Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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



Bending energy of highly elastic membranes. II

Authors: M. G. Hilgers and A. C. Pipkin
Journal: Quart. Appl. Math. 54 (1996), 307-316
MSC: Primary 73K10; Secondary 73G05
DOI: https://doi.org/10.1090/qam/1388018
MathSciNet review: MR1388018
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Abstract: The strain energy per unit area for a deformed sheet of elastic material is estimated by representing the deformation as a power series in the thickness variable. The membrane energy is the lowest-order approximation obtained in this way. Strain-gradient and bending energies appear in the next order of approximation. Neither the membrane energy nor the higher-order approximation satisfy the Legendre-Hadamard material stability condition if the stress is compressive in some direction, so the theories based on either of these approximations can lead to problems with no stable solution. An energy function that does satisfy the material stability conditions is obtained by omitting the strain-gradient term, provided that certain longitudinal moduli are positive. A modified form for which existence of solutions can be guaranteed is proposed.

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

  • [1] M. G. Hilgers and A. C. Pipkin, Elastic sheets with bending stiffness, Quart. J. Mech. Appl. Math. 45, 57-75 (1992)
  • [2] M. G. Hilgers and A. C. Pipkin, Bending energy of highly elastic membranes, Quart. Appl. Math. 50, 389-400 (1992)
  • [3] A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity, 4th edition, Dover, New York, 1944

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DOI: https://doi.org/10.1090/qam/1388018
Article copyright: © Copyright 1996 American Mathematical Society

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