Localization effects in Bammann-Chiesa-Johnson metals with damage delocalization

The presence of softening in the Bammann-Chiesa-Johnson (BCJ) material model presents a major physical drawback: the unlimited localization of strain which results in spurious zero dissipated energy at failure. This difficulty resolves when the BCJ model is modified to incorporate a nonlocal evolution equation for the damage, as proposed by Pijaudier-Cabot and Bazant (1987). The objective of this work is to theoretically assess the ability of such a modified BCJ model to prevent unlimited localization of the strain. To that end, we investigate two localization problems in nonlocal BCJ metals: appearance of a spatial discontinuity of the velocity gradient in a finite, inhomogeneous body, and localization into finite bands. We show that in spite of the softening arising from the damage, no spatial discontinuity occurs in the velocity gradient. Also, we find that the dissipation energy is continuously distributed in nonlocal BCJ metals and therefore can not localize into zones of vanishing volume. As a result, the appearance of any vanishing width adiabatic shear band is impossible in a nonlocal BCJ metal. Finally, we study the finite element (FE) solution of shear banding in a rectangular mesh, deformed in plane strain tension and containing an imperfection, thereby illustrating the effects of imperfections on the localization of the strain.