1,720,988 research outputs found
Concentration Phenomena in the Optimal Design of Thin Rods
No full textIn this paper we analyze the concentration phenomena which occur in thin rods, solving the following optimization problem: a given fraction of elastic material must be distributed into a cylindrical design region with infinitesimal cross section in an optimal way, so that it maximizes the resistance to a given external load. For small volume fractions, the optimal configuration of material is described by a measure which concentrates on 2-rectifiable sets. For some choices of the external charging, the concentration phenomena turn out to be related to some new variants of the Cheeger problem of the cross section of the rod. The same study has already been carried out in the particular case of pure torsion regime in [6]. Here we extend those results by enlarging the class of admissible loads
Elements of Advanced Mathematical Analysis for Physics and Engineering
No full textDeep comprehension of applied sciences requires a solid knowledge of Mathematical Analysis. For most of high level scientific research, the good understanding of Functional Analysis and weak solutions to differential equations is essential. This book aims to deal with the main topics that are necessary to achieve such a knowledge
On Blaschke–Santaló diagrams for the torsional rigidity and the first Dirichlet eigenvalue
Full text linkWe study Blaschke–Santaló diagrams associated with the torsional rigidity and the first eigenvalue of the Laplacian with Dirichlet boundary conditions. We work under convexity and volume constraints, in both strong (volume exactly one) and weak (volume at most one) form. We discuss some topological (closedness, simply connectedness) and geometric (shape of the boundaries, slopes near the point corresponding to the ball) properties of these diagrams, also providing a list of conjectures
Energy release rate and stress intensity factors in planar elasticity in presence of smooth cracks
No full textIn this work we first analyze the singular behavior of the displacement u of a linearly elastic body in dimension 2 close to the tip of a smooth crack, extending the well-known results for straight fractures to general smooth ones. As conjectured by Griffith (Phys Eng Sci 221:163–198, 1921), u behaves as the sum of an H2-function and a linear combination of two singular functions whose profile is similar to the square root of the distance from the tip. The coefficients of the linear combination are the so called stress intensity factors. Afterwards, we prove the differentiability of the elastic energy with respect to an infinitesimal fracture elongation and we compute the energy release rate, enlightening its dependence on the stress intensity factors
On a Cheeger–Kohler-Jobin Inequality
Full text linkWe discuss the minimization of a Kohler-Jobin type scale-invariant functional among open, convex, bounded sets, namelymin {T-2(Omega)(1/N+2) h(1)(Omega) : Omega subset of R-N, open, convex, bounded}where T-2(Omega) denotes the torsional rigidity of a set Omega and h(1)(Omega) its Cheeger constant. We prove the existence of an optimal set and we conjecture that the ball is the unique minimizer. We provide a sufficient condition for the validity of the conjecture, and an application of the conjecture to prove a quantitative inequality for the Cheeger constant. We also show lack of existence for the problem above among several other classes of sets. As a side result we discuss the equivalence of the several definitions of Cheeger constants present in the literature and show a quite general class of sets for which those are equivalent
About the Blaschke-Santaló Diagram of Area, Perimeter and Moment of Inertia
No full textWe study the Blaschke-Santalo diagram associated to the area, the perimeter, and the moment of inertia. We work in dimension 2, under two assumptions on the shapes: convexity and the presence of two orthogonal axis of symmetry. We discuss topological and geometrical properties of the diagram. As a by-product we address a conjecture by Polya, in the simplified setting of double symmetry
A variational method for second order shape derivatives
Full text linkWe consider shape functionals obtained as minima on Sobolev spaces of classical integrals having smooth and convex densities, under mixed Dirichlet.Neumann boundary conditions. We propose a new approach for the computation of the second order shape derivative of such functionals, yielding a general existence and representation theorem. In particular, we consider the p-torsional rigidity functional for p ≥ 2
Crack growth by vanishing viscosity in planar elasticity
Full text linkWe show the existence of quasistatic evolutions in a fracture model for brittle materials by a vanishing viscosity approach, in the setting of planar linearized elasticity. Differently from previous works, the crack is not prescribed a priori and is selected in a class of (unions of) regular curves. To prove the result, it is crucial to analyze the properties of the energy release rate showing that it is independent of the crack extension
On two functionals involving the maximum of the torsion function
Full text linkIn this paper we investigate upper and lower bounds of two shape functionals involving the maximum of the torsion function. More precisely, we consider T(Ω)/(M(Ω)|Ω|) and M(Ω)λ1(Ω), where Ω is a bounded open set of Rd with finite Lebesgue measure |Ω|, M(Ω) denotes the maximum of the torsion function, T(Ω) the torsion, and λ1(Ω) the first Dirichlet eigenvalue. Particular attention is devoted to the subclass of convex sets
Minimization of the eigenvalues of the dirichlet-laplacian with a diameter constraint
Full text linkIn this paper we look for the domains minimizing the hth eigenvalue of the Dirichlet-Laplacian λh with a constraint on the diameter. Existence of an optimal domain is easily obtained and is attained at a constant width body. In the case of a simple eigenvalue, we provide nonstandard (i.e., nonlocal) optimality conditions. Then we address the question of whether the disk is an optimal domain in the plane, and we give the precise list of the 17 eigenvalues for which the disk is a local minimum. We conclude by some numerical simulations showing the 20 first optimal domains in the plane
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