Abstract
Elastic crystals with a microstructure include ferroelectric crystals, ferromagnetic crystals and crystals with internal mechanical degrees of freedom. In recent works concerning the discrete or continuum modelling of the behavior of such elastic crystals, we have been able to delineate ā general descriptive framework in which,using the concepts of solitary waves and solitons, the dynamics of simple structures in domains and walls can be accommodated. To that purpose the notion of Bloch and Néel walls in deformable crystals was introduced in all cases. The general nonlinear mathematical problem obtained concerns a nonlinear hyperbolic dispersive system made of a sine-Gordon equation, or a double sine-Gordon equation, for an internal parameter related to the microstructure, which is nonlinearly coupled to one or two wave equations governing elastic displacements. While exact stable nonlinear solutions of the solitary-wave type can be exhibited in a more or less straightforward manner the problem of the interaction of such wave motions (representing then the collision of walls and the coalescence of domains) and that of the transient motion of such waves when acted upon by an external stimulus (then representing the starting motion of walls) can be tackled only by using more sophisticated methods such as singular perturbations, Whitham's averaged Lagrangian method and those methods familiar in soliton theory (Blacklund transformations, inverse scattering method). This is the concern of the present lecture with an emphasis on perturbations.
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Maugin, G. (1986). Solitons and domain structure in elastic crystals with a microstructure. In: Kröner, E., Kirchgässner, K. (eds) Trends in Applications of Pure Mathematics to Mechanics. Lecture Notes in Physics, vol 249. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0016392
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DOI: https://doi.org/10.1007/BFb0016392
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