Multiscale Modeling in Continuum Mechanics and Structured Deformations
July 15, 2002 — July 19, 2002
Coordinators:
- David R. Owen (Carnegie Mellon University, Pittsburgh Pennsylvania, USA)
- Gianpietro Del Piero (Univ. d. Studi di Ferrara, Ferrara, Italy)
The purpose of the course is to present some recent progress in two active and related areas of continuum mechanics: fracture mechanics and structured deformations.
The course is divided into two parts. The first part deals with the theory of structured deformations, a basic concept that can be used to describe smooth and non-smooth geometrical changes at different length scales, including, for example, both slip and microslip and, of particular interest in the second part of this course, both macroscopic fracture and microfracture.
The foundations of the theory of structured deformations will be presented in a self-contained way. They will be employed in the formulation of field theories for bodies undergoing deformations at different scales. The extension of the theory to deformations expressed by higher order gradients will be considered in detail. The theory will also be applied to the study of single crystals undergoing single or multiple slip.
In the second part of the course, the energetics of isotropic nonlinear elastic bodies will be discussed, followed by a variational formulation for nonlinear crack problems in which the conditions for crack equilibrium or propagation are determined by the asymptotic behavior of the strain and stress fields near the tip of the crack. Finally, an evolutionary model for fracture will be presented within Griffith’s scheme for brittle fracture. It will be accompanied by several examples, solved either by asymptotic methods (for composite plates or beams) or by an accurate finite element scheme. Both techniques are based on the concept of Gamma-convergence.
The course is addressed to researchers in the field of Continuum Mechanics, interested in recent developments of theories for modeling material response, theoretical approaches to fracture mechanics, unified theories for different aspects of material behavior, and solution of problems involving materials with internal microstructure.