A two-phase elastic column and piston model has been developed to model the effects of vibration upon a bed of particulate solids. Computer simulations have been used to study internal and boundary behaviour for small elastic deformations, and predictions have been compared to experimental data where possible. The model predicts the passage of stress waves through the material. Fluid bulk modulus affects interstitial pressure. Relatively incompressible liquids influence the strain distribution, whereas gases have little effect upon internal strain. Fluid permeability plays a role in dissipating pressure. An impermeable radial boundary, such as an enclosing membrane, gives different pressure distributions from a free-standing sample. These effects have been quantified by dimensionless groups, which have practical significance, and the computer predictions are supported by experimental data. Energy dissipation has been modelled by simple viscous damping mechanisms, and hysteresis loops have been produced. Simple elastic systems need not give linear stress-strain relationships due to inertial effects. The rate of internal energy dissipation is enhanced when the variation in the internal properties of the material is increased.