  A closed-form description is proposed to explain nonlinear and slow dynamics
effects exhibited by sandstone bars in longitudinal resonance experiments.
Along with the fast subsystem of longitudinal nonlinear displacements we
examine the strain-dependent slow subsystem of broken intergrain and
interlamina cohesive bonds. We show that even the simplest but
phenomenologically correct modelling of their mutual feedback elucidates the
main experimental findings typical for forced longitudinal oscillations of
sandstone bars, namely, (i) hysteretic behavior of a resonance curve on both
its up- and down-slopes, (ii) linear softening of resonant frequency with
increase of driving level, and (iii) gradual recovery (increase) of resonant
frequency at low dynamical strains after the sample was conditioned by high
strains. In order to reproduce the highly nonlinear elastic features of
sandstone grained structure a realistic non-perturbative form of strain
potential energy was adopted. In our theory slow dynamics associated with the
experimentally observed memory of peak strain history is attributed to
strain-induced kinetic changes in concentration of ruptured inter-grain and
inter-lamina cohesive bonds causing a net hysteretic effect on the elastic
Young's modulus. Finally, we explain how enhancement of hysteretic phenomena
originates from an increase in equilibrium concentration of ruptured cohesive
bonds that are due to water saturation.
