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Key wordsCosserat continuum, the micropolar medium, Riemann wave, bending-torsion tensor, longitudinal thermoelastic wave, elastic rotational wave, evolution of a wave.In this paper we consider the nonlinear thermoelastic plane longitudinal waves and plane nonlinear elastic rotational waves in the model of a geometrically nonlinear micropolar continuum (Cosserat medium). The equation for longitudinal thermoelastic waves is studied in two extreme cases: we consider the medium with low and high thermal conductivity. It has been shown that in these cases the equation for longitudinal thermoelastic waves reduces to the known equations describing the classical Riemann waves and Riemann waves with damping. The peculiarities of propagation of these waves have been analyzed. The equation for rotational waves in the absence of thermoelastic effects, is reduced to a nonlinear ordinary differential equation of the second order for stationary waves. It has been demonstrated that the equation has solutions in the supersonic case in the form of periodic waves. The dependences of the amplitude on the wave number and the relative rotational angle of the stationary rotation wave has been obtained.

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