Waves as a trigger for Multiscale, Multi-Physics Instabilities

Abstract

We propose a non-local, mesoscopic physics approach for coupling multiphysics processes across scales in porous or multiphase media. Mesoscopic physics is here used to describe the probabilistic interactions of discrete phenomena at intermediate scales when the overall generalized thermodynamic stress field is incompatible with accelerations from local thermodynamic feedbacks. The mechanism of instabilities relies on the cross-scale interaction of macro-scale generalized Thermo-Hydro-Mechano-Chemical (THMC) thermodynamic forces that can trigger meso-scale generalized thermodynamic fluxes of another kind. A particular application of self-diffusion dominated meso-scale THMC instabilities is related to the explicit standing wave solutions of the familiar acoustic tensor localization criterion in plasticity theory. These zero-wave speed acceleration waves solutions are here interpreted as solitary waves also known as solitons. Contrary to the classical standing wave theory we admit, however, non-zero travelling wave speed solutions which are related to the cross-diffusion coefficients between different macro- and meso-scale thermodynamic forces and fluxes. These cross-diffusion terms in the 4 x 4 THMC diffusion matrix are shown to lead to multiple diffusional P- and S-wave equations as THMC coupled, time-resolved dynamic solutions of the equation of motion. Uncertainties in the location of meso-scale material instabilities are captured by a wave-scale correlation of probability amplitudes. Cross-diffusional waves have unusual dispersion patterns and, although they assume a quasi-solitary state, they generally do not behave like solitons but show complex interactions when they collide. Their characteristic wavenumber and constant speed define mesoscopic internal material time-space relations entirely controlled by the coefficients of the coupled THMC reaction-cross-diffusion matrix. Collision of cross-diffusion waves can lead to an energy cascade connecting large and small-scales and result in internal material damage. As an extreme endmember of synchronization across all-scales collisions can cause material collapse interpreted mathematically as solid-state turbulence.