A one-dimensional model of a granular medium with internal stresses is considered that represents a chain consisting of elastically interacting ellipse-shaped particles, which possess two translational and one rotational degrees of freedom. In the long-wavelength approximation, the set of nonlinear equations in partial derivatives has been derived that describes propagation of longitudinal, transverse and rotational waves in such a medium. Dependences of the velocities of elastic waves and the nonlinearity coefficients on the sizes of particles and the parameters of interactions between them have been found in the analytical form. In the field of low frequencies, when the rotational degree of freedom of particles can be neglected, the obtained three-mode set degenerates into the two-mode system. Within the scope of this system, areas of modulation instability (self-modulation) of the shear wave of deformation in the presence of a static longitudinal strain have been found according to the Lighthill criterion, and the various types of wave packets have been studied in the case of modulation instability.
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