Large deformation and failure analysis of river embankments subjected to seismic loading

Abstract

A numerical model based on the theory of mixtures is proposed for the nonlinear large deformation and failure analysis of river embankments subjected to large earthquakes. The governing equations are discretized using the finite element method in large deformation regime with the updated Lagrangian description. A cyclic elasto-viscoplastic model is used for describing the rate-dependent behavior of clayey soils, while a cyclic elasto-plastic model is adopted for sand. The nonlinear kinematic hardening and softening due to structural degradation of soil particles are taken into account in the constitutive relations. Efforts have been made to model the failure modes and damage patterns of river embankments observed in the 2011 Tohoku Earthquake. The effects of ground profile, water table, and earthquake motion on the dynamic response and damage pattern of river embankments are particularly emphasized.

Appendix: Cyclic elasto-plastic model for sand

The constitutive model used for the liquefaction analysis of sandy soils must have the potential to simulate the accumulations of strains and pore pressure during cyclic loading. The cyclic elasto-plastic (EP) model used in this study is the extension of the model of Oka et al. [21] to include the structural degradation effects. The overconsolidation boundary surface Eq. (9) is adopted in the EP model as the boundary between the overconsolidated region and the normally consolidated region. Similar to the EVP model, the strain-softening parameter $\sigma^{\prime}_{ma}$, describing the soil degradation due to structural changes, is defined in the model and assumed to decrease with increasing plastic strain by using Eq. (11).

The yield function for changes in the stress ratio is denoted as

$$f_{y1} = \bar{\eta }_{\chi }^{*} - k = 0$$

(43)

in which $k$ is a parameter for controlling the size of the elastic region. The nonlinear kinematic hardening is taken into account using the evolution Eq. (16).

For changes in the mean effective stress, the second yield function can be defined as

$$f_{y2} = M_{m}^{*} \left| {\ln \left( {{{\sigma^{\prime}_{m} } \mathord{\left/ {\vphantom {{\sigma^{\prime}_{m} } {\sigma^{\prime}_{mk} }}} \right. \kern-0pt} {\sigma^{\prime}_{mk} }}} \right) - y_{m}^{*} } \right| - R_{d} = 0$$

(44)

where $y_{m}^{*}$ is the scalar kinematic hardening parameter defined in Eq. (18) and $R_{d}$ is a scalar variable. Since the strains brought about by changes in the mean effective stress are small in the overconsolidated region, the second yield function can be disregarded in this region. For simplicity, only Eq. (43) has been used in the present analysis.

The plastic potential function is defined as

$$g = \bar{\eta }_{\chi }^{*} + \tilde{M}^{*} \ln \left( {{{\sigma^{\prime}_{m} } \mathord{\left/ {\vphantom {{\sigma^{\prime}_{m} } {\sigma^{\prime}_{mp} }}} \right. \kern-0pt} {\sigma^{\prime}_{mp} }}} \right) = 0$$

(45)

where $\sigma_{mp}^{{\prime }}$ controls the size of the plastic potential, and $\tilde{M}^{*}$ is the dilatancy coefficient defined in Eq. (15).

The generalized flow rule for the constitutive model, using the fourth-order isotropic tensor $C_{ijkl}$, is expressed as

$$D_{ij}^{p} = C_{ijkl} \frac{\partial g}{{\partial \sigma^{\prime}_{kl} }}$$

(46)

where $D_{ij}^{p}$ is the plastic stretching tensor. The stress–dilatancy relation is obtained from the generalized flow rule as

$$\frac{{D_{kk}^{p} }}{{{\left(\hat{D}_{ij}^{p}\hat{D}_{ij}^{p}\right)}^{0.5} }} = D^{*} \left( {\tilde{M}^{*} - \bar{\eta }_{\chi }^{*} } \right)$$

(47)

where $D^{*} = {{3a} \mathord{\left/ {\vphantom {{3a} {2b}}} \right. \kern-0pt} {2b}} + 1$ is the dilatancy parameter controlling the ratio of the plastic volumetric stretch $D_{kk}^{p}$ to the deviatoric stretch ${\left(\hat{D}_{ij}^{p}\hat{D}_{ij}^{p}\right)}^{0.5}$. The variation in $D^{*}$ is given by

$$D^{*} = D_{0}^{*} \left( {{{\tilde{M}^{*} } \mathord{\left/ {\vphantom {{\tilde{M}^{*} } {M_{m}^{*} }}} \right. \kern-0pt} {M_{m}^{*} }}} \right)^{{n_{0} }}$$

(48)

where $D_{0}^{*}$ and $n_{0}$ are material parameters. The reduction in the elastic shear modulus is defined as

$$G = \frac{{G_{0}^{{}} }}{{1 + {{\gamma_{(n)\hbox{max} }^{E*} } \mathord{\left/ {\vphantom {{\gamma_{(n)\hbox{max} }^{E*} } {\gamma_{(n)r}^{E*} }}} \right. \kern-0pt} {\gamma_{(n)r}^{E*} }}}}$$

(49)

where $\gamma_{(n)\hbox{max} }^{E*}$ is the maximum accumulated elastic shear between stress reversal points at past cycles, and $\gamma_{(n)r}^{E*}$ is the elastic referential strain controlling the rate of reduction of $G$ with respect to $\gamma_{(n)\hbox{max} }^{E*} .$