Physical constitutive equations for nonlinear acoustics of materials with internal contacts

A physical theory for the acoustics of microdamaged materials is presented based on the analysis of internal contact roughness. As a start, we adopt the Whitehouse and Archard contact mechanics yielding a decomposition of continuously roughs surfaces into a set of spherical asperities of different sizes and heights. Coupled with a model for the force-displacement relationship for microcontacts, taking into account adhesion hysteresis, it provides a description for an individual intergranular contact. We upscale this model using continuum mechanics of solids with cracks and derive a set of equations for the evolution of acoustical waves. We use this model to predict results for nonlinear acoustical experiments and derive amplitude dependences of resonant frequency and harmonics, and slow dynamics phenomena.