Modeling creep and rate effects in structured anisotropic soft clays

Abstract

In simulations of undrained triaxial tests, most soil models fail to capture the effect of post peak strain rate variation. This is due to the fact that no “swelling” is allowed for the viscoplastic volume strain. Imposing such restriction implies that dilative behavior cannot be modeled. Therefore, a model incorporating creep has been formulated using the so-called time resistance concept that uses a single creep parameter determined from an incremental oedometer test. The key feature of the proposed model is the introduction of the time resistance concept on the plastic multiplier rather than on the volumetric viscoplastic strain. This allows the viscoplastic volume strain to be either positive or negative depending on whether the state of the soil is on the “wet” or “dry” side of critical state line. The proposed model is based on an existing elastoplastic model for structured soft clay (S-CLAY1S). The paper gives a description of the constitutive model and the numerical scheme used in the implementation of the model. Capabilities of the model are illustrated with simulations of oedometer and triaxial tests. Results from such analyses show that the model is able to capture essential features of soft clay behavior.

Appendix

From the parameters used as user input, some internal parameters are determined for the model. The internal parameters of the S-CLAY1S model are calculated through input of the remolded reference Young’s modulus and the remolded reference oedometer modulus, as well as $K_{0}^{NC}$. Approximations for the internal parameters, as used in Wheeler et al. [50], are found by assuming that the elastic stiffness is infinite. In oedometer condition it may be shown that the following relation holds:

$$ \left[ {\begin{array}{*{20}c} 1 \\ {{\frac{2}{3}}} \\ \end{array} } \right] \cdot {\frac{{p_{ref} }}{{\left\{ {E_{oed}^{ref} } \right\}_{i} }}} - \left[ {\begin{array}{*{20}c} {F_{K} } & {F_{J} } \\ {F_{J} } & {F_{G} } \\ \end{array} } \right] \cdot {\frac{{p_{ref} }}{{\left\{ {E_{ref} } \right\}_{i} }}} \cdot \left[ {\begin{array}{*{20}c} 1 \\ {\eta_{K0NC} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} 1 \\ {{\frac{{2\left( {\eta_{K0NC} - \alpha_{K0NC} } \right)}}{{M_{f}^{2} - \eta_{K0NC}^{2} }}}} \\ \end{array} } \right] \cdot \zeta_{i} $$

(21)

The term F _J is introduced in case of elastic anisotropy is assumed.

From Eq. (21), we find the two following equations:

$$ \zeta_{i} = p_{ref} \left( {{\frac{1}{{\left\{ {E_{oed}^{ref} } \right\}_{i} }}} - {\frac{{F_{K} + F_{J} \cdot \eta_{K0NC} }}{{\left\{ {E_{ref} } \right\}_{i} }}}} \right) $$

(22)

$$ \alpha_{K0NC} = \eta_{K0NC} - {\frac{{p_{ref} }}{{\zeta_{i} }}}\left( {{\frac{2}{3}} \cdot {\frac{1}{{\left\{ {E_{oed}^{ref} } \right\}_{i} }}} - {\frac{{F_{J} + F_{G} \cdot \eta_{K0NC} }}{{\left\{ {E_{ref} } \right\}_{i} }}}} \right) \cdot {\frac{{M_{f}^{2} - \eta_{K0NC}^{2} }}{2}} $$

$$ \alpha_{K0NC} \approx \eta_{K0NC} - {\frac{{M_{f}^{2} - \eta_{K0NC}^{2} }}{3}}. $$

(23)

In addition, Eq. (12) combined with the value for $K_{0}^{NC}$ gives a ratio between μ _q and μ _v that must be given to make Eq. (23) true.

$$ \begin{aligned} {\frac{{\mu _{q} }}{{\mu _{v} }}} = & {\frac{{\left( {{\frac{3}{4}} \cdot \eta _{{K0NC}} - \alpha _{{K0NC}} } \right) \cdot \left( {M_{f}^{2} - \eta _{{K0NC}}^{2} } \right)}}{{\left( {\alpha _{{K0NC}} - {\frac{1}{3}} \cdot \eta _{{K0NC}} } \right) \cdot 2\left( {\eta _{{K0NC}} - \alpha _{{K0NC}} } \right)}}} \\ & \approx {\frac{{{\frac{9}{2}}\left( {{\frac{{M_{f}^{2} - \eta _{{K0NC}}^{2} }}{3}} - {\frac{1}{4}} \cdot \eta _{{K0NC}} } \right)}}{{2\eta _{{K0NC}} - M_{f}^{2} + \eta _{{K0NC}}^{2} }}} \\ \end{aligned} . $$

(24)

Leoni et al. [22] proposes the following expression for μ _v : From Eq. (12) and considering q = 0, we obtain:

$$ {\frac{{M_{f}^{2} }}{{M_{f}^{2} \cdot \alpha - 2\alpha^{2} \cdot {{{\mu_{q} }}/{{\mu_{v} }}}}}} \cdot d\alpha = - \mu_{v} \cdot d\varepsilon_{v}^{vp} $$

(25)

by integration, we find:

$$ \int\limits_{{\alpha_{K0NC} }}^{{{{\alpha_{K0NC} } \mathord{\left/ {\vphantom {{\alpha_{K0NC} } {\eta_{i} }}} \right. \kern-\nulldelimiterspace} {\eta_{i} }}}} {{\frac{1}{{\alpha - {{{2\alpha^{2} }}/{{M_{f}^{2} }}} \cdot {{{\mu_{q} }}/{{\mu_{v} }}}}}}} \cdot d\alpha = - \mu_{v} \cdot \zeta \cdot \int\limits_{{p^{\prime } }}^{{n_{2} \cdot p^{\prime } }} {{\frac{1}{{p^{\prime } }}}} \cdot dp^{\prime } $$

$$ \mu_{v} = {\frac{1}{{\zeta \cdot \ln \left( {n_{2} } \right)}}} \cdot \ln \left( {{\frac{{n_{1} \cdot M_{f}^{2} - 2 \cdot {{{\mu_{q} }}/{{\mu_{v} }}} \cdot \alpha_{K0NC} }}{{M_{f}^{2} - 2 \cdot {{{\mu_{q} }}/{{\mu_{v} }}} \cdot \alpha_{K0NC} }}}} \right) $$

(26)

for n ₁ = 10 and $n_{2} = \exp \left( {{{{p_{ref} }}/{{\zeta \cdot \left\{ {E_{oed}^{ref} } \right\}_{i} }}}} \right)$.

$$ \mu_{v} = {\frac{{\left\{ {E_{oed}^{ref} } \right\}_{i} }}{{p_{ref} }}} \cdot \ln\left( {{\frac{{10 \cdot M_{f}^{2} - 2 \cdot {{{\mu_{q} }}/{{\mu_{v} }}} \cdot \alpha_{K0NC} }}{{M_{f}^{2} - 2 \cdot {{{\mu_{q} }}/{{\mu_{v} }}} \cdot \alpha_{K0NC} }}}} \right) $$

(27)