Smooth Lagrangian Crack Band Model Based on Spress-Sprain Relation and Lagrange Multiplier Constraint of Displacement Gradient

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 work-conjugate to "sprain." The partial derivatives of the associated sprain energy density yielded in the preceeding study, sets of curvature resisting self-equilibrated 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 third-order 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 strain-plus-sprain 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 degrees-of-freedom yield the set of equilibrium equations of the structure for each loading step. One- and two-dimensional examples of crack growth in fracture specimens are given. It is demonstrated that the simulation results of the three-point 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.