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The Conservation Law $\partial_yu + \partial_x\sqrt{1 - u^2}=0$ and of Fibre-Reinforced Materials

Rustum Choksi
SIAM Journal on Applied Mathematics
Vol. 56, No. 6 (Dec., 1996), pp. 1539-1560
Stable URL: http://www.jstor.org/stable/2102588
Page Count: 22
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The Conservation Law $\partial_yu + \partial_x\sqrt{1 - u^2}=0$ and of Fibre-Reinforced Materials
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Abstract

The conservation law $\partial_yu + \partial_x \sqrt{1 - u^2} = 0$ is found to govern planar deformations of incompressible materials containing a continuous linear distribution of inextensible fibres. Kinematically feasible deformations are discussed, with emphasis on admissibility and the resolution of nonuniqueness. Many of the aspects of hyperbolic conservation laws have direct consequences in the kinematics of these materials, thus providing an illustrative guide to the theory. Alternatively, the study of this conservation law is geometrically motivated by questions on the structure of the set of points above a continuous function curve whose minimum distance to the curve is achieved in several places.

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