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The Quarterly Journal of Mechanics and Applied Mathematics 1981 34(4):415-429; doi:10.1093/qjmam/34.4.415
© 1981 by Oxford University Press
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PLANE TRACTION PROBLEMS FOR INEXTENSIBLE NETWORKS

A. C. PIPKIN

( Division of Applied Mathematics, Brown University, Providence Rhode Island 02912, U.S.A. )

Finite, plane deformations of sheets or nets formed from two intersecting families of fibres are considered. The fibres are treated as inextensible and the network resists shearing. In dead-loading traction boundary-value problems, the vectors representing the directions of the fibres satisfy a pair of coupled integral equations, which are highly nonlinear. Solutions that can represent minimum-energy states are considered. Under conditions on the boundary loads designed to ensure that the deformed network is not folded, and with restriction to sufficiently large loads, it is proved that the integral equations have exactly one solution with no folds. The proof is based on an iterative energy-minimization procedure, which can be used to compute approximate solutions.


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