Zdenek P. Bazant's KTH Solid Mechanics KEYNOTE seminar "Spress-Sprain Crack Band Model Based on Lagrange Multiplier Constraint and Its Verification by Gap Test"

Zdenek Bazant_March_07_2024.pdf (pdf 162 kB)

Abstract. Preceding studies showed that the resistance of a heterogeneous material to the displacement field curvature is the physically most realistic localization limiter for softening damage and fracture. The curvature was characterized by the second gradient of the displacement vector field, which differs from the strain gradient tensor by the material rotation gradient tensor, and was named the ‘sprain’ tensor, whose work-conjugate force variable was called the ‘spress’ tensor. In the preceding work, the partial derivatives of the associated sprain energy density were used to obtain self-equilibrated sets of curvature-resisting nodal sprain forces, some of which had to be applied on nodes adjacent to the finite element. But this led to an enormous computational burden. This burden is eliminated by formulating a finite element with linear shape functions for both the displacement vector and the approximate displacement gradient tensor. The derivatives of the latter then yield the tensor of displacement curvature (or hessian), obviating the need for adjacent sprain forces. The main idea is to use a Lagrange multiplier tensor to constrain the approximate gradients to the actual displacement gradients. A user element for Abaqus is formulated and used to demonstrate mesh-independent crack band growth, capturing the band width variation and smooth damage distribution across the crack band. The spress-sprain energy dissipation is shown to match various distinctive fracture tests of concrete, including the gap test and its simulation by the LDPM (discrete particle) model. Generalization to plastic-hardening metals (e.g., aluminum), which requires distinguishing energy dissipations at the micrometer scale crack front and in the wake of a millimeter scale hardening yielding zone, is presented in closing.