Some estimates for the torsional rigidity of composite rods

Abstract

A well-known problem in elasticity consists in placing two linearly elastic materials (of different shear moduli) in a given plane domain Ω, so as to maximize the torsional rigidity of the resulting rod; moreover, the proportion of these materials is prescribed. Such a problem may not have a classical solution as the optimal design may contain homogenization regions, where the two materials are mixed in a microscopic scale. Then, the optimal torsional rigidity becomes difficult to compute. In this paper we give some different theoretical upper and lower bounds for the optimal torsional rigidity, and we compare them on some significant domains.

On the characterization of some classes of proximally smooth sets

“…Intuitively, the normal distance of a point y ∈ ∂Ω measures how far one can enter into Ω starting at y and moving along the direction of the inner normal before hitting the cut locus; the precise definition is recalled in Section 2. This notion has been considered from different points of views: in [10,36,37] the regularity of the normal distance under different requirements on the boundary has been investigated, along with some applications to Hamilton-Jacobi equations and to PDEs related with granular matter theory; in [13,23,24,25] the normal distance has been exploited in order to study the minimizing properties of the so-called web functions. Let us also mention that, in a previous paper, we proved a roundedness criterion based on the constancy along the boundary of a C 2 domain of a certain function, depending on the normal distance and on the principal curvatures, see [21,Thm.…”

A new symmetry criterion based on the distance function and applications to PDEʼs

“…In view of the convergence property (23), we paraphrase the classical overdetermined Neumann condition for system (7), namely…”

“…Often, symmetry questions arise in problems in which a crucial role is played by the distance function from the boundary of an open bounded domain Ω ⊂ R n , d Ω (x) := dist(x, ∂Ω). This happens for instance when studying PDE's related with mass transportation theory (see [7,13,14]), or minimization problems in the class of so-called web functions, namely functions which only depend on d Ω (see [19,21,22,23,24,25,37]). Symmetry questions in these frameworks, which will be described more precisely below, pushed us to set up a new roundedness criterion, which brings into play the distance function in a more intrinsic way than merely through the boundary curvatures.…”

“…Let us mention that the functions ϕ and λ already appeared in the literature in different contexts: concerning the function ϕ, it is a crucial tool in the proof of the isoperimetric inequality à la Gromov (see e.g. [3, §1.6.8]), and appears also in mathematical models for granular materials [14,16,26]; concerning the function λ, its regularity has been studied in [15,40,42], while some of its applications to variational problems can be found in [18,21,22,23].…”

On reverse Faber-Krahn inequalities

“…Further, using v(δ) and the first eigenfunction of the concentric domain Ω # , they have constructed a test function on Ω whose level sets coincide with interior parallels and the Rayleigh quotient is smaller than ν 1 (Ω # ). In [12], Hersch gave another proof for (2) using a class of test functions known as 'web functions' (the test function that depends only on the distance from Γ 0 ), for more on web functions see [3] and [5] and the references therein.…”