Smooth Crack Band Model—A Computational Paragon Based on Unorthodox Continuum Homogenization

Abstract

The crack band model, which was shown to provide a superior computational representation of fracture of quasibrittle materials (in this journal, May 2022), still suffers from three limitations: (1) The material damage is forced to be uniform across a oneelement wide band because of unrestricted strain localization instability; (2) the width of the fracture process zone is fixed as the width of a single element; and (3) cracks inclined to rectangular mesh lines are represented by a rough zigzag damage band. Presented is a generalization that overcomes all three, by enforcing a variable multielement width of the crack band front controlled by a material characteristic length l0. This is achieved by introducing a homogenized localization energy density that increases, after a certain threshold, as a function of an invariant of the thirdorder tensor of second gradient of the displacement vector, called the sprain tensorη, representing (in isotropic materials) the magnitude of its Laplacian (not expressible as a straingradient tensor). The continuum free energy density must be augmented by additional sprain energy Φ(l0η), which affects only the postpeak softening damage. In finite element discretization, the localization resistance is effected by applying triplets of selfequilibrated inplane nodal forces, which follow as partial derivatives of Φ(l0η). The force triplets enforce a variable multielement crack band width. The damage distribution across the fracture process zone is nonuniform but smoothed. The standard boundary conditions of the finite element method apply. Numerical simulations document that the crack band propagates through regular rectangular meshes with virtually no directional bias.