Asymptotic estimates for the torsional rigidity of rods with thinning cross sections

For every positive regular Borel measure, possibly infinite valued, vanishing on all sets of p-capacity zero, we characterize the compactness of the embedding \({W^{1,p}({\mathbb R}^N)\cap L^p ({\mathbb R}^N,\mu)\hookrightarrow L^q({\mathbb R}^N)}\) in terms of the qualitative behavior of some characteristic PDE. This question is related to the well posedness of a class of geometric inequalities involving the torsional rigidity and the spectrum of the Dirichlet Laplacian introduced by Polya and Szego (Isoperimetric inequalities in mathematical physics, Annals of mathematics studies, 1951). In particular, we prove that finite torsional rigidity of an arbitrary domain (possibly with infinite measure), implies the compactness of the resolvent of the Laplacian.

We study criteria for the finiteness of the constants in integral inequalities generalizing the Poincaré–Friedrichs inequality and Saint-Venant’s variational definition of torsional rigidity. The Rayleigh–Faber–Krahn isoperimetric inequality and the Saint-Venant–Pólya inequality guarantee the existence of finite constants for domains of finite volume. Criteria for the existence of finite constants for unbounded domains of infinite volume were known only in the cases of planar simply connected and spatial convex domains. We generalize and strengthen some known results and extend them to the case when . Here is one of our results. Suppose that and , where is a compact set and is either a planar domain with uniformly perfect boundary or a spatial domain satisfying the exterior sphere condition. Under these assumptions, a finite constant exists if and only if the integral is finite, where is the distance from the point to the boundary of .

The paper deals with the torsional rigidity of homogenous and orthotropic beam with rectangular crosssection. The torsional rigidity of the considered beam is defined in the framework of the Saint-Venant theory ofuniform torsion. Exact and approximate solutions are given to the determination of the torsional rigidity. The shapeof cross section is determined which gives maximum value of the torsional rigidity for a given cross-sectional area.The dependence of torsional rigidity as a function of the ratio shear moduli of beam is also studied.

The Coupled Equation Approach to the Numerical Solution of the Biharmonic Equation by finite Differences. I

There are no proven radiographic guidelines for predicting fracture risk in children and young adults with a benign skeletal lesion. An in vivo diagnostic study was conducted to determine whether a reduction in the load-carrying capacity of a bone measured with quantitative computed tomography was more accurate than current radiographic guidelines for predicting pathologic fracture in patients with a benign skeletal lesion. Eighteen patients who presented with a fracture through a benign skeletal lesion were compared with eighteen patients who had a benign skeletal lesion that had been thought to be at increased risk for fracture on the basis of currently used radiographic criteria but had not fractured over a two-year period. Structural analysis was performed to calculate the resistance of the affected bones to compressive, bending, and torsional loads with use of serial transaxial quantitative computed tomography data obtained along the length of the bone containing the lesion and from homologous cross sections through the contralateral, normal bone. At each cross section, the ratio of the structural rigidity of the affected bone divided by that of the normal, contralateral bone was determined. The cross section with the greatest reduction in compressive, bending, and torsional rigidity was identified as that most likely to fracture. The mean age (and standard deviation) of the thirty-six patients was 12.5 +/- 3.6 years. Twenty lesions were located in the femur; eleven, in the tibia; three, in the humerus; one, in the ulna; and one, in the pelvis. A combination of the minimum bending and torsional rigidities calculated from the tomographic data provided optimal performance in differentiating between the fracture and non-fracture groups (100% sensitivity and 94% specificity). In contrast, plain radiographic criteria demonstrated 28% to 83% sensitivity and 6% to 78% specificity. The combination of bending and torsional rigidity measured noninvasively with quantitative computed tomography was more accurate (97%) for predicting pathologic fracture through benign bone lesions in children than were standard radiographic criteria (42% to 61% accuracy). We believe that this method can provide accurate objective criteria for planning treatment of benign bone lesions and monitoring treatment response.

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. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Bounds for the curvature of the level curve of the torsion function through an arbitrary point in a convex region D, are used to derive improved isoperimetric inequalities for maximum stress, the torsional rigidity and other functionals. These inequalities are exact if D is either a circle or an infinite strip. A similar procedure is used in the clamped membrane problem, and again improved isoperimetric inequalities are derived.

Philosophy & Public AffairsVolume 31, Issue 3 p. 280-316 Poverty, Well-Being, and Gender: What Counts, Who's Heard? SUSAN MOLLER OKIN, SUSAN MOLLER OKIN SUSAN MOLLER OIUN is Marta Sutton Weeks Professor of Political Science at Stanford University. The author of Women in Western Political Thought (Princeton University Press, 1979, second edition 1992), Justice, Gender; and the Family (Basic Books, 1989), and Is Multiculturalism Badfor Women? (Princeton University Press, iggg), she is currently working on gender and development, feminism and multiculturalism, and popular accounts of genetics and evolutionary biology.Search for more papers by this author SUSAN MOLLER OKIN, SUSAN MOLLER OKIN SUSAN MOLLER OIUN is Marta Sutton Weeks Professor of Political Science at Stanford University. The author of Women in Western Political Thought (Princeton University Press, 1979, second edition 1992), Justice, Gender; and the Family (Basic Books, 1989), and Is Multiculturalism Badfor Women? (Princeton University Press, iggg), she is currently working on gender and development, feminism and multiculturalism, and popular accounts of genetics and evolutionary biology.Search for more papers by this author First published: 11 January 2005 https://doi.org/10.1111/j.1088-4963.2003.00280.xCitations: 65 This article discusses recent literature on global poverty and economic development, focussing on Brooke Ackerly, Political Theory and Feminist Social Criticism (Cambridge: Cambridge University Press, 2000); Martha C. Nussbaum, Women and Human Development: The Capabilities Approach (Cambridge: Cambridge University Press, 2000); and Amartya Sen, Development as Freedom (New York: Knopf, 1999). I am very grateful to Pranap Bardhan, Samuel Bowles, Alberto Diaz-Cayeros, Joshua Cohen, Gerald Cohen, Lynn Eden, Jane Mansbridge, John Roemer, Philippe van Parijs, Robert van der Veen, Debra Satz, Hillel Steiner, Myra Strober, Cass Sunstein, and the Editors of Philosophy & Public Affairs for their helpful comments on an earlier draft of this article. Read the full textAboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume31, Issue3July 2003Pages 280-316 RelatedInformation

We develop the theory of torsional rigidity—a quantity routinely considered for Dirichlet Laplacians on bounded planar domains—for Laplacians on metric graphs with at least one Dirichlet vertex. Using a variational characterization that goes back to Pólya, we develop surgical principles that, in turn, allow us to prove isoperimetric-type inequalities: we can hence compare the torsional rigidity of general metric graphs with that of intervals of the same total length. In the spirit of the Kohler-Jobin inequality, we also derive sharp bounds on the ground-state energy of a quantum graph in terms of its torsional rigidity: this is particularly attractive since computing the torsional rigidity reduces to inverting a matrix whose size is the number of the graph’s vertices and is, thus, much easier than computing eigenvalues.

Denote by P(G) the torsional rigidity of a simply connected plane domain G, and by I2(G) the Euclidean moment of inertia of G. In 1995 F.G. Avkhadiev proved that P(G) and I2(G) are comparable quantities in sense of Polya and Szego. Moreover, it was shown that the ratio P(G) /I2(G) belongs to the segment [1, 64]. We investigate the following conjecture P(G) ≥ 3I2(G), where G is a simply connected domain. We prove that the conjecture is true for polygonal domains circumscribed about a circle. For convex domains we show sharp isoperimetric inequalities, which justify the conjecture, in particular, we prove that P(G) > 2I2(G). Some aspects of approximate formulas for P(G) are also discussed.

Certain trigonometric formulae and combinatorial identities are used to derive exact and closed expressions for the stress functions, torsional rigidities and peripheral shearing stresses of certain isotropic cylinders under torsion. The cross sections are bounded by the closed curves \documentclass{article}\pagestyle{empty}\begin{document}$ r = a\left| {\sin \frac{\theta }{n}} \right|^n \left({ - \pi < \theta \mathbin{\lower.3ex\hbox{$\buildrel<\over {\smash{\scriptstyle=}\vphantom{_x}}$}} \pi } \right) $\end{document} where n is a positive integer (n > 1). Complex variable methods are applied to obtain the complex torsion functions, torsional rigidities and shearing stresses when the sections are bounded by the contours \documentclass{article}\pagestyle{empty}\begin{document}$ r = a\cos ^n \frac{\theta }{n}\left({ - \frac{{n\pi }}{2} < \theta \mathbin{\lower.3ex\hbox{$\buildrel<\over {\smash{\scriptstyle=}\vphantom{_x}}$}} \frac{{n\pi }}{2}} \right) $\end{document}, where n is a positive number (0 < n ≦ 2). Numerical results are presented in the form of tables and graphs illustrating the variation of torsional rigidities and the distribution of tangential shearing stresses along the edges of the cross sections. The cardioid and lemniscate of Bernoulli are included as special cases.

Effect of axial restraint in composite bars under nonlinear inelastic uniform torsion by BEM

Postbuckling analysis of beams of arbitrary cross section using BEM

This paper presents the proof of several inequalities by using the technique introduced by Alexandroff, Bakelman, and Pucci to establish their ABP estimate. First, the author gives a new and simple proof of a lower bound of Berestycki, Nirenberg, and Varadhan concerning the principal eigenvalue of an elliptic operator with bounded measurable coefficients. The rest of the paper is a survey on the proofs of several isoperimetric and Sobolev inequalities using the ABP technique. This includes new proofs of the classical isoperimetric inequality, the Wulff isoperimetric inequality, and the Lions-Pacella isoperimetric inequality in convex cones. For this last inequality, the new proof was recently found by the author, Xavier Ros-Oton, and Joaquim Serra in a work where new Sobolev inequalities with weights came up by studying an open question raised by Haim Brezis.

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