Research Paper
Large-deformation geomechanical problems studied by a shear-transformation-zone model using the material point method

https://doi.org/10.1016/j.compgeo.2021.104153Get rights and content

Abstract

A non-equilibrium statistical thermodynamics-based framework for describing elasto-viscoplastic deformations in amorphous materials termed shear-transformation-zone (STZ) theory has been implemented in the material point method (MPM) and applied to the modeling of large-deformation geomechanical problems. Plane strain biaxial shear tests with different loading rates were first performed using the model to emphasize its rate-dependency and to compare its performance with that of a μ(I) rheological model. The model was then utilized to simulate three classical large-deformation geomechanical problems, namely, the granular column collapse, the footing settling on a soft ground, and the soil-pipe interaction. It has been shown from this study that the STZ theory is promising in modeling rate-dependent viscoplastic behaviors of geomaterials, as well as solid–fluid phase transition and strain localization.

Introduction

Numerical simulations of geomechanical problems often entail a high level of complexity resulted primarily from two aspects. Firstly, the mechanical responses of geomaterials are rather complex often dealing with a wide spectrum of phenomena, such as nonlinearity, dilatancy (Li and Dafalias, 2000), anisotropy (Guo and Zhao, 2014), rate-dependency (Jop et al., 2006, Yamamuro et al., 2011, Suescun-Florez and Iskander, 2017), solid–fluid phase transition (Dunatunga and Kamrin, 2015), among others. To develop a constitutive model capable of replicating even part of these phenomena requires considerable efforts. Secondly, a real geomechanical problem usually involves multiphysics processes, interactions among soils, fluids, and structures (Oñate et al., 2011), as well as large deformations of geomaterials. It hence inevitably poses huge challenges for the numerical modeling practices and urgently demands the development of robust numerical algorithms and advanced numerical methods to tackle these challenges.

Specifically, modeling of large deformation, which is common in modern geotechnical engineering designs and applications, such as landslides (Jin et al., 2020), dam breaks, buried anchors (Zhang et al., 2020), and piled foundations (Yang et al., 2020), has proven to be crucial yet challenging. Eulerian method such as the finite volume method (FVM) has been applied in modeling dense rapid granular flows by treating the material as a viscous fluid and adopting a μ(I) rheological model (Lagrée et al., 2011). On the other hand, the solid-like behavior of a geomaterial is typically loading history-dependent and can be more conveniently modeled by the Lagrangian method such as the finite element method (FEM). However, FEM is known to be vulnerable to mesh entanglement under large deformations. Various remeshing techniques have therefore been developed to cure mesh distortion, for example, the arbitrary Lagrangian Eulerian (ALE) (Nazem et al., 2009), the remeshing and interpolation technique with small strain (RITSS) (Hu and Randolph, 1998), and the particle finite element method (PFEM) (Idelsohn et al., 2004, Zhang et al., 2013). Additionally, meshless methods nowadays have also enjoyed many successes in modeling large-deformation geomechanical problems, which mainly include the smoothed particle hydrodynamics (SPH) (Bui et al., 2008), the reproducing kernel particle method (RKPM) (Xie and Wang, 2014), and the material point method (MPM) (Sołowski and Sloan, 2015).

Among the meshless methods above, MPM was originally proposed by Sulsky et al. (1994) based on the extension of the particle-in-cell (PIC) method (Harlow, 1964) and the fluid implicit particle (FLIP) method (Brackbill et al., 1988). It combines the strengths of both the Eulerian and Lagrangian methods. The key idea behind MPM is that the state information of the material such as the mass, the velocity, the stress, and the strain is stored at a set of material points that move following the deformation of the continuum body, while the governing balance of momentum equation is solved on a background mesh usually fixed in space. Using two separate discretization frames allows easy tracking of the evolution of the material history-dependent variables and effective avoiding of possible mesh distortion. These merits of MPM make it particularly attractive for modeling geomaterials undergoing solid–fluid phase transition and large deformations, as exemplified in the review article by Soga et al. (2016). It also fits perfectly in the present study to model three classical problems involving large deformations, namely, the granular column collapse, the footing settling on a soft ground, and the soil-pipe interaction.

For the constitutive model suitable for describing both solid-like and fluid-like geomaterial behaviors, a unified model combining the Drucker-Prager elasto-plasticity with the μ(I) rheology was outlined in Dunatunga and Kamrin (2015) for dry granular materials. With a similar framework extended for multiphase granular materials, large deformations of fluid-sediment mixtures can also be simulated (Baumgarten and Kamrin, 2019). Fang (2010) developed a thermodynamically consistent constitutive model with a pressure-ratio order parameter, in which solid–fluid transition is regarded as a second order phase transition process and described by a kinematic evolution of the order parameter. Prime et al., 2014a, Prime et al., 2014b proposed a model for solid–fluid transition which associates an elasto-plastic relation and a Bingham viscous law, where the phase transition is governed by the second-order work stability criterion. Similar work on solid–fluid phase transition was presented in Zhang et al. (2017) by combining the Tresca and the Bingham models. Vescovi et al. (2020b) developed a constitutive model for granular flows under simple shear conditions by merging the critical state concept of granular solids and the kinetic theory of granular gases, in which the state variable of the material is described by the granular temperature. By applying the mixture theory, the same framework is also able to model the saturated granular flows (Vescovi et al., 2020a).

In this paper, a non-equilibrium statistical thermodynamics-based framework for describing elasto-viscoplastic deformations of amorphous materials (e.g. glasses, foams, suspensions, and granular media), called shear-transformation-zone (STZ) theory, is applied to model geomaterials in their solid and fluid phases, as well as the phase transition. The theory hypothesizes that the irreversible deformation in amorphous materials takes place in localized regions called STZs scattered in a bulk elastic matrix. In this sense, the concept of STZs is similar to that of the dislocations in a crystal structure. These STZ regions undergo configurational rearrangements through flipping between two opposite states relative to the direction of principal shear stress, and their kinematics is described by the evolution of a temperature-like state variable termed compactivity (Lieou and Langer, 2012). Every single STZ transition, or equivalently a locally exchange of nearest neighbor relationships between particles, generates a unit of local plastic strain. The macroscopic plastic strain of the material is accommodated by continuous creation and annihilation of STZs. It is noted that the STZ theory has already been successfully adopted in modeling granular materials to study e.g. the dynamic earthquake rupture (Daub and Carlson, 2010), the grain fragmentation (Lieou et al., 2014a), the stick–slip instability (Lieou et al., 2017), and the strain localization in sheared fault gouges (Ma and Elbanna, 2018).

The present study will adopt an STZ model developed in Ma and Elbanna (2018) to simulate large deformation in geomaterials using the MPM. To the best knowledge of the authors, this will constitute a first attempt for an STZ model to be applied in the large-deformation granular flow scenario. The rest of the paper is organized as follows. The MPM and the STZ formulations are introduced in Section 2, where the performance of the STZ model under plane strain biaxial shear and its comparison with a μ(I) rheological model are also provided. Sections 3-5 present the simulation results and discussion of the granular column collapse test, the footing settling test and the soil-pipe interaction test, respectively. The study is finally concluded in Section 6.

Section snippets

Material point method (MPM)

In MPM, the computational domain is spatially discretized using two frames, i.e. a set of material points and a background mesh. The former is used to discretize the continuum body carrying all the material information, such as the mass, the strain, the stress, and any other internal state variables, and will move with the deformation of the body following the Lagrangian description. The latter, on the other hand, disassociated with the movement of the material (usually fixed in space thus free

Granular column collapse

The simulation setup follows the experimental test done by Nguyen et al. (2017) using a stack of aluminium rods to approximate the plane strain condition. The initial aspect ratio of the column is 1.0 (initial height = initial width = 10 cm) as shown in Fig. 3. The column is discretized into 32 × 32 elements with 4 material points in each element. The mesh discretization is 4 times finer than that used in Liang and Zhao (2019). Note that increasing the number of material points generally could

Conclusions

A nonequilibrium thermodynamics based STZ model has been presented in the study for the first time to model large deformation geomechanical problems with the MPM. The model was first applied in the biaxial shear test to demonstrate its rate-dependent behaviors and compared with the performance of a μ(I) rheological model. Three classical geomechanical problems, namely the granular column collapse, the footing settling on a soft ground, and the soil-pipe interaction were studied with the STZ

CRediT authorship contribution statement

W.L. Li: Conceptualization, Methodology, Data curation, Writing - original draft, Visualization, Investigation. N. Guo: Supervision, Conceptualization, Methodology, Writing - original draft. Z.X. Yang: Supervision, Conceptualization, Funding acquisition, Writing - review & editing. T. Helfer: Software, Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors thank the two anonymous reviewers for their constructive comments. The study is supported by the National Natural Science Foundation of China (Nos. 52078456 and 52020105003). The development of MFront is supported financially by CEA, EDF and Framatome in the framework of the PLEIADES project.

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