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Publication 15-CNA-011

Explicit Formulas For Relaxed Disarrangement Densities Arising From Structured Deformations

Ana Cristina Barroso
Faculdade de Ciencias
Universidade de Lisboa
Departamento de Matematica and CMAF
Lisboa, Portugal
acbarroso@fc.ul.pt

José Matias
Departamento de Matemática
Instituto Superior Técnico
Av. Rovisco Pais, 1049-001 Lisboa, Portugal
jmatias@math.ist.utl.pt

Marco Morandotti
SISSA
Trieste, Italy
marco.morandotti@sissa.it

David R. Owen
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
do04@andrew.cmu.edu

Abstract: 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 := ∇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) |trM|, 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 Baía, Matias, and Santos [1], conforms the roles of (trM)+, (trM)-, and |trM| 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.

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