Calculus of Variations and Geometric Measure Theory

A. C. Barroso - J. Matias - M. Morandotti - D. R. Owen

Explicit Formulas for Relaxed Disarrangement Densities Arising from Structured Deformations

created by morandott on 27 Aug 2015
modified on 19 Jul 2017


Published Paper

Inserted: 27 aug 2015
Last Updated: 19 jul 2017

Journal: Math. Mech. Complex Syst.
Volume: 5
Number: 2
Pages: 163-189
Year: 2017
Doi: 10.2140/memocs.2017.5.163


Structured deformations provide a multiscale geometry that captures the contributions at the macrolevel of both smooth geometrical changes and non-smooth geometrical changes (disarrangements) at submacroscopic levels. For each (first-order) structured deformation (g,G) of a continuous body, the tensor field $G$ is known to be a measure of deformations without disarrangements, and $M:=\nabla g−G$ is known to be a measure of deformations due to disarrangements. The tensor fields $G$ and $M$ together deliver not only standard notions of plastic deformation, but $M$ and its curl deliver the Burgers vector field associated with closed curves in the body and the dislocation density field used in describing geometrical changes in bodies with defects. Recently, Owen and Paroni [13] evaluated explicitly some relaxed energy densities arising in Choksi and Fonseca’s energetics of structured deformations [4] and thereby showed: (1) $(trM)^+$ , the positive part of $trM$, is a volume density of disarrangements due to submacroscopic separations, (2) $(trM)^−$, the negative part of $trM$, is a volume density of disarrangements due to submacroscopic switches and interpenetrations, and (3) $
$, the absolute value of $trM$, is a volume density of all three of these non-tangential disarrangements: separations, switches, and interpenetrations. The main contribution of the present research is to show that a different approach to the energetics of structured deformations, that due to Ba\'{i}a, Matias, and Santos [1], confirms the roles of $(trM)^+$, $(trM)^−$, and $
$ established by Owen and Paroni. In doing so, we give an alternative, shorter proof of Owen and Paroni’s results, and we establish additional explicit formulas for other measures of disarrangements.

Keywords: relaxation, Structured deformations, Disarrangements, Interfacial density, Bulk density, Isotropic vectors