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

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 one-element 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 zig-zag damage band. Presented is a generalization that overcomes all three, by enforcing a variable multi-element 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 third-order 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 strain-gradient 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 self-equilibrated in-plane nodal forces, which follow as partial derivatives of $Φ(l0η)$⁠. The force triplets enforce a variable multi-element crack band width. The damage distribution across the fracture process zone is non-uniform 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.