Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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



Bending energy of highly elastic membranes

Authors: M. G. Hilgers and A. C. Pipkin
Journal: Quart. Appl. Math. 50 (1992), 389-400
MSC: Primary 73K10; Secondary 73G05
DOI: https://doi.org/10.1090/qam/1162282
MathSciNet review: MR1162282
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For a membrane composed of elastic material with strain energy $ W$ per unit initial volume, approximations to the energy per unit initial area are obtained by integrating $ W$ through the thickness. The usual stretching energy $ M\left( {{r_{,a}}} \right)$ is modified by including a bending energy term $ \alpha B\left( {{r_{,a}}, {r_{,ab}}} \right)$ that is quadratic in the second derivatives $ {r_{,ab}}$. If $ W$ is the strain energy function for a stable material, $ M$ need not satisfy the Legendre-Hadamard material stability conditions, but the modified energy $ M + \alpha B$ does satisfy these conditions. The special form that $ B$ takes when the membrane is isotropic is given.

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

  • [1] A. E. Green and W. Zerna, Theoretical Elasticity, Oxford Univ. Press, London and New York, 1954 MR 0064598
  • [2] A. C. Pipkin, The relaxed energy density for isotropic elastic membranes, IMA J. Appl. Math. 36, 85-99 (1986) MR 984460
  • [3] M. G. Hilgers and A. C. Pipkin, Elastic sheets with bending stiffness, QJMAM, forthcoming. MR 1154763
  • [4] J. M. Ball, J. C. Currie, and P. J. Olver, Null Lagrangians, weak continuity, and variational problems of arbitrary order, J. Funct. Anal. 41, 135-174 (1981) MR 615159
  • [5] G. Kirchhoff, Uber das Gleichgewicht und die Bewegung einer elastischen Scheibe, J. Reine Angew. Math. 40, 51-88 (1850) MR 1578677
  • [6] L. M. Graves, The Weierstrass condition for multiple integral variation problems, Duke Math. J. 5, 656-660 (1939) MR 0000099
  • [7] W.-B. Wang and A. C. Pipkin, Inextensible networks with bending stiffness, Quart. J. Mech. Appl. Math. 39, 343-359 (1986)
  • [8] A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity, 4th ed., Dover, New York, 1944 MR 0010851
  • [9] M. G. Hilgers and A. C. Pipkin, Energy-minimizing deformations of elastic sheets with bending stiffness, J. Elast., forthcoming. MR 1220979
  • [10] L. van Hove, Sur l'extension de la condition de Legendre du calcul des variations aux integrales multiples à plusieurs fonctions inconnues, Nederl. Akad. Wetensch. Indag. Math. 50, 18-23 (1947) MR 0020223
  • [11] R. W. Odgen, Non-linear Elastic Deformations, Ellis Horwood, Chichester, 1984
  • [12] J. K. Knowles and Eli Sternberg, On the failure of ellipticity of the equations for finite elastostatic plane strain, Arch. Rational Mech. Anal. 63, 321-336 (1977) MR 0431861

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 73K10, 73G05

Retrieve articles in all journals with MSC: 73K10, 73G05

Additional Information

DOI: https://doi.org/10.1090/qam/1162282
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society