Journal of Elasticity

, Volume 105, Issue 1–2, pp 305–312 | Cite as

A Classical, Nonlinear Thermodynamic Theory of Elastic Shells Based on a Single Constitutive Assumption



By integrating along a thickness-like coordinate, exact two-dimensional equations of motion and a Second Law of Thermodynamics (a Clausius-Duhem Inequality) are derived, without approximation, for a shell-like body. The theory derived is called classical because the only stress measures that appear are resultants and couples. Construction of a Virtual Power Identity automatically produces associated extensional and bending strains that are nonlinear in the deformed position \(\bar{\mathbf {y}}\) of a reference surface and a rotation tensor \(\hbox{\mathversion {bold}$\mathsf{Q}$}\). The only approximations come when the Virtual Power Identity is augmented by heating and an internal energy and the resulting expression taken as the First Law of Thermodynamics (Conservation of Energy) for the shell. A Legendre-Fenchel Transformation is introduced to remove possible ill-conditioning when certain approximations are introduced into the strain-energy density.


Nonlinear theory of elastic shells Thermodymanics 

Mathematics Subject Classification (2000)

74A15 74K25 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Simmonds, J.G.: A simple nonlinear thermodynamic theory of arbitrary elastic beams. J. Elast. 81, 51–62 (2005) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Libai, A., Simmonds, J.G.: The Nonlinear Theory of Elastic Shells, 2nd edn. Cambridge University Press, Cambridge (1998) MATHCrossRefGoogle Scholar
  3. 3.
    Simmonds, J.G.: Rotary inertia in the classical nonlinear theory of shells and the constitutive (non-kinematic) Kirchhoff hypothesis. J. Appl. Mech. 68, 320–323 (2001) ADSMATHCrossRefGoogle Scholar
  4. 4.
    Pietraszkiewicz, W., Szwabowicz, M.L.: Entirely Lagrangian non-linear theory of thin shells. Arch. Mech. 33, 273–288 (1981) MathSciNetMATHGoogle Scholar
  5. 5.
    Pietraszkiewicz, W., Chróścielewski, J., Makowski, J.: On dynamically and kinematically exact theory of shells. In: Shell Structures. Taylor and Francis, London, pp. 163–167 (2005) Google Scholar
  6. 6.
    Eremeyev, V.A., Pietraszkiewicz, W.: Local symmetry group in the general theory of elastic shells. J. Elast. 85, 125–152 (2006) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Coleman, B.D., Noll, W.: The thermodynamics of elastic materials with heat conduction and viscosity. Arch. Ration. Mech. Anal. 51, 1–53 (1963) MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Department of Civil and Environmental EngineeringUniversity of VirginiaCharlottesvilleUSA

Personalised recommendations