Nonlinear dispersion properties of high-frequency waves in the gradient theory of elasticity

The dispersion law ceases to be linear already at ultrasonic frequencies of elastic vibrations of particles as mechanical perturbation waves propagate through the medium. A variant of the continuum model of an elastic medium is proposed which is based on the assumption of pair and triplet potential interaction between infinitely small particles; this allows one to represent the dispersion law with any required accuracy. The corresponding wave equation, which is still linear, can have an arbitrarily large order of partial derivatives with respect to the coordinates. It is suggested that the results of comparing the representations of the dispersion law from the elasticity and solid-state physics viewpoints should be used to determine nonclassical characteristics of the elastic state of the medium. The theoretical conclusions are illustrated with calculations performed for plane waves propagating through aluminum.