Dimension reduction in variational problems, asymptotic development in Γ-convergence and thin structures in elasticity

Abstract

61Anzellotti, G., Baldo, S., and D. Percivale, Dimension reduction in variational problems, asymptotic development in rconvergence and thin structures in elasticity, Asymptotic Analysis 9 (1994) 61-100.We consider families of variational problems :F. over domains [}. whose extension in one or more directions is small compared to the extension in the other directions, and goes to zero while ~ tends to zero. We study then the "variational" convergence of the functionals:F. to a new functional defined on a domain A in a lower dimensional space, where those "dimensions" that were small in [}. disappear. A general framework is presented in the first part of the paper and an application to the elastic rod and the elastic plate is given in the second part.

The Gap Between Linear Elasticity and the Variational Limit of Finite Elasticity in Pure Traction Problems

“…1 prescribed on the boundary of a three-dimensional material body, via a rigorous deduction from the nonlinear elasticity theory. We mention several papers facing issues in elasticity which are connected with the context of our paper: [1], [2], [3], [4], [5] [6], [7], [8], [9], [20] [22], [23], [24], [26], [27], [28], [29]. The present paper focus on the same general question studied in [12], but here we deal with the pure traction problem, i.e.…”

Asymptotic analysis of the Dirichlet fractional Laplacian in domains becoming unbounded

“…The last remark explains why the variant notion of Γ-convergence of [1] fits better to our asymptotic analysis: the restriction of the limit problem to the space H s (ω n ) turns out to be coercive, while its extension by +∞ to the bigger space H s (Ω n+k 1 ) does not.…”

“…The use of convergences based on suitable averages is quite usual in the literature on dimension reduction problems, see e.g. [1,9,12,13]. The next result is a particular case of the more general Theorem 3.1.…”

“…Once again, as well as in the local case (see [5]), a dimension reduction effect is observed: roughly speaking, while the section ω n becomes somehow negligible for ℓ large, the problems (1.2) on Ω n+k ℓ converge to the problem (1.3), which is settled on ω n . To perform our asymptotic analysis we adopt a variant of the notion of De Giorgi's Γ-convergence [11], which has been proposed in [1] to study dimension reduction problems. For general definitions and results about Γ-convergence theory we refer to the monographs [10] by Dal Maso and [2] by Braides.…”

A New Variational Approach to Linearization of Traction Problems in Elasticity

“…On the other hand, if inequality in (1.12) is fulfilled only in a weak sense by the collection of skew symmetric matrices, then still argmin F contains argmin E and min F = min E, but F may have infinitely many minimizing critical points which are not minimizers of E. Therefore, only two cases are allowed: either min F = min E or inf F = −∞; actually the second case arises in presence of compressive surface load. We mention several contributions facing issues in elasticity which are strictly connected with the context of present paper: [1], [2], [3], [4] [5], [6], [7], [8], [19] [21], [22], [24], [25], [26], [27], [28].…”