One-dimensional consolidation in unsaturated soils under cyclic loading

•Consolidation theory in unsaturated soils under cyclic loading is developed.•A boundary-value problem for free drainage conditions is solved analytically.•Closed-form solutions for excess pore water pressure and total settlement are derived.•A dimensionless excitation frequency for unsaturated soils is defined.•Its expected range of values is far greater than that typically encountered in saturated soils.

The one-dimensional consolidation model of poroelasticity of Lo et al. (2014) for an unsaturated soil under constant loading is generalized to include an arbitrary time-dependent loading. A closed-form solution for the pore water and air pressures along with the total settlement is derived by employing a Fourier series representation in the spatial domain and a Laplace transformation in the time domain. This solution is illustrated for the important example of a fully-permeable soil cylinder with an undrained initial condition acted upon by a periodic stress. Our results indicate that, in terms of a dimensionless time scale, the transient solution decays to zero most slowly in a water-saturated soil, whereas for an unsaturated soil, the time for the transient solution to die out is inversely proportional to the initial water saturation. The generalization presented here shows that the diffusion time scale for pore water in an unsaturated soil is orders of magnitude greater than that in a water-saturated soil, mainly because of the much smaller hydraulic conductivity of the former.