Non-equilibrium strain induces hysteresis and anisotropy in the quasi-static and dynamic elastic behavior of sandstones: Theory and experiments

Materials with grain contacts or partially closed cracks exhibit anomalous elastic behavior: hysteresis in quasi-static experiments and slow dynamics in fast dynamic ones. Albeit the behavior in the two cases (which correspond to very different strain ranges) appears different, it should stem from the same physics and thus could be modelled by a universal equation of state. We propose a modification of the standard acoustoelastic theory, introducing the concept of conditioning induced non-equilibrium strain, which allows to predict the evolution of elastic wave velocity in both quasi-static and dynamic ranges, including the velocity anisotropy induced by external uniaxial loading. %In the limit case of a dynamic experiment, the modulus derived from our approach is formally equivalent to the butterfly modulus, i.e. to the sum of classical nonlinear terms (proportional to strain power) plus a butterfly like hysteresis modulus.