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
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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.

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

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