Gibson's equation of soil consolidation in convective coordinate and a method for obtaining solutions

It was believed that the Lagrangian coordinate system is the most appropriate one to be used in finite strain theory of soil consolidation. The disadvantages of using the Eulerian and convective coordinate systems were described in the papers by Gibson et al.. In fact, the convective coordinate of a material point is a function of time and the Lagrangian coordinate, and thus it is an unknown in a consolidation problem. This, as is well-known, will cause mathematical inconvenience. However, in this paper it is shown that, when formulated in a coordinate system which differs from the convective coordinate system only by a function of time, the equation proposed by Gibson et al. becomes linear if the effect of the self-weight of solids and pore fluid are neglected and the function which plays the role of a coefficient of consolidation is chosen properly. Using this fact we can obtain exact solutions to the equation developed by Gibson et al.. In view of the nonlinear character of the finite strain theory of soil consolidation, any exact solutions are important and noteworthy because they frequently reveal interesting phenomena and serve as a benchmark for numerical schemes.