Advertisement

Mathematics of the Masonry–Like model and Limit Analysis

  • M. Šilhavý
Part of the CISM International Centre for Mechanical Sciences book series (CISM, volume 551)

Abstract

These notes present a brief introduction to the mathematics of equilibrium of no–tension (masonry–like) materials. We first review the constitutive equations using the idea that the stress of the no–tension material must be always negative semidefinite. The strain tensor is naturally split into the sum of the elastic strain and fracture strain. The stress depends linearly on the elastic strain via the fourth–order tensor of elasticities. Then we consider a body made of a no–tension material, introduce the loads and the total energy of the deformation with is the sum of the internal energy and the energy of the loads. Then we examine the question whether the total energy is bounded from below. That brings us to the important notion of the strong compatibility of loads. The loads are strongly compatible if they can be equilibrated (in the sense of the principle of virtual work) by a square integrable negative semidefinite stress field. The total energy is bounded from below if and only if the loads are strongly compatible. The notion of strong compatibility of loads is central in the limit analysis and in a strengthened form in the theory of existence of equilibrium states. Roughly speaking, if the loads are strongly compatible, then the body is safe, while otherwise strongly incompatible loads lead to the collapse of the body. To determine whether the loads are strongly compatible, it is not necessary to solve the full displacement problem, it suffices to find the negative semidefinite square integrable stress field, which is independent of the constitutive theory.

Keywords

Limit Analysis Lipschitz Domain Vector Space Versus Order Tensor Masonry Structure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. [Adams & Fournier 2003]
    Adams, R. A.; Fournier, J. J. F. 2003. Sobolev spaces (Second edition). New York: Academic Press.Google Scholar
  2. [Ambrosio et al. 1997]
    Ambrosio, L.; Coscia, A.; Dal Maso, G. 1997. Fine properties of functions with bounded deformation. Arch. Rational Mech. Anal., Vol. 139: 201–238.Google Scholar
  3. [Ambrosio et al. 2000]
    Ambrosio, L.; Fusco, N.; Pallara, D. 2000. Functions of bounded variation and free discontinuity problems. Oxford: Clarendon Press.Google Scholar
  4. [Angelillo 1993]
    Angelillo, M. 1993. Constitutive relations for no-tension materials. M eccanica (Milano), Vol. 28: 195–202.Google Scholar
  5. [Anzellotti 1985]
    Anzellotti, G. 1985. A class of convex non-corecive functionals and masonry-like materials. Ann. Inst. Henri Poincaré, Vol. 2: 261–307.Google Scholar
  6. [Barsotti & Vannucci 2013]
    Barsotti, R.; Vannucci, P. 2013. Wrinkling of Orthotropic Membranes: An Analysis by the Polar Method. J. Elasticity. To appear. DOI 10.1007/s10659-012-9408-zGoogle Scholar
  7. [Del Piero 1989]
    Del Piero, G. 1989. Constitutive equations and compatibility of the external loads for linear elastic masonry-like materials. M eccanica, Vol. 24: 150–162.Google Scholar
  8. [Del Piero 1998]
    Del Piero, G. 1998. Limit analysis and no-tension materials. I nt. J. Plasticity, Vol. 14: 259–271.Google Scholar
  9. [Di Pasquale 1984]
    Di Pasquale, S. 1984. Statica dei solidi murari teorie ed esperienze. Dipartimento di Costruzioni, Pubblicazione n. 27Google Scholar
  10. [Ekeland & Temam 1999]
    Ekeland, I.; Temam, R. 1999. Convex analysis and variational problems. Philadelphia: SIAM.Google Scholar
  11. [Epstein 2002]
    Epstein, M. 2002. From Saturated Elasticity to Finite Evolution, Plasticity and Growth. M athematics and mechanics of solids, Vol. 7: 255–283.Google Scholar
  12. [Fonseca & Leoni 2007]
    Fonseca, I.; Leoni, G. 2007. M odern Methods in the Calculus of Variations: Lp Spaces. New York: Springer.Google Scholar
  13. [Giaquinta & Giusti 1985]
    Giaquinta, M.; Giusti, E. 1985. Researches on the equilibrium of masonry structures. Arch. Rational Mech. Anal., Vol. 88: 359–392.Google Scholar
  14. [Gurtin 1981]
    Gurtin, M. E. 1981. An introduction to continuum mechanics. Boston: Academic Press.Google Scholar
  15. [Lucchesi et al. 1994]
    Lucchesi, M.; Padovani, C.; Pagni, A. 1994. A numerical method for solving equilibrium problems of masonry–like solids. M eccanica, Vol. 24: 175–193.Google Scholar
  16. [Lucchesi et al. 1996]
    Lucchesi, M.; Padovani, C.; Zani, N. 1996. Masonry–like materials with bounded compressive strength. I nt J. Solids and Structures, Vol. 33: 1961–1994.Google Scholar
  17. [Lucchesi et al. 2008a]
    Lucchesi, M.; Padovani, C.; Pasquinelli, G.; Zani, N. 2008. M asonry Constructions: Mechanical Models and Numerical Applications. Berlin: Springer.Google Scholar
  18. [Lucchesi et al. 2010]
    Lucchesi, M.; Padovani, C.; ˇSilhav´y, M. 2010. An energetic view on the limit analysis of normal bodies. Quart. Appl. Math., Vol. 68: 713–746.Google Scholar
  19. [Lucchesi et al. 2006]
    Lucchesi, M.; ˇSilhav´y, M.; Zani, N. 2006. A new class of equilibrated stress fields for no–tension bodies. Journal of Mechanics of Materials and Structures, Vol. 1: 503–539.Google Scholar
  20. [Lucchesi et al. 2008b]
    Lucchesi, M.; ˇSilhav´y, M.; Zani, N. 2008. Integration of measures and admissible stress fields for masonry bodies. Journal of Mechanics of Materials and Structures, Vol. 3: 675–696.Google Scholar
  21. [Lucchesi et al. 2009]
    Lucchesi, M.; ˇSilhav´y, M.; Zani, N. 2009. Equilibrated divergence measure stress tensor fields for heavy masonry bodies. European Journal of Mechanics A/Solids, Vol. 28: 223–232.Google Scholar
  22. [Lucchesi et al. 2011]
    Lucchesi, M.; ˇSilhav´y, M.; Zani, N. 2011. Integration of parametric measures and the statics of masonry panels. Annals of Solid and Structural Mechanics, Vol. 2: 33–44.Google Scholar
  23. [Lucchesi et al. 2012]
    Lucchesi, M.; ˇSilhav´y, M.; Zani, N. 2012. On the choice of functions spaces in the limit analysis for masonry bodies. Journal of Mechanics of Materials and Structures, Vol. 7: 795–836.Google Scholar
  24. [Müller 1999]
    Müller, S. 1999. Variational Models for Microstructure and Phase Transitions. Pp. 85–210 in S. Hildebrandt, M. Struwe (ed.), In Calculus of variations and geometric evolution problems (Cetraro, 1996) Lecture notes in Math. 1713 Springer, Berlin.Google Scholar
  25. [Padovani et al. 2008]
    Padovani, C.; Pasquinelli, G.; ˇSilhav´y, M. 2008. Processes in masonry bodies and the dynamical significance of collapse. M ath. Mech. Solids, Vol. 13: 573–610.Google Scholar
  26. [Rockafellar 1970]
    Rockafellar, R. T. 1970. Convex analysis. Princeton: Princeton University Press.Google Scholar
  27. [Rudin 1970]
    Rudin, W. 1970. Real and complex analysis. New York: McGraw-Hill.Google Scholar
  28. [Šilhavý 2008]
    Šilhavý, M. 2008. Cauchy’s stress theorem for stresses represented by measures. Continuum Mechanics and Thermodynamics, Vol. 20: 75–96.Google Scholar
  29. [Šilhavý 2013]
    Šilhavý, M. 2013. Collapse mechanisms and the existence of equilibrium solutions for masonry bodies. To appear in Mathematics and Mechanics of Solids.Google Scholar
  30. [Temam 1983]
    Temam, R. 1983. Problémes mathématiques en plasticité. Paris: Gauthier–Villars.Google Scholar
  31. [Temam & Strang 1980]
    Temam, R.; Strang, G. 1980. Functions of Bounded Deformation. Arch. Rational Mech. Anal., Vol. 75: 7–21.Google Scholar

Copyright information

© CISM, Udine 2014

Authors and Affiliations

  • M. Šilhavý
    • 1
  1. 1.Institute of Mathematics of the AV ČRPrague 1Czech Republic

Personalised recommendations