A nonlinear granular medium with particle rotation: A one-dimensional model

Abstract

Equations of nonlinear acoustics are derived from the micromechanical representation of a granular medium as a system of elastically interacting particles possessing translational and rotational degrees of freedom. The structure of the equations is invariant with respect to the shape and size of the particles. The changes in the latter affect only the coefficients in the equations. The inclusion of microrotations and moment interactions of particles leads to the formation of a new type of waves in the medium—microrotational waves. Their dispersion properties are similar to those of spin waves propagating in a magnetoelastic medium. In the low-frequency approximation, the microrotational waves disappear, and the equation describing the transverse waves acquires a term with quadratic nonlinearity. The latter provides an explanation for the generation of the second shear harmonic that is observed in real solids contrary to the predictions of the nonlinear theory of elasticity, which prohibits such phenomena.