Bifurcation into a localized mode from non-uniform periodic deformations around a periodic pattern of voids

• A unit cell model with periodic boundary conditions is used to analyze a material with a periodic pattern of voids.
• Bifurcation into a shear band is studied, using also a cell model inside the band.
• The bifurcation conditions on the band interface are only satisfied on the average.
• Under shear loading the voids may collapse to micro-cracks prior to bifurcation.
• First bifurcation may occur at a local maximum, followed by further load increase.

Bifurcation into a shear band is studied for a porous ductile material subject to a combination of shear loading and tensile or compressive loading in different directions. The material with a periodic array of voids is studied by numerical solutions for a plane strain unit cell model with fully periodic boundary conditions. The fundamental pre-bifurcation solution has been studied before, with focus on ductile fracture under conditions of low stress triaxiality. In the previous studies it has been shown that voids in shear are flattened out to micro-cracks, which rotate and elongate until interaction with neighboring micro-cracks gives coalescence. These failure mechanisms are included in the present study, but here the focus is on the possibility that failure may occur earlier, if bifurcation leads to a shear band crossing over many cells, where the plastic strains inside the band will grow very large, while the overall strains in the material will not increase any further. The unit cell analysis with full periodicity is used both inside and outside the band to find the average behavior in the two material regions. This does not allow for point-wise satisfaction of compatibility and equilibrium along the interface between the two regions, but these conditions can be satisfied on the average. The bifurcation analysis includes determination of the direction along which a shear band is first critical.