Available online at www.sciencedirect.com International Journal of Non-Linear Mechanics 39 (2004) 1289–1299 Non-linear bell-shaped and kink-shaped strain waves in microstructured solids A.V. Porubov a; b , F. Pastrone c ; * a Ioe Physical Technical Institute of the Russian Academy of Sciences, St.Petersburg 194021, Russia b Institute for High-Performance Computing and Data Bases, St.Petersburg 194291, Russia c Dipartimento di Matematica, Universit a di Torino, Via C. Alberto 10, Torino 10123, Italy Received 1 May 2003; accepted 25 September 2003 Abstract The evolution of nite-amplitude strain waves is studied in a medium with microstructure when dissipation and energy input are taken into account. The governing non-linear equation for longitudinal strain waves is obtained in the one-dimensional case. The propagation and attenuation or amplication of bell-shaped and kink-shaped waves, whose parameters are dened in an explicit form through the parameters of the microstructured medium, are studied. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Non-linear strain waves; Microstructure 1. Introduction The classical theory of elasticity cannot account for eects caused by the microstructure of a mate- rial. The theory of microstructures have been devel- oped recently, see [1–6] and references therein quoted. However, general models usually contain large num- ber of parameters whose values are unlikely to be determined. Simplied, phenomenological modelling allows the lowering of the parameters set but even in this case a few works can be mentioned with the data on the microstructure parameters, see, e.g. [7]. Lack of data prevents use of the microstructure models in practice. Strain waves may be useful in developing a * Corresponding author. Dipartimento di Matematica, Univer- sit a di Torino, Via C. Alberto 10, Torino 10123, Italy. Tel.: +39-11-670-2820; fax: +3911-670-2878. E-mail address: pastrone@dm.unito.it (F. Pastrone). suitable method in order to estimate the microparam- eters since shape, amplitude and velocity of the strain wave can carry information about the microstructure. For this purpose, the most interesting are the waves that keep their shape and velocity on propagation. The celebrated example is the bell-shaped strain solitary wave. Its permanent shape is supported by the balance between non-linearity and dispersion. The summary of recent achievements in bell-shaped solitary waves modelling in an elastic wave-guides may be found in [8,9]. Moreover, strain solitary waves were recently observed in an elastic rod and in a plate [9]. The inuence of dissipation/energy input may be described by various methods, see [10] and references therein. It results in an amplication/attenuation of the wave up to innity/zero unless saturation mecha- nisms come into play. When a dissipative bell-shaped solitary wave of a permanent form propagates, this process may be described in an explicit form 0020-7462/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2003.09.002