Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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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
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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  • 1. Gianpietro Del Piero, Constitutive equation and compatibility of the external loads for linear elastic masonry-like materials, Meccanica 24 (1989), no. 3, 150–162 (English, with Italian summary). MR 1037468, 10.1007/BF01559418
  • 2. Del Piero, G. Limit analysis and no-tension materials. Int. J. Plasticity, 14 (1998) 259-271.
  • 3. Di Pasquale, S. Statica dei solidi murari teorie ed esperienze. (1984) Dipartimento di Costruzioni, Università di Firenze, Pubblicazione n. 27.
  • 4. Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR 0257325
  • 5. Morton E. Gurtin, An introduction to continuum mechanics, Mathematics in Science and Engineering, vol. 158, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 636255
  • 6. Heyman, J. The stone skeleton. Internat. J. Solids Structures, 2 (1966) 249-279.
  • 7. Lucchesi, M.; Šilhavý, M.; Zani, N. Singular equilibrated stress fields for no-tension panels. Lecture notes in applied and computational mechanics, vol. 23. Springer, 2005, 255-265.
  • 8. Lucchesi, M.; Šilhavý, M.; Zani, N. Stress state for heavy masonry panels. Proceedings of the ``Colloquium Lagrangianum,'' Venezia, 2004. Springer (to appear).
  • 9. Lucchesi, M.; Šilhavý, M.; Zani, N. Stress fields for axisymmetric no-tension bodies. Proceedings of XVII$ ^{th}$ AIMETA Congress, Florence, 2005.
  • 10. Lucchesi, M.; Šilhavý, M.; Zani, N. A new class of equilibrated stress fields for no-tension bodies. Journal of Mechanics of Materials and Structures, 1 (2006) 503-539.
  • 11. Lucchesi, M.; Zani, N. On the collapse of masonry panel. Proceedings of VII$ ^{th}$ International Seminar on Structural Masonry for Developing Countries, Belo Horizonte, Brazil, 2002.
  • 12. Lucchesi, M.; Zani, N. Some explicit solutions to equilibrium problem for masonry like bodies. Structural Engineering and Mechanics, 16 (2003) 295-316.
  • 13. Lucchesi, M.; Zani, N. Stati di sforzo per pannelli costituiti da materiale non resistente a trazione. Proceedings of XVI$ ^{th}$ AIMETA Congress, Ferrara, 2003.
  • 14. Polito, L. Equivalenti lineari di equazioni non lineari alle derivate parziali e loro soluzione con dati iniziali. 2001. ADIA 2001-6, Dipartimento di Ingegneria Aerospaziale, Università di Pisa
  • 15. M. Šilhavý, Cauchy’s stress theorem and tensor fields with divergences in 𝐿^{𝑝}, Arch. Rational Mech. Anal. 116 (1991), no. 3, 223–255. MR 1132761, 10.1007/BF00375122
  • 16. Miroslav Šilhavý, The mechanics and thermodynamics of continuous media, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1997. MR 1423807
  • 17. M. Šilhavý, Divergence measure fields and Cauchy’s stress theorem, Rend. Sem. Mat. Univ. Padova 113 (2005), 15–45. MR 2168979

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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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