Localization-induced band and cohesive model

A localization-induced cohesive model has been proposed for shear band evol ution, crack growth, and fracture. Strain gradient theory has been applied to establish the criterion of the onset of localization and the governing e quation in the post-bifurcation stage. Analytical solutions in one-dimensio nal case are used to establish the "traction-separation" law, in which stra in gradient and material intrinsic length scale present strong effects. In addition, the solution predicts a finite width for the localization-induced band. It is observed that a larger length scale contributes to the growth of a larger width of localization region and separation for softening mater ials. The proposed model provides a procedure to establish the fracture tou ghness analytically since the material length scale is taken into account. From the traction-separation analysis, it is found that damage decreases se paration, whereas an increase in material length scale increases the openin g displacement: however, the traction-normalized opening displacement curve s (with respect to the material length scale) are identical. Based on the m ethodology of multiple scale analysis in meshfree method, a computational a pproach has been proposed to enrich the one-dimensional traction-separation law to define fracture.