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/1963/34492