Stress Lock-In

As pointed out before, the crack band analysis for mode I and elements aligned with the crack path gives, in general, good results. Various details of numerical modeling, however, deserve attention. Leibengood, Darwin and Dodds (1986) modeled the crack band by square elements with four integration points straddling the line of symmetry. They showed that the results for the crack band and the discrete crack closely match each other if the cracking directions at the integration points within the finite elements are forced to be parallel to each other and to the symmetry line of the crack band. But if this parallelism is not enforced, the cracks form at different orientations at each integration point of the same element, as shown in Fig. 8.6.4a. Then the response predicted by the crack band model is somewhat stiffer than that predicted by the discrete crack model, even if the element sides are parallel to the true crack.

The reason for this behavior is that the integration points lying out of the symmetry plane sense the shear components, and so the cracks form at an angle. This results in spurious fracturing strain components parallel to the crack path, which cause large strains in the neighboring elements parallel to the main crack path, which results in overall stiffening. Although the crack growth is actually in mode I, the problem is aggravated because, at the integration points, the loading is interpreted as mixed mode. This is manifested by a spurious sensitivity of the solution to the shear retention factor /3S, which, for mode 1, should be nil. A solution to this problem is to determine the crack normal and the cracking strain on the average or at a single integration point at the center of the element. This is actually the only way consistent with the hypothesis that the cracking strain is uniformly distributed over the element.

The foregoing problem is related to the phenomenon known as stress lock-in, that appears in mixed mode problems, when the crack grows skew to the mesh (Rots 1988, 1989). The stress lock-in consists
in the effect that the elements near the crack band remain stressed after a nearly complete failure of the elements in the crack band, because the inelastic strain in the band induces stresses in the neighbors, as sketched in Fig. 8.6.4b. Rots’ (1988,1989) results indicate that the degree of stress lock-in is very sensitive to the shear retention factor, as illustrated by the results in Fig. 8.6.4c, corresponding to a double cantilever beam tested by Kobayashi et al. (1985). The fixed crack must be used with a zero shear retention factor to get a better approximation. In the example shown, the smeared crack model with the best behavior is the rotating crack model, but in other cases (single-notch sheared beams, for example), the fixed craek with a zero shear retention factor may be better (Rots 1988, 1989).

The only effective solution to this problem (keeping classical finite elements) is to first run the calcula­tions with a standard mesh to get the approximate crack path, and then remesh to get mesh lines aligned with the crack path and run the calculation again (see also Section 8.7.4).