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Nonlinear dynamics of oriented elastic solids

II. Propagation of solitons

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Abstract

Based on a continuum model for oriented elastic solids the set of nonlinear dispersive equations derived in Part I of this work allows one to investigate the nonlinear wave propagation of the soliton type. The equations govern the coupled rotation-displacement motions in connection with the linear elastic behavior and large-amplitude rotations of the director field. In the one-dimensional version of the equations and for two simple configurations an exhaustive study of solitons is presented. We show that the transverse and/or longitudinal elastic displacements are coupled to the rotational motion so that solitons, jointly in the rotation of the director and the elastic deformations, are exhibited. These solitons are solutions of a system of linear wave equations for the elastic displacements which are nonlinearly coupled to a sine-Gordon equation for the rotational motion. For each configuration, the solutions are numerically illustrated and the energy of the solitions is calculated. Finally, some applications of the continuum model and the related nonlinear dynamics to several physical situations are given and additional more complex problems are also evoked by way of conclusion.

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  1. Laboratoire de Modélisation en Mécanique, associé au C.N.R.S., Université Pierre et Marie Curie, Tour 66-4 place Jussieu, F-75252, Paris Cedex 05, France

    J. Pouget & G. A. Maugin

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  1. J. Pouget
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Pouget, J., Maugin, G.A. Nonlinear dynamics of oriented elastic solids. J Elasticity 22, 157–183 (1989). https://doi.org/10.1007/BF00041109

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