Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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

   
 
 

 

An energetic view on the limit analysis of normal bodies


Authors: Massimiliano Lucchesi, Cristina Padovani and Miroslav Šilhavý
Journal: Quart. Appl. Math. 68 (2010), 713-746
MSC (2000): Primary 74H20; Secondary 49Q15, 74H25, 74H35, 74H40
DOI: https://doi.org/10.1090/S0033-569X-2010-01182-2
Published electronically: September 17, 2010
MathSciNet review: 2761241
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This note presents a limit analysis for normal materials based on energy minimization. The class of normal materials includes some of those used to model masonry structures, namely, no-tension materials and materials with bounded compressive strength; it also includes the Hencky plastic materials. Considering loads $ \mathfrak{L}(\lambda)$ that depend affinely on the loading multiplier $ \lambda\in\mathbb{R},$ we examine the infimum $ I_0(\lambda)$ of the potential energy $ I(\boldsymbol{u},\lambda)$ over the set of all admissible displacements $ \boldsymbol{u}.$ Since $ I_0(\lambda)$ is a concave function of $ \lambda$, the set $ \Lambda$ of all $ \lambda$ with $ I_0(\lambda)>-\infty$ is an interval. Each finite endpoint $ \lambda_{\mathrm{c}} \in\mathbb{R}$ of $ \Lambda$ is called a collapse multiplier, and we interpret the loads corresponding to $ \lambda_{\mathrm{c}} $ as the loads at which the collapse of the structure occurs. We show that the standard definition of collapse based on the collapse mechanism does not capture all situations: the collapse mechanism is sufficient but not necessary for the collapse. We then examine the validity of the static and kinematic theorems of limit analysis under the present definition. We show that the static theorem holds unconditionally while the kinematic theorem holds for Hencky plastic materials and materials with bounded compressive strength. For no-tension materials it generally does not hold; a weaker version is given for this class of materials.


References [Enhancements On Off] (What's this?)

  • 1. Adams, R. A., Sobolev Spaces, Academic Press, New York, 1975. MR 0450957 (56:9247)
  • 2. Anzellotti, G., A class of convex non-coercive functionals and masonry-like materials, Ann. Inst. Henri Poincaré 2 (1985), 261-307. MR 801581 (87f:49056)
  • 3. Del Piero, G., Constitutive equations and compatibility of the external loads for linear elastic masonry-like materials, Meccanica 24 (1989), 150-162. MR 1037468 (91i:73004)
  • 4. Del Piero, G., Limit analysis and no-tension materials, Int. J. Plasticity 14 (1998), 259-271.
  • 5. Di Pasquale, S., New trends in the analysis of masonry structures, Meccanica 27 (1992), 173-184.
  • 6. Ekeland, I.; Témam, R., Convex analysis and variational problems, North-Holland, Amsterdam, 1976. MR 0463994 (57:3931b)
  • 7. Fonseca, I.; Leoni, G., Modern Methods in the Calculus of Variations: $ L\sp p$ Spaces, Springer, New York, 2007. MR 2341508 (2008m:49001)
  • 8. Giaquinta, M.; Giusti, G., Researches on the equilibrium of masonry structures, Arch. Rational Mech. Anal. 88 (1985), 359-392. MR 781597 (86h:73012)
  • 9. Gurtin, M. E., An introduction to continuum mechanics, Academic Press, Boston, 1981. MR 636255 (84c:73001)
  • 10. Lucchesi, M.; Padovani, C.; Pagni, A., A numerical method for solving equilibrium problems of masonry-like solids, Meccanica 24 (1994), 175-193.
  • 11. Lucchesi, M.; Padovani, C.; Zani, N., Masonry-like materials with bounded compressive strength, Int J. Solids and Structures 33 (1996), 1961-1994. MR 1384902 (97a:73067)
  • 12. Lucchesi, M.; Padovani, C.; Pasquinelli, G.; Zani, N., Masonry Constructions: Mechanical Models and Numerical Applications, Springer, Berlin, 2008.
  • 13. Lucchesi, M.; Zani, N., Some explicit solutions to equilibrium problem for masonry like bodies, Structural Engineering and Mechanics 16 (2003), 295-316.
  • 14. Nečas, J., Equations aux Derivées Partielles, Presses de l'Universite de Montréal, 1965.
  • 15. Nečas, J.; Hlaváček, I., Mathematical theory of elastic and elastic-plastic bodies: An introduction, Elsevier Scientific Publishing, Amsterdam, 1981. MR 600655 (82h:73002)
  • 16. Padovani, C.; Pasquinelli, G.; Šilhavý, M., Processes in masonry bodies and the dynamical significance of collapse, Math. Mech. Solids 13 (2008), 573-610. MR 2442288
  • 17. Témam, R., Problémes mathématiques en plasticité, Gauthier-Villars, Paris, 1983.
  • 18. Témam, R.; Strang, G., Duality and relaxation in the variational problems of plasticity, J. Mécanique 19 (1980), 493-527. MR 595981 (82i:73017a)

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC (2000): 74H20, 49Q15, 74H25, 74H35, 74H40

Retrieve articles in all journals with MSC (2000): 74H20, 49Q15, 74H25, 74H35, 74H40


Additional Information

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

Cristina Padovani
Affiliation: Istituto di Scienza e Tecnologie dell’Informazione “Alessandro Faedo”, Consiglio Nazionale delle Ricerche, Via G. Moruzzi, 1, San Cataldo 56124 Pisa, Italy
Email: Cristina.Padovani@isti.cnr.it

Miroslav Šilhavý
Affiliation: Institute of Mathematics of the AV ČR, Žitná 25, 115 67 Prague 1, Czech Republic
Email: silhavy@math.cas.cz

DOI: https://doi.org/10.1090/S0033-569X-2010-01182-2
Keywords: Collapse, limit analysis, minimum energy
Received by editor(s): February 20, 2009
Published electronically: September 17, 2010
Additional Notes: The research of M. Šilhavý was supported by the CNR Short-term Mobility project “Giustificazione energetica dell’analisi limite di costruzioni in muratura” and by Dipartimento di Costruzioni, Università di Firenze. The support is gratefully acknowledged
Article copyright: © Copyright 2010 Brown University

American Mathematical Society