Bending energy of highly elastic membranes

Hilgers, M. G.; Pipkin, A. C.

doi:10.1090/qam/1162282

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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