A multi-scale strain-localization analysis of a layered strip with debonding interfaces

The paper is focused on the multi-scale modeling of shear banding in a two-phase linear elastic period- ically layered strip with damaging interfaces. A two-dimensional layered strip is considered subjected to transverse shear and is assumed to have a finite thickness along the direction of the layers and an infinite extension along the direction perpendicular to layering. The strip is analyzed as a second-gradient con- tinuum resulting from a second-order homogenization procedure developed by the Authors, here special- ized to the case of layered materials. This analysis is also aimed to understand the influence on the strain localization and post-peak structural response of the displacement boundary conditions prescribed at the strip edges. To this end, a first model representative of the strip with warping allowed at the edges is ana- lyzed in which the strain localization process is obtained as a results of a bifurcation in analogy to the approach by Chambon et al. (1998). A second model is analyzed in which the warping of the edge is inhibited and the damage propagates from the center of the specimen without exhibiting bifurcation phenomena. For this latter case the effects of a possible interaction between the shear band and the boundary shear layer are considered, which are influenced mainly by the characteristic lengths of the model and the strip length. For realistic values of the relevant parameters it is shown that the boundary conditions have a small effects on the elastic response and on the overall strength of the model. Con- versely, the boundary conditions have a significant effect on the shear band location, the post-peak response and the structural brittleness. Since the model parameters directly depend on the material microstructure as a result of the homogenization process, both the extension of the shear band and the occurrence of snap-back in the post-peak phase may be controlled in terms of the constitutive param- eters and of the geometry of the phases.