Dimensional reduction for energies with linear growth involving the bending moment

Abstract

Abstract\ud A Γ -convergence analysis is used to perform a 3D–2D dimension reduction of variational problems with linear growth. The\ud adopted scaling gives rise to a nonlinear membrane model which, because of the presence of higher order external loadings inducing\ud a bending moment, may depend on the average in the transverse direction of a Cosserat vector field, as well as on the deformation\ud of the mid-plane. The assumption of linear growth on the energy leads to an asymptotic analysis in the spaces of measures and of\ud functions with bounded variation

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

“…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.…”

“…One of the main features of our setting is a non-standard scaling of the applied forces, which induce bending-torsion moments in the limit model. 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.…”

A relaxation result in B V × L p for integral functionals depending on chemical composition and elastic strain

“…We study separately the cases 1 < p < ∞, p = ∞ and discuss briefly the case p = 1 in the Appendix. Comparing the results in [5] (where the assumptions allow to invoke De La Vallé-Poussin Criterion) with the true linear growth setting as in [3].…”

“…Arguing as in [3,Lemma 4.7] one can prove that the set function J 1 (v, u, ·), defined as in (A.1) for all the open subsets of , is the trace on these latter sets of a Radon measure absolutely continuous with respect to L N + |Du| + |v|. Then the rest of the proof can be obtained as in [3] by providing a lower bound and an upper bound, that we will sketch in the sequel, just emphasizing the main differences.…”

“…Then the rest of the proof can be obtained as in [3] by providing a lower bound and an upper bound, that we will sketch in the sequel, just emphasizing the main differences.…”