1 Introduction

Void ratio (or porosity) is the most widely used index by geotechnical engineers and soil mechanists to characterize the “state” of granular soils. The celebrated critical state soil mechanics framework uses void ratio to define critical state which further distinguishes dense and loose soils [36]. Void ratio (e) informs engineers how well the soil is packed [3] and its potential for liquefaction [33]. On the other hand, fabric structure characterized by the directional statistics of particles, voids, and contact normal vectors is also tightly related to the macro-behaviour of granular soils including anisotropy [2, 60], non-coaxiality [8], critical state [23], and liquefaction [49]. However, despite both void ratio and fabric tensors are quantitative descriptors of soil internal structure, their interconnections are seldomly studied. A specific question is, can the fabric data be used to deduce the void ratio of granular materials? First of all, it is straightforward to see that void ratio is proportional to the hydrostatic component of void-based fabric tensors [24]. Such clear relation does not exist for particle- or contact-based fabric tensors. Many studies in the field of granular physics, powder technology, and chemical engineering have been devoted to establishing a relation between e and the coordination number (Z), i.e., the first invariant of the non-normalized contact fabric tensor. For monodisperse granular assemblies, an e–Z relation can be pinned down by considering several idealized packings including cubic (Z = 6, e = 0.9099), orthorhombic (Z = 8, e = 0.6540), tetragonal–sphenoidal (Z = 10, e = 0.4533), and rhombohedral (Z = 12, e = 0.3503). A number of empirical e–Z equations have been also proposed for bi-disperse or polydisperse granular assemblies based on experiments and DEM simulations of gravitational stable or compressed specimens [9, 17, 31, 40, 42, 55].

For general stress paths that involve shearing, the relation between Z and e is no longer unique. This can be demonstrated by considering an undrained test: if each e uniquely corresponds to one Z value, Z must be a constant during undrained shearing, since e is kept constant. This is not supported by DEM experiments, showing that undrained samples can liquefy where Z drops sharply [10, 29, 47]. It is thus inferred that e should at least also be a function of the fabric anisotropy (F), or equivalently the second invariant of contact fabric tensor. Rothenburg and Kruyt [37] have similarly pointed out that the relationship between Z and e is affected by the anisotropy of contact orientations. Kruyt [20] showed that Z evolves with both volumetric and shear strains, implying that e may be a function of both Z and fabric anisotropy. Huang et al. [15] shows that the critical state e–Z relation is not unique and is dependent on the intermediate principal stress ratio b, and the variation of e is apparently related to F which is sensitive to b. They clarified that the increase of F with b is the cause of non-uniqueness of e–Z relation at critical state. In an attempt to integrate fabric tensor in the constitutive modelling of sand, Zhang et al. [59] suggested that a relation between e and soil fabric is imperative to unify the classical notion of critical state defined in the e–p–q space [36] (p is the mean effective stress; q is the deviatoric stress) and the recently discovered critical fabric surface (CFS) in the principal fabric space [52]. Hence, the fabric–void ratio relationship is worth a systematic investigation for better understanding and modelling of the micro–macro-behaviour of granular soils.

The objective of this paper is to study the linkage between void ratio and the non-normalized second-rank contact fabric tensor for granular soils. Towards this goal, a series of three-dimensional (3D) DEM simulations consists of consolidation, undrained, and drained monotonic triaxial tests are conducted (Sect. 2). The fabric tensor and the void ratio data are plotted in the e–Z–F space, through which a unique fabric–void ratio surface (FVS) is identified (Sect. 3). The FVS is then mathematically represented and further validated by additional DEM tests with various initial void ratios (Sect. 4). Finally, an undrained cyclic triaxial tests is performed to examine whether the proposed FVS can capture the fabric–void ratio data for samples experiencing stress reversals (Sect. 5). The significance of proposed FVS is discussed in Sect. 6. The main conclusions and possible future extensions of this work are discussed at the end (Sect. 7).

2 Methodology

2.1 DEM configuration

The open-source program YADE [39] is used in this study to carry out all DEM simulations. Consolidation and triaxial tests are simulated employing periodical boundary conditions on a granular representative volume (REV) made of 10,000 sphere particles with a grain size distribution shown in Fig. 1. The choice of 10,000 particles is made to balance the computational efficiency and the representativeness of the granular specimen. Other DEM studies on granular REVs have used similar or fewer particles [21, 41, 62]. All particles are randomly generated without contacts in a 3 × 3 × 3 cm3 box and then isotropically or anisotropically consolidated prior to shearing. Linear elastic contact law is adopted with the particle normal stiffness kn and tangential stiffness ks being kn/d = ks/d = 100 MPa, where d is the particle diameter. The normal and tangential interparticle forces between two particles with stiffness kn1, ks1 and kn2, ks2 are calculated by Fn = Kn δn = kn1 ∙ kn2/(kn1 + kn2) δn and Fs = Ks δs = ks1 ∙ ks2/(ks1 + ks2) δs, where Kn, δn and Ks, δs are the stiffness and displacement of the contact in the normal and tangential direction. The interparticle friction is modelled by the Coulomb’s law with the friction coefficient set to μ = 0.5, a typical value for quartz sand [47, 61].

Fig. 1
figure 1

Grain size distribution of the DEM specimen used in this study

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The simulation is conducted under quasi-static condition where the influence of particle mass (inertia) is negligible, so that the density scaling technique can be adopted to reduce the computation cost [28, 29, 44, 54]. Specifically, the critical timestep of the system is related to the minimum particle size and elastic wave propagation speed by \(\Delta t_{{{\text{cr}}}} = \min \left( {R_{i} \sqrt {{{\left( {\rho_{{\text{g}}} } \right)_{i} } \mathord{\left/ {\vphantom {{\left( {\rho_{{\text{g}}} } \right)_{i} } {E_{i} }}} \right. \kern-\nulldelimiterspace} {E_{i} }}} } \right)\) [39], where subscript “i” represents the ith particle, R the particle radius, ρg the particle density, and E the elastic modulus. By scaling the particle density from ρg = 2.65 × 103 kg to 2.65 × 106 kg, Δtcr is increased by a factor of \(\sqrt {1000} = 31.62\), allowing the computation to accelerate by 31.62 times. For triaxial test simulations, the inertia number \(I = \dot{\varepsilon }\overline{d}\sqrt {{{\rho_{{\text{g}}} } \mathord{\left/ {\vphantom {{\rho_{{\text{g}}} } p}} \right. \kern-\nulldelimiterspace} p}}\) should be much less than 10–3 to ensure the quasi-static condition [4, 12, 27, 32], where \(\dot{\varepsilon }\) is the strain rate, \(\overline{d}\) the average particle diameter, and p the mean effective stress. Through a parametric study, a strain rate of 0.05 s−1 is selected to ensure the maximum \(I \ll 10^{ - 3}\), such that the quasi-static requirement is satisfied.

In DEM, a common method to control the initial void ratio of granular specimens is to adjust the initial friction coefficient μ0 when generate and initially compact the spheres before the official test program starts [11, 54, 57]. The use of a lower value of μ0 eases the particle rearrangement during initial compaction and thus leads to a denser specimen, and vice versa. Note that the value of μ0 must be less than or equal to μ; otherwise, the sample will have sudden collapse at the moment when μ0 is updated to μ. Since we are interested in the consolidation data in this study, it is necessary to update μ0 to μ at the very early stage of compaction, such that this operation does not interference with the consolidation and triaxial data which must reflect the behaviour of soils with μ = 0.5. The procedure adopted in this study is the following: spheres are sparsely generated in a cubic regime according to the designated grain size distribution (Fig. 1) with μ0 selected between 0 and 0.5; the periodic boundaries are moved inwards to isotropically consolidate the particles to p = 5 kPa which is far less than the consolidation stresses pc = 50–2000 kPa studied in this work; μ0 is then updated to μ = 0.5 and the remaining consolidation and triaxial shearing are conducted following the ordinary procedure.

2.2 Experiment design

Three series of DEM experiments are performed in this study. In the first series, μ0 = 0.3 is used to create an initially medium-dense packing e0 = 0.6437. Note that, throughout this paper, notation e0 refers to the void ratio of the specimen at isotropic p = 50 kPa. The medium-dense samples are isotropically or anisotropically consolidated under various stress ratios η0 = 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6 and 0.7, as shown in Fig. 2. This is to probe the fabric–void ratio relation of granular packings during consolidation. For isotropically consolidated specimens, undrained (CIU) and drained (CID) triaxial tests are performed at various confining stresses pc to allow the stress ratio η evolve from 0 to critical value. These are designed to cover a wide range of intermediate states in the fabric–void ratio space. Various intermediate principal strain and stress ratios with be = 0, 0.25, 0.5, 0.75, 1 and b = 0, 0.25, 0.5, 0.75, 1 are used in the CIU and CID tests to check the potential Lode-angle dependency of the fabric–void ratio relation. Here, be = (ε2 − ε3)/(ε1 − ε3) and b = (σ2 − σ3)/(σ1 − σ3), where (ε1, ε2, ε3) and (σ1, σ2, σ3) are the major, intermediate, and minor principal strains and stresses, respectively. The testing details of the consolidation, CIU and CID tests are summarized in Table 1.

Fig. 2
figure 2

Stress paths of consolidation tests

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Table 1 Summary of consolidation, CIU, and CID tests
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To examine the effect of initial void ratio e0 on the fabric–void ratio relation, another series of consolidation and triaxial tests on specimens prepared at different μ0 values are performed. They are named C_e0, CIU_e0, and CID_e0 tests, respectively (Table 2). These sample are prepared with μ0 = 0.2 or 0.5 to obtain an initially dense (e0 = 0.6174) or loose (e0 = 0.6701) state, respectively.

Table 2 Summary of C_e0, CIU_e0, and CID_e0 tests
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Finally, an isotropically consolidated undrained cyclic triaxial (CIUC) test is performed to probe the fabric–void ratio relation under complex stress paths involving loading reversals. The CIUC test is conducted on the dense specimen (μ0 = 0.2 and e0 = 0.6174). Cyclic loading is applied after consolidation (pc = 300 kPa) in a stress-controlled manner with maximum deviatoric stress qmax = 150 kPa. After cyclic loading for N = 20 cycles, the specimen is monotonically sheared until reaching critical state.

A total of 137 simulations are conducted in this study, including eight consolidation tests, 40 CIU tests, 40 CID tests, 16 C_e0 tests, 16 CIU_e0 tests, 16 CID_e0 tests, and 1 CIUC tests.

2.3 Fabric tensor definition

We focus on contact-based characterization of soil fabric. For a given granular assembly, the directional distribution of contacts is given by

$$\rho \left( {\mathbf{n}} \right) = \frac{{2N_{{\text{c}}} }}{{N_{{\text{p}}} }}\overline{\rho }\left( {\mathbf{n}} \right),$$
(1)

where n is the unit contact normal vector; \(\overline{\rho }\left( {\mathbf{n}} \right)\) is the distribution density; Np the number of particles, and Nc the number of contacts. The integration of ρ(n) overall direction gives the coordination number, Z

$$\int\limits_{4\pi } {\rho \left( {\mathbf{n}} \right)} {\text{d}}\Omega = \frac{{2N_{{\text{c}}} }}{{N_{{\text{p}}} }} = Z,$$
(2)

where Ω ∈ [0, 4π] is the solid angle. Kanatani [18] defined three kinds of fabric tensors, with the first kind expressed as

$$G_{ij} = \int\limits_{4\pi } {\rho \left( {\mathbf{n}} \right)n_{i} } n_{j} {\text{d}}\Omega ,$$
(3)

where ni with i = 1, 2, 3 is the component of the contact normal. It is straightforward to see that the trace of Gij is exactly the coordination number, i.e., Gkk = Z and the discretized form of Gij is

$$G_{ij} = \frac{2}{{N_{p} }}\sum\limits_{\alpha = 1}^{{N_{{\text{c}}} }} {n_{i}^{(\alpha )} n_{j}^{(\alpha )} } .$$
(4)

On the other hand, it is useful to approximate ρ(n) by a 2nd-order tensor

$$\rho \left( {\mathbf{n}} \right) \approx \frac{1}{4\pi }E_{ij} n_{i} n_{j} ,$$
(5)

where the Eij is the fabric tensor of the second kind [18]. It can be shown that the mean spherical part of Eij is Z (i.e., Z = Ekk/3) by integrating Eq. (5) over all directions. By multiplying Eq. (5) with nknl and integrating over Ω ∈ [0, 4π], the relation between Eij and Gij is obtained as

$$E_{ij} = \frac{15}{2}\left( {G_{ij} - \frac{1}{5}G_{kk} \delta_{ij} } \right).$$
(6)

Finally, the fabric tensor of the third kind is simply the deviatoric part of Eij which also has a linear relationship with the deviatoric part of Gij

$$F_{ij} = E_{ij} - \frac{1}{3}E_{kk} \delta_{ij} = \frac{15}{2}\left( {G_{ij} - \frac{1}{3}G_{kk} \delta_{ij} } \right) = \frac{15}{2}G^{\prime}_{ij} ,$$
(7)

where the superscript' means the deviatoric part. Substituting Eq. (7) into Eq. (5) gives

$$\rho \left( {\mathbf{n}} \right) \approx \frac{1}{4\pi }\left( {Z + F_{ij} n_{i} n_{j} } \right),$$
(8)

which can be viewed as the spherical harmonic expansion of ρ(n) truncated to the second order. The normalized fabric tensors of the first, second, and third kind can be obtained with the same mathematical procedure with respect to \(\overline{\rho }({\mathbf{n}})\) instead of \(\rho ({\mathbf{n}})\). Most of the previous DEM studies [11, 38, 63] have reported their fabric data in terms of normalized fabric tensors to focus on fabric anisotropy. However, the information of Z is lost in this representation, which thus cannot reveal the full picture of fabric evolution for granular materials undergoing deformation. In analogous to using p and q where p = σkk/3 and \(q = \sqrt {(3/2)\sigma^{\prime}_{ij} \sigma^{\prime}_{ij} }\) to represent the stress state of soil specimens, here we use the first two invariants of the non-normalized contact fabric tensor Eij, namely Z = Ekk/3 and \(F = \sqrt {\left( {3/2} \right)F_{ij} F_{ij} }\), to monitor the fabric evolution and to correlate with void ratio in each test.

3 Fabric–void ratio surface

3.1 Consolidation tests

The consolidation lines in the e–p plane are plotted in Fig. 3a. The normal consolidation line (NCL) is best fitted by \(e = \Gamma - \lambda \left( {{p \mathord{\left/ {\vphantom {p {p_{{\text{a}}} }}} \right. \kern-\nulldelimiterspace} {p_{{\text{a}}} }}} \right)^{\xi }\) with pa = 101.3 kPa (the atmosphere pressure), Γ = 0.6555, λ = 0.01832, and ξ = 0.7628. The consolidation lines of η0 = 0.1–0.3 tests almost coincide with the NCL, and the lines of η0 ≥ 0.4 tests become lower when η0 increases. The observation that the consolidation line locates lower for specimens consolidated at higher η0 is consistent with previous findings [6, 19, 34]. The consolidation lines are plotted in the e–Z plane in Fig. 3b. Data from different η0 tests deviate with each other at the beginning of consolidation, but converges as Z increases. This proves that the e–Z relation is non-unique for frictional granular materials, as speculated in the Introduction session. Figure 3c shows the fabric paths of consolidation tests in the Z–F plane. It is evident that higher η0 lead to overall stronger fabric anisotropy during consolidation. Another observation is that F decreases with the increase of Z for η0 ≥ 0.4 tests, indicating reduced fabric anisotropy under high confining stresses. This is expected, as higher confining stress creates stronger and more connected force networks to support the same stress anisotropy with a weaker contact anisotropy. By combining Fig. 3a–c, the fabric–void ratio relation of consolidation tests in e–Z–F space is shown in Fig. 3d. More data are needed to probe the fabric–void ratio states in between the consolidation lines to tell whether a unique surface can be constructed.

Fig. 3
figure 3

a Consolidation lines in e–p plane, b consolidation lines in e–Z plane, c fabric paths in Z–F plane, and d fabric–void ratio paths in e–Z–F space

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3.2 Undrained (CIU) tests

The stress paths and stress–strain curves of CIU test under triaxial compression (be = 0) and extension (be = 1) conditions are presented in Fig. 4. It is observed that several tests under small pc are liquefied, while others reach critical state at around axial strain of 20–30%. The slopes of the critical state line (CSL) in p–q space for compression and extension tests are Mc = 0.77 and Me = 0.61, respectively.

Fig. 4
figure 4

Stress and strain evolution of CIU tests: a stress of be = 0 tests, b stress–strain of be = 0 tests, c stress of be = 1 tests, and d stress–strain of be = 1 tests

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Figure 5a, b shows the fabric paths of triaxial compression and extension tests, respectively. The initial states of all samples are nearly isotropic with F ≈ 0 and a Z value of 3.5–6. Upon shearing, fabric anisotropy F starts to develop accompanied by the decrease of Z. For liquefied specimens, their fabric paths quickly evolve towards the origin at the onset of liquefaction as marked by the dash lines. As axial strain keeps increasing, the stresses of liquefied specimens remain nearly zero (i.e., p ≈ q ≈ 0 kPa), while their fabric structures start to rebuild as manifested by the development of a fabric path in the low Z regime, which appears to be independent of the initial condition of the specimen. The fabrics of liquefied samples finally reach steady state somewhere along this unique curve. Regarding the minimum value of Z during liquefaction, Nguyen et al. [29] found that the Z of liquefied samples directly evolve to a steady-state value 3.91 instead of dropping to 0 first, which is different from the observations of Gu et al. [10], Wang and Wei [47], Wen and Zhang [52], and the current paper. Wang et al. [48] showed that the value of Z drops to ~ 1 instead of 0 upon liquefaction. More studies on the fabric structure and coordination number of liquefied frictional granular materials are needed to clarify these inconsistencies.

Fig. 5
figure 5

Fabric evolution of CIU tests: a Z–F relation of be = 0 tests, b Z–F relation of be = 1 tests, and c fabric–void ratio paths of be = 0–1 tests

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The fabric paths of non-liquefied specimens never drop below a threshold coordination number (Zth) and reached to some critical-state fabric (Zc, Fc) when sheared to large strain levels. Connecting the liquefied fabric path with the series of critical-state fabric data, it is possible to construct a critical-state line in the Z–F plane that attracts the fabric states of both liquefied and non-liquefied specimens when sheared to large strain levels (Fig. 5a, b). This line can be mathematically represented by the Gunary equation \(F = Z/\left( {a_{1} + a_{2} \sqrt Z + a_{3} Z} \right)\) where (a1 = 1.05, a2 = − 1.17, a3 = 0.755) for be = 0 tests and (a1 = 0.82, a2 = − 0.58, a3 = 0.49) for be = 1 tests. These envelopes are in fact projections of the more general critical fabric surface (CFS) in the principal fabric (E1–E2–E3) space, as recently proposed by Wen and Zhang [52].

Figure 5c plots all the fabric and void ratio data obtained from CIU tests in e–Z–F space. Starting from the NCL shown in Fig. 3d, specimens with different pc and be are sheared under undrained condition which enforces a constant void ratio during shearing. The tests with pc = 50 and 100 kPa are liquefied, while other tests with higher pc are not. It is observed that the fabric–void ratio data seem to fall in two apparent regimes separated by Zth. Data in Z < Zth belong to liquefied specimens and Z > Zth for non-liquefied tests. It is also observed that for tests with pc ≥ 300 kPa, their fabric–void ratio relations are almost independent of the shear mode be, despite that the values of Z and F at critical state are be-dependent (see Fig. 5a, b). For liquefied specimens with pc = 50–100 kPa, the fabric–void ratio data under different be condition also coincide with each other prior to liquefaction and roughly collapse into one line in the Z < Zth regime after liquefaction.

3.3 Drained (CID) tests

The CID tests results for b = 0 and b = 1 are shown in Fig. 6a–b and Fig. 6c–d, respectively. It is observed from Fig. 6a, c that the critical state stress ratio is Mc = 0.77 for compression tests and Me = 0.61 for extension tests, which is consistent with the CIU tests results shown in Fig. 4a, c. For the drained fabric paths in Fig. 6b, d, anisotropy F evolves from near zero to a peak and then drops to the critical state value. These critical fabric values again fall on the same CFS identified previously in the undrained tests (Fig. 5a, b). This again proves that CFS is independent of liquefaction or drainage conditions and can serve as a universal attractor for fabric state upon continuous shearing [52]. Figure 6e plots the fabric data against the evolving void ratios obtained from CID tests. Similar to Fig. 5c, it is found that the fabric path in CID tests is independent of the shear mode b except near the critical-state values.

Fig. 6
figure 6

Evolution of CID tests: a stress–strain of b = 0 tests, b fabric paths of b = 0 tests, c stress–strain of b = 1 tests, d fabric paths of b = 1 tests, and e fabric–void ratio paths of b = 0–1 tests

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Figure 7 combines the fabric–void ratio data obtained from consolidation, CIU, and CID tests presented in Figs. 3d, 5c, and 6e. An astonishing finding is that all the fabric–void ratio data for non-liquefied specimens visually collapse into one single surface which shall be referred to as the non-liquefied (NL) fabric–void ratio surface (FVS). The “state” of the sample (characterized by fabric and void ratio) simply travels along this surface via different paths when subjected to monotonic shearing. It is also noted that the post-liquefaction data from CIU tests collapse into another surface (or line) in the low Z (< Zth) regime and exhibits certain degree of scattering. Based on the observations in Fig. 5a, b, this surface can be regarded as the CFS for liquefied soils extended vertically along the void ratio axis, which will be referred to as the liquefied (L) FVS. The L-FVS data only expand within a small range of e near the upper bound of NL-FVS in the e–Z–F space, since only very loose packing is liquefied in this series of CIU tests.

Fig. 7
figure 7

Compiled fabric–void ratio data from consolidation, CIU, and CID tests

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Note that the current fabric–void ratio relation is investigated in the e–Z–F space and the effect of the third invariant of Eij or the fabric Lode angle θE is omitted. This appears to be a reasonable assumption, since the shear mode be or b is observed to have little effect on the evolution of fabric–void ratio data in CIU and CID tests prior to reaching the critical state (see Figs. 5c and 6e), and there is not much data scattering around the FVS (see the next section for a quantitative evaluation) identified in the e–Z–F space including the near critical state regime (see Fig. 7). Therefore, θE will not be considered when we construct a model of the fabric–void ratio relation in the following.

4 Mathematical description and validation

We shall pursue a mathematical description of the FVS to evaluate the quality of the data correlation and to facilitate the integration of such surface in constitutive models for sands such as the critical fabric theory proposed by Zhang et al. [59]. In addition, the uniqueness of FVS will be validated using data from samples prepared to different initial densities (i.e., different e0).

4.1 Mathematical description

For the non-liquefied portion of the FVS, a good starting point is the Z–n equation proposed by O’hern et al. [31] for isotropically or oedometrically compressed granular assemblies

$$Z - Z_{r} = h\left( {n - n_{r} } \right)^{\varphi } ,$$
(9)

where n is the porosity; h and φ are material parameters; Zr is the coordination number at a reference porosity nr which is usually taken at the jamming point. Taking Eq. (9) as a reference for \(e = f\left( {Z,F} \right)\) at F = 0, the mathematical expression of the NL-FVS is proposed as

$$e = e_{{\text{r}}} + h_{{\text{r}}} \left( {Z - Z_{{\text{r}}} } \right)^{\zeta } + A_{1} F + A_{2} F^{2} ,$$
(10)

where a second-order polynomial in terms of F is added to consider the effect of fabric anisotropy; er and Zr are, respectively, the void ratio and coordination number taken at the reference point; hr, ζ, A1 and A2 are material parameters. Equation (10) is best fitted to the NL-FVS data with parameters er = 0.6641, Zr = 3.258, hr =  − 0.02898, ζ = 2.0, A1 = − 0.02627, and A2 = 0.004352 with the accuracy quantified by R2 = 0.9852 (Fig. 8).

Fig. 8
figure 8

Mathematical description of the non-liquefied fabric–void ratio surface

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For the liquefied portion (i.e., Z ≤ Zth) of the FVS, we first examine the data on the Z–F plane, as shown in Fig. 9a, given all data in this portion come from CIU tests conducted at similar void ratios. The fitted CFS curves for be = 0 and 1 in Fig. 5a, b are also plotted here in Fig. 9a. It is clear that the CFS is dependent on the shear mode, which is consistent with the observation of Wen and Zhang [52] who inspected the shape of CFS in the principal fabric space. For simplicity, here, we adopt an averaged critical fabric curve with expression

$$F = {Z \mathord{\left/ {\vphantom {Z {\left( {0.9937 - 1.044\sqrt Z + 0.7070Z} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {0.9937 - 1.044\sqrt Z + 0.7070Z} \right)}}$$
(11)

to represent this data cluster in the fabric–void ratio space (Fig. 9b). By doing so, we have hypothesized that averaged critical fabric curve or the L-FVS is independent of void ratio as long as the sample is fully liquefied. This assumption shall be further tested with more CIU tests conducted at a wider range of void ratios in the next section. For now, the L-FVS fitting gives a R2 value of 0.9568, as shown in Fig. 9b.

Fig. 9
figure 9

Mathematical description of the liquefied fabric–void ratio surface in a Z–F plane and b e–Z–F space

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The small scattering (R2 = 0.9852 for NL-FVS and R2 = 0.9568 for L-FVS) supports the existence of a unique FVS linking the void ratio and the first two invariants of contact fabric tensor for samples subjected to monotonic consolidation, undrained, and drained triaxial shearing. This surface exhibits weak dependency on the shear mode in both liquefied and non-liquefied regimes. Comparing to the many e–Z models developed for gravity-filled granular packings in powder technology [1, 9, 45, 58], our proposed FVS depicts a more complete picture by incorporating the effect of fabric anisotropy on the density of granular assemblies. The new FVS concept is therefore applicable for conditions involving anisotropic consolidation and triaxial shearing which are relevant for soil mechanics applications, providing a new perspective to analyse the internal structure of granular soils.

4.2 Validation

The FVS in Figs. 8 and 9 is developed exclusively based on medium-dense samples prepared with μ0 = 0.3. To validate the uniqueness and the robustness of the FVS, additional tests including C_e0, CIU_e0 and CID_e0 tests using relatively dense (prepared with μ0 = 0.2) and loose (prepared with μ0 = 0.5) samples are conducted, as summarized in Table 2.

Figure 10 present the critical state data of CIU, CID, CIU_e0, and CID_e0 tests on the e–p and the Z–p planes. The conventional CSL in the e–p plane can be well represented by the power-law function of Li and Wang [25] as \(e_{{\text{c}}} = 0.630 - 0.00365\left( {p/p_{{\text{a}}} } \right)^{1.202}\) where pa = 101.3 kPa. The critical-state Z–p data also collapse into a single curve fitted by \(Z_{{\text{c}}} = 2.779 + 0.679(p/p_{{\text{a}}} )^{0.441}\). These results confirm that the CSL in the e–Z–p space is independent of the sample’s initial void ratio, shear mode, and drainage conditions, which is consistent with the previous findings [10, 30, 52]. Figure 11a compares the NCLs for the dense, medium dense, and loose samples in the e–p plane. Figure 11b plots the corresponding NCLs in e–Z plane, or in other words, the fabric–void ratio relation for isotropically consolidated samples (F ≈ 0). It is observed that the e–Z curves are closely located in a narrow band (despite some slight variations in the dense and loose regime) in contrast to the distinct curves in the e–p plane.

Fig. 10
figure 10

Critical-state lines of CIU, CID, CIU_e0, and CID_e0 tests in a e–p plane and b Z–p plane

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Fig. 11
figure 11

NCLs of specimens with different initial density in a e–p plane and b e–Z plane

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The previously constructed NL-FVS (Eq. (10)) and the fabric–void ratio data from the new C_e0, CIU_e0 and CID_e0 tests are plotted together in Fig. 12a. It can be observed that the new data qualitatively falls on the same surface. For a specific fabric–void ratio data (Zdata, Fdata, edata), its corresponding e on the FVS can be calculated by e(Zdata, Fdata) using Eq. (10) and is denoted as eFVS. The comparison of edata and eFVS is then shown in Fig. 12b. The small scattering (R2 = 0.9700) quantitatively validates the uniqueness of the NL-FVS with respect to the initial densities of the specimens. It is noted that the eFVS deviates slightly from the edata in the very dense regime (around edata = 0.45). This deviation could be due to several limitations of this study. First, the contact model is linear elastic which might be reasonable for low confining pressures, but cannot represent real contacts (nonlinear, pressure-dependent) at high-pressure levels. The error due to this idealization therefore shows up at dense packing regime where high confining stress is applied. Second, the present study uses the first two invariants of Eij (represented by Z and F) to correlate with void ratio. This approach neglects the information represented by θE, as well as the higher order information contained in the full contact distribution function. Future enrichments of FVS considering the above may remove the deviations between edata and eFVS.

Fig. 12
figure 12

Validation of NL-FVS by C_e0, CIU_e0, and CID_e0 tests: a FVS and fabric–void ratio data; b comparison between the void ratio data and those calculated by the FVS equation

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The data from liquefied specimens in CIU_e0 test and the mathematical L-FVS (Fig. 9b) are plotted together in the Z–F plane (Fig. 13a) and in the fabric–void ratio space (Fig. 13b). The agreement between the data and the mathematical L-FVS (Eq. (11)) is quite well as evidenced by R2 = 0.9806. The observation that the liquefied fabric data of all undrained tests with different e0 can be represented by the same Z–F curve confirms our earlier hypothesis that the critical fabric curve is independent of void ratio in the liquefied regime, and thus validated the L-FVS proposed in Fig. 9b and Eq. (11). It is also observed in Fig. 13b that Z and F for denser soils evolve to larger values in the liquefied regime. This feature can be utilized to precisely locate the threshold Z (Zth) that separates the liquefied and the non-liquefied states, i.e., by observing the maximum Z (Zmax) of the specimen with the minimum void ratio among all liquefied tests. The Zmax of all liquefied undrained tests presented in Figs. 9 and 13 is Zmax = 2.73 from the CIU_e0 test with e = 0.6171 and pc = 50 kPa. This means that the value of Zth must be ≥ 2.73. On the other hand, it is interesting to note that the critical state Zc at p = 0 calculated by the CSL equation in Fig. 10b is Zc(p = 0) = 2.779, which is quite close to Zth. It is therefore reasonable to infer that the intersection Zc(p = 0) in the power-law equation that fits the critical state Z–p data is not just a fitting parameter but has the physical meaning of the threshold Z that distinguishes the liquefied and non-liquefied soils.

Fig. 13
figure 13

Validation of L-FVS by CIU_e0 test data in a Z–F plane and b e–Z–F space

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5 Undrained cyclic (CIUC) test

We have demonstrated that the states (fabric, void ratio) of granular material travel along a single FVS during monotonic loading including consolidation, CIU, and CID tests. There are two aspects remain to be addressed: (1) it is unclear whether the same FVS works for stress paths involving loading reversals; (2) at the moment of static liquefaction, the fabric–void ratio data jump from the NL-FVS to the L-FVS and then stays on it, but it is not clear whether this sudden transition happens during cyclic liquefaction and whether liquefied states can evolve from the L-FVS back to the NL-FVS as shear strain accumulates. To answer these questions, a CIUC test is conducted on a dense sample with pc = 300 kPa, qmax = 150 kPa, and e = 0.5911. The specimen is subjected to a two-stage loading. The first stage is cyclic triaxial loading which stops when the number of cycles N reaches 20. After this, the specimen is monotonically sheared through triaxial compression to the critical state.

Figure 14a–d shows the stress–strain curve, the stress path, the fabric path, and the fabric–void ratio path from CIUC tests, respectively. It is observed from Fig. 14a, b that the specimen exhibits the typical cyclic liquefaction/mobility behaviour. In Fig. 14c, the fabric evolution path drifts to lower Z values during cyclic loading and jumps to the origin as soon as the sample liquefies. As shear continues, the fabric path evolves partially along the L-FVS and re-enters the non-liquefied regime (Z > Zth = 2.73 regime) where the sample regains some shear strength. The same phenomenon can be better visualized in the fabric–void ratio space in Fig. 14d. It is clear that the “butterfly” stress loop in Fig. 14b corresponds to a closed fabric path circulating between the L-FVS and the non-liquefied regime.

Fig. 14
figure 14

Evolution of CIUC test in: a stress–strain plane, b stress plane, c fabric plane, and d fabric–void ratio space

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Figure 14c, d confirms that the fabric–void ratio data during cyclic loading approximately stay on the L-FVS after liquefaction, but does not stay on the NL-FVS before liquefaction. If the descriptor of soil’s inner structure is sufficient, it should fully quantify the “memory” of the soil and exhibit a one-to-one relation with the macroscopic properties of the soil. We suspect that the non-uniqueness of FVS for monotonically and cyclically loaded specimens can be removed by considering higher order fabric information which is not reflected in the 2nd-rank fabric tensor. Along this line, examining the full directional distribution function of contact normal may reveal some unique microstructural features of cyclically loaded granular materials.

The monotonic loading after the cyclic stage takes the specimen to its critical state and is marked by blue dot lines and red stars in Fig. 14. It is observed that the monotonic shearing eventually brings the cyclically loaded specimens back to the NL-FVS with the squared error (SE) = 3.73 × 10–5, confirming again the FVS identified in Fig. 8 could serve as a reference surface for granular materials under monotonic shearing, although it does not uniquely relate void ratio and fabric for cyclically loaded specimens.

6 Significance of FVS

With the FVS identified, a “so what” question naturally follows. We would like to make the case here that FVS can change how we conceive the constitutive theories and models of granular soils. Specifically, the fabric–void ratio relation proposed here connects two important state variables (void ratio and fabric), implying that the previously observed critical state line in the conventional e–p–q space [36] and fabric space [61] is not independent to each other. Indeed, the authors have recently proposed a constitutive framework, namely the critical fabric theory [59], where fabric is treated as the single internal state variable of granular soils. In this framework, the only criterion that judges whether a soil has reached critical state is to see if its fabric state converged to the critical fabric surface [52]. As the fabric evolves towards this CFS, the other state variables (e–p–q) approach their apparent critical state because of some geometrical or microstructural relations that link fabric with the void ratio and stress state of granular soils. To exercise this logic, let us combine the CFS (i.e., F(Z) relation) under an arbitrary shear mode (e.g., compression or extension as shown in Fig. 5 or Fig. 6), a fabric–void ratio relation (FVS shown in Fig. 8), and a critical state Z–p relation (shown in Fig. 10b). A mathematical expression between e and p can be derived by

$$e = e_{{\text{r}}} + h_{{\text{r}}} \left[ {Z(p) - Z_{{\text{r}}} } \right]^{\zeta } + A_{1} F\left( {Z(p)} \right) + A_{2} F^{2} \left( {Z(p)} \right),$$
(12)

where parameters er, Zr, hr, ζ, A1, and A2 are already calibrated in Fig. 8 and presented after Eq. (10). The obtained equation is plotted against the DEM critical-state data (Fig. 10a) in e–p plane in Fig. 15. It is apparent that the derived CSL coincides with the CSL data, supporting that the classical critical-state theory can be equivalently framed in terms of a critical fabric relation and a fabric–void ratio relation. It is worth to note that, in the original model [59], the fabric–void ratio relation is taken from O’hern et al. [31] which only depends on the coordination number drawn from isotropic or oedometric compression tests (shown as Eq. (9)). The current study shows that this relation needs to be updated by the new FVS shown in Fig. 8 for general stress paths involving shear and compression.

Fig. 15
figure 15

Comparison between the derived CSL in e–p plane and the DEM critical-state data in Fig. 10a

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In a more general context, many elastoplastic/hypoplastic models start to acknowledge soil fabric [14, 23, 26, 35, 43, 50, 56] and are often formulated within the framework of anisotropic critical state theory (ACST) [23, 26, 35, 50, 56]. In these models, the CSL in e–p–q space and the critical fabric are treated as separate conditions for attaining critical state, and thus, the stress-dilatancy relation and fabric evolution laws are often proposed separately. The new FVS identified in this study, however, suggests that the evolution of fabric and void ratio (or dilatancy) are intrinsically coupled especially under monotonic loading. This finding should inspire more sensible and realistic stress–dilatancy–fabric relations in future elastoplastic models based on ACST.

It should be noted that the current FVS expressed in void ratio e and the first two invariants of second-order fabric tensor Z and F can only be used in modelling monotonic triaxial tests. The fabric–void ratio paths for cyclically loaded specimens do not collapse on the same FVS, but are only attracted by it when monotonically sheared after the cyclic loading. We speculate that higher order fabric tensors may contain some extra structural information for cyclically loaded specimens that are not captured by the second-order tensor studied in the current work. Establishing enriched descriptions of soil fabric based on contact, particle, and void vectors could be the first step in the follow-up studies along this line. In addition, further validation and investigation of FVS needs not only DEM simulations [10, 29, 46, 47, 51] but also advanced laboratory tests equipped with in-situ X-ray microtomography and other advanced imaging techniques [5, 7, 13, 16, 22, 53].

7 Concluding remarks

The relationship between the contact fabric tensor Eij and the void ratio of granular soils is studied by investigating the e–Z–F data of 137 DEM numerical tests. Among them, 8 consolidation and 80 true triaxial tests data are used for identifying a fabric–void ratio surface, and the rest 49 tests are used to verify the uniqueness of the FVS. A cyclic triaxial test is employed to check whether the FVS observed from monotonic shearing is applicable when stress reversal is included.

The results of the monotonic tests show that the usual e–Z relation of granular packings is significantly influenced by the fabric anisotropy F. Such dependency can be conveniently represented by a parametric surface, namely the fabric–void ratio surface, in the e–Z–F space. The FVS consists of two separate parts with each describing the fabric–void ratio relation for liquefied and non-liquefied soils. When static liquefaction happens, the fabric path suddenly jumps from the non-liquefied portion of the FVS to the liquefied portion of the FVS. The threshold coordination number (Zth) that separates the liquefied and non-liquefied specimens is found to be Zth = 2.73. The proposed FVS is validated through an independent series of consolidation and triaxial tests with different initial void ratios.

The fabric path of cyclically loaded specimen does not travel along the NL-FVS established on the monotonic test results. This suggests that the two invariants of the 2nd-rank fabric tensor cannot completely tell the differences of soil structures induced by recent stress reversals. After cyclic loading, sufficient monotonic loading can remove the effect of recent stress history and thus recover the fabric–void ratio relation depicted by the FVS.

Revealing connections between the microstructural attributes and the macroscale behaviours of granular is an ongoing endeavour for years. This study identifies an important reference surface for correlating the void ratio with the fabric structure of granular materials and helps better understand the structural evolution of granular soils during compaction and liquefaction. Future extensions of this work include: (1) incorporating higher order fabric information to seek for a unique fabric–void ratio relation for both monotonically and cyclically loaded specimens; (2) investigating the fabric state of liquefied soils and understand its transition near the jamming point; (3) implementing the concept of FVS in fabric-centred constitutive theories of granular soils to unify the descriptions of critical state in terms of void ratio and fabric tensors.