture mechanics. According to this concept, the stress in a finite element is limited by the tensile strength of the material, /t’. After the strength limit is reached, the stress in the finite element must decrease. In the initial practice, the stress was assumed to drop suddenly to zero, but it was soon realized that better and more realistic results are usually obtained if the stress is reduced gradually, i. e., the material is assumed to exhibit gradual strain softening (Scanlon 1971; Lin and Scordelis 1975); see Fig. 1.2.1. The concept of sudden or gradual strain-softening, though, proved to be a mixed blessing. After this concept had been implemented in large finite element codes and widely applied, it was discovered that the convergence properties are incorrect and the calculation results are unobjective as they significantly depend on the analyst’s choice of the mesh (Bazant 1976, 1983; Bazant and Cedolin 1979, 1980, 1983; Bazant and Oh 1983a; Darwin 1985; Rots et al. 1985).

This problem, known as spurious mesh sensitivity, can be illustrated, for example, by the rectangular panel in Fig. 1.2.2a, which is subjected to a uniform vertical displacement at the top boundary. A small region near the center of the left side is assumed to have a slightly smaller strength than the rest of the panel, and consequently a smeared crack band starts growing from left to right. The solution is obtained by incremental loading with two finite element meshes of very different mesh sizes, as shown (Fig 1.2.2b, e). Stability check indicates that cracking must always localize in this problem into a band of single-element width at the cracking front. Typical numerical results for this as well as other similar problems are illustrated in Fig. 1.2.2d—f. In the load-deflection diagram (Fig. 1.2.2d), one can see that the

displacement

Figure 1.2.3 Load-deflection curves with and without yielding plateau (adapted from ЛС1 Committee 446 1992).

peak load as well as the post-peak softeffing strongly depends on the mesh size, the peak load being roughly proportional to h~where h is the element size. Plotting the load vs. the length of the crack hand, one again finds large differences (Fig. 1.2.2e). The energy that is dissipated due to cracking decreases with the refinement of the mesh (Fig. 1.2.21), and converges to zero as h —> 0, which is, of course, physically unacceptable.

The only way to avoid the foregoing inanifestations of unobjectivity is some form of fracture mechanics or nonlocal model. By specifying the energy dissipated by cracking per unit length of the crack or the crack band, the overall energy dissipation is forced to be independent of the element subdivision (see the horizontal dashed line in Fig. 1.2.21), and so is (he maximum load.