Localization and Mesh Objectivity

Consider a homogeneous bar of initial length L (Fig. 8.2.1a) made of a material whose stress-strain curve (uniaxial) is assumed to exhibit softening, as sketched in Fig. 8.2.1b. Because of the hypothesis of homogeneity, we can imagine the bar to be subdivided in N identical shorter bars (N being arbitrary) which then act as N equal elements coupled in series, as sketched in Fig. 8.2.1c. We have seen in the previous section that when N elements are coupled in series, the strain localizes after the peak in only one of them, so that the resulting а-є curves look similar to that in Fig. 8.1.4b.

Therefore, the postpeak softening of the bar depends totally on the assumed subdivision, as indicated in Fig. 8.2. Id. This has two direct consequences: on purely mechanical grounds, the result is absurd because the physical result cannot depend on the imagined subdivision; on numerical grounds, it implies that the result one would obtain by using finite elements would completely depend on the number of elements or element size. This is a subjective choice of the analyst, and, thus, is not an objective property, as pointed out by Bazant (1976). This last property is referred to as lack of mesh objectivity, or as spurious mesh sensitivity.

Keeping the numerical point of view, we must realize that the response of the foregoing model is reached upon infinite mesh refinement, i. e., for N •> oo. This means that strain localization is predicted to occur only within one infinitely thin element, i. e., an element of infinitely small length and volume. Now, is that consistent with the principles of thermodynamics’? Yes, it is. We stated in the previous section that among many possible equilibrium paths, the actual one is that in which the second-order complementary work fi2W* at constant SP is maximum. Since, in the softening branch, SP is negative, the maximum 82Ж occurs for the path with the largest (positive) inverse slope. As shown in the figure, this slope does indeed correspond to considering infinitely many elements (Fig. 8.2.Id).

*

Л further implication is that any variable that is ultimately bounded by the length or the volume will vanish. (We say that a variable ф is ultimately bounded by volume V if ф < MV, for some finite M > 0.) This is so for the inelastic displacement and the energy dissipation after the peak. We say that the corresponding physical quantities have measure zero. Let us take a closer look at this problem for one special, yet important case.