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Fractional-order modelling of state-dependent non-associated behaviour of soil without using state variable and plastic potential
- Yifei Sun^{1}Email authorView ORCID ID profile and
- Changjie Zheng^{2}
https://doi.org/10.1186/s13662-019-2040-5
© The Author(s) 2019
- Received: 11 November 2018
- Accepted: 22 February 2019
- Published: 1 March 2019
Abstract
It was found that the constitutive behaviour of granular soil was dependent on its density and pressure (i.e. material state). To capture such state dependence, a variety of state variables were empirically proposed and introduced into the existing plastic potential functions, which inevitably resulted in the complexity and meaninglessness of some model parameters. The purpose of this study is to theoretically investigate the state-dependent non-associated behaviour of granular soils without using predefined plastic potential and state variable. A novel state-dependent non-associated model for granular soils is mathematically developed by incorporating the stress-fractional operator into the bounding surface plasticity. Unlike previous studies using empirical state variables, the soil state and non-associativity in this study are considered via analytical solution, where a state-dependent plastic flow rule and the corresponding hardening modulus without using additional plastic potentials are obtained. Possible mathematical connection with a well-known empirical state variable is also discussed. The non-associativity between plastic flow and loading directions as well as material hardening is found to be controlled by the fractional-derivative order. To validate the proposed approach, a series of drained and undrained triaxial test results of different granular soils are simulated and compared, where a good agreement between the model predictions and the corresponding test results is observed.
Keywords
- Fractional plasticity
- Constitutive relations
- State dependence
- Granular soils
1 Introduction
It has been widely acknowledged that the strength and deformation behaviour of granular soil, such as sand and rockfill, is significantly dependent on its density and pressure (material state) [1]. Before the proper consideration of state dependence during constitutive modelling, different model parameters were often required for modelling the stress-strain behaviour of granular soils with different initial densities or subjected to different confining pressures [2–4].
For the purpose of better understanding and unified constitutive modelling of granular soils, a variety of different empirical state variables have been suggested, such as the state ratio of the difference between the threshold and current void ratios to the difference between the threshold and critical void ratios [5], the ratio of the current to critical void ratios [6], the disturbance in disturbed state concept [7], the stress ratio of the current to critical mean effective stresses [8] and the most widely used state variable (ψ) defined by the difference between the current and critical state void ratios [9]. It can be found that developing a reasonable state-dependent non-associated plastic flow rule (or stress-dilatancy relationship) has been of the utmost importance in recent years. One of the popular approaches was to modify the existing stress-dilatancy equations, for example, the Cam-clay (CC) stress-dilatancy equation [10] and Rowe’s stress-dilatancy equation [11], by empirically incorporating ψ [12–16]. Undeniably, this approach can always substantially improve the model performance; however, the empirical correlation between the state variable and the existing constitutive parameters would inevitably result in more model parameters. It was found that the fractional mechanics was an efficient way to capture the relaxation [17], diffusion [18–20], and stress-strain [14, 21] behaviour of materials. To reduce the number of model parameters without the loss of modelling capability, Sun and Shen [21] proposed a non-associated plastic flow rule for granular soil by simply conducting fractional-order derivatives of the yielding surface, where the obtained vector (plastic flow direction) was no longer normal to the yielding surface, even without using an additional plastic potential. This non-normality increased as the fractional order (α) decreased [14, 22, 23]. To consider the state dependence, the state-dependent fractional plasticity model was then proposed [14] by empirically incorporating ψ, which however made the parameters of the obtained stress-dilatancy equation lack physical meaning.
This study attempts to theoretically investigate the state-dependent non-associated stress-strain behaviour of granular soils. A state-dependent non-associated constitutive model without using any empirical state variables and plastic potentials is developed by using strict mathematics. Instead of modelling the dependence of soil state by empirically incorporating state variables, analytical derivations of the state-dependent plastic flow rule and the associated hardening rule are presented. As the fractional derivative is defined in integral form, the soil state is captured through the integrating range from the lower limit (current stress state) to the upper limit (critical stress state). This paper is divided into four main parts: Sect. 2 defines the basic constitutive relations and the relevant fractional derivative used in this study; Sect. 3 develops a novel state-dependent constitutive model without using plastic potential, where the state-dependent fractional plastic flow rule is analytically derived; Sect. 4 presents the identification and sensitivity analysis of model parameters; Sect. 5 provides the model validation against a series of laboratory test results of different granular soils; Sect. 6 concludes the study. For the sake of simplicity, all the derivations and discussions in this study are limited to homogenous and isotropic materials.
2 Notations and definitions
2.1 Constitutive relations
2.2 Fractional derivative and yielding surface
3 State-dependent fractional model
3.1 State-dependent plastic flow
3.2 Possible mathematical connection with \(I_{p}\)
3.3 Bounding surface and loading direction
Note that detailed derivations of Eqs. (25) and (26) can be found in Sun and Shen [21] and Sun et al. [49], thus not repeated here for simplicity.
3.4 Hardening modulus
4 Parameter identification and sensitivity analysis
There are totally nine parameters (\(M _{c}\), λ, \(e_{\varGamma } \), c, α, \(h _{1}\), \(h _{2}\), κ, ν) in the proposed state-dependent model, which can be all determined from traditional triaxial tests. Detailed elaborations on parameter identification and sensitivity analysis are given below.
5 Model validation
Xiao et al. [12] reported a series of drained triaxial test results of Tacheng rockfill with different initial void ratios. The material mainly consisted of sub-angular to rounded particles with a median diameter (\(d _{50}\)) of 23 mm and a coefficient of uniformity (\(C _{u}\)) of 5.4. Samples were prepared by layered compaction to have a diameter around 300 mm and a height around 600 mm. Initial void ratios and the corresponding confining pressures can be found in Figs. 8–11 and thus not repeated here. It is observed from Figs. 8–11 that even without using the state variable and plastic potential function, the proposed model can well simulate with state-dependent constitutive behaviour of Tacheng rockfill subjected to different initial states (confining pressures and void ratios). The strain hardening and softening behaviour as well as the corresponding volumetric contraction and dilation behaviour of Tacheng rockfill can be all reasonably captured.
6 Conclusions
- (1)
Without using predefined state variables and plastic potential functions, a novel state-dependent stress-dilatancy equation was analytically derived by using the fractional-order plasticity theory. Dependence of the non-associated flow on material state was modelled through rigorous mathematical definition of the fractional stress gradient.
- (2)
Possible mathematical connections between the proposed state-dependent dilatancy equation and the state pressure index by Wang et al. [8] were also discussed, where the dependence of state pressure on the stress-dilatancy phenomenon of granular soil was analytically proved.
- (3)
The extent of non-associativity and hardening modulus were influenced by the material state via the vertical and horizontal distances from the current stress state to the corresponding critical stress state in the \(p' - q\) plane.
- (4)
With the increase of the fractional order, the predicted peak stress decreased while the volumetric dilation increased. Samples modelled by a higher fractional order reached critical state more quickly; the transition from the strain softening behaviour to strain hardening behaviour also increased with the increase of the fractional order.
- (5)
Model parameters can be all determined from traditional triaxial test results. It was found that the proposed state-dependent fractional plasticity model can well capture the stress-strain behaviour of different granular soils subjected to a variety of loading conditions.
Declarations
Acknowledgements
The authors would like to thank Prof. Yannis F. Dafalias for his invaluable suggestions and Prof. Wen Chen for his lifelong inspiration.
Availability of data and materials
All the data in this study were collected from published literatures, which were appropriately cited.
Funding
The financial support provided by the National Natural Science Foundation of China (Grant Nos. 41630638, 51679068), the National Key Basic Research Program of China (“973” Program) (Grant No. 2015CB057901) and the China Postdoctoral Science Foundation (Grant No. 2017M621607) is appreciated.
Authors’ contributions
The first author formulated the main ideas and equations of the paper, the second author helped to prepare the figures. All the authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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