A unified constitutive model for cemented/non-cemented soils under monotonic and cyclic loading

Abstract

In this paper, an elastoplastic constitutive model is proposed to uniformly describe the mechanical behavior of cemented/ non-cemented soil under different loading conditions. A state variable characterizing the degree of cementation is introduced into the yield function of the cyclic mobility model, and an evolution rule is proposed to describe the degradation of cementation and structure. Since the cyclic mobility model is developed based on the concepts of subloading surface and superloading surface, the newly proposed model inherits both its advantages and can systematically describe the effects of cementation, structure, overconsolidation, and stress-induced anisotropy on the mechanical behavior of soil. Compared with cyclic mobility model, this proposed model adds only one state variable associated with cementation, which has clear physical meaning and can be determined by uniaxial compression or tension test. The capability of the proposed model has been validated carefully through a series of tests such as isotropic, uniaxial, and triaxial tests under monotonic and cyclic loading conditions.

Appendix A

$$\frac{\partial f}{{\partial p}} = \frac{1}{p + \delta } + \frac{{{{\partial \left( {\left( {\eta^{*\delta } } \right)^{2} } \right)} \mathord{\left/ {\vphantom {{\partial \left( {\left( {\eta^{*\delta } } \right)^{2} } \right)} {\partial p}}} \right. \kern-\nulldelimiterspace} {\partial p}}}}{{M^{2} - \zeta^{2} + \left( {\eta^{*\delta } } \right)^{2} }} = \frac{{M^{2} - \left( {\eta^{\delta } } \right)^{2} }}{{\left( {M^{2} - \zeta^{2} + \left( {\eta^{*\delta } } \right)^{2} } \right)\left( {p + \delta } \right)}}$$

(A-1)

$$\frac{\partial f}{{\partial S_{ij} }} = \frac{{3\hat{\eta }_{ij}^{\delta } }}{{\left( {M^{2} - \zeta^{2} + \left( {\eta^{*\delta } } \right)^{2} } \right)\left( {p + \delta } \right)}}$$

(A-2)

$$\frac{\partial f}{{\partial \sigma_{ij} }} = \frac{\partial f}{{\partial S_{ij} }} + \frac{\partial f}{{\partial p}}\frac{{\delta_{ij} }}{3} = \frac{{3\hat{\eta }_{ij}^{\delta } + \frac{{\delta_{ij} }}{3}\left( {M^{2} - \left( {\eta^{\delta } } \right)^{2} } \right)}}{{\left( {M^{2} - \zeta^{2} + \left( {\eta^{ * \delta } } \right)^{2} } \right)\left( {p + \delta } \right)}}$$

(A-3)

$$\frac{\partial f}{{\partial \varepsilon_{v}^{p} }} = - \frac{{\left( {\frac{R}{{R^{ * } }} \cdot \tilde{p}_{m0} \cdot \exp \left( {\frac{{\varepsilon_{v}^{p} }}{{C_{p} }}} \right)} \right)\frac{1}{{C_{p} }}}}{{\left( {\frac{R}{{R^{ * } }} \cdot \tilde{p}_{m0} \cdot \exp \left( {\frac{{\varepsilon_{v}^{p} }}{{C_{p} }}} \right)} \right) + \delta }} = - \frac{{p_{m} }}{{\left( {p_{m} + \delta } \right)C_{p} }}$$

(A-4)

$$\frac{\partial f}{{\partial \beta_{ij}^{{}} }} = \frac{{\partial \ln \left( {\frac{{M^{2} - \zeta^{2} + \left( {\eta^{ * \delta } } \right)^{2} }}{{M^{2} - \zeta^{2} }}} \right)}}{{\partial \beta_{ij}^{{}} }} = \frac{{ - 3M^{2} \hat{\eta }_{ij}^{\delta } + 3\left( {\eta^{ * \delta } } \right)^{2} \beta_{ij}^{{}} + 3\zeta^{2} \hat{\eta }_{ij}^{\delta } }}{{\left( {M^{2} - \zeta^{2} + \left( {\eta^{ * \delta } } \right)^{2} } \right)\left( {M^{2} - \zeta^{2} } \right)}}$$

(A-5)

$$\frac{\partial f}{{\partial R}} = - \frac{{\left( {\frac{1}{{R^{ * } }}\tilde{p}_{m0} \cdot \exp \left( {\frac{{\varepsilon_{v}^{p} }}{{C_{p} }}} \right)} \right)}}{{\left( {\frac{R}{{R^{ * } }} \cdot \tilde{p}_{m0} \cdot \exp \left( {\frac{{\varepsilon_{v}^{p} }}{{C_{p} }}} \right)} \right) + \delta }} = - \frac{{p_{m} }}{{\left( {p_{m} + \delta } \right)R}}$$

(A-6)

$$\frac{\partial f}{{\partial R^{ * } }} = \frac{{\frac{R}{{\left( {R^{ * } } \right)^{2} }} \cdot \tilde{p}_{m0} \cdot \exp \left( {\frac{{\varepsilon_{v}^{p} }}{{C_{p} }}} \right)}}{{\left( {\frac{R}{{R^{ * } }} \cdot \tilde{p}_{m0} \cdot \exp \left( {\frac{{\varepsilon_{v}^{p} }}{{C_{p} }}} \right)} \right) + \delta }} = \frac{{p_{m} }}{{\left( {p_{m} + \delta } \right)R^{ * } }}$$

(A-7)

$$\frac{\partial f}{{\partial \delta }} = \frac{1}{p + \delta } + \frac{{ - 3\hat{\eta }_{ij}^{\delta } \eta_{ij}^{\delta } }}{{\left( {M^{2} - \zeta^{2} + \left( {\eta^{ * \delta } } \right)^{2} } \right)\left( {p + \delta } \right)}} - \frac{1}{{\left( {p_{m} + \delta } \right)}}$$

(A-8)

$$\frac{\partial f}{{\partial \varepsilon_{v}^{p} }}d\varepsilon_{v}^{p} = - \Lambda \frac{{p_{m} }}{{\left( {p_{m} + \delta } \right)}}\frac{{M^{2} - \left( {\eta^{\delta } } \right)^{2} }}{{C_{p} \left( {M^{2} - \zeta^{2} + \left( {\eta^{ * \delta } } \right)^{2} } \right)\left( {p + \delta } \right)}}$$

(A-9)

$$\frac{\partial f}{{\partial \beta_{ij} }}d\beta_{ij} = \Lambda \frac{{\sqrt 6 Mb_{r} \left( {M - \zeta } \right)\left( {\eta^{ * \delta } } \right)^{2} \left( { - 2M^{2} + 3\eta_{ij}^{\delta } \beta_{ij}^{{}} } \right)}}{{C_{p} \left( {M^{2} - \zeta^{2} + \left( {\eta^{ * \delta } } \right)^{2} } \right)^{2} \left( {M^{2} - \zeta^{2} } \right)\left( {p + \delta } \right)}}$$

(A-10)

$$\begin{aligned}\frac{\partial f}{{\partial R}}\text{d}R &= \left[ {\Lambda \frac{{ - mM\ln R\left[ {\frac{{\left( {{p \mathord{\left/ {\vphantom {p {p_{N} }}} \right. \kern-\nulldelimiterspace} {p_{N} }}} \right)^{2} }}{{1 + \left( {{p \mathord{\left/ {\vphantom {p {p_{N} }}} \right. \kern-\nulldelimiterspace} {p_{N} }}} \right)^{2} }}} \right]T }}{{C_{p} \left( {M^{2} - \zeta^{2} + \left( {\eta^{ * \delta } } \right)^{2} } \right)\left( {p + \delta } \right)}} - R\frac{{\eta^{\delta } }}{M}\frac{\partial f}{{\partial \beta_{ij}^{{}} }}d\beta_{ij}^{{}} } \right]\\ &\quad \times\frac{\bar{p}_{m}}{(p_{m}+\delta)},T=\sqrt {6\left( {\eta^{ * \delta } } \right)^{2} + \frac{1}{3}\left( {M^{2} - \left( {\eta^{\delta } } \right)^{2} } \right)^{2} }\end{aligned}$$

(A-11)

$$\frac{\partial f}{{\partial R^{ * } }}dR^{ * } = \Lambda \frac{{p_{m} aM\left( {1 - R^{ * } } \right)\sqrt {6\left( {\eta^{ * \delta } } \right)^{2} + \frac{1}{3}\left( {M^{2} - \left( {\eta^{\delta } } \right)^{2} } \right)^{2} } }}{{\left( {p_{m} + \delta } \right)C_{p} \left( {M^{2} - \zeta^{2} + \left( {\eta^{ * \delta } } \right)^{2} } \right)\left( {p + \delta } \right)}}$$

(A-12)

$$\frac{\partial f}{{\partial \delta }}\text{d}\delta = \left[ {\frac{{\left( {p - p_{m} } \right)}}{{\left( {p + \delta } \right)\left( {p_{m} + \delta } \right)}} + \frac{{3\hat{\eta }_{ij}^{\delta } \eta_{ij}^{\delta } }}{{\left( {M^{2} - \zeta^{2} + \left( {\eta^{ * \delta } } \right)^{2} } \right)\left( {p + \delta } \right)}}} \right]\times \frac{{\delta_{0} }}{{1 - R_{o}^{ * } }}\text{d}R^{ * }$$

(A-13)