00431nas a2200133   4500008004100000245004100041210004000082260001000122100002700132700001700159700002200176700002000198856007900218       2016                        en  d00aSecond-order structured deformations0 aSecondorder structured deformations  bSISSA1 aBarroso, Ana, Cristina1 aMatias, Jose1 aMorandotti, Marco1 aOwen, David, R.  uhttps://www.math.sissa.it/publication/second-order-structured-deformations02252nas a2200145   4500008004100000245009600041210006900137260001000206520175300216100002701969700001701996700002202013700002002035856005102055       2015                        en  d00aExplicit formulas for relaxed disarrangement densities arising from structured deformations0 aExplicit formulas for relaxed disarrangement densities arising f  bSISSA3 aStructured 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.1 aBarroso, Ana, Cristina1 aMatias, Jose1 aMorandotti, Marco1 aOwen, David, R.  uhttp://urania.sissa.it/xmlui/handle/1963/3449200685nas a2200121   4500008004100000245007200041210006900113260001000182520028100192100001700473700002200490856005100512       2015                        en  d00aHomogenization problems in the Calculus of  Variations: an overview0 aHomogenization problems in the Calculus of Variations an overvie  bSISSA3 aIn 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.1 aMatias, Jose1 aMorandotti, Marco  uhttp://urania.sissa.it/xmlui/handle/1963/3445500862nas a2200133   4500008004100000245008700041210006900128260001000197520041000207100001700617700002200634700002200656856005000678       2014                        en  d00aHomogenization of functional with linear growth in the context of A-quasiconvexity0 aHomogenization of functional with linear growth in the context o  bSISSA3 aThis 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.1 aMatias, Jose1 aMorandotti, Marco1 aSantos, Pedro, M.  uhttp://urania.sissa.it/xmlui/handle/1963/7436