Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



A note on equilibrated stress fields for no-tension bodies under gravity

Authors: M. Lucchesi, M. Silhavy and N. Zani
Journal: Quart. Appl. Math. 65 (2007), 605-624
MSC (2000): Primary 74G70; Secondary 49Q15
DOI: https://doi.org/10.1090/S0033-569X-07-01052-5
Published electronically: October 16, 2007
MathSciNet review: 2370353
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Abstract: We study the equilibrium problem for two-dimensional bodies made of a no-tension material under gravity, subjected to distributed or concentrated loads on their boundary. Admissible and equilibrated stress fields are interpreted as tensor-valued measures with distributional divergence represented by a vector-valued measure, as developed by the authors of the present paper. Such stress fields allow us to consider stress concentrations on surfaces and lines. Working in $ \mathbb{R}^n,$ we calculate the weak divergence of a stress field that is asymptotically of the form $ \vert{\text{\bfseries {\textit{x}}}} \vert^{-n+1}{\text{\bfseries {\textit{T}... ...({\text{\bfseries {\textit{x}}}} /\vert{\text{\bfseries {\textit{x}}}} \vert)$ for $ {\text{\bfseries {\textit{x}}}} \to\mathbf{0}$ on a region that is asymptotically a cone with vertex $ \mathbf{0}$. Such stress fields arise as parts of our solutions for two-dimensional panels. Proceeding to problems in dimension two, we first determine an admissible equilibrated solution for a half-plane under gravity that underlies two subsequent solutions for rectangular panels. For the latter we give solutions for three types of loads.

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Additional Information

M. Lucchesi
Affiliation: Dipartimento di Costruzioni, Università di Firenze, Piazza Brunelleschi 6, 50121 Firenze, Italia
Email: massimiliano.lucchesi@unifi.it

M. Silhavy
Affiliation: Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italia
Address at time of publication: Mathematical Institute of the AV ČR, Žitná 25, 115 67 Prague 1, Czech Republic
Email: silhavy@math.cas.cz

N. Zani
Affiliation: Dipartimento di Costruzioni, Università di Firenze, Piazza Brunelleschi 6, 50121 Firenze, Italia
Email: nicola.zani@unifi.it

DOI: https://doi.org/10.1090/S0033-569X-07-01052-5
Keywords: Masonry panels, equilibrium, divergence measures
Received by editor(s): February 3, 2006
Published electronically: October 16, 2007
Additional Notes: The authors thank the referee for helpful comments on the previous version of the paper. The research of M. Šilhavý was supported by a grant of MIUR “Variational theory of microstructure, semiconvexity, and complex materials.” The support is gratefully acknowledged.
Article copyright: © Copyright 2007 Brown University

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