### Stress-Sprain Crack Band Model Based on Lagrange Multiplier Constraint of Displacement Gradient

McCormick - Mechanical Engineering (ME)

11:00 AM A230, Technological InstituteEVENT DETAILS

Prof. Zdeněk P Bažant

Walter P. Murphy Professor of Civil and Environmental Engineering

Northwestern University

A preceding seminar on 1/12/23 argued 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 includes the material rotation gradient, and was named the 'sprain' tensor (a force variable work-conjugate to 'sprain' tensor). Initially, the partial derivatives of the associated sprain energy density were used to obtain self-equilibrated sets of curvature-resisting nodal sprain forces. Some forces acted on nodes adjacent to the finite element, which led to enormous computational burden. This burden is now 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 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.

Born and educated in Prague (Ph.D. 1963), Bažant joined the Northwestern faculty in 1969, where he has been W.P. Murphy Professor since 1990 and simultaneously McCormick Institute Professor since 2002, and Director of Center for Geomaterials (1981-87). He has been inducted to the NAS, NAE, AAAS, Royal Soc. London and 8 other national academies, and has received the Austrian Cross of Honor for Science and the Art I. Class from the President of Austria, among many other awards and 9 honorary doctorates. He has authored nine books on structural stability, fracture, its probability, and concrete creep and hygrothermal effects. In 2019 Stanford University citation survey, he ranked worldwide no.1 in CE and no.2 in engineering overall. In 2015, ASCE established the Bažant Medal for Failure and Damage Prevention and in 2023, ASME established the Bažant Medal for contributions to mechanics.

TIME Thursday December 14, 2023 at 11:00 AM - 12:00 PM

LOCATION A230, Technological Institute map it

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CONTACT Jeremy Wells jeremywells@northwestern.edu

CALENDAR McCormick - Mechanical Engineering (ME)