Propagation and interactions of nonlinear shear waves in a discrete lattice

Abstract

We study nonlinear shear waves in a discrete lattice using as a one-dimensional model an internally stressed mass-spring chain of which each particle represents a crystal plane. The dynamics of the lattice proves to be dominated by geometrical nonlinearity along with dispersion, which allows for the existence of solitary waves even in the absence of nonlinearity in the constitutive relation of the medium. We present analytic expressions for kink and envelope solitary waves, the latter being polarized either circularly or linearly, and we discuss the effects of mutual interactions. The collisions of envelope waves are treated analytically in a semidiscrete approximation using a perturbative method. The results, in agreement with numerical simulations, show that circularly polarized envelopes behave like solitons in the lattice while linearly polarized ones undergo variations of polarization and should be considered as bound states of two circularly polarized waves.