The propagation of longitudinal and shear waves in a damaged medium wi
th microstructure is investigated. In order to describe the behavior o
f an elastic medium with microfractures, the internal energy is specif
ied in the form of a Myasnikov-Lyakhovskii potential. The presence of
microstructure is taken into account with the aid of the Leroux hypoth
eses. The resulting mathematical model is strongly nonlinear and can d
escribe stationary waves. Dependencies relating the primary wave param
eters to the extent of material fracturing are determined. It is shown
that these dependencies may be used as a basis for developing an acou
stic technique to test the fracturing of materials.