Abstract
In nonclassical mechanics it is natural to deal with the problem of the propagation of nonlinear waves in solids with different internal structural scales (Engelbrecht, J., Pastrone, F., Braun, M., and Berezovski, A. Hierarchy of waves in nonclassical materials. In Universality of Nonclassical Nonlinearity (Delsanto, P. P., ed.). Springer, 2007, 29–48). The choice of suitable microstrain functions of the microdisplacements, of their time derivatives as strain velocities, allows us to obtain the field equations via a variational principle (see Pastrone, F., Cermelli, P., and Porubov, A. V. Nonlinear waves in 1-D solids with microsctructure. Mater. Phys. Mech., 2004, 7, 9–16; Casasso, A. and Pastrone, F. Wave propagation in solids with vectorial microstructures. Wave Motion, doi: 10.1016/j.wavemoti.2009.12.006; Porubov, A. V., Pastrone, F., and Maugin, G. A. Selection of two-dimensional nonlinear strain waves in micro-structured media. C. R. Acad. Sci. Paris, 2004, Ser. I 337, 513–518) in three different cases: one-dimensional solids with two different microscales, two-dimensional solids with microstructures, and plane granular media. In all cases the hierarchical structure of equations due to the scales in materials is evident.
Highlights
The problem of complexity revealed its importance in continuum mechanics and, in particular, in nonlinearly elastic structures
As extensively described in [5], the cornerstones for describing dynamic processes of microstructured materials at intensive and high-speed deformations are the following: (i) nonclassical theory of continua able to account for internal scales; (ii) hierarchical structure of waves due to the scales in materials; (iii) nonlinearities caused by large deformation and character of stress–strain relations
The field equations obtained in the previous sections can be used to study the propagation of nonlinear waves, as done in the papers [1,4,5,9,10,11]
Summary
The problem of complexity revealed its importance in continuum mechanics and, in particular, in nonlinearly elastic structures. In many cases the mathematical theory does not provide suggestions on how to perform experiments in order to exhibit the existence, consistency, and influence of the microstructure over the macrobody. In this sense we can talk of nonclassical, nonlinear elasticity, in the sense that some of the pillars of the classical exact theory of elasticity are relaxed. (i) nonclassical theory of continua able to account for internal scales; (ii) hierarchical structure of waves due to the scales in materials; (iii) nonlinearities caused by large deformation and character of stress–strain relations. In terms of wave characteristics, there are many physical effects due to the microstructure and its possible structural changes in the wave field.
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