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Flow liquefaction instability prediction using finite elements

Abstract

In this paper, a mathematical criterion based on bifurcation theory is developed to predict the onset of liquefaction instability in fully saturated porous media under static and dynamic loading conditions. The proposed liquefaction criterion is general and can be applied to any elastoplastic constitutive model. Since the liquefaction criterion is only as accurate as the underlying constitutive model utilized, the modified Manzari–Dafalias model is chosen for its accuracy, relative simplicity and elegance. Moreover, a fully implicit return mapping algorithm is developed for the numerical implementation of the Manzari–Dafalias model, and a consistent tangent operator is derived to obtain optimal convergence with finite elements. The accuracy of the implementation is benchmarked against laboratory experiments under monotonic and cyclic loading conditions, and a qualitative boundary value problem. The framework is expected to serve as a tool to enable prediction of liquefaction occurrence in the field under general static and dynamic conditions. Further, the methodology can help advance our understanding of the phenomenon in the field as it can clearly differentiate between unstable behavior, such as flow liquefaction, and material failure, such as cyclic mobility.

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Correspondence to José E. Andrade.

Appendices

Appendix 1

The coefficient matrices in the discretized governing Eq. (4) are given by

$$ \begin{aligned} {\mathbf{M}}_{u} & = \int\limits_{\Upomega } {{\mathbf{N}}_{u}^{\text{T}} \rho {\mathbf{N}}_{u} {\text{d}}\Upomega } \\ {\mathbf{M}}_{w} & = \int\limits_{\Upomega } {\nabla {\mathbf{N}}_{{p_{w} }}^{{ {\text{T}}}} \varvec{ }\rho_{w} \left( {{\mathbf{k}}/\mu_{w} } \right){\mathbf{N}}_{u} {\text{d}}\Upomega } \\ {\mathbf{Q}} & = \int\limits_{\Upomega } {{\mathbf{B}}^{\text{T}} \alpha {\mathbf{m}} {\mathbf{N}}_{{p_{w} }} {\text{d}}\Upomega } \\ {\mathbf{C}} & = \int\limits_{\Upomega } {{\mathbf{N}}_{{p_{w} }}^{\text{T}} \left( {\frac{\alpha - n}{{K_{\text{s}} }} + \frac{n}{{K_{w} }}} \right){\mathbf{N}}_{{p_{w} }} {\text{d}}\Upomega } \\ {\mathbf{H}} & = \int\limits_{\Upomega } {\nabla {\mathbf{N}}_{{p_{w} }}^{{ {\text{T}}}} \left( {{\mathbf{k}}/\mu_{w} } \right)\nabla {\mathbf{N}}_{{p_{w} }} {\text{d}}\Upomega } \\ \end{aligned} $$
(42)

and the external force and flux vectors are given by

$$ \begin{aligned} {\mathbf{F}}_{u} & = \int\limits_{\Upomega } {{\mathbf{N}}_{u}^{\text{T}} \rho {\mathbf{b}}\,{\text{d}}\Upomega } + \int\limits_{{\Upgamma _{t} }} {{\mathbf{N}}_{u}^{\text{T}} {\bar{\mathbf{t}}} {\text{d}}\Upgamma } \\ {\mathbf{F}}_{w} & = \int\limits_{\Upomega } {\nabla {\mathbf{N}}_{{p_{w} }}^{\text{T}} \rho_{w} \left( {{\mathbf{k}}/\mu_{w} } \right){\mathbf{b}} {\text{d}}\Upomega } - \int\limits_{{\Upgamma _{{q_{w} }} }} {{\mathbf{N}}_{{p_{w} }}^{\text{T}} \bar{q}_{w} {\text{d}}\Upgamma } \\ \end{aligned} $$
(43)

where m is the identity vector defined as \( {\mathbf{m}} = \left[ {\begin{array}{*{20}c} 1 & 1 & 1 & 0 & 0 & 0 \\ \end{array} } \right]^{\text{T}} ,\,{\bar{\mathbf{t}}} \) is the prescribed traction imposed on boundary \( \Upgamma _{t} \) and \( \bar{q}_{w} \) is the prescribed outflow imposed on the permeable boundary \( \Upgamma _{{q_{w} }} . \)

Appendix 2

The components of the local Jacobian matrix \( {\mathbf{J}}^{i} \) are given by

$$ \begin{aligned} \partial {\mathbf{r}}_{{\varepsilon^{e} }}^{i} /\partial {\varvec{\upvarepsilon}}^{e,i} & = {\bf 1} + \Updelta L^{i} \partial {\mathbf{R}}^{i} /\partial {\varvec{\upvarepsilon}}^{e,i} \\ \partial {\mathbf{r}}_{{\varepsilon^{e} }}^{i} /\partial {\varvec{\upalpha}}^{i} & =\Updelta L^{i} \partial {\mathbf{R}}^{i} /\partial {\varvec{\upalpha}}^{i} \\ \partial {\mathbf{r}}_{{\varepsilon^{e} }}^{i} /\partial {\mathbf{z}}^{i} & =\Updelta L^{i} \partial {\mathbf{R}}^{i} /\partial {\mathbf{z}}^{i} \\ \partial {\mathbf{r}}_{{\varepsilon^{e} }}^{i} /\partial\Updelta L^{i} & = {\mathbf{R}}^{i} \\ \partial {\mathbf{r}}_{\alpha }^{i} /\partial {\varvec{\upvarepsilon}}^{e,i} & = -\Updelta L^{i} \left( {2/3} \right)\left( {{\mathbf{b}}^{i}\otimes \partial h^{i} /\partial {\varvec{\upvarepsilon}}^{e,i} + h^{i} \partial {\mathbf{b}}^{i} /\partial {\varvec{\upvarepsilon}}^{e,i} } \right) \\ \partial {\mathbf{r}}_{\alpha }^{i} /\partial {\varvec{\upalpha}}^{i} & = {\bf 1} -\Updelta L^{i} \left( {2/3} \right)\left( {{\mathbf{b}}^{i}\otimes \partial h^{i} /\partial {\varvec{\upalpha}}^{i} + h^{i} \partial {\mathbf{b}}^{i} /\partial {\varvec{\upalpha}}^{i} } \right) \\ \partial {\mathbf{r}}_{\alpha }^{i} /\partial {\mathbf{z}}^{i} & = {\bf 0} \\ \partial {\mathbf{r}}_{\alpha }^{i} /\partial\Updelta L^{i} & = - \left( {2/3} \right)h^{i} {\mathbf{b}}^{i} \\ \partial {\mathbf{r}}_{z}^{i} /\partial {\varvec{\upvarepsilon}}^{e,i} & = c_{z}\Updelta L^{i} \left( { - H\left( { - D^{i} } \right)\left( {z_{ {\rm max} } {\mathbf{n}}^{i} + {\mathbf{z}}^{i} } \right)\otimes \partial D^{i} /\partial {\varvec{\upvarepsilon}}^{e,i} + \langle- D^{i}\rangle \varvec{ }z_{ {\rm max} } \partial {\mathbf{n}}^{i} /\partial {\varvec{\upvarepsilon}}^{e,i} } \right) \\ \partial {\mathbf{r}}_{z}^{i} /\partial {\varvec{\upalpha}}^{i} & = c_{z}\Updelta L^{i} \left( { - H\left( { - D^{i} } \right)\left( {z_{ {\rm max} } {\mathbf{n}}^{i} + {\mathbf{z}}^{i} } \right)\otimes \partial D^{i} /\partial {\varvec{\upalpha}}^{i} + \langle- D^{i}\rangle \varvec{ }z_{ {\rm max} } \partial {\mathbf{n}}^{i} /\partial {\varvec{\upalpha}}^{i} } \right) \\ \partial {\mathbf{r}}_{z}^{i} /\partial {\mathbf{z}}^{i} & = {\bf 1} + c_{z}\Updelta L^{i} \left( { - H\left( { - D^{i} } \right)\left( {z_{ {\rm max} } {\mathbf{n}}^{i} + {\mathbf{z}}^{i} } \right)\otimes \partial D^{i} /\partial {\mathbf{z}}^{i} + \langle- D^{i}\rangle \varvec{ }{\bf 1}} \right) \\ \partial {\mathbf{r}}_{z}^{i} /\partial\Updelta L^{i} & = c_{z} \langle- D^{i}\rangle \left( {z_{ {\rm max} } {\mathbf{n}}^{i} + {\mathbf{z}}^{i} } \right) \\ \partial {\text{r}}_{{\Updelta L}}^{i} /\partial {\varvec{\upvarepsilon}}^{e,i} & = {\mathbf{n}}^{i}\cdot \partial {\mathbf{s}}^{i} /\partial {\varvec{\upvarepsilon}}^{e,i} - \left( {{\varvec{\upalpha}}^{i}\cdot {\mathbf{n}}^{i} + \sqrt {2/3} m} \right)\partial p^{i} /\partial {\varvec{\upvarepsilon}}^{e,i} \\ \partial {\text{r}}_{{\Updelta L}}^{i} /\partial {\varvec{\upalpha}}^{i} & = - p^{i} {\mathbf{n}}^{i} \\ \partial {\text{r}}_{{\Updelta L}}^{i} /\partial {\mathbf{z}}^{i} & = {\bf 0} \\ \partial {\text{r}}_{{\Updelta L}}^{i} /\partial\Updelta L^{i} & = 0 \\ \end{aligned} $$
(44)

in which

$$ \begin{aligned} \partial {\mathbf{R}}^{i} /\partial {\varvec{\upvarepsilon}}^{e,i} & = {\mathbf{n}}^{i} \otimes \partial B^{i} /\partial {\varvec{\upvarepsilon}}^{e,i} + B^{i} \partial {\mathbf{n}}^{i} /\partial {\varvec{\upvarepsilon}}^{e,i} + \left( {\left( {{\mathbf{n}}^{i} } \right)^{2} - \left( {1/3} \right){\mathbf{m}}} \right) \otimes \partial C^{i} /\partial {\varvec{\upvarepsilon}}^{e,i} + 2C^{i} {\mathbf{n}}^{i} \\ & \quad \cdot \partial {\mathbf{n}}^{i} /\partial {\varvec{\upvarepsilon}}^{e,i} - \left( {1/3} \right){\mathbf{m}} \otimes \partial D^{i} /\partial {\varvec{\upvarepsilon}}^{e,i} \\ \partial {\mathbf{R}}^{i} /\partial {\varvec{\upalpha}}^{i} & = {\mathbf{n}}^{i} \otimes \partial B^{i} /\partial {\varvec{\upalpha}}^{i} + B^{i} \partial {\mathbf{n}}^{i} /\partial {\varvec{\upalpha}}^{i} + \left( {\left( {{\mathbf{n}}^{i} } \right)^{2} - \left( {1/3} \right){\mathbf{m}}} \right) \otimes \partial C^{i} /\partial {\varvec{\upalpha}}^{i} + 2C^{i} {\mathbf{n}}^{i} \\ & \quad \cdot \partial {\mathbf{n}}^{i} /\partial {\varvec{\upalpha}}^{i} - \left( {1/3} \right){\mathbf{m}} \otimes \partial D^{i} /\partial {\varvec{\upalpha}}^{i} \\ \partial {\mathbf{R}}^{i} /\partial {\mathbf{z}}^{i} & = - \left( {1/3} \right){\mathbf{m}} \otimes \partial D^{i} /\partial {\mathbf{z}}^{i} \\ \partial h^{i} /\partial {\varvec{\upvarepsilon}}^{e,i} & = h^{i} \left( {\partial {b_{0}^{i}} /\partial {\varvec{\upvarepsilon}}^{e,i} - h^{i} \left( {{\varvec{\upalpha}}^{i} - {\varvec{\upalpha}}_{\text{in}} } \right) \cdot \partial {\mathbf{n}}^{i} /\partial {\varvec{\upvarepsilon}}^{e,i} } \right)/b_{0}^{i} \\ \partial h^{i} /\partial {\varvec{\upalpha}}^{i} & = - \left( {h^{i} } \right)^{2} \left( {{\mathbf{n}}^{i} + \left( {{\varvec{\upalpha}}^{i} - {\varvec{\upalpha}}_{\text{in}} } \right) \cdot \partial {\mathbf{n}}^{i} /\partial {\varvec{\upalpha}}^{i} } \right)/b_{0}^{i} \\ \partial {\mathbf{b}}^{i} /\partial {\varvec{\upvarepsilon}}^{e,i} & = \sqrt {2/3} \left( {{\mathbf{n}}^{i} \otimes \partial \alpha_{\theta }^{b,i} /\partial {\varvec{\upvarepsilon}}^{e,i} + \alpha_{\theta }^{b,i} \partial {\mathbf{n}}^{i} /\partial {\varvec{\upvarepsilon}}^{e,i} } \right) \\ \partial {\mathbf{b}}^{i} /\partial {\varvec{\upalpha}}^{i} & = \sqrt {2/3} \left( {{\mathbf{n}}^{i} \otimes \partial \alpha_{\theta }^{b,i} /\partial {\varvec{\upalpha}}^{i} + \alpha_{\theta }^{b,i} \partial {\mathbf{n}}^{i} /\partial {\varvec{\upalpha}}^{i} } \right) - {\bf 1} \\ \partial D^{i} /\partial {\varvec{\upvarepsilon}}^{e,i} & = \left( {A_{0} H\left( {{\mathbf{z}}^{i} \cdot {\mathbf{n}}^{i} } \right)\left( {{\mathbf{d}}^{i} \cdot {\mathbf{n}}^{i} } \right)\varvec{ }{\mathbf{z}}^{i} - A_{d}^{i} {\varvec{\upalpha}}^{i} } \right) \cdot \partial {\mathbf{n}}^{i} /\partial {\varvec{\upvarepsilon}}^{e,i} + \sqrt {2/3} A_{d}^{i} \partial \alpha_{\theta }^{d,i} /\partial {\varvec{\upvarepsilon}}^{e,i} \\ \partial D^{i} /\partial {\varvec{\upalpha}}^{i} & = \left( {A_{0} H\left( {{\mathbf{z}}^{i} \cdot {\mathbf{n}}^{i} } \right)\left( {{\mathbf{d}}^{i} \cdot {\mathbf{n}}^{i} } \right)\varvec{ }{\mathbf{z}}^{i} - A_{d}^{i} {\varvec{\upalpha}}^{i} } \right) \cdot \partial {\mathbf{n}}^{i} /\partial {\varvec{\upalpha}}^{i} + A_{d}^{i} \left( {\sqrt {2/3} \partial \alpha_{\theta }^{d,i} /\partial {\varvec{\upalpha}}^{i} - {\mathbf{n}}^{i} } \right) \\ \partial D^{i} /\partial {\mathbf{z}}^{i} & = A_{0} H\left( {{\mathbf{z}}^{i} \cdot {\mathbf{n}}^{i} } \right)\left( {{\mathbf{d}}^{i} \cdot {\mathbf{n}}^{i} } \right)\varvec{ }{\mathbf{n}}^{i} \\ \partial {\mathbf{n}}^{i} /\partial {\varvec{\upvarepsilon}}^{e,i} & = \left( {\partial {\mathbf{s}}^{i} /\partial {\varvec{\upvarepsilon}}^{e,i} - {\varvec{\upalpha}}^{i} \otimes \partial p^{i} /\partial {\varvec{\upvarepsilon}}^{e,i} - {\mathbf{n}}^{i} \otimes \left( {{\mathbf{n}}^{i} \cdot \partial {\mathbf{s}}^{i} /\partial {\varvec{\upvarepsilon}}^{e,i} - \left( {{\varvec{\upalpha}}^{i} \cdot {\mathbf{n}}^{i} } \right)\partial {p^{i}} /\partial {\varvec{\upvarepsilon}}^{e,i} } \right)} \right)\varvec{ }/\left| {{\mathbf{s}}^{i} - p^{i} {\varvec{\upalpha}}^{i} } \right| \\ \partial {\mathbf{n}}^{i} /\partial {\varvec{\upalpha}}^{i} & = - p^{i} \left( {{\bf 1} - {\mathbf{n}}^{i} \otimes {\mathbf{n}}^{i} } \right)/\left| {{\mathbf{s}}^{i} - p^{i} {\varvec{\upalpha}}^{i} } \right| \\ \partial p^{i} /\partial {\varvec{\upvarepsilon}}^{e,i} & = - {{K}}^{i} {\mathbf{m}} \\ \partial {\mathbf{s}}^{i} /\partial {\varvec{\upvarepsilon}}^{e,i} & = - {{K}}^{i} /\left( {2p^{i} } \right)\Updelta {\mathbf{s}}^{i} \otimes{\mathbf{m}} + 2{{G}}^{i} \left( {{\bf 1} - \left( {1/3} \right){\mathbf{m}} \otimes {\mathbf{m}}} \right) \\ \end{aligned} $$
(45)

where \( {\mathbf{1}} \) is the 6 × 6 identity matrix, H is the Heaviside step function whose value is unity for positive arguments and zero otherwise, the symbol ǀ ǀ denotes the \( L_{2} \) norm of a vector, and the symbol \( \otimes \) denotes the dyadic product of two vectors.

Appendix 3

The components of \( \partial {\mathbf{r}}/\partial {\varvec{\upvarepsilon}} \) are given by

$$ \begin{aligned} \partial {\mathbf{r}}_{{\varepsilon^{e} }} /\partial {\varvec{\upvarepsilon}} & = - {\bf 1} +\Updelta L\partial {\mathbf{R}}/\partial {\varvec{\upvarepsilon}} \\ \partial {\mathbf{r}}_{\alpha } /\partial {\varvec{\upvarepsilon}} & = -\Updelta L\left( {2/3} \right)\left( {{\mathbf{b}} \otimes \partial h/\partial {\varvec{\upvarepsilon}} + h\,\partial {\mathbf{b}}/\partial {\varvec{\upvarepsilon}}} \right) \\ \partial {\mathbf{r}}_{z} /\partial {\varvec{\upvarepsilon}} & = c_{z}\Updelta L\left( { - H\left( { - D} \right)\left( {z_{ \hbox{max} } {\mathbf{n}} + {\mathbf{z}}} \right) \otimes \partial D/\partial {\varvec{\upvarepsilon}} + \langle - D\rangle \varvec{ }z_{ \hbox{max} } \partial {\mathbf{n}}/\partial {\varvec{\upvarepsilon}}} \right) \\ \partial {\text{r}}_{{\Updelta L}} /\partial {\varvec{\upvarepsilon}} & = {\mathbf{n}} \cdot \partial {\mathbf{s}}/\partial {\varvec{\upvarepsilon}} - \left( {{\varvec{\upalpha}} \cdot {\mathbf{n}} + \sqrt {2/3} m} \right)\partial p/\partial {\varvec{\upvarepsilon}} \\ \end{aligned} $$
(46)

in which

$$ \begin{aligned} \partial {\mathbf{R}}/\partial {\varvec{\upvarepsilon}} & = {\mathbf{n}} \otimes \partial B/\partial {\varvec{\upvarepsilon}} + B\,\partial {\mathbf{n}}/\partial {\varvec{\upvarepsilon}} + \left( {{\mathbf{n}}^{2} - \left( {1/3} \right){\mathbf{m}}} \right) \otimes \partial C/\partial {\varvec{\upvarepsilon}} + 2C{\mathbf{n}} \cdot \partial {\mathbf{n}}/\partial {\varvec{\upvarepsilon}} - \left( {1/3} \right){\mathbf{m}} \otimes \partial D/\partial {\varvec{\upvarepsilon}} \\ \partial h/\partial {\varvec{\upvarepsilon}} & = h\left( {\partial b_{0} /\partial {\varvec{\upvarepsilon}} - h\left( {{\varvec{\upalpha}} - {\varvec{\upalpha}}_{\text{in}} } \right) \cdot \partial {\mathbf{n}}/\partial {\varvec{\upvarepsilon}}} \right)/b_{0} \\ \partial {\mathbf{b}}/\partial {\varvec{\upvarepsilon}} & = \sqrt {2/3} \left( {{\mathbf{n}} \otimes \partial \alpha_{\theta }^{b} /\partial {\varvec{\upvarepsilon}} + \alpha_{\theta }^{b} \partial {\mathbf{n}}/\partial {\varvec{\upvarepsilon}}} \right) \\ \partial D/\partial {\varvec{\upvarepsilon}} & = \left( {A_{0} H\left( {{\mathbf{z}} \cdot {\mathbf{n}}} \right)\left( {{\mathbf{d}} \cdot {\mathbf{n}}} \right)\varvec{ }{\mathbf{z}} - A_{d} {\varvec{\upalpha}}} \right) \cdot \partial {\mathbf{n}}/\partial {\varvec{\upvarepsilon}} + \sqrt {2/3} A_{d} \partial \alpha_{\theta }^{d} /\partial {\varvec{\upvarepsilon}} \\ \partial {\mathbf{n}}/\partial {\varvec{\upvarepsilon}} & = \left( {\partial {\mathbf{s}}/\partial {\varvec{\upvarepsilon}} - {\varvec{\upalpha}} \otimes \partial p/\partial {\varvec{\upvarepsilon}} - {\mathbf{n}} \otimes \left( {{\mathbf{n}} \cdot \partial {\mathbf{s}}/\partial {\varvec{\upvarepsilon}} - \left( {{\varvec{\upalpha}} \cdot {\mathbf{n}}} \right)\partial p/\partial {\varvec{\upvarepsilon}}} \right)} \right)/\left| {{\mathbf{s}} - p{\varvec{\upalpha}}} \right| \\ \partial p/\partial {\varvec{\upvarepsilon}} & = \left( {4.97 + e} \right)/\left( {2.97 - e} \right){{K}}\Updelta \varepsilon_{\text{v}}^{e} {\mathbf{m}} \\ \partial {\mathbf{s}}/\partial {\varvec{\upvarepsilon}} & = \left( {4.97 + e} \right)/\left( {2.97 - e} \right)\left( { - 1 + {{K}}\Updelta \varepsilon_{\text{v}}^{e} /\left( {2p} \right)} \right)\Updelta {\mathbf{s}} \otimes {\mathbf{m}} \\ \end{aligned} $$
(47)

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Mohammadnejad, T., Andrade, J.E. Flow liquefaction instability prediction using finite elements. Acta Geotech. 10, 83–100 (2015). https://doi.org/10.1007/s11440-014-0342-z

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Keywords

  • Finite element analysis
  • Fully implicit return mapping algorithm
  • Granular materials
  • Liquefaction instability
  • Manzari–Dafalias plasticity model
  • Static and dynamic liquefaction