Dimension Reduction in the Context of Structured Deformations

Abstract

Dedicated to our friend and colleague Graça Carita, who left us far too soon.Abstract. 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.

Structured Deformations of Continua: Theory and Applications

“…Moreover, in this case, another relaxation procedure is available in the literature: in [17] a relaxation that simultaneously defines a 2d energy on structured deformations is studied. We prove [6,Section 5] that the left-and right-hand paths provide the same relaxed energy densities, whereas those computed using the central path in Fig. 2 are lower.…”

Upscaling and spatial localization of non-local energies with applications to crystal plasticity

“…After the presentation of the iterated limiting procedure carried out in Sections 4.1 and 4.2, a legitimate question is whether the two operations commute, namely, whether we obtain the same result if we reverse the order in which the two limits are taken: first letting r → 0 + and then letting n → ∞. The problem is a relevant one in the scientific community and a similar question was studied in [30] for a problem of dimension reduction in the context of structured deformations. In the following few lines, we give a brief explanation of why in the present case a commutability result does not hold.…”

Bending-torsion moments in thin multi-structures in the context of nonlinear elasticity

“…These forces were introduced in [6] (also see [5]) concerning the membrane case, then adapted to the string case in [18] and also to the membrane case in the BV setting in [3] and in the Orlicz-Sobolev setting in [37,38]. We also refer the work [9] for a related study in the context of structured deformations. Interestingly, besides similar bending effects as those derived in [3,5,6,9,18,37,38], we observe here a fine interaction between the non-standard forces and the junction of the multi-structure.…”

“…We also refer the work [9] for a related study in the context of structured deformations. Interestingly, besides similar bending effects as those derived in [3,5,6,9,18,37,38], we observe here a fine interaction between the non-standard forces and the junction of the multi-structure. Moreover, we assume that our structure satisfies a deformation condition on a suitable part of its boundary that goes beyond the clamped case, which is the case commonly assumed in the literature.…”

“…As it will become clear later on, the presence of the terms G a and G b , of the same order of f a and f b , respectively, will induce the appearance of bending-torsion moments terms in the limit model. As we mentioned before, this approach was considered before in [3,5,6,9,18,37,38] for thin structures but not multi-structures.…”