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

Abstract

Abstract A preceding 2023 study argued that the resistance of a heterogeneous material to the displacement field curvature is the physically most 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’. In this study, the computational obstacles using nodal sprain forces in the previous study are overcome by using finite elements with linear shape functions for both the displacement vector and for an approximate displacement gradient tensor. The actual gradient calculated from the nodal displacement vectors is constrained to the approximate gradient by means of a Lagrange multiplier tensor. The gradient tensor of the approximate gradient tensor then represents the third-order approximate 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 conditions of stationary values of the total free energy of the structure with respect all these unknowns yields the set of equilibrium equations of the structure for each loading step. Examples of crack growth in fracture specimens are given. It is demonstrated that the simulation results are independent of the orientation of a regular square mesh, and capture the width variation of the crack band, and converge as the finite elements are refined.