A mathematical treatment of one-dimensional soil consolidation
Terzaghi's conception of the nature of one-dimensional soil consolidation [1] is shown to lead to a non-linear differential equation. A dimensional analysis of this equation and the boundary conditions of the standard consolidation test [2] gives a more general explanation of a well known linear relationship between the total consolidation U(t) after a time t and tl/'\ By linearizing the equation in a general manner, an expression is obtained for U{t) which includes secondary consolidation terms. Two solutions of the linearized equation are obtained; the first for the standard consolidation test and the second for consolidation under a boundary load increasing uniformly with time. Introduction. A medium of clay saturated with water and subjected to a constant pressure, continues to contract in volume over a long period of time. Terzaghi, in his Erdbaumechanik [3], put forward a diffusion theory to explain the time dependent nature of this volume contraction. The following three ideas are the basis of his treatment and will be taken as the starting point of this paper. (1) The medium is imagined as a porous water saturated skeleton of particles of negligible compressibility, so that a change in the volume of any region of the skeleton equals the volume of water displaced through its boundary. (2) Stresses applied to the boundary of the clay produce a hydrostatic head h in the pore water and a stress at each point of the skeleton of such a direction and magnitude as to maintain static equilibrium at all instants of time. (3) It is finally assumed that spatial variations in h cause the pore water to diffuse through the skeleton in accordance with Darcy's law, which equates the quantity of water flowing through any small . plane surface a in unit time to the product of the area of a, the rate of change of h with distance along the normal to a and a constant K called the permeability. By considering the problem of the consolidation of clay in one direction only, the need for a tensor description of the stress distribution is avoided. The one-dimension problem is defined by the assumption that the fluid flow and the motion of the skeleton particles are in one direction only and that all relevant physical properties are constant at any instant of time on any plane with a normal along this direction. The stress function is then the pressure P(z, t) which the skeleton in the neighborhood of the plane z can support at time t and, like K, will depend on the physical state of the solid matter