Application of a Stress-Fractional Model for Capturing the Stress-Strain Behavior of Granular Soils

Granular soils usually serve as the ﬁller material in geotechnical engineering. This study presents the development and application of a stress-fractional model for granular soils with diﬀerent initial material states. To capture the plastic loading and ﬂow behaviors, a subloading surface with the fractional stress gradient is used. The developed model contains twelve parameters which can be determined through triaxial tests. To validate the developed model, the well-documented test results of Firoozkuh No. 161 sand and crushed basalt are simulated and discussed. It is found that the stress-fractional model can reasonably simulate the undrained and drained behaviors of granular soils consolidated with diﬀerent densities and mean eﬀective pressures.


Introduction
Granular soils, e.g., ballast and sand, are often encountered or used in engineering practice. For example, ballast usually serves as an essential track construction layer to bear the repeated moving load transmitted by the train. During the whole operation period, a rail track usually experiences a large number of repeated train loads. Sand can be used as a filter material. Accurate prediction of the corresponding maintenance periods necessitates the development of an advanced constitutive model that captures ballast or sand deformation [1]. Although traditional elastoplastic constitutive models have been investigated widely and successfully applied in many fields, more effort is required to realistically describe the stress-strain relationship of granular soil, which was observed to be state-dependent and nonassociated. In classical plasticity theory, the dependence of associative or nonassociative plastic flow of granular soil on the material state can be mathematically elaborated by using a loading/ bounding/reference surface and a plastic potential surface with frictional parameters dependent on the material state. For example, a series of elegant constitutive models have been developed for predicting the instability and flow liquefaction behavior of Firoozkuh sand [2] as well as other sands.
However, different from the modelling techniques often used in traditional plasticity, a new modelling framework based on the fractional plasticity (FP) has been developed and implemented recently [3][4][5][6], in which the plastic potential was not necessarily used. However, the previous FP approach was established based on granular soil and the soil-structure interface [7], using the loading/ yielding surface.
is study attempts to develop a statedependent constitutive model for granular soil by using the fractional plasticity and subloading surface concepts [8]. For simplicity, conventional stress and strain notations in soil mechanics, with compression positive and extension negative, are used. In addition, this study is limited to homogenous materials under isotropic or anisotropic consolidation.

Constitutive Model
In elastoplastic theory [3,9], the tensorial incremental plastic strain, ε where m kl , n ij , and K p are the plastic loading tensor, plastic flow tensor, and plastic modulus, respectively, and i, j, k, and l are indices running from 1 to 3. is the Macaulay bracket. σ kl ′ is the tensorial incremental effective stress. In this study, m kl defines a unit normal at the subloading surface (f ), and n ij defines the plastic strain rate direction. For the purpose of obtaining the analytical solution, Sun et al. [9] suggested to use the following fractional derivatives for modelling: where the fractional order, α, ranges between n and n − 1, with the integer n > 0 [3]. D denotes partial differentiation. a stands for the integral limit in equations (3a) and (3b).
Depending on the relative position of a with regard to σ ′ , one should decide whether equation (3a) or (3b) should be used. en, m kl and n ij can be expressed as follows [10]: where δ ij is the Kronecker delta; the tensorial deviatoric stress, s ij , can be defined as s ij � σ ij ′ − p ′ δ ij ; the mean effective stress (p ′ ) can be defined as p ′ � σ ij ′ δ ij /3. q � ������ � 3s ij s ij /2 is the deviatoric stress. Although m kl and n ij incorporate the contributions from the deviatoric stress components, (3s kl /(2q) or 3s ij /(2q)) and the volumetric parts (d f δ kl /3 or d g δ ij /3), equations (4) and (5) assume a Drucker-Prager locus in the π− plane. d f and d g can be defined as In addition, using the consistency condition and H � LH, K p can be determined as where H is the law for isotropic hardening. Combining equations (1), (4), (5), and (8), the details of FP have now been provided.
In this study, the modified subloading surface [11] is used: where c is the material constant. M is the slope of the critical state line (CSL) in the p ′ − q plane. M � M c for compression, and M � M e for extension. p 0 ′ defines the location of the subloading surface. Combining equation (9) with equations (6) and (7), one has where p c ′ � p a [(e Γ − e)/λ] 1/ξ is the stress at the critical state; p a � 100 kPa is the atmospheric pressure; c � 0 is used when developing equation (11). e Γ , λ, and ξ are material constants describing the CSL in the e − p ′ plane. e is the current void ratio.
where t � +1 for loading and t � − 1 for unloading. e plastic modulus can be obtained as where h 0 � (h 1 − h 2 e)(1 + e)exp(βψ), in which β, h 1 , and h 2 are material constants, and ψ � e − e Γ + λ(p c ′ /p a ) ξ ; η in is the stress ratio at the last stress reversal; M p � Mexp(− βψ) is the stress ratio obtained at the peak state of granular soil under shearing. Moreover, incremental elastic strain (ε e ij ) is considered using Hooke's law: with ] being the tensorial elastic compliance and ] being [12], where F(e) � (2.97 − e) 2 /(1 + e) and G 0 is a material constant. It is noted that methods on how to determine the model constants from laboratory tests can be found in Sun et al. [3]. e values of the model constants for simulating railroad ballast and sand are shown in Table 1.

Numerical Implementation
In this section, the numerical methodology of the proposed FP model is given. Strain-controlled loading and the subsequent stress evolution are used in the numerical scheme, where there are five main procedures: (1) For step j (� 1, 2, 3, . . ., n), input the stress and strain conditions (p ′ j , q j , η j , e j , and ψ j ) and calculate the jth plastic loading and flow directions by using equations (4) and (5).
(2) en, calculate the hardening modulus (H j ) as well as the elastoplastic compliance matrix (C j ) at step j by using equations (1) and (11). (3) Furthermore, compute the strain increments induced by axial loading (ε j ) under drained loading. (4) e induced stress increments (σ ′ j ) can then be back-calculated using equation (14). (5) Update the jth total stress and strain matrices after obtaining the strain and stress increments by using the following: Loading and unloading are judged by the dot product of the loading and incremental stress tensors, i.e., m ij σ ij ′ ≥ 0 for loading and m ij σ ij ′ < 0 for unloading. For more details, one can refer to Sun et al. [13]. Subroutines using the above methodology are then implemented in MATLAB for model simulation.

Model Application
is section provides the model application in simulating the real stress-strain behavior of ballast and sand subjected to element tests. e well-documented test data of crushed basalt [14] and Firoozkuh sand [2] are simulated and compared. Figure 1 shows the simulation results for the monotonic behavior of ballast with different initial material states [14]. All the laboratory tests were carried out on the saturated crushed basalt with a specimen size of 300 mm in diameter and 600 mm in height. For more details on test materials and test setup, please refer to Salim and Indraratna [14]. It is observed from Figure 1 that the proposed mathematical approach can well capture the stress-strain response of ballast, especially the stress dilation and strain hardening/softening induced by different material states. Figures 2 and 3 provide the model simulations of the cyclic behavior of ballast with different initial states. In Figure 2, an initial void ratio (e 0 ) of 0.65 and a confining pressure (σ 3 ′ ) of 50 kPa are used for simulation. It can be observed that the shearing strain amplitude (δε 1 ) influences the cyclic behavior of ballast. If a small strain amplitude with higher frequency was applied, ballast would exhibit much more volumetric contraction and arrive at a higher strength more quickly, which indicates that a proper previbration with small strain amplitude and higher frequency can be a benefit for accelerating the ballast settlement.

Ballast.
In Figure 3, e 0 of 0.65 and δε 1 of 6% are used for simulation. It can be observed that the initial confining pressure has a significant influence on the cyclic behavior of ballast. If a weak confinement (lower σ 3 ′ ) was applied, the material would exhibit much higher volumetric dilation and lower strength, which indicates the instability of the ballast layer. However, if a stronger confinement was applied, higher strength and enhanced densification of ballast can be achieved. Figures 4-6 show the model simulations of the undrained behavior of Firoozkuh No. 161 sand with different initial material states provided by Lashkari et al. [2]. e Firoozkuh No. 161 sand was reported to have a coefficient of uniformity of 1.9 and a mean particle size of 0.24. Test samples were 5 cm in diameter and 10 cm in height. Figure 4  sand. e effective stress paths are reasonably simulated. Figure 5 shows the model simulations of the undrained behavior of medium-dense Firoozkuh No. 161 sand, where a good agreement between the model simulations and the test results is observed. e predicted effective stresses first arrived at the phase transformation states and then mobilised towards the critical state. In addition to the test results under isotropic consolidation, Figure 6 shows the model simulation of the Firoozkuh No. 161 sand subjected to initially anisotropic consolidation. It can be found that the liquefaction and partial liquefaction behaviors of Firoozkuh No. 161 sand are well simulated.

Conclusions
is study presented an application of a stress-fractional model to simulate the constitutive behavior of granular soils, i.e., railroad ballast and Firoozkuh sand, under undrained or drained loads. Details of the model development and model implementation were provided. e main findings can be summarized as follows: (1) e developed model contained twelve parameters, which can be identified through laboratory tests.

Data Availability
e test data used to support the findings of this study are included within the article.