 Research
 Open Access
 Published:
Fractionalorder modelling of statedependent nonassociated behaviour of soil without using state variable and plastic potential
Advances in Difference Equations volume 2019, Article number: 83 (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 statedependent nonassociated behaviour of granular soils without using predefined plastic potential and state variable. A novel statedependent nonassociated model for granular soils is mathematically developed by incorporating the stressfractional operator into the bounding surface plasticity. Unlike previous studies using empirical state variables, the soil state and nonassociativity in this study are considered via analytical solution, where a statedependent plastic flow rule and the corresponding hardening modulus without using additional plastic potentials are obtained. Possible mathematical connection with a wellknown empirical state variable is also discussed. The nonassociativity between plastic flow and loading directions as well as material hardening is found to be controlled by the fractionalderivative 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.
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 stressstrain behaviour of granular soils with different initial densities or subjected to different confining pressures [2,3,4].
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 statedependent nonassociated plastic flow rule (or stressdilatancy relationship) has been of the utmost importance in recent years. One of the popular approaches was to modify the existing stressdilatancy equations, for example, the Camclay (CC) stressdilatancy equation [10] and Rowe’s stressdilatancy equation [11], by empirically incorporating ψ [12,13,14,15,16]. 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 [18,19,20], and stressstrain [14, 21] behaviour of materials. To reduce the number of model parameters without the loss of modelling capability, Sun and Shen [21] proposed a nonassociated plastic flow rule for granular soil by simply conducting fractionalorder 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 nonnormality increased as the fractional order (α) decreased [14, 22, 23]. To consider the state dependence, the statedependent fractional plasticity model was then proposed [14] by empirically incorporating ψ, which however made the parameters of the obtained stressdilatancy equation lack physical meaning.
This study attempts to theoretically investigate the statedependent nonassociated stressstrain behaviour of granular soils. A statedependent nonassociated 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 statedependent 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 statedependent constitutive model without using plastic potential, where the statedependent 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.
Notations and definitions
Constitutive relations
Following the conventional assumption in soil mechanics, compressive stress and strain are considered as positive while the extensive ones are negative. All the stress mentioned in this study is effective stress obtained by using Terzhgi’s effective stress principle. In the elastoplastic model, there are four main parts, i.e. the elastic stiffness tensor (E), plastic loading tensor (m), plastic flow tensor (n) and hardening modulus (H). Accordingly, the total strain (\(\varepsilon _{ij}\)) can be decomposed into the following elastic (\(\varepsilon _{ij}^{e}\)) and plastic (\(\varepsilon _{ij}^{p}\)) parts:
where \(i, j = 1, 2, 3\). Δ indicates increment, while the superscripts e and p imply the elastic and plastic components, respectively. Based on Hooke’s law, the incremental elastic strain tensor (\(\Delta \varepsilon _{ij}^{e}\)) can be correlated to the incremental effective stress tensor (\(\Delta \sigma '_{ij}\)) by
where \(E _{\mathit{ijkl}}\) denotes the fourthorder elastic stiffness tensor, which can be defined as follows [24]:
in which \(\delta _{ij}\) is the Kronecker delta. K and G are the bulk and shear moduli, respectively, which can be expressed by using [25]
where κ is the gradient of the swelling line in the \(e  \ln p'\) plane; e is the current void ratio of the sample; ν is Poisson’s ratio. \(p' = \sigma '_{ij}\delta _{ij} / 3\) is the mean effective principal stress. In addition, the generalised shear stress \(q = \sqrt{3 / 2s_{ij}s_{ij}}\), in which \(s_{ij} = \sigma '_{ij}  \sigma '_{kk}\delta _{ij} / 3\), is the deviatoric stress tensor. The corresponding volumetric strain (\(\varepsilon _{v}\)) can be defined as \(\varepsilon _{v} = \varepsilon _{ij}\delta _{ij}\), while the generalised shear strain \(\varepsilon _{s} = \sqrt{2 / 3e_{ij}e_{ij}}\), where \(e_{ij} = \varepsilon _{ij}  \varepsilon _{v}\delta _{ij}\), is the deviatoric strain tensor.
In plasticity approach, the incremental plastic strain tensor (\(\Delta \varepsilon _{ij}^{p}\)) can be related to incremental effective stress tensor such that [26]
where the plastic loading tensor (\(m _{ij}\)) is normal to the yielding surface and thus can be determined by conducting firstorder derivative of the yielding surface; the plastic flow tensor (\(n _{ij}\)) is nonnormal to the yielding surface for nonassociative geomaterials and can be determined by conducting fractionalorder derivative of the yielding surface, as shown in Sumelka and Nowak [23] and Sun et al. [27]; the hardening modulus (H) for granular soils involves both the size hardening and a relative position of the yielding surface with respect to the bounding surface, and it will be defined later. Combining Eqs. (2)–(6), the following elastoplastic constitutive relation can be given:
Fractional derivative and yielding surface
There are many different definitions of fractional derivatives [17,18,19,20, 28, 29], where each of them is in integral form and somehow complex. Therefore, only some of the functions, for example, the powerlaw function, have certain analytical solution, whereas the rest need numerical approximation [30, 31]. Following the fractional plasticity [14, 22, 32], the following wellknown Caputo’s leftsided (Eq. (8)) and rightsided (Eq. (9)) fractional derivatives [33, 34] are used in this study:
where D (\(= \partial ^{\alpha } /\partial \sigma ^{\prime \,\alpha } \)) denotes partial derivation to obtain the fractional stress gradient on the yielding function f. It is noted that Caputo’s fractional derivative has singular kernel, which may not be applicable in a more specific case, e.g. capturing material heterogeneities and structures with different scales. A possible solution of this limit is to propose a new fractional derivative without singular kernel, for example, the Caputo–Fabrizio derivative [35] and Yang–Srivastava–Machado derivative [28]. It is worthwhile pointing out that the Yang–Srivastava–Machado fractional derivative [28] is a wellknown extension of the Riemann–Liouville fractional derivative with singular kernel, which has great potential in solving local thermal and mechanical problems. However, in this study, only mechanical phenomena related to plasticity are considered; therefore, the original Caputo’s fractional derivative [33, 34] is sufficient and effective [35]. Following Sun et al. [27, 36], the modified Camclay (MCC) yielding function is used
where the critical state stress ratio M can be expressed as [37]
in which \(M _{c}\) denotes the critical state stress ratio for triaxial compression; c denotes the ratio between the critical state stress ratio for triaxial extension (\(M _{e}\)) and \(M _{c}\). The Lode angle (θ) is expressed as \(\theta = \arccos \{ 9s_{ij}s_{jk}s _{ki} / [ 2(3s_{lr}s_{lr} / 2)^{3 / 2} ] \} / 3\). \(\theta \in [  \pi / 6,\pi / 6]\), where \(\theta = \pi / 6\) corresponds to triaxial compression, whereas \(\theta =  \pi / 6\) corresponds to triaxial extension. \(p'_{0}\) controls the size (hardening) of the yielding surface. Furthermore, in Eq. (9), the gamma function is defined as follows: \(\varGamma (x) = \int _{0}^{\infty } e^{  \tau } \tau ^{x  1}\,d \tau \). α is the fractional order and should be at least less than two due to the thermodynamic restriction for a positive plastic dissipation [27]. As discussed in Sun and Shen [21], α represents the extent of nonassociativity between plastic flow and loading directions as well as the degree of state dependence of material deformation. \(\sigma '\) is the current effective stress, while \(\sigma '_{c}\) is the corresponding criticalstate stress. The use of \(\sigma '_{c}\) for integral limit is attributed to the basic assumption from the critical state soil mechanics [10]: upon shearing, soil would deform continuously and finally approach the critical state represented by the critical state lines (CSLs) in the \(e  \ln p'\) and \(p'  q\) planes, respectively, as follows:
where \(p_{r}\) is the unit pressure; λ is the gradient of the CSL in the \(e  \ln p'\) plane. \(e_{\varGamma } \) denotes the intercept of the CSL at \(p'= 1\mbox{ kPa}\). \(p'\) and \(p'_{c}\) are the mean effective principal stresses at the current and the corresponding critical states, respectively, while q and \(q_{c}\) are the deviator stresses at the current and corresponding critical states, respectively. It should be noted that in this study we connect the current stress point \((p', q)\) with the corresponding CSL by using the critical state stress ratio M as shown in Fig. 1. Before soil reaches the critical state, there are three possible locations of the current stress point in relation to the CSL in the \(p'  q\) plane, i.e. on the “dry” side (above), on the “wet” side (below) and on the CSL. If the current state is on the CSL, then \(p' = p'_{c}\) and \(q = q_{c}\), where only one criticalstate stress point is indicated, and the material is currently undergoing critical state flow. However, in the rest two cases, it is observed that \(p'_{c}\) and \(q_{c}\) should represent two independent criticalstate stress points (A and B) on the CSL. As shown in Fig. 1, stress point A implies a soil with dilation trend, while stress point B represents a soil with contraction trend. \(p'_{c}\) is obtained by extending a horizontal parallel line from the current stress points (A or B) with regard to the \(p'\)axis and intersecting the CSL, while \(q _{c}\) is obtained by intersecting a vertical parallel line extended from the current stress points (A or B) with regard to the qaxis. The relative location of the current stress point to the critical state line defines which equation, (8) or (9), will be used. More specifically, if \(p' > p'_{c}\) or \(q > q_{c}\), Eq. (8) should be used and vice versa. However, as demonstrated in [36], even if different positions (or definitions) of the current stress point (or fractional derivatives) were used, a unique statedependent plastic flow rule was obtained without using any plastic potential and empirical state parameters in the end.
Statedependent fractional model
Statedependent plastic flow
In this study, the statedependent fractional plastic flow tensor (\(n _{ij}\)) is defined as
where \(d _{g}\) is the stressdilatancy ratio, which has several different definitions by different researchers, for example, the CC expression [26, 38], elliptic expression [12, 39], Rowe’s expression [6, 40], etc. However, no matter which kind of dilatancy equation was used in the classical plasticity model, an empirical state index was necessarily incorporated for the unified modelling of statedependent behaviour of geomaterials subjected to a wide range of densities and pressures [3]. In this study, a threedimensional statedependent dilatancy ratio without using empirical state index and plastic potential can be derived by following Sun et al. [27, 36], where the unified stressdilatancy ratio (\(d _{g}\)) can be obtained by using Eqs. (8) and (9) to perform fractionalorder derivatives on the MCC surface such that
where the material flow is found to be influenced by several factors, including the Lode angle (θ) via M, current stress \((p', q)\) and criticalstate stress \((p'_{c}, q _{c})\); most importantly, it is also determined by the stress distances from the current state to the corresponding critical state (\(p'  p'_{c}\) and \(q  q _{c}\)). It is easy to find that \(d_{g} = 0\) at critical state where \(p' = p'_{c}\) and \(q = q_{c}\), indicating no plastic volumetric strain. The fractional gradient on MCC, by using Eqs. (8) and (9), intrinsically considers the state information of soil during shearing. Therefore, unlike classical isotropic and anisotropic plasticity models [41,42,43,44,45] where the influence of material state on plastic flow requires the empirical incorporation of a state parameter (e.g. ψ), a statedependent plastic flow is mathematically developed in this study. By using Eq. (15), the effect of α on the stressdilatancy behaviour of granular soil during shearing can be obtained. As shown in Sun et al. [36],the stressdilatancy curve exhibited clockwise rotation coupled with a downward shifting as α increased. It reduces to the classical MCC stressdilatancy model [10] with \(\alpha = 1\).
It should be also noted that there are only two chances for the stressdilatancy ratio \(d_{g} = 0\). One is at the critical state and the other is at the phase transformation state. At critical state where \(p' = p'_{c}\) and \(q = q_{c}\), the stressdilatancy ratio is automatically equal to zero, indicating no plastic volumetric strain. At the phase transformation state, \(d _{g} = 0\) can be ensured by a proper value of α, which can be defined by rearranging Eq. (16) as follows:
in which \(p'_{ct} = p_{r}\exp ((e_{\varGamma }  e_{t})/\lambda )\); \(e_{t}\), \(p'_{t}\) and \(\eta _{t}\) are the void ratio, effective mean principal stress and stress ratio, respectively, at the phase transformation state. Due to the use of the vertical (\(q  q _{c}\)) and horizontal (\(p'  p'_{c}\)) distances through α, the phase transformation line is not fixed but changed with the initial material state. For further clarification, an analogy of the function of α with that of the state parameter (m) in Li and Dafalias [1] can be made, where m was used to capture the statedependent plastic flow and ensured the variation of the phase transformation line with material state. Therefore, a statedependent plastic flow is analytically developed.
Possible mathematical connection with \(I_{p}\)
Another interesting finding on the mathematical connection with the empirical state variable \(I _{p}\) [8] can be made by further substituting Eq. (13) into Eq. (15). Then, the above stressdilatancy equation can be rearranged as follows:
where \(I_{p} = p'/p_{c}\) is the state pressure index defined by Wang et al. [8]. A mathematical connection between the derived stressdilatancy ratio and the empirical state pressure index \(I _{p}\) is observed. Unlike other studies [46] that empirically incorporated \(I _{p}\), the current study reflects the dependence of state pressure from the natural mathematical derivation. However, it should be noted that the relation between \(d _{g}\) and \(I _{p}\) in particular ties this model to other models [1, 13,14,15, 41, 47] that used \(I _{p}\) or ψ [9] with the added advantage of deriving such relation by conducting fractional derivative of a specific (e.g. MCC) yield surface. Does this connection also generally exist when using other yielding surfaces? Further studies need to be carried out. Nevertheless, it should be emphasized that due to the integral definition of the fractional derivative, the fractional approach proposed in this study intrinsically considers the state information from current state to critical state, as indicated in the initial definition of the fractional derivatives shown in Eqs. (8) and (9). One can also theoretically derive a statedependent stressdilatancy equation by conducting fractional derivative of other yielding surfaces.
Bounding surface and loading direction
For the sake of simplicity, the bounding surface (f̄) is assumed to have the same shape as the yielding surface, i.e.
where \(\bar{p}'_{0}\) is the intercept of the bounding surface with the abscissa, controlling the size of the bounding surface. The image stress point \((\bar{p}', \bar{q})\) on the bounding surface can be expressed by employing a scalar ρ as
in which the stress ratio η can be defined by using the radial mapping rule [39] as follows:
In bounding surface plasticity [48], the loading tensor (\(m _{ij}\)) is normal to the bounding surface and therefore can be obtained by conducting firstorder derivative of the bounding surface function as follows:
where the loading ratio \(d _{f}\) is formulated as
Further substituting Eqs. (19)–(21) into Eq. (18), the scalar (ρ) which determines the image stress point can be obtained as follows:
In addition, the position of the initial bounding surface (\(\bar{p}'_{0i}\)) can be further obtained by intersecting the normal compression and swelling lines in the \(e  p'\) plane as follows:
where \(e _{0}\) is the initial void ratio prior to shearing. \(p'_{ic}\) is the initial confining pressure. The evolution of the bounding surface (\(\bar{p}'_{0}\)) can be further obtained as follows:
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.
Hardening modulus
According to Dafalias [48], the hardening modulus H is determined by both the size (hardening) of the bounding surface and the distance between the loading and bounding surfaces. H was also observed to depend on the material state where a state variable was usually empirically incorporated, for example, in Wang et al. [8]. It is usually decomposed into two components [12, 39, 47]:
where \(H_{b}\) is determined by applying consistency condition on the bounding surface that experiences isotropic hardening:
It is easy to find that \(H _{b}\) is statedependent due to the dependence of \(d _{g}\) on material state. \(H_{\delta } \) is related to the ratio between the distance (δ) from the current stress point to the image stress point and the distance (\(\delta _{\max } \)) from the stress origin to the image stress point [21, 39]:
where \(h_{0} = h_{1}e  h_{2}\); \(h _{1}\) and \(h _{2}\) are material constants. As expected, the hardening modulus \(H = + \infty \) at the load onset where \(\delta _{\max }  \delta \to 0\), \(\eta = 0\) and \(\dot{\eta } = 0\), indicating a state of no plastic strain. \(H = 0\) at the critical state where \(p' = p'_{c}\), \(q = q_{c}\), \(\bar{p}'_{0} = p'_{0}\).
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 statedependent model, which can be all determined from traditional triaxial tests. Detailed elaborations on parameter identification and sensitivity analysis are given below.
Similar to the classical plasticity models, there are four critical state parameters, i.e. \(M _{c}\), λ, \(e_{\varGamma } \) and c. The critical state stress ratio (\(M _{c}\)) is determined by measuring the gradient of the critical state line in the \(p'  q\) plane. With the increase of \(M _{c}\), the predicted shear strength of granular soil increases while the volumetric dilation decreases, as shown in Fig. 2. Note that the other model parameters used for simulations in Fig. 2 are: \(\lambda = 0.057\), \(e_{\varGamma } = 0.64\), \(c = 1\), \(\alpha = 1.12\), \(h _{1} = h _{2} = 2.5\), \(\kappa = 0.0013\) and \(\nu = 0.25\).
λ and \(e_{\varGamma } \) are the critical state parameters in the \(e  \ln p'\) plane. λ can be obtained by measuring the gradient of the critical state line in the \(e  \ln p'\) plane, while \(e_{\varGamma } \) can be determined by the intercept of the critical state line at \(p' = 1\). c can be determined by further conducting a triaxial extension test to obtain the critical state stress ratio (\(M _{e}\)) for triaxial extension and then \(c = M _{e} /M _{c}\). As shown in Fig. 3, with the increase of λ, the predicted peak stress and volumetric dilation decrease. However, as shown in Fig. 4, with the increase of \(e_{\varGamma } \), the predicted peak stress and volumetric dilation increase. Note that the other model parameters used for simulations in Figs. 3 and 4 are: \(M _{c} = 1.7\), \(c = 1\), \(\alpha = 1.12\), \(h _{1} = h _{2} = 2.5\), \(\kappa = 0.0013\) and \(\nu = 0.25\).
There is one new parameter, the fractional order α, that is distinct from the ones in the traditional plasticity model. It controls the statedependent plastic flow of granular soil. Thus, it can be obtained by using the stressdilatancy ratio at the phase transformation state, i.e. Equation (16). As shown in Fig. 5, with the increase of α, the predicted peak stress decreases while the volumetric dilation increases. Samples modelled by a higher α reach critical state more quickly; a transition from the strain softening behaviour to strain hardening behaviour is also observed when α increases from 1.0 to 1.4. Note that the other model parameters used for simulations in Fig. 5 are: \(M _{c} = 1.7\), \(c = 1\), \(\lambda = 0.057\), \(e_{\varGamma } = 0.64\), \(h _{1} = h _{2} = 2.5\), \(\kappa = 0.0013\) and \(\nu = 0.25\).
The hardening parameter \(h _{0}\) is correlated with the void ratio through \(h _{1}\) and \(h _{2}\). As illustrated in Li and Dafalias [1] as well as Sun and Shen [21], they can be obtained by fitting the \(\varepsilon _{1}  q\) relationship of samples with different void ratios. The effect of \(h _{0}\) on the stressstrain behaviour of granular soil can be found in Fig. 6, where an increasing shear strength coupled with an increasing volumetric dilation is found. Note that the other model parameters used for simulations in Fig. 6 are: \(M _{c} = 1.7\), \(c = 1\), \(\lambda = 0.057\), \(e_{\varGamma } = 0.64\), \(\alpha = 1.12\), \(\kappa = 0.0013\) and \(\nu = 0.25\).
The elastic constant κ can be determined by measuring the gradient of the swelling line in the \(e  \ln p'\) plane. As observed in Fig. 7, with the increase of κ, the peak deviator stress increases, while the initial deviator stress increases more rapidly with a lower κ. In addition, a distinct difference is observed in the relationship between \(\varepsilon _{v}\) and \(\varepsilon _{1}\). At the initial loading stage where \(\varepsilon _{1}\) is small, a large volumetric contraction is observed for soils with high κ. However, the effect of κ on \(\varepsilon _{v}\) gradually becomes insignificant when \(\varepsilon _{1}\) increases, where the plastic parts of \(\varepsilon _{1}\) and \(\varepsilon _{v}\) become dominant.
Poisson’s ratio ν defines the lateral deformation ability of the material and can be obtained by using the following equation during the initial loading stage [12]:
The effect of ν was found to be limited and thus not presented here for simplicity. However, according to the theory of elasticity, a higher lateral deformation could be observed with increasing ν. Detailed values of each model parameter used for predicting the stressstrain behaviour of different granular soils are listed in Table 1. It is noted that for simulating triaxial tests with \(\theta = \pi /6\), c will not be used.
Model validation
This section demonstrates the model’s ability to capture the statedependent constitutive behaviour of different granular soils, including rockfill and sand with different initial states. A series of triaxial tests results of Tacheng rockfill [12], Sacramento River sand [51] and Ottawa sand [50] are simulated in Figs. 8–14. Drained stressstrain behaviour of Tacheng rockfill [12] with four different initial void ratios is simulated in Figs. 8–11, while Figs. 12–14 present the model predictions of the drained and undrained triaxial behaviour of Sacramento River sand [51] and Ottawa sand [50]. It is noted that the all the test results are represented by discrete data points, while continuous lines are used for model predictions.
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 subangular 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 statedependent 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.
Lee and Seed [51] conducted a comprehensive study on the drained and undrained triaxial behaviour of Sacramento River sand under different initial conditions. The Sacramento River sand mainly consisted of subround quartz aggregates and occasional shell fragments with a \(d _{50}\) of 0.22 mm and a \(C _{u}\) equal to 1.45. The minimum and maximum void ratios were tested to be 0.61 and 1.03, respectively. Samples were prepared to have an initial diameter around 42.67 mm and a height around 103.63 mm. The initial void ratio of 0.87 is used for simulating the drained test results, whereas the initial void ratios of 0.86, 0.86, 0.86 and 0.85 are used for simulating undrained tests carried out under the confining pressures of 98, 294, 490 and 1069 kPa, respectively. Figure 12 shows the model simulations of the drained constitutive behaviour of Sacramento River sand, where a good agreement between the model simulations and the corresponding test results can be observed. Figure 13 shows the model simulations of the undrained behaviour of Sacramento River sand subjected to different initial conditions. As the axial strain increases, the simulated deviator stresses increase until reaching critical (steady) state flow, which is in good agreement with the corresponding test results, which further validates the newly developed statedependent stressdilatancy relationship.
Figure 14 presents the simulation results of undrained constitutive behaviour of uniform Ottawa sand [50] that mainly consisted of round/subround quartz aggregates. Test results of Ottawa sand with the initial void ratios of 0.793, 0.793 and 0.805 under the respective corresponding confining pressures of 348, 475 and 550 kPa are simulated. It is found that the model predictions match well with the test results. The simulated deviator stress increases initially until reaching a peak value and then decreases significantly at the critical state, indicating a state of static liquefaction of the material. Samples with the same initial void ratios approach the same deviator stress with further shearing, which can be all reasonably characterised by the proposed fractional plasticity model.
Conclusions
It was found that the stressstrain behaviour of granular soil was statedependent. To capture such state dependence, many sophisticated constitutive models have been proposed by incorporating empirical state variables into additional plastic potential functions. To simplify the modelling approach, a novel statedependent fractional plasticity model without using predefined state variable and plastic potential was proposed by conducting fractionalorder derivatives of the yielding function. Detailed identifications and sensitivity analysis of model parameters were then carried out. To further validate the developed model, a series of drained and undrained test results of different granular soils were simulated and discussed. The main findings are summarized as follows:

(1)
Without using predefined state variables and plastic potential functions, a novel statedependent stressdilatancy equation was analytically derived by using the fractionalorder plasticity theory. Dependence of the nonassociated flow on material state was modelled through rigorous mathematical definition of the fractional stress gradient.

(2)
Possible mathematical connections between the proposed statedependent dilatancy equation and the state pressure index by Wang et al. [8] were also discussed, where the dependence of state pressure on the stressdilatancy phenomenon of granular soil was analytically proved.

(3)
The extent of nonassociativity 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 statedependent fractional plasticity model can well capture the stressstrain behaviour of different granular soils subjected to a variety of loading conditions.
References
Li, X., Dafalias, Y.: Dilatancy for cohesionless soils. Geotechnique 50(4), 449–460 (2000). https://doi.org/10.1680/geot.2000.50.4.449
Dafalias, Y.F., Taiebat, M.: SANISANDZ: zero elastic range sand plasticity model. Geotechnique 66(12), 999–1013 (2016). https://doi.org/10.1680/jgeot.15.P.271
Javanmardi, Y., Imam, S.M.R., Pastor, M., Manzanal, D.: A reference state curve to define the state of soils over a wide range of pressures and densities. Geotechnique 68(2), 95–106 (2018). https://doi.org/10.1680/jgeot.16.P.136
Yang, J., Li, X.: Statedependent strength of sands from the perspective of unified modeling. J. Geotech. Geoenviron. Eng. 130(2), 186–198 (2004). https://doi.org/10.1061/(ASCE)10900241(2004)130:2(186)
Ishihara, K.: Liquefaction and flow failure during earthquakes. Geotechnique 43(3), 351–451 (1993). https://doi.org/10.1680/geot.1993.43.3.351
Wan, R., Guo, P.: A simple constitutive model for granular soils: modified stressdilatancy approach. Comput. Geotech. 22(2), 109–133 (1998). https://doi.org/10.1016/S0266352X(98)000044
Desai, C.S.: Mechanics of Materials and Interfaces: The Disturbed State Concept. CRC Press, Boca Raton (2000)
Wang, Z., Dafalias, Y., Li, X., Makdisi, F.: State pressure index for modeling sand behavior. J. Geotech. Geoenviron. Eng. 128(6), 511–519 (2002). https://doi.org/10.1061/(ASCE)10900241(2002)128:6(511)
Been, K., Jefferies, M.G.: A state parameter for sands. Geotechnique 35(2), 99–112 (1985). https://doi.org/10.1016/01489062(85)902633
Schofield, A., Wroth, P.: Critical State Soil Mechanics. McGrawHill, London (1968)
Rowe, P.W.: The stressdilatancy relation for static equilibrium of an assembly of particles in contact. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 269(1339), 500–527 (1962). https://doi.org/10.1098/rspa.1962.0193
Xiao, Y., Liu, H., Chen, Y., Jiang, J.: Bounding surface model for rockfill materials dependent on density and pressure under triaxial stress conditions. J. Eng. Mech. 140(4), 04014002 (2014). https://doi.org/10.1061/(ASCE)EM.19437889.0000702
Mortara, G.: A constitutive framework for the elastoplastic modelling of geomaterials. Int. J. Solids Struct. 63, 139–152 (2015). https://doi.org/10.1016/j.ijsolstr.2015.02.047
Sun, Y., Xiao, Y.: Fractional order plasticity model for granular soils subjected to monotonic triaxial compression. Int. J. Solids Struct. 118–119, 224–234 (2017). https://doi.org/10.1016/j.ijsolstr.2017.03.005
Najma, A., Latifi, M.: Predicting flow liquefaction, a constitutive model approach. Acta Geotech. 12(4), 793–808 (2017). https://doi.org/10.1007/s114400160517x
Tasiopoulou, P., Gerolymos, N.: Constitutive modelling of sand: a progressive calibration procedure accounting for intrinsic and stressinduced anisotropy. Geotechnique 66(9), 754–770 (2016). https://doi.org/10.1680/jgeot.15.P.284
Yang, X.: New rheological problems involving general fractional derivatives with nonsingular powerlaw kernels. Proc. Rom. Acad., Ser. A: Math. Phys. Tech. Sci. Inf. Sci. 19(1), 45–52 (2018)
Saad, K.M., Baleanu, D., Atangana, A.: New fractional derivatives applied to the Korteweg–de Vries and Korteweg–de Vries–Burger’s equations. Comput. Appl. Math. 37(4), 5203–5216 (2018)
Saad, K.M., Atangana, A., Baleanu, D.: New fractional derivatives with nonsingular kernel applied to the Burgers equation. Chaos 28(6), 063109 (2018). https://doi.org/10.1063/1.5026284
Saad, K., AlSharif, E.H.: Analytical study for time and timespace fractional Burgers’ equation. Adv. Differ. Equ. 2017(1), 300 (2017). https://doi.org/10.1186/s1366201713580
Sun, Y., Shen, Y.: Constitutive model of granular soils using fractional order plastic flow rule. Int. J. Geomech. 17(8), 04017025 (2017). https://doi.org/10.1061/(ASCE)GM.19435622.0000904
Sumelka, W., Nowak, M.: Nonnormality and induced plastic anisotropy under fractional plastic flow rule: a numerical study. Int. J. Numer. Anal. Methods Geomech. 40(5), 651–675 (2016). https://doi.org/10.1002/nag.2421
Sumelka, W., Nowak, M.: On a general numerical scheme for the fractional plastic flow rule. Mech. Mater. 116, 120–129 (2018). https://doi.org/10.1016/j.mechmat.2017.02.005
Heidarzadeh, H., Oliaei, M.: Development of a generalized model using a new plastic modulus based on bounding surface plasticity. Acta Geotech. 13(4), 925–941 (2018). https://doi.org/10.1007/s1144001705990
Wheeler, S.J., Näätänen, A., Karstunen, M., Lojander, M.: An anisotropic elastoplastic model for soft clays. Can. Geotech. J. 40(2), 403–418 (2003). https://doi.org/10.1139/t02119
Pastor, M., Zienkiewicz, O.C., Chan, A.H.C.: Generalized plasticity and the modelling of soil behaviour. Int. J. Numer. Anal. Methods Geomech. 14(3), 151–190 (1990). https://doi.org/10.1002/nag.1610140302
Sun, Y., Gao, Y., Zhu, Q.: Fractional order plasticity modelling of statedependent behaviour of granular soils without using plastic potential. Int. J. Plast. 102, 53–69 (2018). https://doi.org/10.1016/j.ijplas.2017.12.001
Yang, X., Srivastava, H.M., Machado, J.A.T.: A new fractional derivative without singular kernel: application to the modelling of the steady heat flow. Therm. Sci. 20(2), 753–756 (2016)
Yang, X., Gao, F., Machado, J.A.T., Baleanu, D.: A new fractional derivative involving the normalized sinc function without singular kernel. Eur. Phys. J. Spec. Top. 226, 3567–3575 (2017)
Sun, Y., Gao, Y., Chen, C.: Criticalstate fractional model and its numerical scheme for isotropic granular soil considering statedependence. Int. J. Geomech. 13(9), 04018202 (2018). https://doi.org/10.1061/(ASCE)GM.19435622.0001353
Yang, X.J., Gao, F., Ju, Y., Zhou, H.W.: Fundamental solutions of the general fractionalorder diffusion equations. Math. Methods Appl. Sci. 41(18), 9312–9320 (2018). https://doi.org/10.1002/mma.5341
Sumelka, W.: Fractional viscoplasticity. Mech. Res. Commun. 56, 31–36 (2014). https://doi.org/10.1016/j.mechrescom.2013.11.005
Caputo, M.: Linear models of dissipation whose Q is almost frequency independentII. Geophys. J. Int. 13(5), 529–539 (1967). https://doi.org/10.1111/j.1365246X.1967.tb02303.x
Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Mathematics in Science and Engineering, vol. 198. Academic Press, San Diego (1998)
Caputo, M., Fabrizio, M.: A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1, 73–85 (2015). https://doi.org/10.12785/pfda/010201
Sun, Y., Gao, Y., Shen, Y.: Mathematical aspect of the statedependent stressdilatancy of granular soil under triaxial loading. Geotechnique 69(2), 158–165 (2018). https://doi.org/10.1680/jgeot.17.t.029
Sheng, D., Sloan, S.W., Yu, H.S.: Aspects of finite element implementation of critical state models. Comput. Mech. 26(2), 185–196 (2000). https://doi.org/10.1007/s004660000166
Kan, M.E., Taiebat, H.A.: Application of advanced bounding surface plasticity model in static and seismic analyses of Zipingpu Dam. Can. Geotech. J. 53(3), 455–471 (2015). https://doi.org/10.1139/cgj20150120
Bardet, J.P.: Bounding surface plasticity model for sands. J. Eng. Mech. 112(11), 1198–1217 (1986). https://doi.org/10.1061/(ASCE)07339399(1986)112:11(1198)
Yu, H., Khong, C., Wang, J.: A unified plasticity model for cyclic behaviour of clay and sand. Mech. Res. Commun. 34(2), 97–114 (2007). https://doi.org/10.1016/j.mechrescom.2006.06.010
Yin, Z.Y., Chang, C.S., Hicher, P.Y., Karstunen, M.: Micromechanical analysis of kinematic hardening in natural clay. Int. J. Plast. 25(8), 1413–1435 (2009). https://doi.org/10.1016/j.ijplas.2008.11.009
Yin, Z.Y., Chang, C.S., Hicher, P.Y.: Micromechanical modelling for effect of inherent anisotropy on cyclic behaviour of sand. Int. J. Solids Struct. 47(14–15), 1933–1951 (2010). https://doi.org/10.1016/j.ijsolstr.2010.03.028
Dafalias, Y.F.: Must critical state theory be revisited to include fabric effects? Acta Geotech. 11(3), 479–491 (2016). https://doi.org/10.1007/s1144001604410
Leoni, M., Karstunen, M., Vermeer, P.A.: Anisotropic creep model for soft soils. Geotechnique 58(3), 215–226 (2008). https://doi.org/10.1680/geot.2008.58.3.215
Seidalinov, G., Taiebat, M.: Bounding surface SANICLAY plasticity model for cyclic clay behavior. Int. J. Numer. Anal. Methods Geomech. 38(7), 702–724 (2014). https://doi.org/10.1002/nag.2229
Xiao, Y., Liu, H., Chen, Y., Jiang, J.: Bounding surface plasticity model incorporating the state pressure index for rockfill materials. J. Eng. Mech. 140(11), 04014087 (2014). https://doi.org/10.1061/(ASCE)EM.19437889.0000802
Chen, Q., Indraratna, B., Carter, J., Rujikiatkamjorn, C.: A theoretical and experimental study on the behaviour of lignosulfonatetreated sandy silt. Comput. Geotech. 61, 316–327 (2014). https://doi.org/10.1016/j.compgeo.2014.06.010
Dafalias, Y.F.: Bounding surface plasticity. I: mathematical foundation and hypoplasticity. J. Eng. Mech. 112(9), 966–987 (1986). https://doi.org/10.1061/(ASCE)07339399(1986)112:9(966)
Sun, Y., Indraratna, B., Carter, J.P., Marchant, T., Nimbalkar, S.: Application of fractional calculus in modelling ballast deformation under cyclic loading. Comput. Geotech. 82, 16–30 (2017). https://doi.org/10.1016/j.compgeo.2016.09.010
Sasitharan, S., Robertson, P.K., Sego, D.C., Morgenstern, N.R.: Stateboundary surface for very loose sand and its practical implications. Can. Geotech. J. 31(3), 321–334 (1994). https://doi.org/10.1139/t94040
Lee, K.L., Seed, H.B.: Drained strength characteristics of sands. J. Soil Mech. Found. Div. 93(6), 117–141 (1967)
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.
Author information
Authors and Affiliations
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.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
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.
About this article
Cite this article
Sun, Y., Zheng, C. Fractionalorder modelling of statedependent nonassociated behaviour of soil without using state variable and plastic potential. Adv Differ Equ 2019, 83 (2019). https://doi.org/10.1186/s1366201920405
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s1366201920405
Keywords
 Fractional plasticity
 Constitutive relations
 State dependence
 Granular soils