Kinetic energy of highly elastic membranes
Authors:
M. G. Hilgers and A. C. Pipkin
Journal:
Quart. Appl. Math. 55 (1997), 791-800
MSC:
Primary 73K10; Secondary 73C50, 73G05
DOI:
https://doi.org/10.1090/qam/1486549
MathSciNet review:
MR1486549
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Abstract: A theory of elastic sheets with bending stiffness has been proposed in which the strain energy density of the sheet includes a dependence on the second-order derivatives. To study the motion of such sheets, a kinetic energy is required that is accurate to the same order. This is obtained by representing the deformation as a power series in the thickness variable. The lowest-order approximation yields the standard membrane kinetic energy. The next order includes a velocity gradient term. A particularly simple physical interpretation for the additional term is obtained. Furthermore, the matrices involved in this term are shown to possess desirable properties, which can be utilized in future analysis.
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M. M. Balaban, A. E. Green, and P. M. Naghdi, Simple force multipoles in the theory of deformation surfaces, J. Math. Phys. 8, no. 5, 1026–1036 (1967)
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- M. G. Hilgers, Dynamics of elastic sheets with bending stiffness, Quart. J. Mech. Appl. Math. 50 (1997), no. 4, 525–543. MR 1488383, DOI https://doi.org/10.1093/qjmam/50.4.525
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- A. E. H. Love, A treatise on the Mathematical Theory of Elasticity, Dover Publications, New York, 1944. Fourth Ed. MR 0010851
M. G. Hilgers and A. C. Pipkin, Elastic sheets with bending stiffness, Quart. J. Mech. Appl. Math. 45, 57–75 (1992)
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)
M. G. Hilgers and A. C. Pipkin, Energy-minimizing deformations of elastic sheets with bending stiffness, J. Elasticity 31, 125–139 (1993)
M. G. Hilgers and A. C. Pipkin, Bending energy of highly elastic membranes, Quart. Appl. Math. 50, 389–400 (1992)
M. G. Hilgers and A. C. Pipkin, Bending energy of highly elastic membranes II, Quart. Appl. Math. 54, 307–316 (1996)
M. M. Balaban, A. E. Green, and P. M. Naghdi, Simple force multipoles in the theory of deformation surfaces, J. Math. Phys. 8, no. 5, 1026–1036 (1967)
J. L. Ericksen, Plane infinitesimal waves in homogeneous elastic plates, J. Elasticity 3, 161–167 (1973)
M. G. Hilgers, Dynamics of elastic sheets with bending stiffness, Quart J. Mech. Appl. Math. (to appear)
M. G. Hilgers, Plane infinitesimal waves in elastic sheets with bending stiffness, Math. Mech. Solids (to appear)
A. E. H. Love, A treatise on the mathematical theory of elasticity, 4th edition, Dover, New York, 1944
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Article copyright:
© Copyright 1997
American Mathematical Society