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.
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Abstract Free, publiclyaccessible full text available March 1, 2025 
Free, publiclyaccessible full text available April 1, 2025

Motivated by the extraordinary strength of nacre, which exceeds the strength of its fragile constituents by an order of magnitude, the fishnet statistics became in 2017 the only analytically solvable probabilistic model of structural strength other than the weakestlink and fiberbundle models. These two models lead, respectively, to the Weibull and Gaussian (or normal) distributions at the largesize limit, which are hardly distinguishable in the central range of failure probability. But they differ enormously at the failure probability level of 10−6 , considered as the maximum tolerable for engineering structures. Under the assumption that no more than three fishnet links fail prior to the peak load, the preceding studies led to exact solutions intermediate between Weibull and Gaussian distributions. Here massive Monte Carlo simulations are used to show that these exact solutions do not apply for fishnets with more than about 500 links. The simulations show that, as the number of links becomes larger, the likelihood of having more than three failed links up to the peak load is no longer negligible and becomes large for fishnets with many thousands of links. A differential equation is derived for the probability distribution of nottoolarge fishnets, characterized by the size effect, the mean and the coefficient of variation. Although the largesize asymptotic distribution is beyond the reach of the Monte Carlo simulations, it can by illuminated by approximating the largescale fishnet as a continuum with a crack or a circular hole. For the former, instability is proven via complex variables, and for the latter via a known elasticity solution for a hole in a continuum under antiplane shear. The fact that rows or enclaves of link failures acting as cracks or holes can form in the largescale continuum at many random locations necessarily leads to the Weibull distribution of the large fishnet, given that these cracks or holes become unstable as soon they reach a certain critical size. The Weibull modulus of this continuum is estimated to be more than triple that of the central range of small fishnets. The new model is expected to allow spinoffs for printed materials with octet architecture maximizing the strength–weight ratio.more » « lessFree, publiclyaccessible full text available January 1, 2025