# Alternative Analytical Solution of Consolidation at Constant Rate of Deformation

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## 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.

## Keywords

Clays Consolidation Compressibility CRD consolidation CRS consolidation## Notation

- CRD
Constant rate of deformation

- IL
Incremental loading

- \(c_{v}\)
Coefficient of consolidation

*e*Void ratio

*F*Dimensionless factor from Eq. 13

- \(H_{0}\)
Initial total depth of the test specimen

*H*Current total depth of the test specimen

*k*Coefficient of permeability

- \(P_{a}\)
Atmospheric pressure

- \(R_{u}\)
Pore water pressure ratio

*t*Time

- \(T_{v}\)
Time factor

*u*Pore water pressure within the specimen

- \(u_{b}\)
Pore water pressure at the base of the test specimen

- \(u_{\text {avg}}\)
Average pore water pressure

- \(U_{\text {lin}} = u/P_{a}\)
Non dimensional pore water pressure factor for linear material

- \(U_{\text {ln}} = \ln (1-u/\sigma )\)
Non dimensional pore water pressure factor for nonlinear material

*U*Non dimensional pore water pressure factor from Eq. 11

*V*Non dimensional term from Eq. 31

*W*Non dimensional term from Eq. 31

*z*Depth

*Z*Non dimensional depth factor

- \(\alpha\)
Ratio of \(u_{\text {avg}}\) and \(u_{b}\)

- \(\gamma\)
A constant term defined in Eq. 11

- \(\gamma _{\text {lin}}\)
A constant term defined in Eq. 9

- \(\gamma _{\text {nl}}\)
A constant term defined in Eq. 10

- \(\gamma _{w}\)
Unit weight of water

- \(\sigma ^{'}\)
Vertical effective stress

- \(\sigma ^{'}_{\text {avg}}\)
Average effective stress

- \((\sigma ^{'}_{\text {lin}})_{\text {avg}}\)
Average effective stress for linear material

- \((\sigma ^{'}_{\text {nl}})_{\text {avg}}\)
Average effective stress for nonlinear material

## Notes

### Acknowledgements

Financial Support from Department of Science and Technology (DST) India through Grants SERB/FTP/ETA-0041/2014 is gratefully acknowledged. Any opinions, findings, and conclusions or recommendations expressed in this material are those of authors and do not necessarily reflect the views of DST.

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