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.
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A preceding 2023 study argued that the resistance of a heterogeneous material to the curvature of the displacement field is the most physically realistic localization limiter for softening damage. The curvature was characterized by the second gradient of the displacement vector field, which includes the material rotation gradient, and was named the “sprain” tensor, while the term “spress” is here proposed as the force variable workconjugate to “sprain.” The partial derivatives of the associated sprain energy density yielded in the preceeding study, sets of curvature resisting selfequilibrated nodal sprain forces. However, the fact that the sprain forces had to be applied on the adjacent nodes of a finite element greatly complicated the programming and extended the simulation time in a commercial code such as abaqus by almost two orders of magnitude. In the present model, Smooth Lagrangian Crack Band Model (slCBM), these computational obstacles are here overcome by using finite elements with linear shape functions for both the displacement vector and for an approximate displacement gradient tensor. A crucial feature is that the nodal values of the approximate gradient tensor are shared by adjacent finite elements. The actual displacement gradient tensor calculated from the nodal displacement vectors is constrained to the approximate displacement gradient tensor by means of a Lagrange multiplier tensor, either one for each element or one for each node. The gradient tensor of the approximate gradient tensor then represents the approximate thirdorder displacement curvature tensor, or Hessian of the displacement field. Importantly, the Lagrange multiplier behaves as an externally applied generalized moment density that, similar to gravity, does not affect the total strainplussprain energy density of material. The Helmholtz free energy of the finite element and its associated stiffness matrix are formulated and implemented in a user’s element of abaqus. The conditions of stationary values of the total free energy of the structure with respect to the nodal degreesoffreedom yield the set of equilibrium equations of the structure for each loading step. One and twodimensional examples of crack growth in fracture specimens are given. It is demonstrated that the simulation results of the threepoint bend test are independent of the orientation of a regular square mesh, capture the width variation of the crack band, the damage strain profile across the band, and converge as the finite element mesh is refined.
more » « less Award ID(s):
 2029641
 NSFPAR ID:
 10525829
 Publisher / Repository:
 ASME
 Date Published:
 Journal Name:
 Journal of Applied Mechanics
 Volume:
 91
 Issue:
 3
 ISSN:
 00218936
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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