Estimation of Critical State Parameters from One-dimensional Consolidation and Triaxial Compression Tests

Abstract

Determination of critical state parameters are essential, when critical state soil mechanics based soil models are used for the analysis. Isotropic consolidation test and triaxial tests with accurate measurements of volume change and pore pressure are essential for the direct measurement of critical state parameters. However, in the conventional geotechnical engineering practice, isotropic compression tests are usually not conducted but only one-dimensional consolidation test are performed. The usual parameters reported in most of the soil investigation reports are the shear strength parameters c′ and ϕ′ and the consolidation parameters such as compression index (C _c ) and the recompression index (C _r ) apart from, the overburden pressure σ ₀′ and the corresponding void ratio e₀. The critical state parameters are seldom reported in most of the soil investigation reports. This paper explains a procedure to estimate the critical state parameters, in the absence of isotropic compression test data, based on one-dimensional consolidation tests and triaxial compression tests, using the equations of Modified Cam clay state boundary surface. Comparison of the calculated values from the suggested method with the measured values, on three reconstituted soils and one undisturbed soil, lends support to the validity of the proposed procedure.

Appendix: Prediction of Stress–Strain Response Using Modified Cam Clay Model

As the behavior under triaxial stress state is only considered, the stresses and strains are expressed in terms of the corresponding deviatoric and volumetric components. $dp^{\prime}$ and $dq$ are the increments in mean normal effective stress and deviatoric stress respectively; $d\varepsilon_{v}$ and $d\varepsilon_{d}$ are the corresponding total volumetric and shear strain increments, respectively.

• (1) The final mean effective stress corresponding to the initial specific volume ($v_{0}$) for the case of undrained shearing can be obtained using Eq. (15), for different effective confining pressures. A number of increments ($dp^{\prime}$) between the initial mean effective stress, ($p_{0}^{\prime }$) and final mean effective stress ($p_{f}^{\prime }$) are taken and for each increment the corresponding strain increments will be calculated.

  $$ p_{f}^{\prime } = \exp \left( {\frac{{\Upgamma - v_{0} }}{\lambda }} \right) $$

  (15)

• (2) For each increment of the mean effective stress ($p^{\prime}$), the yield surface is updated in the second step. In the case of Modified Cam Clay model only isotropic hardening is considered and the evolution of yield surface depends on the variation of the hardening parameter ($p_{0}^{\prime }$). The value of hardening parameter for the succeeding step (ith step) can be obtained using the following Eq. (16).

  $$ (p_{c}^{\prime } )_{i} = (p_{c}^{\prime } )_{i - 1} \left( {\frac{{p_{i - 1}^{\prime } }}{{p_{i}^{\prime } }}} \right)^{\kappa /(\lambda - \kappa )} $$

  (16)

• (3) The value of deviatoric stress corresponding to the mean normal stress in each step can be calculated from the equation of yield surface as given below.

  $$ q = Mp^{\prime}\sqrt {\frac{{p_{c}^{\prime } }}{{p^{\prime}}} - 1} $$

  (17)

• (4) Elastic volumetric strain increments is calculated based on the Eq. (18). It should be noted that the value of $d\varepsilon_{v}^{e}$ will be negative as $dp^{\prime}$ will be always negative.

  $$ d\varepsilon_{v}^{e} = \frac{\kappa }{{v_{0} }}\frac{{dp^{\prime}}}{{p^{\prime}}} $$

  (18)

• (5) For the undrained shearing case the total volumetric strain increment will be always zero and hence the plastic volumetric strain increment can be obtained as in Eq. (19)

  $$ d\varepsilon_{v}^{p} = - d\varepsilon_{v}^{e} = \frac{ - \kappa }{{v_{0} }}\frac{{dp^{\prime}}}{{p^{\prime}}} $$

  (19)

• (6) In this step the plastic shear strain increment can be calculated from the equation of flow rule for the Modified Cam Clay model as in Eq. (20)

  $$ d\varepsilon_{d}^{p} = d\varepsilon_{v}^{p} \frac{2\eta }{{M^{2} - \eta^{2} }} $$

  (20)

• (7) The elastic shear strain increment can be calculated based on Eq. (21), where G is the shear modulus.

  $$ d\varepsilon_{d}^{e} = \frac{dq}{3G} $$

  (21)

• (8) The total shear strain increment can be calculated by adding the volumetric and shear strain increments as shown in Eq. (22). The total shear strain values at each step can be obtained by cumulative summation of the shear strain increments. For the undrained case the axial strain will have the same value as the shear strain because the volumetric strain will be always zero.

  $$ d\varepsilon_{d} = d\varepsilon_{d}^{e} + d\varepsilon_{d}^{p} $$

  (22)

• (9) The pore pressure at each increment can be calculated by subtracting the effective mean normal stress from the total mean normal stress as shown in Eq. (23). The total mean normal stress for each increment can be obtained based on the equation of the total stress path as given in Eq. (24).

  $$ du = p - p^{\prime} $$

  (23)
  $$ p = p_{0}^{\prime } + \frac{q}{3} $$

  (24)