Structured Deformations and the Microgeometry of Continua

Carnegie Mellon University, Pittsburgh PA

Investigators

Abstract

DMS Award Abstract Award #: 0102477 PI: Owen, David Institution: Carnegie Mellon University Program: Applied Mathematics Program Manager: Catherine Mavriplis Title: Structured Deformations and the Microgeometry of Continua Structured deformations provide a multiscale geometry and kinematics for describing the effects at the macrolevel of both smooth and non-smooth deformations occurring at submacroscopic levels. The proposed research focuses on (i) applying this recently developed multiscale geometry to refine and improve important continuum descriptions of single crystals undergoing slipping on more than one crystallographic system, (ii) identifying via a relaxation technique from the calculus of variations a geometrical variable that measures the volume swept out by submacroscopic vacancies and understanding the connection between this variable and different types of dislocation movements, and (iii) obtaining relations that complete the system of balance laws and constitutive relations that arise when a continuous body undergoes structured deformations and motions. This research bears on the fields of materials science and engineering and on the mathematical study of geometry at multiple length and time scales. The geometrical changes in a deforming paper clip or other metallic body that appear continuous and smooth to the naked eye often appear to be jerky and abrupt when viewed in a microscope. Similarly, the smooth stretching of a plastic sandwich bag or other thin plastic film appears differently at macroscopic and submacroscopic scales. The most successful theories for studying macroscopic changes in a body - paper clip, sandwich bag, or otherwise -- brought about by applied forces or heat sources are called field theories. The success of field theories rests first on our physical understanding of how a particular substance, e.g., the particular metal comprising the paper clip or the particular plastic constituting the sandwich bag, influences the detailed form of the field theories and, second, on our mathematical understanding of the equations that comprise the field theory. The research undertaken in this project is part of an ongoing program to enrich these field theories by permitting them to describe and predict the behavior of a body that changes shape differently at macroscopic and submacroscopic length scales. The desired outcome of this research is refined, improved field theories that permit more accurate simulations of the behavior of technologically important materials. Date: June 25, 2001