Alternative Analytical Solution of Consolidation at Constant Rate of Deformation

Abstract

In this paper, we present an alternative solution of consolidation at constant rate of deformation (CRD). Governing equation of CRD consolidation is formulated considering excess pore water pressure as primary variable. Formulation of the governing equation is done for soil as linear as well as nonlinear materials. An analytical solution to the governing equation is derived which consists of transient state and steady state components. The solution derived herein yields profile of excess pore water pressure distribution with depth at any time during the test. The present paper analyzes the excess pore water pressure distribution profile during CRD consolidation and factors affecting it in detail. Analysis of the obtained solution indicates the existence of moving boundary condition within the test specimen at early stage of the test where transient state condition prevails. It also indicates that the pore water pressure ratio affects significantly the excess pore water distribution profile, in turn, average value of pore water pressure and average effective stress in the specimen. An expression for consolidation parameters for steady state condition are derived using the present solution. Further, evolution of excess pore water pressure at the base of CRD consolidation test specimen during the test is predicted for normally consolidated as well as for over consolidated soils at different rates of deformation.

Appendix I: Analytical Solution

The governing equation shown in Eq. 11 is non homogeneous parabolic partial differential equation. Therefore, the solution to the governing equation can be split into two parts, namely steady state part and transient state part, as follows

$$\begin{aligned} U(Z, T_{v}) = V(Z) + W(Z, T_{v}) \end{aligned}$$

(31)

where V(Z) and $W(Z, T_{v})$ are the steady state part and transient state part of the solution, respectively. Substitution of U from Eq. 31 into Eq. 11 yields

$$\begin{aligned} \frac{d^{2}V}{dZ^{2}}+\frac{\partial ^{2}W}{\partial Z^{2}} = \frac{\partial W}{\partial T_{v}} - \gamma \end{aligned}$$

(32a)

subject to the initial condition,

$$\begin{aligned} W(Z, 0) = -V(Z) \end{aligned}$$

(32b)

and the boundary conditions

$$\begin{aligned} V(0)+W(0,T_{v}) = 0 \end{aligned}$$

(32c)

$$\begin{aligned} \frac{dV}{dZ}(1,T_{v})+\frac{\partial W}{\partial Z}(1, T_{v}) = 0 \end{aligned}$$

(32d)

Equation 32a can be further decomposed into two equations by writing non homogeneous and homogeneous part separately, which are as follows,

$$\begin{aligned} \frac{d^{2}V}{dZ^{2}} + \gamma = 0 \end{aligned}$$

(33a)

subject to boundary condition

$$\begin{aligned} V(0) = 0 \end{aligned}$$

(33b)

$$\begin{aligned} \frac{dV}{dZ}(1,T_{v}) = 0 \end{aligned}$$

(33c)

and

$$\begin{aligned} \frac{\partial ^{2}W}{\partial Z^{2}} = \frac{\partial W}{\partial t} \end{aligned}$$

(34a)

subject to boundary condition

$$\begin{aligned} W(0,T_{v}) = 0 \end{aligned}$$

(34b)

$$\begin{aligned} \frac{\partial W}{\partial Z}(1,T_{v}) = 0 \end{aligned}$$

(34c)

$$\begin{aligned} W(Z, 0) = -V(Z) \end{aligned}$$

(34d)

By solving Eq. 33a, we get the following expression for V,

$$\begin{aligned} V(Z) = -\frac{\gamma }{2}Z^{2}+\gamma Z \end{aligned}$$

(35)

The solution to W can be obtained by applying the method of separation of variables,

$$\begin{aligned} W(Z,T_{v}) = F(Z)G(T_{v}) \end{aligned}$$

(36)

Substitution of W from Eq. 36 into Eq. 34a yields

$$\begin{aligned} G'+\mu ^2G = 0 \end{aligned}$$

(37)

and

$$\begin{aligned} F''+\mu ^{2}F = 0 \end{aligned}$$

(38a)

$$\begin{aligned} F(0) = 0 \end{aligned}$$

(38b)

$$\begin{aligned} \frac{dF}{dZ}(Z = 1) = 0 \end{aligned}$$

(38c)

Solving the Eq. 38a for F, we get

$$\begin{aligned} F(Z) = A\cos \mu Z+B\sin \mu Z \end{aligned}$$

(39)

where A and B are constant. Application of boundary conditions, shown in Eqs. 38b and 38c, yields

$$\begin{aligned} \mu _{n} = (2n+1)\frac{\pi }{2}, \quad n = 0,1,2,3,....... \end{aligned}$$

(40)

This gives the solution to G as,

$$\begin{aligned} G = G_{n}(T_{v}) = \exp \left[ \left( -\frac{(2n+1)\pi }{2}\right) ^{2}T_{v}\right] \end{aligned}$$

(41)

Thus, the family of solution to W can be written as,

$$\begin{aligned} W_{n}(Z,T_{v}) = F_{n}(Z)G_{n}(T_{v}) = b_{n}\sin \left( \frac{(2n+1)\pi }{2}Z\right) \exp \left[ -\left( \frac{(2n+1)\pi }{2}\right) ^{2}T_{v}\right] \end{aligned}$$

(42)

where $b_{n}$ is a constant. By applying initial condition, $W(Z,0) = -V(Z)$, following expression for V(Z) can be obtained,

$$\begin{aligned} -V(Z) = \sum _{n = 0}^{\infty } b_{n}\sin \left( \frac{(2n+1)\pi }{2}Z\right) \end{aligned}$$

(43)

Series shown in Eq. 42 is an orthogonal family of functions, therefore the coefficient, $b_{n}$ can be determined as

$$\begin{aligned} b_{n} = 2\int _{0}^{1}-V(Z) \sin \left( \frac{(2n+1)\pi }{2}Z\right) dZ \end{aligned}$$

(44)

This leads to,

$$\begin{aligned} b_{n} = -\frac{16\gamma }{(2n+1)^{3}\pi ^{3}} \end{aligned}$$

(45)

Thus, general solution to Eq. 11 is

$$\begin{aligned} U(Z,T_{v})&= \left( -\frac{\gamma }{2}Z^{2}+\gamma Z \right) \nonumber \\&\quad - 16\gamma \sum _{n = 0}^{\infty } \left[ \left( \frac{1}{(2n+1)\pi }\right) ^{3}\sin \left( \frac{(2n+1)\pi }{2}Z\right) \exp \left( -\left( \frac{(2n+1)\pi }{2}\right) ^{2}T_{v}\right) \right] \end{aligned}$$

(46)