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 [2,3,4].

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 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 [12,13,14,15,16]. 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 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

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:

$$ \Delta \varepsilon _{ij} = \Delta \varepsilon _{ij}^{e} + \Delta \varepsilon _{ij}^{p}, $$
(1)

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

$$ \Delta \sigma '_{ij} = E_{ijkl}\Delta \varepsilon _{kl}^{e}, $$
(2)

where \(E _{\mathit{ijkl}}\) denotes the fourth-order elastic stiffness tensor, which can be defined as follows [24]:

$$ E_{ijkl} = ( K - 2G / 3 )\delta _{ij}\delta _{kl} + G ( \delta _{ik}\delta _{jl} + \delta _{il}\delta _{jk} ), $$
(3)

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]

$$\begin{aligned}& K = \frac{1 + e}{\kappa } p', \end{aligned}$$
(4)
$$\begin{aligned}& G = \frac{3(1 - 2\nu )}{2 + 2\nu } K, \end{aligned}$$
(5)

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]

$$ \Delta \varepsilon _{ij}^{p} = \frac{1}{H}n_{ij}m_{kl} \Delta \sigma '_{kl}, $$
(6)

where the plastic loading tensor (\(m _{ij}\)) is normal to the yielding surface and thus can be determined by conducting first-order derivative of the yielding surface; the plastic flow tensor (\(n _{ij}\)) is non-normal to the yielding surface for non-associative geomaterials and can be determined by conducting fractional-order 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:

$$ \Delta \sigma '_{ij} = \biggl[ E_{ijkl} - \frac{E_{ijct}n_{ct}E_{klrs}m _{rs}}{H + m_{ct}E_{ctab}n_{ab}} \biggr]\Delta \varepsilon _{kl}. $$
(7)

2.2 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 power-law function, have certain analytical solution, whereas the rest need numerical approximation [30, 31]. Following the fractional plasticity [14, 22, 32], the following well-known Caputo’s left-sided (Eq. (8)) and right-sided (Eq. (9)) fractional derivatives [33, 34] are used in this study:

$$\begin{aligned}& {}_{\sigma '_{c}}D_{\sigma '}^{\alpha } f\bigl(\sigma ' \bigr) = \frac{1}{\varGamma (n - \alpha )} \int _{\sigma '_{c}}^{\sigma '} \frac{f^{(n)}(\chi )\,d \chi }{(\sigma ' - \chi )^{\alpha + 1 - n}},\quad \sigma ' > \sigma '_{c}, \end{aligned}$$
(8)
$$\begin{aligned}& {}_{\sigma '}D_{\sigma '_{c}}^{\alpha } f\bigl(\sigma ' \bigr) = \frac{( - 1)^{n}}{ \varGamma (n - \alpha )} \int _{\sigma '}^{\sigma '_{c}} \frac{f^{(n)}( \chi )\,d\chi }{(\chi - \sigma ')^{\alpha + 1 - n}},\quad \sigma '_{c} > \sigma ', \end{aligned}$$
(9)

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 well-known 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 Cam-clay (MCC) yielding function is used

$$ f = \bigl(2p' - p'_{0}\bigr)^{2} + \biggl( \frac{2q}{M} \biggr)^{2} - p_{0} ^{\prime 2} = 0, $$
(10)

where the critical state stress ratio M can be expressed as [37]

$$ M = M_{c} \biggl[ \frac{2c^{4}}{1 + c^{4} - (1 - c^{4})\sin (3\theta )} \biggr]^{1 / 4} $$
(11)

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 non-associativity 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 critical-state 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:

$$\begin{aligned}& p'_{c} = p_{r}\exp \biggl( \frac{e_{\varGamma } - e}{\lambda } \biggr), \end{aligned}$$
(12)
$$\begin{aligned}& q_{c} = q + M\bigl(p' - p'_{c} \bigr), \end{aligned}$$
(13)

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 critical-state 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 critical-state 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 q-axis. 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 state-dependent plastic flow rule was obtained without using any plastic potential and empirical state parameters in the end.

Figure 1
figure 1

Schematic show of the possible location of current stress point

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3 State-dependent fractional model

3.1 State-dependent plastic flow

In this study, the state-dependent fractional plastic flow tensor (\(n _{ij}\)) is defined as

$$ n_{ij} = \frac{1}{\sqrt{1 + d_{g}^{2}}} \biggl[ \frac{d_{g}}{3} \delta _{ij} + \frac{3s_{ij}}{2q} \biggr], $$
(14)

where \(d _{g}\) is the stress-dilatancy 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 state-dependent behaviour of geomaterials subjected to a wide range of densities and pressures [3]. In this study, a three-dimensional state-dependent dilatancy ratio without using empirical state index and plastic potential can be derived by following Sun et al. [27, 36], where the unified stress-dilatancy ratio (\(d _{g}\)) can be obtained by using Eqs. (8) and (9) to perform fractional-order derivatives on the MCC surface such that

$$ \begin{aligned}[b] d_{g} &= - \frac{{}_{p'}D_{p'_{c}}^{\alpha } f(p')}{{}_{q_{c}}D_{q} ^{\alpha } f(q)} = - \frac{{}_{p'_{c}}D_{p'}^{\alpha } f(p')}{{}_{q}D _{q_{c}}^{\alpha } f(q)} \\ &= M^{1 + \alpha } \frac{(p' - p'_{c}) + (2 - \alpha )(p'_{c} - p'_{0}/2)}{(q - q_{c}) + (2 - \alpha )q_{c}}, \end{aligned} $$
(15)

where the material flow is found to be influenced by several factors, including the Lode angle (θ) via M, current stress \((p', q)\) and critical-state 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 state-dependent plastic flow is mathematically developed in this study. By using Eq. (15), the effect of α on the stress-dilatancy behaviour of granular soil during shearing can be obtained. As shown in Sun et al. [36],the stress-dilatancy curve exhibited clockwise rotation coupled with a downward shifting as α increased. It reduces to the classical MCC stress-dilatancy model [10] with \(\alpha = 1\).

It should be also noted that there are only two chances for the stress-dilatancy 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 stress-dilatancy 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:

$$ \alpha = \frac{2M^{2}p'_{ct} - 2\eta _{t}^{2}p'_{t}}{2M^{2}p'_{ct} - M ^{2}p'_{t} - \eta _{t}^{2}p'_{t}}, $$
(16)

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 state-dependent plastic flow and ensured the variation of the phase transformation line with material state. Therefore, a state-dependent plastic flow is analytically developed.

3.2 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 stress-dilatancy equation can be rearranged as follows:

$$ \begin{aligned}[b] d_{g} &= M^{1 + \alpha } \frac{(I_{p} - 1) + (2 - \alpha ) [ 1 - I _{p}/2 - I_{p}\eta ^{2} / (2M^{2}) ]}{(1 - I_{p})M + (2 - \alpha ) [ q/p'_{c} + M(I_{p} - 1) ]} \\ &= M^{\alpha } \frac{ [ \alpha /2 - (1 - \alpha /2)\eta ^{2} / M ^{2} ]I_{p} + (1 - \alpha )}{(1 - I_{p}) + (2 - \alpha ) [ q/Mp'_{c} + (I_{p} - 1) ]}, \end{aligned} $$
(17)

where \(I_{p} = p'/p_{c}\) is the state pressure index defined by Wang et al. [8]. A mathematical connection between the derived stress-dilatancy 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 state-dependent stress-dilatancy equation by conducting fractional derivative of other yielding surfaces.

3.3 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.

$$ \bar{f} = \bigl(2\bar{p}' - \bar{p}'_{0} \bigr)^{2} + \biggl( \frac{2\bar{q}}{M} \biggr)^{2} - \bar{p}_{0}^{\prime 2} = 0, $$
(18)

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

$$\begin{aligned}& \bar{p}' = \rho \bar{p}'_{0} \end{aligned}$$
(19)
$$\begin{aligned}& \bar{q} = \rho \eta \bar{p}'_{0}, \end{aligned}$$
(20)

in which the stress ratio η can be defined by using the radial mapping rule [39] as follows:

$$ \eta = \frac{q}{p'} = \frac{\bar{q}}{\bar{p}'}. $$
(21)

In bounding surface plasticity [48], the loading tensor (\(m _{ij}\)) is normal to the bounding surface and therefore can be obtained by conducting first-order derivative of the bounding surface function as follows:

$$ m_{ij} = \frac{1}{\sqrt{1 + d_{f}^{2}}} \biggl[ \frac{d_{f}}{3} \delta _{ij} + \frac{3s_{ij}}{2q} \biggr], $$
(22)

where the loading ratio \(d _{f}\) is formulated as

$$ d_{f} = \frac{M^{2} - \eta ^{2}}{2\eta }. $$
(23)

Further substituting Eqs. (19)–(21) into Eq. (18), the scalar (ρ) which determines the image stress point can be obtained as follows:

$$ \rho = \frac{1}{1 + (\eta /M)^{2}}. $$
(24)

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:

$$ \bar{p}'_{0i} = 2p_{r}\exp \biggl[ \frac{e_{\varGamma } - e_{0} - \kappa \ln p'_{ic}}{\lambda - \kappa } \biggr], $$
(25)

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:

$$ \bar{p}'_{0} = \bar{p}'_{0i}\exp \biggl( \frac{1 + e_{0}}{\lambda - \kappa } \varepsilon _{v}^{p} \biggr). $$
(26)

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

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]:

$$ H = H_{b} + H_{\delta }, $$
(27)

where \(H_{b}\) is determined by applying consistency condition on the bounding surface that experiences isotropic hardening:

$$ H_{b} = - \frac{\partial \bar{f}}{\partial \bar{p'}_{0}}\frac{\partial \bar{p'}_{0}}{\partial \varepsilon _{v}^{p}}\frac{n_{v}}{ \Vert \frac{ \partial \bar{f}}{\partial \bar{\boldsymbol{\sigma }}} \Vert } = \frac{1 + e_{0}}{\lambda - \kappa } \frac{\bar{p'}M_{c}^{2}d_{g} / \sqrt{d _{g}^{2} + 1}}{\sqrt{(2\rho - 1)^{2}M_{c}^{4} + 4\rho ^{2}\eta ^{2}}}. $$
(28)

It is easy to find that \(H _{b}\) is state-dependent 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]:

$$ H_{\delta } = h_{0}p'\frac{1 + e_{0}}{\lambda - \kappa } \frac{\delta }{\delta _{\max } - \delta }, $$
(29)

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 \(p' = p'_{c}\), \(q = q_{c}\), \(\bar{p}'_{0} = p'_{0}\).

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.

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 \(\nu = 0.25\).

Figure 2
figure 2

Effect of \(M _{c}\) on the stress-strain behaviour of granular soil

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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 \(\nu = 0.25\).

Figure 3
figure 3

Effect of λ on the stress-strain behaviour of granular soil

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Figure 4
figure 4

Effect of \(e_{\varGamma } \) on the stress-strain behaviour of granular soil

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There is one new parameter, the fractional order α, that is distinct from the ones in the traditional plasticity model. It controls the state-dependent plastic flow of granular soil. Thus, it can be obtained by using the stress-dilatancy 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 \(\nu = 0.25\).

Figure 5
figure 5

Effect of α on the stress-strain behaviour of granular soil

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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 stress-strain 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 \(\nu = 0.25\).

Figure 6
figure 6

Effects of \(h _{1}\) and \(h _{2}\) on the stress-strain behaviour of granular soil

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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.

Figure 7
figure 7

Effect of κ on the stress-strain behaviour of granular soil

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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]:

$$ \nu \approx - \frac{\varepsilon _{3}}{\varepsilon _{1}}. $$
(30)

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 stress-strain 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.

Table 1 Model parameters
Full size table

5 Model validation

This section demonstrates the model’s ability to capture the state-dependent 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 stress-strain 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.

Figure 8
figure 8

Model predictions of the drained behaviour of Tacheng rockfill with \(e _{0} = 0.189\) (data from Xiao et al. [12])

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Figure 9
figure 9

Model predictions of the drained behaviour of Tacheng rockfill with \(e _{0} = 0.244\) (data from Xiao et al. [12])

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Figure 10
figure 10

Model predictions of the drained behaviour of Tacheng rockfill with \(e _{0} = 0.285\) (data from Xiao et al. [12])

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Figure 11
figure 11

Model predictions of the drained behaviour of Tacheng rockfill with \(e _{0} = 0.317\) (data from Xiao et al. [12])

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Figure 12
figure 12

Model predictions of the drained behaviour of Sacramento River sand [51]

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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.

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 sub-round 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 state-dependent stress-dilatancy relationship.

Figure 13
figure 13

Model predictions of the undrained behaviour of Sacramento River sand [51]

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Figure 14 presents the simulation results of undrained constitutive behaviour of uniform Ottawa sand [50] that mainly consisted of round/sub-round 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.

Figure 14
figure 14

Model predictions of the undrained behaviour of Ottawa sand [50]

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6 Conclusions

It was found that the stress-strain behaviour of granular soil was state-dependent. 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 state-dependent fractional plasticity model without using predefined state variable and plastic potential was proposed by conducting fractional-order 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. (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. (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. (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. (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. (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.