Geometrically Sufficient Disarrangements and Measures of Vacancy Movement

David R. Owen
Department of Mathematical Sciences
Carnegie Mellon University

Multiscale geometrical changes typically result in a discrepancy between, on the one hand, smooth macroscopic deformation as measured by the deformation gradient F and, on the other hand, contributions at the macrolevel from smooth submacroscopic deformation, as measured by a second tensor field G. Alone, this discrepancy generally would result in a loss of injectivity of transplacements and in an associated interpenetration of matter. The theory of structured deformations has shown that, when submacroscopic volume changes detG do not exceed macroscopic volume changes detF, injectivity can be maintained by means of superposed piecewise rigid deformations that act at the smallest length scale so as to avoid interpenetration of matter. The submacroscopic distortions and piecewise rigid deformations produce non-smooth but injective geometrical changes (disarrangements) whose contributions to the macroscopic deformation gradient F appear in the difference F - G. In this talk, I will describe joint work with Roberto Paroni that identifies an explicit formula in terms of F and G for how much "switching" occurs submacroscopically via piecewise rigid deformations at the smallest length scale. Such switching can be realized in a real material via the movement of vacancies. The principal tool employed in this research is a characterization due to Choksi and Fonseca of the bulk part of a relaxed interfacial energy.