Propagation of acoustic pulses in material with hysteretic nonlinearity

Evolution equations for propagation of both unipolar and bipolar acoustic pulses are derived by using hysteretic stress-strain relationships. Hysteretic stress-strain loops that incorporate quadratic nonlinearity are derived by applying the model of Preisach-Mayergoyz space for the characterization of structural elements in a micro-inhomogeneous material. Exact solutions of the nonlinear evolution equations predict broadening in time and reduction in amplitude of a unipolar finite-amplitude acoustic pulse. In contrast with some earlier theoretical predictions, the transformation of the pulse shape predicted here satisfies the law of "momentum" conservation (the "equality of areas" law in nonlinear acoustics of elastic materials). A bipolar pulse of nonzero momentum first transforms during its propagation into a unipolar pulse of the same duration. This process occurs in accordance with the "momentum" conservation law and without formation of shock fronts in the particle velocity profile.