A note on equilibrated stress fields for no-tension bodies under gravity
Authors:
M. Lucchesi, M. Šilhavý 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
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
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 $|\mathbfit{x}|^{-n+1} \mathbfit{T}_0(\mathbfit{x} /|\mathbfit{x}|)$ for $\mathbfit{x} \to \mathbf {0}$ 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.
- Gianpietro Del Piero, Constitutive equation and compatibility of the external loads for linear elastic masonry-like materials, Meccanica–J. Italian Assoc. Theoret. Appl. Mech. 24 (1989), no. 3, 150–162 (English, with Italian summary). MR 1037468, DOI https://doi.org/10.1007/BF01559418
8a Del Piero, G. Limit analysis and no-tension materials. Int. J. Plasticity, 14 (1998) 259–271.
5 Di Pasquale, S. Statica dei solidi murari teorie ed esperienze. (1984) Dipartimento di Costruzioni, Università di Firenze, Pubblicazione n. 27.
- Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR 0257325
- 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
HY Heyman, J. The stone skeleton. Internat. J. Solids Structures, 2 (1966) 249–279.
LSZ 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.
LSZ2 Lucchesi, M.; Šilhavý, M.; Zani, N. Stress state for heavy masonry panels. Proceedings of the “Colloquium Lagrangianum,” Venezia, 2004. Springer (to appear).
LSZ33 Lucchesi, M.; Šilhavý, M.; Zani, N. Stress fields for axisymmetric no-tension bodies. Proceedings of XVII$^{th}$ AIMETA Congress, Florence, 2005.
LSZ44 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.
2 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.
1 Lucchesi, M.; Zani, N. Some explicit solutions to equilibrium problem for masonry like bodies. Structural Engineering and Mechanics, 16 (2003) 295–316.
3 Lucchesi, M.; Zani, N. Stati di sforzo per pannelli costituiti da materiale non resistente a trazione. Proceedings of XVI$^{th}$ AIMETA Congress, Ferrara, 2003.
N1 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
- 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, DOI https://doi.org/10.1007/BF00375122
- Miroslav Šilhavý, The mechanics and thermodynamics of continuous media, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1997. MR 1423807
- M. Šilhavý, Divergence measure fields and Cauchy’s stress theorem, Rend. Sem. Mat. Univ. Padova 113 (2005), 15–45. MR 2168979
4 Del Piero, G. Constitutive equations and compatibility of the external loads for linear elastic masonry-like materials. Meccanica, 24 (1989) 150–162.
8a Del Piero, G. Limit analysis and no-tension materials. Int. J. Plasticity, 14 (1998) 259–271.
5 Di Pasquale, S. Statica dei solidi murari teorie ed esperienze. (1984) Dipartimento di Costruzioni, Università di Firenze, Pubblicazione n. 27.
F Federer, H. Geometric measure theory. Springer, New York, 1969.
GB1 Gurtin, M. E. An introduction to continuum mechanics. Academic Press, Boston, 1981.
HY Heyman, J. The stone skeleton. Internat. J. Solids Structures, 2 (1966) 249–279.
LSZ 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.
LSZ2 Lucchesi, M.; Šilhavý, M.; Zani, N. Stress state for heavy masonry panels. Proceedings of the “Colloquium Lagrangianum,” Venezia, 2004. Springer (to appear).
LSZ33 Lucchesi, M.; Šilhavý, M.; Zani, N. Stress fields for axisymmetric no-tension bodies. Proceedings of XVII$^{th}$ AIMETA Congress, Florence, 2005.
LSZ44 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.
2 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.
1 Lucchesi, M.; Zani, N. Some explicit solutions to equilibrium problem for masonry like bodies. Structural Engineering and Mechanics, 16 (2003) 295–316.
3 Lucchesi, M.; Zani, N. Stati di sforzo per pannelli costituiti da materiale non resistente a trazione. Proceedings of XVI$^{th}$ AIMETA Congress, Ferrara, 2003.
N1 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
S91 Šilhavý, M. Cauchy’s stress theorem and tensor fields with divergences in $L^ p$. Arch. Rational Mech. Anal., 116 (1991) 223–255.
book Šilhavý, M. The Mechanics and Thermodynamics of Continuous Media. Springer, Berlin, 1997.
S05 Šilhavý, M. Normal traces of divergence measure vectorfields on fractal boundaries. 2005. (Preprint, Dipartimento di Matematica, University of Pisa, October 2005.)
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC (2000):
74G70,
49Q15
Retrieve articles in all journals
with MSC (2000):
74G70,
49Q15
Additional Information
M. Lucchesi
Affiliation:
Dipartimento di Costruzioni, Università di Firenze, Piazza Brunelleschi 6, 50121 Firenze, Italia
Email:
massimiliano.lucchesi@unifi.it
M. Šilhavý
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
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