Research Paper
Stress-fractional model with rotational hardening for anisotropic clay

https://doi.org/10.1016/j.compgeo.2020.103719Get rights and content

Abstract

Anisotropic clays are very common in practical engineering, where their constitutive relations were usually developed based on classical plasticity theory with associated flow rule. However, the softening responses of clays under compression following anisotropic consolidation can be non-associated and state-dependent. Fractional derivative is defined based on an integral form that can consider the material state and nonassociativity from the initial mathematical definition. In this study, a new anisotropic constitutive method for clay is developed by enriching the fractional plasticity with combined isotropic and rotational hardening. The developed model can consider the effect of evolving material anisotropy on the state-dependent non-associated behaviour of clay. To validate the approach, undrained and drained test results of different clays, are simulated and discussed, where a good agreement between the model simulations and test results is observed. It is also found that the material anisotropy keeps evolving with shearing, until reaching the final value at critical state.

Introduction

Anisotropically consolidated clays are often encountered in practical geotechnical engineering. The stress and strain responses of anisotropic clays are often very complex due to the anisotropy generated during consolidation and sedimentation, etc. (Kutter and Sathialingam, 1992, Shi et al., 2020, Shi et al., 2019). When anisotropic soil was subjected to undrained shearing, complete or partial loss of the shear strength would occur when the positive excess pore water pressure accumulated, which finally triggered undrained instability of soil. To understand such complex behaviour and instruct further engineering design, numerous experimental and analytical studies had been carried out (Gong et al., 2018, Liu et al., 2014, Liu et al., 2018, Liu et al., 2020, Jin et al., 2019, Wang et al., 2019, Zhang et al., 2020). For example, a series of triaxial tests were performed on saturated clay, where the effect of various initial states on clay deformation were investigated (Banerjee and Yousif, 1986). Zhang et al. (2020) developed a long short‐term memory (LSTM) neural network model for describing the cyclic behaviour of geomaterials. Jin and Yin (2020) proposed an enhanced backtracking search algorithm for optimised constitutive modelling of soils, where a practical tool for parameter identification was developed. Yao et al., 2009, Yao et al., 2014) proposed a unified hardening model by modifying the Cam-clay model (Schofield and Wroth, 1968) for capturing the hardening/softening and liquefaction behaviour of both granular and soft soils. Despite of the elegant model performances, the initial structural anisotropy and the subsequent degradation of soil still needs further representation. To capture the anisotropic behaviour of soil, Dafalias (Dafalias, 1986) proposed a well-known anisotropic plasticity model by introducing a coupling volumetric and deviatoric plastic strain rates into the rate of plastic work expression. The resulting plastic potential surface also served as a yield surface for associative plasticity, which was a rotated and distorted ellipse that controlled by a fabric scalar; however, an unique critical state line may not be reached due to the frozen of the yielding surface by the volumetric plastic strain at η = M (η: stress ratio, M: critical-state stress ratio). To resolve this limitation, Wheeler et al. (2003) developed an alternative rotational hardening law by considering contribution from both volumetric and shear straining; then, a general framework for modelling the anisotropic constitutive behaviour of clays subjected to K0-consolidation was established. This work was then used by many other researchers in modelling complex behaviour of geomaterials under different loading conditions, such as the time-dependent triaxial loads (Leoni et al., 2008), temperature-dependent loads (Wang et al., 2008, Wang et al., 2016), and multiaxial loads (Yin et al., 2010). However, similar to Wheeler et al. (Wheeler et al., 2003), these models usually assumed an associated flow rule for modelling the anisotropic behaviour of different soils, which had been proved by experimental evidences (Venda Oliveira and Lemos, 2014) that it was not adequate enough for soils experiencing both hardening and softening. To solve this problem, a series of non-associated plasticity models by assuming two different plastic potential and yielding functions were developed (Dafalias et al., 2006, Taiebat and Dafalias, 2008). Similar approaches can be also found in Chen and Yang (2019) where the drained and undrained behaviour of over-consolidated clays were simulated.

Instead of using additional plastic potential function, Sun and his co-workers (Sun et al., 2020, Sumelka and Nowak, 2016, Qu et al., 2019) developed a novel fractional plasticity approach by using stress-fractional gradient at the yielding surface. Compared to other critical state-based models (Xiao et al., 2019, Indraratna et al., 2014, Yin et al., 2013, Yin et al., 2018), this approach did not require the use of an additional plastic potential and a state index, e.g., ψ, but can still capture the state-dependent non-associated behaviour of granular soils. However, the current fractional plasticity models were mainly based on a symmetric yielding surface with pure isotropic hardening and thus cannot characterise the anisotropic behaviour of clays without further modification. Therefore, in order to enrich the current fractional plasticity, an attempt is made to develop a state-dependent non-associated anisotropic model for predicting the drained and undrained behaviour of clay, by incorporating rotational hardening. Instead of modelling the dependence of load state by incorporating state parameter, analytical derivations of the state-dependent plastic flow rule is presented by following the previous work (Sun et al., 2020, Sumelka and Nowak, 2016, Qu et al., 2019).

This paper is divided into four main parts: Section 2 defines the fractional derivatives and critical state used in this study; Section 3 develops a novel fractional anisotropic constitutive model, where the state-dependent anisotropic plastic flow rule is analytically derived; Section 4 presents the identification of model parameters; Section 5 validates the proposed model by simulating a series of laboratory test results of different clays; conclusions are made in Section 6. Basic stress–strain notations and constitutive relations are provided in the Appendix.

Due to its integral form of definition, the fractional derivative of a function, e.g., yielding function (f), at point σ' always has two kinds of definitions. One is the left-sided derivative, the other is the right-sided derivative. In real-world physics involving time variation, the left-sided fractional derivative is often used to emphasise the impact from past events. Take the Caputo’s left-sided fractional derivative (Caputo and Fabrizio, 2015, Caputo and Fabrizio, 2016) for example:σc'Dσ'αf(σ')=1Γ(n-α)σ'σc'f(n)(χ)dχ(σ'-χ)α+1-n,σ'>σc'where D (=α/σ'α) denotes partial derivation; Γ denotes gamma function. α(n-1,n], is the fractional order and n is a positive integer. n = 1 or 2, thus, 0<α<2. σc' is the lower integral limit (e.g., time or stress), indicating the impact of past history on current state (σc'σ'). Moreover, the corresponding right-sided fractional derivative reflects the potential influence of future events. For example, the Caputo’s right-sided fractional derivative:σ'Dσc'αf(σ')=(-1)nΓ(n-α)σc'σ'f(n)(χ)dχ(χ-σ')α+1-n,σc'>σ'where σc' in this case is the upper limit for integration. It is easy to find that Eq. (2) takes into account the impact of future event on current state (σ'σc'). Note that the event considered here is the critical state which the soil would finally approach in future after sufficient shearing. The integral limit for fractional derivative measures how the current state is approaching the critical state and how such distance between these two states influences soil behaviour. In fractional plasticity (Sun et al., 2020, Sumelka, 2014), we adopt such salient advantages of the fractional derivative in capturing history or state dependent phenomenon. Instead of distinguishing the impact of past or future events on soil deformation, we suggest to measure the relative position between the current stress state and critical stress state. Then, σ' and σc' are the current effective stress and the corresponding critical-state stress, respectively. As shown in Fig. 1, before the soil can reach the final critical state, there should be three possible positions of the current stress state regarding the corresponding critical state in the p'-q plane (p': mean effective principal stress, q: deviator stress). The first one is located at the “dry” side of the critical state line (CSL) (Point A), where dilation may occur; the second one is located at the “wet” side of the CSL (Point B), where only contraction occurs; the third one is just at the CSL, where the critical state behaviour of soil takes places. It can be stated that the relative position of the current stress state to the critical state determines whether Eqs. (1) or (2) will be used. It is noted that when the current stress is at the CSL, fractional derivatives defined by Eqs. (1), (2) will automatically reduce to the same value at the stress point of interest. In the later work of Sun et al. (2020);α was found to be slightly dependent on the initial material state. Thus, for better model performance, it can be expressed byα=expk(1-p'/p0')where k is a material constant; p0 is the size of the yielding surface, as shown in Fig. 1. Then, for clays at the initial anisotropic consolidation or isotropic-consolidation state, the fractional order α=1. However, at the subsequent shearing stage, α increased, representing the development of non-associativity and the degree of state-dependence of material deformation.

Based on Schofield and Wroth (Schofield and Wroth, 1968), the following critical state lines in the e-lnp' and p'-q planes for clay are used:pc'=prexpeΓ-e/λqc=q+tM(p'-pc')where t = +1 for compression and t = -1 for extension; pr(=1kPa) is the unit pressure; λ is the gradient of the CSL in the e-lnp' plane; eΓ denotes the intercept of the CSL at p'=1kPa; e is the current void ratio; p'and pc' are the mean effective principal stresses at the current and the corresponding critical states, respectively; q and qc are the deviator stresses at the current and the corresponding critical states, respectively. Note that for sand subjected to a wide range of initial states, the CSL will be a curved line instead of a straight line. In this case, Eq. (4) can be hardly used. The extension of this model for considering sand with wide range of initial states can be done by using a curved CSL, e.g., in Russell and Khalili (2004), and an alternative yielding surface as shown in SANISAND (Taiebat and Dafalias, 2008, Dafalias and Taiebat, 2016). However, this is not within the scope of this study.

Section snippets

Loading direction

As materials can be anisotropically consolidated before being subjected to external loading, the following anisotropic yielding function (f) suggested by Dafalias (1986) is used:f=32sij-p'βijdsij-p'βijd-M2-32βijdβijdp0'-p'p'=0where the critical state stress ratio, M, can be expressed as (Sheng et al., 2000):M=Mc2c41+c4+(1-c4)sin(3θ)1/4

in which Mc [=6sinϕc/(3-sinϕc)] denotes the critical state stress ratio for triaxial compression, and ϕc is the critical state friction angle; c [=(3-sinϕc)/(3+sin

Application

In this section, we present the application of the enriched anisotropic fractional plasticity approach for capturing undrained and drained behaviour of different clays (Banerjee and Yousif, 1986, Surarak et al., 2012, Zervoyannis, 1982). Details on test material and test setup can be found in each relevant literature (Banerjee and Yousif, 1986, Surarak et al., 2012, Zervoyannis, 1982) and thus not repeated here for simplicity. Comparisons between the model predictions and test results are shown

Conclusions

Natural clays in the field were usually consolidated in an anisotropic condition, where the difference between vertical and horizontal stresses would cause measurable anisotropy within the soil. When subjected to transient external loads, such soils could experience drained dilatation and undrained instability, which would result in the failure of the upper-bearing facility. To capture the drained and undrained behaviour of anisotropically consolidated clay, an anisotropic fractional plasticity

CRediT authorship contribution statement

Yifei Sun: Conceptualization, Methodology, Writing - original draft. Chen Chen: Data curation, Software, Visualization. Yufeng Gao: Supervision, Project administration, Funding acquisition, 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.

Acknowledgement

The first author would like to thank Prof. Wen Chen and Prof. Yang Xiao for the invaluable inspiration. The financial support provided by the National Natural Science Foundation of China (Grant No. 51890912) and the Alexander Von Humboldt Foundation, Germany are appreciated.

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