Concepts in the mechanics of thin structures

  • Paolo Podio-Guidugli
Part of the CISM International Centre for Mechanical Sciences book series (CISM, volume 503)


This writing is meant to restitute — a year later, to the best of my recollection — the contents of a series of six lectures I gave in Udine, at the Centro Internazionale di Scienze Meccaniche, within the framework of the Course on Classical and Advanced Theories of Thin Structures: Mechanical and Mathematical Aspects (June 5–9 2006). For this reason, I have made an effort to keep my presentation style informal and colloquial, even when, here and there, I have added some complementing material.

There are three parts. In the Premiss, I try and explain why and how, to my taste, the mechanics of thin structures should be presented; in particular, I introduce the direct and deductive approaches and I collect a few bits of history of the latter, in its two main variants, the asymptotic method and the method of internal constraints. Section 2 is devoted to an exposition of the method of formal scaling, a unified approach to classic rod and plate theories that is an outgrowth of the method of internal constraints. Finally, in Section 3, I discuss how to best approximate the stress field in a linearly elastic structure-like body, by exploiting the freedom in the choice of reactive stress fields that maintain the constraints.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. I. Aganović, J. Tambaca and Z. Tutek. Derivation and justification of the models of rods and plates from linearized three-dimensional micropolar elasticity. J. Elasticity, 84:131–152, 2006.zbMATHCrossRefMathSciNetGoogle Scholar
  2. S.S. Antman. Nonlinear Problems of Elasticity. Springer-Verlag, 1995.Google Scholar
  3. S.S. Antman and R.S. Marlow. Material constraints, Lagrange multipliers, and compatibility. Applications to rod and shell theories. Arch. Rational Mech. Anal., 116:257–299, 1991.zbMATHCrossRefMathSciNetGoogle Scholar
  4. F. Bourquin and P.G. Ciarlet. Modeling and justification of eigenvalue problems for junctions between elastic structures. J. Funct. Anal. 87: 392–427, 1989.zbMATHCrossRefMathSciNetGoogle Scholar
  5. A.-L. Cauchy. De la pression ou tension dans un corps solide. Ex. de Math., 2:42–56, 1827.Google Scholar
  6. A.-L. Cauchy. Sur l’équilibre et le mouvement d’une plaque solide. Ex. de Math., 8:328–355, 1829.Google Scholar
  7. P.G. Ciarlet. A justification of the von Kármán equations. Arch. Rational Mech. Anal., 73:349–389, 1980.zbMATHCrossRefMathSciNetGoogle Scholar
  8. P.G. Ciarlet. Recent progresses in the two-dimensional approximation of three-dimensional plate models in nonlinear elasticity. In Numerical Approximation of Partial Differential Equations (E.L. Ortiz Ed.), pages 3–19. North-Holland, 1987.Google Scholar
  9. P.G. Ciarlet. Plates and Junctions in Elastic Multi-Structures: An Asymptotic Analysis. Springer-Verlag, 1990.Google Scholar
  10. P.G. Ciarlet. Mathematical Elasticity. Vol. II: Theory of Plates. North-Holland, 1997.Google Scholar
  11. P.G. Ciarlet. Mathematical Elasticity. Vol. III: Theory of Shells. North-Holland, 2000.Google Scholar
  12. P.G. Ciarlet. An Introduction to Differential Geometry with Applications to Elasticity. Springer, 2005.Google Scholar
  13. P.G. Ciarlet and P. Destuynder. A justification of the two-dimensional linear plate model. J. Mécanique, 18:315–344, 1979.zbMATHMathSciNetGoogle Scholar
  14. P.G. Ciarlet, H. Le Dret and R. Nzengwa. Junctions between three-dimensional and two-dimensional linearly elastic structures. J. Math. Pures Appl., 68:261–295, 1989.zbMATHMathSciNetGoogle Scholar
  15. F. Davì. Sul modello monodimensionale di un corpo elastico allungato. In Atti IX Congr. AIMETA, pages 137–140, 1989.Google Scholar
  16. A. DiCarlo, P. Podio-Guidugli, and W.O. Williams. Shells with thickness distension. Int. J. Solids Structures, 38:1201–1225, 2001.zbMATHCrossRefGoogle Scholar
  17. G. Kirchhoff. Uber das gleichgewicht und die bewegung einer elastischen scheibe. J. reine angew. Math., 40:51–88, 1850.zbMATHGoogle Scholar
  18. W.T. Koiter and J.G. Simmonds. Foundations of shell theory. Dept. Mech. Engrg., Delft Uni. Tech. Report 473, 1972.Google Scholar
  19. M. Lembo. The membranal and flexural equations of thin elastic plates deduced exactly from the three-dimensional theory. Meccanica, 24:93–97, 1989.zbMATHCrossRefGoogle Scholar
  20. M. Lembo and P. Podio-Guidugli. Plate theory as an exact consequence of three-dimensional elasticity. European J. Mech., A/Solids, 10:1–32, 1991.MathSciNetGoogle Scholar
  21. M. Lembo and P. Podio-Guidugli. Internal constraints, reactive stresses, and the Timoshenko beam theory. J. Elasticity, 65:131–148, 2001.zbMATHCrossRefMathSciNetGoogle Scholar
  22. M. Lembo and P. Podio-Guidugli. How to use reactive stresses to improve plate-theory approximations of the stress field in a linearly elastic platelike body. Int. J. Solids Structures, 44 (5):1337–1369, 2007.zbMATHCrossRefGoogle Scholar
  23. B. Miara. Justification of the asymptotic analysis of elastic plates, I. The linear case. Asymptotic Analysis, 9:47–60, 1994a.zbMATHMathSciNetGoogle Scholar
  24. B. Miara. Justification of the asymptotic analysis of elastic plates, II. The non-linear case. Asymptotic Analysis, 9:119–134, 1994b.zbMATHMathSciNetGoogle Scholar
  25. B. Miara and P. Podio-Guidugli. Une approche formelle unifiée des theories classiques de plaques et poutres. C.R. Acad. Sci. Paris, Ser. I, 343: 675–678, 2006.zbMATHMathSciNetGoogle Scholar
  26. B. Miara and P. Podio-Guidugli. Deduction by scaling: a unified approach to classic plate and rod theories. Asymptotic Analysis, 51 (2):113–131, 2007.zbMATHMathSciNetGoogle Scholar
  27. P.M. Naghdi. The theory of shells and plates. In Handbuch der Physik VIa/2, pages 425–640. Springer Verlag, 1972.Google Scholar
  28. P. Nardinocchi and P. Podio-Guidugli. The equations of Reissner-Mindlin plates obtained by the method of internal constraints. Meccanica, 29: 143–157, 1994.zbMATHCrossRefGoogle Scholar
  29. P. Nardinocchi and P. Podio-Guidugli. Angle plates. J. Elasticity, 63:19–53, 2001.zbMATHCrossRefMathSciNetGoogle Scholar
  30. R. Paroni, P. Podio-Guidugli and G. Tomassetti. The Reissner-Mindlin plate theory via γ-convergence. C.R. Acad. Sci. Paris, Ser. I, 343:437–440, 2006.zbMATHMathSciNetGoogle Scholar
  31. R. Paroni, P. Podio-Guidugli and G. Tomassetti. A justification of the Reissner-Mindlin plate theory through variational convergence. Analysis and Applications, 5:165–182, 2007.zbMATHCrossRefMathSciNetGoogle Scholar
  32. P. Podio-Guidugli. An exact derivation of the thin plate equation. J. Elasticity, 22:121–133, 1989.zbMATHCrossRefMathSciNetGoogle Scholar
  33. P. Podio-Guidugli. Constraint and scaling methods to derive shell theory from three-dimensional elasticity. Riv. Mat. Univ. Parma, 16:73–83, 1990.zbMATHMathSciNetGoogle Scholar
  34. P. Podio-Guidugli. Lezioni sulla Teoria Lineare dei Gusci Elastici Sottili. Masson, 1991.Google Scholar
  35. P. Podio-Guidugli. Scaling and constraints in the mechanics of linearly elastic structures. Unpublished manuscript, 1995.Google Scholar
  36. P. Podio-Guidugli. Old and new invariance methods in continuum mechanics. In Proc. Modern Group Analysis VI (N.H. Ibragimov & F.M. Mahomed Eds.), pages 41–52. New Age Int. Pub., 1997.Google Scholar
  37. P. Podio-Guidugli. Some recent results on Saint-Venant problem. In Atti dei Convegni Lincei N. 140, “Il Problema di de Saint-Venant: Aspetti Teorici e Applicativi”, pages 35–45. Accademia Nazionale dei Lincei, 1998.Google Scholar
  38. P. Podio-Guidugli. A new quasilinear model for plate buckling. J. Elasticity, 71:157–182, 2003.zbMATHCrossRefMathSciNetGoogle Scholar
  39. S.-D. Poisson. Mémoire sur l’équilibre et le mouvement des corps élastiques. Mém. Acad. Sci. Paris, 3:357–570, 1828.Google Scholar
  40. C. Truesdell. A First Course in Rational Continuum Mechanics, Vol. 1, 2nd Ed. Academic Press, 1991.Google Scholar

Copyright information

© CISM, Udine 2008

Authors and Affiliations

  • Paolo Podio-Guidugli
    • 1
  1. 1.Dipartimento di Ingegneria CivileUniversità di Roma TorVergataRomaItaly

Personalised recommendations