Calculus of Variations and Geometric Measure Theory

G. Carita - J. Matias - M. Morandotti - D. R. Owen

Dimension reduction in the context of structured deformations

created by morandott on 09 Sep 2017
modified on 05 Oct 2018


Published Paper

Inserted: 9 sep 2017
Last Updated: 5 oct 2018

Journal: Journal of Elasticity
Volume: 133
Number: 1
Pages: 1--35
Year: 2018
Doi: 10.1007/s10659-018-9670-9


In this paper we apply both the procedure of dimension reduction and the incorporation of structured deformations to a three-dimensional continuum in the form of a thinning domain. We apply the two processes one after the other, exchanging the order, and so obtain for each order both a relaxed bulk and a relaxed interfacial energy. Our implementation requires some substantial modifications of the two relaxation procedures. For the specific choice of an initial energy including only the surface term, we compute the energy densities explicitly and show that they are the same, independent of the order of the relaxation processes. Moreover, we compare our explicit results with those obtained when the limiting process of dimension reduction and of passage to the structured deformation is carried out at the same time. We finally show that, in a portion of the common domain of the relaxed energy densities, the simultaneous procedure gives an energy strictly lower than that obtained in the two-step relaxations.

Keywords: relaxation, dimension reduction, Structured deformations, integral representation of functionals, explicit formulas