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Explicit formulas for relaxed disarrangement densities arising from structured deformations

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dc.contributor.author Barroso, Ana Cristina
dc.contributor.author Matias, Jose
dc.contributor.author Morandotti, Marco
dc.contributor.author Owen, David R.
dc.date.accessioned 2015-08-28T10:11:19Z
dc.date.available 2015-08-28T10:11:19Z
dc.date.issued 2015
dc.identifier.uri http://preprints.sissa.it/xmlui/handle/1963/34492
dc.description.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 Ba\'{i}a, Matias, and Santos [1], confirms 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. en_US
dc.language.iso en en_US
dc.publisher SISSA en_US
dc.relation.ispartofseries SISSA;37/2015/MATE
dc.subject Structured deformations en_US
dc.subject relaxation en_US
dc.subject disarrangements en_US
dc.subject interfacial density en_US
dc.subject bulk density en_US
dc.subject isotropic vectors en_US
dc.title Explicit formulas for relaxed disarrangement densities arising from structured deformations en_US
dc.type Preprint en_US
dc.miur.area 1 en_US
dc.contributor.area Mathematics en_US
dc.relation.firstpage 1 en_US
dc.relation.lastpage 17 en_US


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