The strength reduction method in clay hypoplasticity

doi:10.1016/j.compgeo.2020.103687

Computers and Geotechnics

Volume 126, October 2020, 103687

https://doi.org/10.1016/j.compgeo.2020.103687 Get rights and content

Abstract

An evaluation of slope stability using the finite element method is often performed as a complementary analysis to the plastic deformation analysis. The slope stability is typically evaluated by the reduction of the strength parameters $c$ and $φ$ of the Mohr-Coulomb model and it is represented by the so-called factor of safety. It is much less common to use the strength reduction method in combination with more advanced constitutive models and, for some modelling frameworks, it has actually not yet been developed. This leads to the need for advanced model substittion by the Mohr-Coulomb model and a further recalibration of the constitutive model parameters. In this paper, a new strength reduction method was developed for the models based on hypoplasticity, in particular to the hypoplastic model for clays. In addition to a simple reduction of the strength variables, a subsequent return of the deviatoric stress to the asymptotic state boundary surface is implemented so that all the integration points remain within the so-called asymptotic state boundary surface. Predictions using the newly developed method are thoroughly tested from the quantitative as well as qualitative point of view and compared with the predictions obtained from the Mohr-Coulomb model.

Introduction

Various methods can be employed to evaluate the factor of safety (FS) in slope stability analysis. The limit equilibrium (LE) analysis provides an estimation of FS along the predefined failure surfaces of a circular (Bishop, 1955, Petterson, 1955) or polygonal (Morgenstern and Price, 1965, Spencer, 1967, Sarma, 1973) shape. The slope itself is considered as a rigid body with the strength defined by the Mohr-Coulomb parameters $φ$ and c. Due to the nature of LE, the failure surface corresponding to the lowest FS is determined by the iterative procedure. This, however, does not ensure that the solution represents the slip surface of a minimal factor of safety. The limit analysis (LA) can provide estimations of the upper and lower bounds for the energetic and static equilibrium and, with the aid of the Finite Element Method (FEM), these bounds can be determined even the complex problems (Lyamin and Sloan, 2002a, Lyamin and Sloan, 2002b). A relatively novel approch represented by the Material Point Method (Sulsky et al., 1995) was also introduced into the stability analysis (Zabala and Alonso, 2011).

In finite element method, slope stability is quantified by the Strength Reduction method (SR), which is commonly used in combination with the Mohr-Coulomb model (M–C). This method was initially developed by Zienkiewicz et al. (1975) and has since been extensively studied (Tschuchnigg et al., 2015, Yang et al., 2012), including a solution dependency on both the FEM mesh refinement and elasto-plastic flow rule (Tschuchnigg et al., 2015). Further modifications of the SR can be also found in Tschuchnigg et al., 2015, Dyson and Tolooiyan, 2018. The advantage of connecting staged construction modelling with stability analysis without the need to use additional tools while preserving the stress and variables history is significant.

Even though the M–C model is capable of predicting the stress state at failure, the prediction of the strain response is rather unsatisfactory. Advanced elasto-plastic constitutive soil model theories, such as bounding surface plasticity (Dafalias, 1986), and other modelling frameworks including hypoplasticity (Wu and Kolymbas, 1990) and barodesy (Kolymbas, 2012) were proposed for both sand (Von Wolffersdorff, 1996, Kolymbas et al., 2009) and clay (Mašín, 2013, Medicus and Fellin, 2017) soils. These were further enhanced for various model characteristics to capture intergranular strain stiffness (Niemunis and Herle, 1997), anisotropy (Mašín, 2014) or viscosity (Jerman and Mašín, 2020) in order to overcome the difficulties. These models can account for the nonlinear strain response as well as, state and loading history dependency of soils. The strength reduction method for these advanced constitutive models is, however, usually not avaliable. Consequently, if the advanced constitutive model is used during staged construction and a stability analysis is required, the M–C model has to be employed and a recalibration of the parameters is necessary.

Recently, an example of the strength reduction method for the clay barodesy model (Medicus and Fellin, 2017) was proposed in Schneider-Muntau et al. (2018), giving an option for dealing with the stability analysis while using advanced non-elasto-plastic constitutive models. This paper is dedicated to the strength reduction method applied to the hypoplastic model for clay (Mašín, 2013). In the following, the usual sign convention of continuum mechanics (compression negative) is adopted throughout. In line with Terzaghi’s principle of effective stress, all stresses are effective stresses.

Section snippets

Strength reduction using the Mohr-Coulomb model

The stress state of the M–C model is limited in the principal stress space by the yield surface F of a hexagonal pyramid shape. Since the M–C is an elastic – perfectly plastic model, once the loading path reaches the stress state on the yield surface the model exhibits a plastic strain response. In the integration point, the reduction of the strength parameters from their initial values $φ_{1}$ (internal friction angle) and $c_{1}$ (cohesion) to their reduced counterparts $φ_{2}$ and $c_{2}$ directly reduces the

Hypoplastic clay model

The hypoplastic clay model (H–C) (Mašín, 2013) is one of the advanced constitutive models currently available in geotechnical software. It is capable of predicting nonlinear soil behaviour and captures soil hardening as well as post peak softening. The model behaviour is defined by five parameters closely resembling the parameters of the Cam-Clay model (Roscoe and Burland, 1968). The parameters $λ^{*}, κ^{*}$ and N are defined in $\ln p \times \ln (e + 1)$ space, see Fig. 2a. The parameter N represents the value of $\ln ($

Strength reduction in clay hypoplasticity

As already mentioned, advanced constitutive models, including hypoplastic models, satisfactorily predict the soil behaviour and characteristics, such as soil hardening and softening during monotonic loading. The strength reduction developed in this article is split into two parts: The method of critical friction angle reduction $φ_{c}$ is described first and the stress return to the ASBS is then elaborated.

Testing in single-element model

The proposed procedure was tested in the simulation of the element test. An example of undrained triaxial simulation is illustrated in Fig. 9, which is divided into two parts: Firstly, the strength reduction is conducted simultaneously with the triaxial loading; the reduction was executed from the critical state friction angle $φ_{c} = 30 °$ down to $φ_{c} = 15 °$ . Subsequently, the strength reduction was cancelled while the triaxial loading continued to imposE further deformation. The reduction stopped at an

umat implementation

As mentioned above, the strength reduction method was implemented in the umat format of ABAQUS and executed in the PLAXIS finite element code through an interface. The reduction of the critical state friction angle $φ_{c}$ is mapped to auxiliary quantity t (time) as seen in Fig. 10. Since not only a time increment $Δ t^{i}$ but also the exact value of the time $t_{r}^{i}$ at any given step i is provided by the FEM software, the value of the reduced friction angle $φ_{c, r}^{i}$ at the step i can be directly evaluated as $φ_{c,}$

Study of the strength reduction method

The reliability of the strength reduction method was further tested in the PLAXIS 2D software. Three FEM models of different geometries were created with slope inclinations $β = 7.67 °, β = 19.08 °$ and $β = 30.46 °$ , see Fig. 11. In addition, a finer FEM mesh was adopted in the areas associated with the development of the slip surface. To fully test the developed method, models of embankments of loose soil as well as excavation in overconsolidated soil were tested.

Different densities of FEM mesh were used

Conclusions

This study demonstrated the quantitative and qualitative predictions of a newly developed strength reduction method for the hypoplastic clay model (Mašín, 2013). The method adopts a strength parameter reduction, combined with an analytical formulation of the stress projection back to the ASBS in the octahedral plane of stress at a constant Lode angle. It was demonstrated that this stress reduction is a necessary component of the strength reduction method since the lack of the stress return

CRediT authorship contribution statement

Tomáš Kadlíček: Conceptualization, Methodology, Software, Validation, Formal analysis, Data curation, Writing - original draft. David Mašín: Conceptualization, Methodology, Writing - review & editing, Supervision, Project administration, Funding acquisition.

Declaration of Competing Interest

The authors declared that there is no conflict of interest.

Acknowledgement

Financial support from a research grants LTACH19028 of the Czech Ministry of Education, Youth and Sports and grant 17-21903S by the Czech Science Foundation is greatly appreciated. The second author acknowledges institutional support by the Center for Geosphere Dynamics (UNCE/SCI/006).

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Citation Excerpt :
This makes the evaluation of the factor of safety based on the approach proposed by Nitzsche and Herle (2020) more accurate but also significantly slower compared to other stress-based approaches, for instance the ones by Zou et al. (1995) or Farias and Naylor (1998). Besides the concept presented by Nitzsche and Herle (2020), stability analyses taking into consideration advanced constitutive models have also been conducted via SRFEA, for instance using clay hypoplasticity (Kadlíček and Mašín, 2020) or barodesy (Schneider-Muntau et al., 2018; Tschuchnigg et al., 2019). However, these studies (Nitzsche and Herle, 2020; Kadlíček and Mašín, 2020; Schneider-Muntau et al., 2018; Tschuchnigg et al., 2019) have not been extended and applied to slopes subjected to earthquake loading.
Assessment of the seismic slope stability in terms of a factor of safety (FoS) is often addressed using pseudo-static approaches neglecting material-induced failure and the role of pore-fluids. On the other hand, sophisticated constitutive models that capture these effects can usually not be considered in strength reduction analyses. In this study, the concept of strain-dependent slope stability presented by Nitzsche and Herle (2020) was enhanced to enable the analysis of slope stability problems under earthquake loading. This allowed to apply advanced constitutive models to both, the dynamic and the stability analysis. The applicability of the concept was shown by analysis of FoS and failure mechanisms for a water-saturated opencast mine slope subject to earthquake loading. To capture the non-linear material behavior, the hypoplastic model with intergranular strain was used. Machine learning algorithms adopted to the problem at hand provided accurate approximations of FoS and failure mechanism while reducing computational costs by 2–3 orders of magnitude.
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We further evaluate stress paths in principal stress space, investigate different overconsolidation ratios and evaluate and interpret possible soil states in the framework of Critical State Soil Mechanics. The outcomes of this studies can therefore also be relevant for plane-strain applications of clay hypoplasticity as carried out by e.g. Nitzsche and Herle (2014), Stutz et al. (2017), Kadlíček and Mašín (2020) and Jerman and Mašín (2021). For a summary of the symbols and notation that we have adopted, please refer to Appendix A.
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Research PaperThe strength reduction method in clay hypoplasticity

Abstract

Introduction

Section snippets

Strength reduction using the Mohr-Coulomb model

Hypoplastic clay model

Strength reduction in clay hypoplasticity

Testing in single-element model

umat implementation

Study of the strength reduction method

Conclusions

CRediT authorship contribution statement

Declaration of Competing Interest

Acknowledgement

Computers and Geotechnics

Computers and Geotechnics

Computer Physics Communications

Computers and Geotechnics

Mechanics of Materials

The use of slip circles in the stability analysis of earth slopes

Geotechnique

Bounding surface plasticity. I: Mathematical foundation and hypoplasticity

Journal of Engineering Mechanics

Optimisation of strength reduction finite element method codes for slope stability analysis

Innovative Infrastructure Solutions

Barodesy: a new hypoplastic approach

International Journal for Numerical and Analytical Methods in Geomechanics

Upper bound limit analysis using linear finite elements and non-linear programming

International Journal for Numerical and Analytical Methods in Geomechanics

Lower bound limit analysis using non-linear programming

International Journal for Numerical Methods in Engineering

A hypoplastic constitutive model for clays

International Journal for Numerical and Analytical Methods in Geomechanics

Research Paper
The strength reduction method in clay hypoplasticity