### 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
det*F*, 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.