Implementation of a Single Hardening Constitutive Model for 3D Analysis of Earth Dams

Abstract

In the practice of geotechnical engineering, reliable information about soil properties are often available in terms of strength parameters but data on the deformability characteristics of the materials are known to a lesser extent. For situations where the analysis of soil resistance is a condition of the project, the traditional Mohr-Coulomb model is usually applied, but there are several other situations where the anticipated estimate of the level of deformations induced by the project is an essential engineering information. In this research the single hardening model proposed by Lade and Kim was implemented in the FLAC 3D finite difference program in order to investigate the three-dimensional geomechanical behavior of earth dams during the construction stage. Numerical analyses were carried out considering 2D and 3D models to study the influence of the intermediate stress on the geomechanical behavior of earth dams. The computed results were also compared with those obtained using the traditional Mohr-Coulomb model to put in evidence the advantages using the single hardening constitutive model when analyzing geotechnical problems.

Appendix

Derivatives of the plastic potential

The derivatives are computed considering the chain rule as follows:

$$ \frac{{\partial g_{p} }}{\partial \sigma } = \frac{{\partial g_{p} }}{{\partial I_{1} }}\frac{{\partial I_{1} }}{\partial \sigma } + \frac{{\partial g_{p} }}{{\partial \sigma_{2} }}\frac{{\partial I_{2} }}{\partial \sigma } + \frac{{\partial g_{p} }}{{\partial I_{3} }}\frac{{\partial I_{3} }}{\partial \sigma } $$

where

$$ \frac{{\partial g_{p} }}{{\partial I_{1} }} = \left( {\psi_{1} \left( {\mu + 3} \right)\frac{{I_{1}^{2} }}{{I_{3} }} - \left( {\mu + 2} \right)\frac{{I_{1} }}{{I_{2} }} + \frac{{\mu \psi_{2} }}{{I_{1} }}} \right)\left( {\frac{{I_{1} }}{{p_{a} }}} \right)^{\mu } $$

$$ \frac{{\partial g_{p} }}{{\partial I_{2} }} = \frac{{I_{1}^{2} }}{{I_{2}^{2} }}\left( {\frac{{I_{1} }}{{p_{a} }}} \right)^{\mu } $$

$$ \frac{{\partial g_{p} }}{{\partial I_{3} }} = - \psi_{1} \frac{{I_{1}^{3} }}{{I_{3}^{2} }}\left( {\frac{{I_{1} }}{{p_{a} }}} \right)^{\mu } $$

Derivatives of the yield function

The derivatives of the yield function are obtained in a similar manner:

$$ \frac{{\partial f'_{p} }}{\partial \sigma } = \frac{{\partial f'_{p} }}{{\partial I_{1} }}\frac{{\partial I_{1} }}{\partial \sigma } + \frac{{\partial f'_{p} }}{{\partial \sigma_{2} }}\frac{{\partial I_{2} }}{\partial \sigma } + \frac{{\partial f^{\prime}_{p} }}{{\partial I_{3} }}\frac{{\partial I_{3} }}{\partial \sigma } $$

where

$$ \frac{{\partial f_{p}^{'} }}{{\partial I_{1} }} = \left( {\frac{3 + h}{{I_{1} }} + \frac{\partial q}{{\partial I_{1} }}} \right)f_{p}^{'} + \frac{{I_{1} }}{{I_{2} }}\left( {\frac{{I_{1} }}{{p_{a} }}} \right)^{h} e^{q} $$

$$ \frac{{\partial f_{p}^{'} }}{{\partial I_{2} }} = \frac{{I_{1}^{2} }}{{I_{2}^{2} }}\left( {\frac{{I_{1} }}{{p_{a} }}} \right)^{h} e^{q} $$

$$ \frac{{\partial f_{p}^{'} }}{{\partial I_{3} }} = f_{p}^{'} \frac{\partial q}{{\partial I_{3} }} - \psi_{1} \frac{{I_{1}^{3} }}{{I_{3}^{2} }}\left( {\frac{{I_{1} }}{{p_{a} }}} \right)^{h} e^{q} $$

in which

$$ \frac{\partial q}{{\partial I_{1} }} = \frac{\alpha }{{\eta_{1} \left( {1 - \left( {1 - \alpha } \right)S} \right)^{2} }}\left( {\frac{{mS\eta_{1} }}{{I_{1} }} + \frac{{3I_{1}^{2} }}{{I_{3} }}\left( {\frac{{I_{1} }}{{p_{a} }}} \right)^{m} } \right) $$

$$ \frac{\partial q}{{\partial I_{3} }} = \frac{\alpha }{{\eta_{1} \left( {1 - \left( {1 - \alpha } \right)S} \right)^{2} }}\frac{{I_{1}^{3} }}{{I_{3}^{2} }}\left( {\frac{{I_{1} }}{{p_{a} }}} \right)^{m} $$

Derivatives of the hardening modulus

$$ \frac{{\partial f_{p} }}{{\partial W_{p} }} = \frac{{\partial f_{p}^{''} }}{{\partial W_{p} }} = \frac{1}{{\rho \left( {Dp_{a} } \right)^{1/\rho } }}W_{p}^{1/\rho - 1} $$