Concepts in the mechanics of thin structures
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.
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
S.S. Antman. Nonlinear Problems of Elasticity
. Springer-Verlag, 1995.Google Scholar
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
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
A.-L. Cauchy. De la pression ou tension dans un corps solide. Ex. de Math.
, 2:42–56, 1827.Google Scholar
A.-L. Cauchy. Sur l’équilibre et le mouvement d’une plaque solide. Ex. de Math.
, 8:328–355, 1829.Google Scholar
P.G. Ciarlet. A justification of the von Kármán equations. Arch. Rational Mech. Anal.
, 73:349–389, 1980.zbMATHCrossRefMathSciNetGoogle Scholar
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
P.G. Ciarlet. Plates and Junctions in Elastic Multi-Structures: An Asymptotic Analysis
. Springer-Verlag, 1990.Google Scholar
P.G. Ciarlet. Mathematical Elasticity. Vol. II: Theory of Plates
. North-Holland, 1997.Google Scholar
P.G. Ciarlet. Mathematical Elasticity. Vol. III: Theory of Shells
. North-Holland, 2000.Google Scholar
P.G. Ciarlet. An Introduction to Differential Geometry with Applications to Elasticity
. Springer, 2005.Google Scholar
P.G. Ciarlet and P. Destuynder. A justification of the two-dimensional linear plate model. J. Mécanique
, 18:315–344, 1979.zbMATHMathSciNetGoogle Scholar
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
F. Davì. Sul modello monodimensionale di un corpo elastico allungato. In Atti IX Congr. AIMETA
, pages 137–140, 1989.Google Scholar
A. DiCarlo, P. Podio-Guidugli, and W.O. Williams. Shells with thickness distension. Int. J. Solids Structures
, 38:1201–1225, 2001.zbMATHCrossRefGoogle Scholar
G. Kirchhoff. Uber das gleichgewicht und die bewegung einer elastischen scheibe. J. reine angew. Math.
, 40:51–88, 1850.zbMATHGoogle Scholar
W.T. Koiter and J.G. Simmonds. Foundations of shell theory. Dept. Mech. Engrg., Delft Uni. Tech. Report 473, 1972.Google Scholar
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
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
M. Lembo and P. Podio-Guidugli. Internal constraints, reactive stresses, and the Timoshenko beam theory. J. Elasticity
, 65:131–148, 2001.zbMATHCrossRefMathSciNetGoogle Scholar
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
B. Miara. Justification of the asymptotic analysis of elastic plates, I. The linear case. Asymptotic Analysis
, 9:47–60, 1994a.zbMATHMathSciNetGoogle Scholar
B. Miara. Justification of the asymptotic analysis of elastic plates, II. The non-linear case. Asymptotic Analysis
, 9:119–134, 1994b.zbMATHMathSciNetGoogle Scholar
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
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
P.M. Naghdi. The theory of shells and plates. In Handbuch der Physik VIa/2
, pages 425–640. Springer Verlag, 1972.Google Scholar
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
P. Nardinocchi and P. Podio-Guidugli. Angle plates. J. Elasticity
, 63:19–53, 2001.zbMATHCrossRefMathSciNetGoogle Scholar
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
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
P. Podio-Guidugli. An exact derivation of the thin plate equation. J. Elasticity
, 22:121–133, 1989.zbMATHCrossRefMathSciNetGoogle Scholar
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
P. Podio-Guidugli. Lezioni sulla Teoria Lineare dei Gusci Elastici Sottili
. Masson, 1991.Google Scholar
P. Podio-Guidugli. Scaling and constraints in the mechanics of linearly elastic structures. Unpublished manuscript, 1995.Google Scholar
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
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
P. Podio-Guidugli. A new quasilinear model for plate buckling. J. Elasticity
, 71:157–182, 2003.zbMATHCrossRefMathSciNetGoogle Scholar
S.-D. Poisson. Mémoire sur l’équilibre et le mouvement des corps élastiques. Mém. Acad. Sci. Paris
, 3:357–570, 1828.Google Scholar
C. Truesdell. A First Course in Rational Continuum Mechanics
, Vol. 1
, 2nd Ed.
Academic Press, 1991.Google Scholar