%0 Report
%D 2016
%T Second-order structured deformations
%A Ana Cristina Barroso
%A Jose Matias
%A Marco Morandotti
%A David R. Owen
%I SISSA
%G en
%1 35497
%2 Mathematics
%4 1
%$ Submitted by Lucio Lubiana (lubiana@sissa.it) on 2016-07-08T11:42:55Z
No. of bitstreams: 1
Morandott-2oStD.pdf: 385787 bytes, checksum: 31ae44cb708c09f82aba2f6ea5e73248 (MD5)
%0 Report
%D 2015
%T Explicit formulas for relaxed disarrangement densities arising from structured deformations
%A Ana Cristina Barroso
%A Jose Matias
%A Marco Morandotti
%A David R. Owen
%X 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.
%I SISSA
%G en
%U http://urania.sissa.it/xmlui/handle/1963/34492
%1 34687
%2 Mathematics
%4 1
%$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2015-08-28T10:09:53Z
No. of bitstreams: 1
SISSA_Preprint_37_2015_MATE.pdf: 315599 bytes, checksum: 2bd676c192205d3a2edd7c48bf53c11c (MD5)
%0 Report
%D 2015
%T Homogenization problems in the Calculus of Variations: an overview
%A Jose Matias
%A Marco Morandotti
%X In this note we present a brief overview of variational methods to solve homogenization problems. The purpose is to give a first insight on the subject by presenting some fundamental theoretical tools, both classical and modern. We conclude by mentioning some open problems.
%I SISSA
%G en
%U http://urania.sissa.it/xmlui/handle/1963/34455
%1 34598
%2 Mathematics
%4 1
%$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2015-03-18T12:11:41Z
No. of bitstreams: 1
surveyhom1.pdf: 297678 bytes, checksum: be0ffd687370fdf3bf6a57d4f6dc61e4 (MD5)
%0 Report
%D 2014
%T Homogenization of functional with linear growth in the context of A-quasiconvexity
%A Jose Matias
%A Marco Morandotti
%A Pedro M. Santos
%X This work deals with the homogenization of functionals with linear growth in the context of A-quasiconvexity. A representation theorem is proved, where the new integrand function is obtained by solving a cell problem where the coupling between homogenization and the A-free condition plays a crucial role. This result extends some previous work to the linear case, thus allowing for concentration effects.
%I SISSA
%G en
%U http://urania.sissa.it/xmlui/handle/1963/7436
%1 7528
%$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2014-10-07T06:20:26Z
No. of bitstreams: 1
Mat-Mor-San-14.pdf: 352795 bytes, checksum: de708e4c922915ea1f523eb23165cb83 (MD5)