DEM investigation on flow instability of particulate assemblies under coupling between volumetric and axial strains

Abstract

The discrete element method (DEM) is employed to investigate the impact of coupling between volumetric and axial strains on the flow liquefaction vulnerability of 3D cubic particulate specimens. The virtual testing program conducted here encompasses a wide range of initial states and varying degrees of coupling between volumetric and axial strains. Utilizing data obtained from DEM simulations, the evolution of micro- and macroscale variables, including coordination number, contact fabric anisotropy, redundancy index, strong force networks, invariants of the effective stress tensor, and excess pore-water pressure, is examined. Results from DEM tests indicate that coupling expansive volumetric strain with axial strain leads to a gradual loosening of the load bearing microstructure, a decrease in coordination number, and a faster change in contact anisotropy. DEM simulations demonstrate that the triggering of flow liquefaction instability is followed by a sudden increase in contact fabric anisotropy and abrupt drops in coordination number and redundancy index. Moreover, a detailed analysis of the findings suggests that the stress ratio at the onset of post-peak softening decreases with increasing expansive volumetric strains.

1 Introduction

Loose sands subjected to constant volume (undrained) conditions may exhibit flow instability, characterized by post-peak loss of shear strength, significant permanent shear deformations, and excessive accumulation of pore-water pressure (e.g., studies by Been and Jefferies [5], Chu et al. [9], and Chu and Wanatowski [11]). Laboratory findings have indicated that granular soil in a loose state, positioned above the critical state line (CSL) in the e versus p′ plane, is particularly prone to flow instability. Conversely, dense sand located below the CSL in this plane demonstrates continuous strain hardening without signs of flow instability.

Geotechnical design typically relies on strength parameters obtained from either fully drained or fully undrained tests. This approach is grounded in the belief that drained and undrained tests encapsulate the widest range of potential responses. However, real-world conditions seldom align with these extremes; instead, they tend to fall within a spectrum of partial drainage. This entails concurrent alterations in both pore volume and pore-water pressure. In particular, the traditional investigation of sand flow instability has focused on conventional tests such as constant volume triaxial, simple shear, and hollow cylinder torsion shear tests (e.g., [4, 8,9,10,11, 20, 31, 32, 36, 57, 60, 61, 69]). However, recent dynamic centrifuge studies have underscored the unrealistic nature of assuming negligible volume change (commonly referred to as undrained or isochoric conditions) in earthquake time scales due to void ratio redistribution caused by pore-water migration and soil heterogeneity (e.g., [1, 19, 41, 66]).

Instances where the assumption of negligible volume change does not hold, such as in the foundations of wind turbines subjected to high cyclic loading from wind, waves, and storm forces, highlight the practical relevance of considering the coupling between volumetric and axial strains [2]. With the increasing development of offshore wind turbines, which feature shorter drainage paths compared to conventional gravity-based structures, there is a growing need to investigate this aspect of granular soil behavior [59]. To delve deeper into this phenomenon, researchers have devised specialized fully automated triaxial tests where controlled volume changes are imposed during shear, revealing that even small dilative (expansive) volume changes can significantly intensify the potential for flow instability. Conversely, contractive volume changes tend to mitigate the susceptibility to flow instability (e.g., [24, 30, 45, 46, 56, 62]). More recent studies employing critical state-based constitutive models have corroborated these findings, showing that expansive volume changes resulting from pore-water inflow increase the potential for flow instability, while pore-water outflow weakens it (e.g., [18, 25, 30, 32, 60]).

In recent years, the discrete element method (DEM) introduced by Cundall and Strack [12] has demonstrated its unique ability as a versatile tool for synchronized investigation of the micro- as well as macroscale mechanical response of granular media (e.g., [7, 13, 15,16,17, 27, 34, 37, 38, 38, 40, 42, 44, 48, 50, 51, 53, 55, 58, 67, 68, 71]). For instance, in the simulation of drained and undrained tests carried out in the conventional and true triaxial apparatuses, clear correlations between asymptotic coordination number, invariants of various fabric tensors at the critical state, and invariants of the effective stress tensor have been reported in the literature (e.g., [13, 14, 16, 17, 33, 40, 43, 44, 50, 55, 67, 71]).

As aforementioned, during and after earthquakes, pore-water flux in sand elements can cause volume changes, impacting flow instability. The persistence of coupling between volumetric and shear strains prevents sand elements from reaching the critical state. Despite a few studies using DEM to investigate flow instability triggering [46, 50, 65], the literature lacks diverse coupled strain paths, and attention to micromechanical quantities.

Herein, 3D-DEM simulations are incorporated to investigate the influence of coupling between volumetric and axial strains on the flow instability of particulate assemblies. To this aim, this paper is organized as follows. Micro- and macroscale variables are introduced first. Then, the mechanical behaviors of 34 assemblies under uncoupled (i.e., 24 constant volume, and 28 drained) triaxial tests are simulated. Thereafter, impacts of linear and transient patterns of coupling between the volumetric and axial strains on mobilization of shear strength, effective stress path, variation of void ratio, accumulation of equivalent excess pore-water pressure, variation of coordination number, evolution of anisotropy, and change in redundancy index are investigated. Through analyzing the data of a total of 213 simulations with coupling between the volumetric and axial strains, impacts of linear coupling coefficient, limiting volume change, and pace of volume change on stress ratio at the onset of post-peak softening are studied.

2 DEM modeling

2.1 Definition of macroscopic and microscopic variables

In this paper, for any arbitrary vectors a and b, tensor product of two vectors is ${\mathbf{a}} \otimes {\mathbf{b}} = a_{i} b_{j}$. For second-order tensors x and y, ${\text{tr}}\,{\mathbf{x}} = x_{ii}$ and ${\text{dev}}\,{\mathbf{x}} = {\mathbf{x}} - [(1/3)\,{\text{tr}}\,{\mathbf{x}}]\,{\mathbf{1}}$ in which 1 denotes the second-order identity tensor. $\left\| {\,{\mathbf{x}}\,} \right\| = \sqrt {{\mathbf{x:x}}}$ wherein ${\mathbf{x:y}} = x_{ij} y_{ij}$.

An average granular stress tensor (i.e., ${\varvec{\sigma }}$) on boundaries of a representative volume of a system consisting of a large number of discrete particles is [13, 14, 23, 50, 54, 71]:

$${\varvec{\sigma }} = \frac{1}{V}\,\sum\limits_{{c \in N_{{\text{c}}} }} {\,{\mathbf{l}}^{c} \otimes \;\;{\mathbf{f}}^{c} }$$

(1)

wherein V and N_c are total volume and total number of contacts in the representative volume, l^c is the branch vector connecting the centers of two contacting particles, and f^c is the contact force at contact c. The first and second invariants of ${{\varvec{\sigma }}}$, mean principal effective stress (i.e., p′) and shear stress (i.e., q), are:

$$p^{\prime } = \frac{1}{3}{\text{tr}}\,{{\varvec{\sigma }}};q = \sqrt{\frac{3}{2}} \;\left\| {\,{\text{dev}}\,{\varvec{\sigma }}\,} \right\|$$

(2)

For any cubic particulate assemblies, the second-rank strain tensor can be calculated from $\varepsilon_{ij} = (\partial u_{i} /\partial X_{j} + \partial u_{j} /\partial X_{i} )/2$ wherein ${\mathbf{u}}\;[ = (\begin{array}{*{20}c} {u_{1} } & {u_{2} } & {u_{3} } \\ \end{array} )]$ represents displacement vector and X_i [with $i \in \left\{ {\,1,\;2,\;3} \right\}$] stands for coordination (see Fig. 1 for the coordination system). Herein, the normal strains are calculated based on displacements at the boundary walls of the cubic granular assembly:

$$\varepsilon_{i} = \frac{{2\,\Delta H_{i} }}{{H_{0} + H_{i} }}\;{\text{in}}\,{\text{which}}\;i \in \left\{ {1,2,3} \right\}$$

(3)

wherein H₀ represents the initial dimensions of identical sides of the cubic specimen (see Fig. 1). In Eq. (3), ε_i, H_i, and $\Delta H_{i} \;[ = H_{0} - H_{i} ]$ stand, respectively, for normal strain, current dimension, and change in dimension of the cubic particulate specimen along i-axis.

Fig. 1

Particle size distribution used in preparation of DEM particulate assemblies and 3D view of a particulate assembly

Hence, the first and second invariants of strain tensor, the volumetric strain (i.e., ε_v) and the shear strain (i.e., ε_q), become [14, 50, 71]:

$$\begin{aligned} \varepsilon_{v} &= \frac{2\Delta V}{{V_{o} + \Delta V}}\\ \varepsilon_{q} &= \sqrt{\frac{2}{3}} \parallel {\text{dev }}{\varvec{\varepsilon}}\parallel = \frac{\sqrt 2 }{3}\sqrt {\left( {\varepsilon_{1} - \varepsilon_{2} } \right)^{2} + \left( {\varepsilon_{1} - \varepsilon_{3} } \right)^{2} + \left( {\varepsilon_{2} - \varepsilon_{3} } \right)^{2} } \\ \end{aligned}$$

(4)

A second-order fabric tensor (i.e., F) is adopted to characterize the evolving contact anisotropy in assemblies of discrete spheres [14, 22, 23, 26, 37, 38, 43, 44, 50, 52, 71]:

$${\mathbf{F}} = \frac{1}{{N_{{\text{c}}} }}\sum\limits_{k = 1}^{{N_{c} }} {{\mathbf{n}}^{k} \otimes {\mathbf{n}}^{k} }$$

(5)

wherein n^k is unit vector normal to the kth contact plane. The second invariant of F, i.e., ${\text{dev }}{\mathbf{F}}[ = {\mathbf{F}} - ({\text{tr}}{\mathbf{F}}/3){\mathbf{1}}]$, is frequently used as a scalar measure of anisotropy in granular media [14, 39, 43, 50, 52, 70]:

$$\left\| {{\text{dev }}{\mathbf{F}}} \right\| = \sqrt {{\text{dev }}{\mathbf{F:}}{\text{dev }}{\mathbf{F}}} = \sqrt {{\mathbf{F:F}} - \frac{1}{3}({\text{tr}}\,{\mathbf{F}})^{2} }$$

(6)

The coordination number (CN), the average contact number per particle, is an appropriate index of mechanical stability in particulate systems [14, 34, 42, 48]:

$${\text{CN}} = \frac{{2N_{{\text{c}}} }}{{N_{{\text{p}}} }}$$

(7)

wherein N_p and N_c are, respectively, the total number of particles and total number of contacts in the representative volume V. A contact commences once two particles touch each other, and accordingly, N_c was multiplied by 2 in Eq. (7).

In 3D assemblies of discrete particles, the total number of elastic constrains associated with N_c contacts is $(3 - 2f_{{\text{s}}} )N_{{\text{c}}}$ wherein f_s is the ratio of total sliding (say plastic) contacts to the total number of contacts of note, in a sliding contact, the mobilized tangential contact force reached its ultimate value which is dictated by both the normal force and inter particle friction angle, μ, at the contact point. On the other hand, $6N_{{\text{p}}}$ degrees of freedom exist for 3D assemblies of N_p discrete particles. Excluding particles with zero active contacts (i.e., $N_{{\text{p}}}^{0}$), Kruyt and Rothenburg [28], Zhou et al. [71] and Huang et al. [17] suggested redundancy index (i.e., I_R) in terms of the ratio of the total number of constraints to the total number of degrees of freedom in particulate assembly:

$$I_{{\text{R}}} = \left( {\frac{{3 - 2f_{{\text{s}}} }}{6}} \right)\;\frac{{N_{{\text{c}}} }}{{N_{{\text{p}}} - N_{{\text{p}}}^{0} }} = \left( {\frac{{3 - 2f_{{\text{s}}} }}{12}} \right)\frac{{{\text{CN}}}}{{1 - (N_{{\text{p}}}^{0} /N_{{\text{p}}} )}}$$

(8)

A transition from $I_{{\text{R}}} > 1$ to $I_{{\text{R}}} < 1$ points to a shift from situations in which the total number of constraints are higher than the total degrees of freedom (say hyperstatic regime) to other situations wherein the total number of constraints is lower than the total degrees of freedom (say hypostatic regime). Accordingly, assemblies with $I_{{\text{R}}} < 1$ are potentially unstable and I_R = 1 can be considered as the onset of instability.

2.2 Specimen preparation

The DEM simulations have been executed using the commercial code PFC3D [21]. The mathematical foundation of DEM can be found in Cundall & Strack [12]. The adopted particle size distribution for the particulate assemblies of this study is shown in Fig. 1. The specimen preparation scheme suggested by Gu et al. [13] was followed; that is, specimens with a total of 8438 spherical particles were first generated in a 5[mm] × 5[mm] × 5[mm] inner cubic space between orthogonal boundary walls for imposing external forces and displacements. A cloud of randomly distributed particles was initially positioned in the inner space between boundary walls. Then, among different existing rules for particle–particle and particle–wall interactions, the one with linear force–displacement contact law was employed. Afterward, the interparticle friction coefficient (denoted as μ) is set to zero, and the particles undergo cycling until they attain a mechanical ratio of 10⁻⁵. This ratio is calculated as the average value of the unbalanced force magnitude divided by the average value of the sum of the magnitudes of the contact forces. Imposing equal inward movement of walls, an isotropic stress state with confining stress of 10 [kPa] was imposed on the specimens. To generate specimens with a wide range of initial void ratios, relatively large μ values (i.e., 0.7–1.0 [-]) during this stage led to the generation of loose specimens, whereas dense specimens were produced via facilitation of particles sliding through assuming low μ values (e.g., 0.05–0.3 [-]). Thereafter, μ was set to 0.5 [-] and each specimen was brought to its prescribed target confining stress (greater than 10 [kPa]) and cycled once more to reach equilibrium. To ensure the rigidity of wall boundaries, the value of wall stiffness was chosen to be higher than particle stiffness. The parameters used in simulations are listed in Table 1. Table 2 provides a comprehensive list of the initial samples utilized in the simulations, along with their corresponding interparticle friction angles. In the preparation stage (confining stress of 10 [kPa]), varying friction angles were intentionally utilized to generate diverse initial sample configurations. However, minor fluctuations in void ratio are anticipated due to the dynamic interaction of particles, wherein some contacts remain stationary, creating spaces for smaller particles to fill.

Table 1 Parameters used in DEM simulations in shearing stage

Table 2 Initial data of samples and interparticle friction angles at different stages of preparation

3 Coupling of strains

3.1 Mathematical definition

Two different patterns for coupling between the volumetric and axial strains, i.e., “linear” and “transient”, are considered here. In the linear pattern, the volumetric strain increases linearly with the axial strain through:

$$\varepsilon_{v} = \zeta \,\,\varepsilon_{1}$$

(9)

wherein $\varepsilon_{1} \,( > 0)$ is normal strain along the 1-axis in Fig. 1 and $\varsigma$ is the proportionality factor. $\zeta \, > 0$ corresponds to contractive (say $\varepsilon_{v} \, > 0$) coupling with continuously decreasing void ratio per ε₁ and $\zeta \, < 0$ refers to expansive (say $\varepsilon_{v} \, < 0$) coupling wherein void ratio increases constantly with ε₁. Of note, $\zeta \, = 0$ dictates the constant volume (i.e., isochoric or undrained) condition.

In practical applications, several factors exert physical limitations on the maximum achievable volumetric strain. These factors encompass soil heterogeneity, the confined extent of the liquefied zone, and the prevailing field drainage conditions. The behavior of volumetric strain exhibits a range of diverse patterns. In certain scenarios, it showcases a linear correlation with axial strain, while in others, it experiences nonlinear variations before eventually stabilizing at a constant volume state. This nonlinearity arises due to the inherent nature of filter boundaries, as they allow water to pass in proportion to the incremental pressure changes. Experimental evidence by Suzuki et al. [59] substantiates that the coupling between volumetric and axial strain induces nonlinear alterations in volumetric strain, which eventually reach a constant value as axial strain increases. The observed variability in this behavior can be influenced by two key factors: the rate of change in volumetric strain (R), which is contingent upon filter permeability and the rate of loading, and the ultimate magnitude of volumetric strain ($\varepsilon_{v\infty }$), representing the final volumetric strain state. Therefore, an idealized strain path with transient ε_v–ε₁ coupling leading to a limited volumetric strain is utilized here as follows:

$$\varepsilon_{v} = \varepsilon_{v\infty } \,\left[ {\;1 - \exp ( - R\,\varepsilon_{1} )} \right]$$

(10)

in which R is a positive parameter controlling the pace of ε_v with $\varepsilon_{1} \,( > 0)$ and ε_v∞ is the ultimate (i.e., limiting) volumetric strain at extremely large axial strains [say $\varepsilon_{1} \, \to + \infty$]. In Eq. (10), $\varepsilon_{{{\text{v}}\infty }} \, > 0$ causes a contractive transient ε_v–ε₁ coupling, whereas $\varepsilon_{{{\text{v}}\infty }} \, < 0$ refers to expansive transient ε_v–ε₁ coupling. The slope of the ε_v versus ε₁ curve (i.e., $\partial \varepsilon_{v} /\partial \varepsilon_{1}$) from Eq. (10) begins from $R\,\varepsilon_{v\infty }$ at $\varepsilon_{1} = 0$ and gradually approaches zero (i.e., undrained condition) as ε₁ becomes very large. Typical curves visualizing the linear and transient ε_v–ε₁ couplings are, respectively, illustrated in parts “a” and “b” of Fig. 2.

Fig. 2

Typical strain paths studied here: a linear ε_v–ε₁ coupling with ζ = − 1.0, − 0.50, − 0.25, − 0.10, − 0.05, 0.0, + 0.25, + 0.50, and + 1.0; and b transient ε_v–ε₁ coupling with R = 0.75 and ε_v∞ = − 2.50, − 1.50, − 0.90, − 0.60, − 0.30, 0, + 0.30, + 0.60, and + 0.90 [%]

3.2 Consequence of ε _v–ε ₁ coupling: an energy-based interpretation

Adopting a minimum active to passive energy ratio criterion, Rowe [49] obtained a stress-dilation law for assemblies of unbreakable frictional particles sheared under triaxial compression (TXC) condition:

$$\frac{{\sigma_{1}^{\prime } }}{{\sigma_{3}^{\prime } }} = \tan^{2} \left( {\frac{\pi }{4} + \frac{{\varphi_{{{\text{cs}}}}^{\prime } }}{2}} \right)\;\left( {1 - \frac{{\dot{\varepsilon }_{v} }}{{\dot{\varepsilon }_{1} }}} \right)$$

(11)

wherein $\sigma_{1}^{\prime }$ and $\sigma_{3}^{\prime }$ are, respectively, the major (along the 1-axis in Fig. 1) and minor (along the 3-axis in Fig. 1) principal effective stresses, and $\varphi_{{{\text{cs}}}}^{\prime }$ is the critical state friction angle. Recalling that $\sigma_{1}^{\prime } = p^{\prime } + (2/3)\,q$, $\sigma_{3}^{\prime } = p^{\prime } - (1/3)\,q$, and $\sin \varphi_{{{\text{cs}}}}^{\prime } = 3M/(6 + M)$ [of note, M is slope of the CSL in the q-p′ plane] hold in TXC shear, rearrangement of terms in Eq. (11) with some ordinary algebra operations lead to the following equation for mobilization of stress ratio (i.e., $\eta$) with $\dot{\varepsilon }_{v} /\dot{\varepsilon }_{1}$:

$$\eta = \frac{q}{{p^{\prime } }} = 3\;\frac{{\;\left( {\frac{{\sigma_{1}^{\prime } }}{{\sigma_{3}^{\prime } }} - 1} \right)\;}}{{\;\left( {\frac{{\sigma_{1}^{\prime } }}{{\sigma_{3}^{\prime } }} + 2} \right)\;}} = \frac{{M - \left( {1 + \frac{2}{3}M} \right)\;\left( {\frac{{\dot{\varepsilon }_{v} }}{{\dot{\varepsilon }_{1} }}} \right)}}{{1 - \frac{1}{3}\;\left( {1 + \frac{2}{3}M} \right)\;\left( {\frac{{\dot{\varepsilon }_{v} }}{{\dot{\varepsilon }_{1} }}} \right)\,}}$$

(12)

Equation (12) signifies that $\dot{\varepsilon }_{v} /\dot{\varepsilon }_{1}$, commonly referred as dilation in the soil mechanics literature (e.g., Bolton [6]), plays a central role in mobilization of η in the unbreakable frictional granular soils under shear. Following treatments of Wan and Guo [63] and Li and Dafalias [35], M may be replaced by either $M^{ * } = M\,(e/e_{cs} )^{\theta }$ or $M^{ * } = M\,\exp (m\psi )$, respectively, in order to further refine predictive capacity of Eq. (12) for granular media of dissimilar states with respect to the CSL wherein θ and m are positive material parameters. e and e_cs are the current and critical state void ratios and $\psi = e - e_{{{\text{cs}}}}$ is the state parameter of Been and Jefferies [4].

For the conventional drained tests wherein ε_v and ε₁ are uncoupled (ε₁ is applied and ε_v is the free response of soil specimen), ε_v is nullified gradually and simultaneously, $\eta$ approaches toward M with the increase in ε₁ [say $(\dot{\varepsilon }_{v} /\dot{\varepsilon }_{1} ) \to 0$, $\eta \to M$ and $e \to e_{{{\text{cs}}}} \;({\text{equally}}\;\,\psi \to 0)$ as $\varepsilon_{1} \, \to + \infty$]. For the linear ε_v–ε₁ coupling (see Eq. (9)), $\dot{\varepsilon }_{v} /\dot{\varepsilon }_{1} \;[ = \zeta ]$ is kept fixed during shear, and thus, $\eta$ approaches steadily to $[M^{ * } - \zeta \,(1 + 2M^{ * } /3)]/[1 - \zeta \,(1 + 2M^{ * } /3)/3]$ as $\varepsilon_{1} \, \to + \infty$. This means that $\eta$ does not approach M and the critical state is not achieved owing to the linear ε_v–ε₁ coupling unless $\zeta \, = 0$ [i.e., constant volume condition] is prescribed. Under the constant volume condition, $e \to e_{{{\text{cs}}}} \;({\text{say}}\;\,\psi \to 0)$, $M^{ * } \to M$, and accordingly, $\eta \to M$ happen as $\varepsilon_{1} \, \to + \infty$. Differentiation of Eq. (10) yields $(\dot{\varepsilon }_{v} /\dot{\varepsilon }_{1} ) = R\,(\varepsilon_{v\infty } - \varepsilon_{v} )$, and accordingly, $\eta$ is calculated from $[M^{ * } - R(\varepsilon_{v\infty } - \varepsilon_{v} )\,(1 + 2M^{ * } /3)]/[1 - R(\varepsilon_{v\infty } - \varepsilon_{v} )\,(1 + 2M^{ * } /3)/3]$ for the transient ε_v–ε₁ coupling. In this case, the critical state is achieved asymptotically as $\varepsilon_{v} \, \to \varepsilon_{v\infty }$. Based on these, it can be concluded that ε_v–ε₁ coupling can prevent [under linear coupling] or cause a major delay [under transient coupling] in reaching the actual critical state in granular media. Therefore, the critical state frame is adopted in the following sections as a benchmark. The macro- and microscale responses of particulate assemblies sheared under linear and transient ε_v–ε₁ coupling are then compared to those attained from identical assemblies sheared under the constant volume condition in the critical state frame.

4 Macro- and microscale behavior

4.1 Uncoupled TXC paths

Been and Jefferies [5] put forward evidence for a strong connection between sand behavior and initial state parameter (i.e., $\psi_{0}$):

$$\psi_{0} = e_{0} - e_{{{\text{cs}}}}$$

(13)

wherein e₀ and e_cs are the initial and critical state void ratios, respectively. Sands are in dense state and exhibit non-flow strain hardening when $\psi_{0} < \psi_{\lim }$ (in which $\psi_{\lim }$ = − 0.05 to zero), whereas loose sands with $\psi_{0} > \psi_{\lim }$ have a tendency to post-peak softening and flow instability under constant volume (undrained) shear. Values for e₀, e_cs, and $\psi_{0}$ are reported in Table 2 and final state parameters of drained and undrained tests reported in Table S1 (see supplementary materials).

The constant volume behavior of four dense ($\psi_{0}$ = − 0.033 to − 0.136) assemblies exhibits non-flow strain-hardening behavior up to a reasonably unique critical state in the q versus p′ [Fig. 3a] and q versus ε_q [Fig. 3b] planes. In Fig. 3c, CN for the specimens with $p_{{\text{c}}}^{\prime }$ = 400 and 600 [kPa] descends steadily with ε_q from 6.25 and 6.15 at ε_q = 0 toward CN_cs ≈ 5.5 at the end of the simulations. For the specimen with $p_{{\text{c}}}^{\prime }$ = 50 and 300 [kPa], CN passes transient minimums at ε_q ≈ 2.25 [%] and 7.25 [%], respectively, and then improves gradually to reach CN_cs ≈ 5.5 [−] asymptotically. ||dev F|| versus ε_q curves in Fig. 3d indicate that the decrease in $\psi_{0}$ amplifies the peak in ||dev F||; however, all ||dev F|| versus ε_q curves approach gradually toward a unique ||dev F||_cs ≈ 0.088 [-] at large shear strains.

Fig. 3

Simulations for the constant volume TXC behavior of four dense particulate assemblies: a q versus ε_q; b q versus p′; c CN versus ε_q; and d ||dev F|| versus ε_q

For three loose ($\psi_{0}$ = − 0.012 to − 0.018) assemblies sheared under the constant volume condition in Fig. 4a, b, flow instability triggered within the range ε_q ≈1.0–1.75 [%], and thereafter, temporary minimum post-peak shear strengths were attained around ε_q ≈ 6.0–7.25 [%] beyond which the q versus ε_q curves improved with ε_q. Due to insufficient shear in Fig. 4a, q did not became plateau prior to the end of simulations, and thus, none of the tests reached the actual critical state. Flow instability of loose sands triggers once the effective stress path intersects instability line obtained from connecting the origin of the q versus p′ plane to the transient constant volume peak shear strength [2, 3, 11, 29, 30, 57, 64]. In Fig. 4b, $\eta_{{{\text{IL}}}}$ ≈0.50 is attained for the slope of the instability line. Temporary minimums for CN in Fig. 4c happened around ε_q ≈ 6.0–8.5 [%], and thereafter, CN improved with ε_q. Contrary to that for the dense assemblies in Fig. 3d, ||dev F|| versus ε_q curves for the loose assemblies in Fig. 4d did not possess any concrete peak and they gradually rose with ε_q to reach ||dev F||_cs ≈0.12. The state-dependent variation of the ||dev F|| versus ε_q curves in Figs. 3d and 4d fortify the notion that mobilization of the ||dev F|| depends on $\psi$.

Fig. 4

Simulations for the constant volume TXC behavior of three loose particulate assemblies: a q versus ε_q; b q versus p′; c CN versus ε_q; and d ||dev F|| versus ε_q

Li and Wang [36] suggested that the critical state line in granular soils can be expressed through:

$$q_{{{\text{cs}}}} = M\,p_{{{\text{cs}}}}^{\prime } ;e_{{{\text{cs}}}} = \,\Gamma - \lambda \,\left( {\frac{{p_{{{\text{cs}}}}^{\prime } }}{{p_{{{\text{ref}}}} }}} \right)^{\,\xi }$$

(14)

wherein M, $\Gamma$, $\lambda$, and $\xi$ are the CSL parameters and p_ref = 101 [kPa] is a reference stress. To construct the CSL, in addition 34 TXC tests were performed. Using the ultimate states of these TXC tests, the critical state parameters were determined at the end of the simulations (i.e., p′, q, and e remained constant upon further shear). The parameters were found to be: M = 0.796 (corresponding to $\varphi_{cs}^{\prime } = \sin^{ - 1} \left[ {3M/(6 + M)} \right] = 20.6^{ \circ }$), $\Gamma$ = 0.741, $\lambda$ = 0.00019, and $\xi$ ≈1. The CSL of the assemblies in the q versus p′ and e versus p′ planes is presented in Fig. 5a, b, respectively.

Fig. 5

CSL of the particulate assemblies in the: a q versus p′; and b e versus p′ planes

4.2 Strain paths with coupling between the volumetric and axial strains

Figure 6 illustrates the mechanical behaviors of nine initially identical dense (e₀ = 0.629 [-], $\psi_{0}$ = − 0.093 [-]) particulate assemblies subjected to linear ε_v–ε₁ coupling under $\zeta$ = − 1.0, − 0.5, − 0.25, − 0.1, − 0.05, 0, + 0.25, + 0.5, and + 1.0 with $p_{c}^{\prime }$ = 100 [kPa]. In fact, the progressive densification/loosening of the assemblies load carrying microstructures caused by the contractive/expansive volume change results in a full spectrum of behaviors from non-flow strain hardening (see tests under $\zeta$ = 0, + 0.25, + 0.50, and + 1.0), to post-peak strain softening with flow instability (see tests under $\zeta$ = − 1.0, − 0.50, − 0.25, − 0.10, and − 0.05) in Fig. 6a–d. Although the initially identical assemblies were in dense state (since $\psi_{0} = - 0.093 < 0$ for all specimens) in the beginning, progressive weakening of load bearing microstructure in the tests with $\zeta$ = − 0.05 and − 0.10 halted early strain hardening and led to partial loss of the shear strength beyond ε_q = 10 [%]. Even a forceful flow instability-induced complete loss of shear strength occurred when specimens were imposed to strong expansive linear ε_v–ε₁ coupling under $\zeta$ = − 0.25, − 0.50 and − 1.0 [see Fig. 6a–d]. On the contrary, contractive ε_v–ε₁ coupling under linear ε_v–ε₁ coupling with $\zeta$ =  + 0.25, + 0.50 and + 1.0 [see Fig. 6a–d] resulted in continuous strengthening of strain hardening compared to that of the specimen sheared under the constant volume condition (say $\zeta$ = 0). Figure 6b, c shows that only state of the specimen sheared under $\zeta$ = 0 condition reached the CSL. Recalling the discussions given in Sect. 3.2, the linear ε_v–ε₁ coupling prevents reaching the critical state. Curves drawn in Fig. 7d indicate the strong influence of the linear ε_v–ε₁ coupling on the accumulation of the equivalent pore-water pressure calculated from $\Delta u = \sigma_{3} - \sigma_{3}^{\prime }$. In Fig. 6d, complete loss of shear strength (say $\Delta u = p_{c}^{\prime }$) due to flow instability happens sooner or later for all specimens sheared under $\zeta < 0$ depending on the magnitude of $\zeta$; however, $\Delta u < 0$ was accumulated for the cases with $\zeta > 0$ following continuous strengthening of load bearing microstructure.

Fig. 6

Simulations for the influence of linear coupling under ζ = − 1.0, − 0.5, − 0.25, − 0.1, − 0.05, 0, + 0.25, + 0.5, and + 1.0 on the behaviors of nine dense (e₀ = 0.629) particulate assemblies with $p_{{\text{c}}}^{\prime }$ = 100 [kPa]: a q versus ε_q; b q versus p′; c e versus ε_q; d Δu versus ε_q; e CN versus ε_q; and f ||dev F|| versus ε_q

Fig. 7

Simulations for the influence of linear coupling under ζ = − 1.0, − 0.5, − 0.25, − 0.05, 0, + 0.10 + 0.25, and + 0.5 on the behaviors of eight loose (e₀ = 0.718) particulate assemblies with $p_{{\text{c}}}^{\prime }$ = 50 [kPa]: a q versus ε_q; b q versus p′; c e versus ε_q; d Δu versus ε_q; e CN versus ε_q; and f ||dev F|| versus ε_q

To have a better perspective of the overall strength of the microstructure, curves for CN versus ε_q are depicted in Fig. 6e. For the specimen sheared under $\zeta$ = 0, the gradual drop of CN begins from 5.8 at ε_q = 0 until CN = 4.9 [-] was reached around ε_q = 4.0 [%], and thereafter, CN improved steadily to reach 5.3 at the critical state. The complete suppression of shear strength for the specimens sheared under $\zeta$ = − 0.25, − 0.50 and − 1.0 in Fig. 6a agrees with the quick drop of CN from 5.8 to 3.3 [-] in Fig. 6e, and accordingly, the complete loss of internal static stability occurred due to the significant deficiency of active contacts as caused by loosening of the load carrying microstructure.

In the shear strain range studied here, CN in the specimens sheared under $\zeta$ = − 0.05 and − 0.10 are lower than that under the $\zeta$ = 0 condition; however, a sudden loss of active contacts is expected for these two specimens at larger shear strains. On the other hand, volumetric contraction in the specimens sheared under $\zeta$ =  + 0.25, + 0.50 and + 1.0 increased CN progressively leading to very stable microstructures as supported by strong strain-hardening responses in Fig. 6a.

The simulated behaviors for eight initially loose (e₀ = 0.718 [-], $\psi_{0}$ = − 0.013 [-]) particulate assemblies subjected to the linear ε_v–ε₁ coupling under $\zeta$ = − 1.0, − 0.50, − 0.25, − 0.05, 0.0, + 0.10, + 0.25 and + 0.50 from $p_{c}^{\prime }$ = 50 [kPa] are illustrated in Fig. 7. The loose specimen sheared under $\zeta$ = 0 condition suffered from a long-standing limited flow instability initiating from the onset of the post-peak strain softening around ε_q≈1% until ε_q ≈ 7% where the specimen strain hardening instigated [see Fig. 7a, b]. Following the progressive loosening of load bearing microstructure associated with expansive volumetric strains [see Fig. 7a–c], all the specimens sheared under $\zeta < 0$ demonstrated permanent complete loss of shear strength and full accumulation of excess pore-water pressure (say $\Delta u = p_{{\text{c}}}^{\prime } = 50\;[{\text{kPa}}]$) in the post-peak regime of the behavior [see Fig. 7d]. Sudden drop of CN from 4.3 [-] to values less than 3 [-] and simultaneously, quick rises in ||dev F|| in Fig. 7e, f point to complete disintegration of the load bearing microstructure for the specimens sheared with $\zeta < 0$. However, in the $\zeta$ = 0 case, CN attained a minimum of 3.2 [-] around ε_q = 3.5 [%] where the specimen lost more than 90 [%] of its peak shear strength, and then, CN improved to CN_cs ≈ 3.4 [-] upon further shearing. On the other hand, contractive volume changes caused mild to strong non-flow responses and accumulation of negative $\Delta$u to occur in the very loose specimens under $\zeta$ =  + 0.10, + 0.25 and + 0.50 in Fig. 7a–d.

Conversely, expansive ε_v–ε₁ coupling intensified initial decrease in p′, slowed down mobilization of q with ε_q, and accumulation of $\Delta$u with ε_q. However, continuous decline in the ultimate shear strength was not the sole outcome of decreasing ε_v∞ and the specimens with ε_v∞ = − 1.50 and − 2.50 [%] suffered from post-peak limited flow instability which was intensified significantly by decreasing ε_v∞. In Fig. 8c, all e versus p′ curves ended on the CSL obtained in Fig. 5b. For the assembly sheared under ε_v∞ = 0 in Fig. 8e, CN suddenly dropped from 5.85 at ε_q≈0 to a minimum of 4.85 [-] around ε_q = 4.75 [%], and then, it improved asymptotically toward 5.35 [-] as ε_q → ∞. The initial loss in CN and peak in ||dev F|| [see Figs. 8e and 9f] decreased with the increase in ε_v∞ from − 2.50 to + 0.90 [%]. However, the minimums of CN were achieved within the range ε_q from 2 to 5 [%], while peaks of ||dev F|| were observed around ε_q ≈ 10 [%]. The latter observations may be attributed to relieve in contact rearrangement under dilative volumetric strains.

Fig. 8

Simulations for the influence of exponential coupling under R = 0.75, and ε_v∞ = − 2.50, − 1.50, − 0.90, − 0.60, − 0.30, 0, + 0.30, + 0.60, and + 0.90 [%] on the behaviors of nine dense (e₀ = 0.629) particulate assemblies with $p_{{\text{c}}}^{\prime }$ = 100 [kPa]: a q versus ε_q; b q versus p′; c e versus ε_q; d Δu versus ε_q; e CN versus ε_q; and f ||dev F|| versus ε_q

Fig. 9

Simulations for the influence of exponential coupling under R = 0.75, and ε_v∞ = −0.90, − 0.60, − 0.30, 0, + 0.30, + 0.60, and + 0.90 [%] on the behaviors of seven medium-loose (e₀ = 0.706) particulate assemblies with $p_{{\text{c}}}^{\prime }$ = 100 [kPa]: a q versus ε_q; b q versus p′; c e versus ε_q; d Δu versus ε_q; e CN versus ε_q; and f ||dev F|| versus ε_q

Results of similar simulations for initially medium-loose (e₀ = 0.706, $p_{c}^{\prime }$ = 100 [kPa], $\psi_{0}$ = − 0.025) assemblies subjected to transient ε_v–ε₁ coupling with ε_v∞ = − 0.90, − 0.6, − 0.30, 0, + 0.30, + 0.60, and + 0.90 [%] under R = 0.75 are depicted in Fig. 9. Owing to the increase in e₀, the specimen sheared under ε_v∞ = 0 showed limited flow response which was followed by a weak strain hardening. Analogous to the previous case, contractive volumetric strains stabilized the mechanical behavior of the assemblies through the increase in the shear strength and decrease in the initial positive $\Delta$u. On the other hand, the assemblies sheared under ε_v∞ = − 0.60 and − 0.90 lost their shear strength almost completely within the shear strain range 5–12 [%]. Also for this case, CN improved slightly with the contractive volumetric strains. On the contrary, CN dropped abruptly for two specimens subjected to dilative volumetric strains with ε_v∞ = − 0.60 and − 0.90 [-] following loosening of microstructure.

Figure 10 indicates the effect of the initial confining pressure ($p_{0}$) on the micro- and macroscale responses of soil assemblies, which were sheared with the same initial void ratio of $e_{0}$ ≈ 0.678 (samples A-13, A-16, and A-28, as outlined in Table 2). In Fig. 10a, representing assemblies under linear coupling, a clear dependency of shear strength on the initial confining pressure is evident for all samples at shear strain levels up to 12 [%]. However, as the strain level increases, all curves converge to a maximum value of shear strength, regardless of the initial confining pressure, and solely dependent on the value of ζ. The ultimate shear strength at higher strain levels (ε_q = 40 [%]) is primarily influenced by ζ. In Fig. 10b, it is evident that all specimens with ζ = 0.05, sheared under different initial confining pressures, display a non-flow strain-hardening response. However, reducing ζ from 0.05 to − 0.1 intensifies the tendency toward a contractive response before reaching the phase transformation state. At ζ = − 0.1, the effective stress paths do not intersect the phase transformation line at all. Instead, a purely contractive behavior is observed, with the mean effective pressure reaching a zero-stress state at high shear strains. According to Fig. 10c, it is observed that the initial CN is nearly identical for samples with the same ζ. However, similarly to the observations made for the shear stress, the value of CN is also influenced by the initial confining pressure at low strain levels (below 12 [%]), converging to an ultimate value at higher strain levels. Moreover, as the shear strain increases, ζ becomes the determining factor also for the ultimate CN value. Samples with lower $p_{0}$ have lower coordination numbers, as shown in Fig. 10c, and higher ║dev F║ values, as depicted in Fig. 10d, at strain levels up to 12 [%]. As the shear strain amplifies, the influence of ζ becomes more pronounced and ultimately decides the final value of ║dev F║ as well. Figure 10e–h present simulation results of the behaviors of particulate assemblies under transient coupling paths. A qualitative comparison between Fig. 10e–h and a–d reveals similar effects of $p_{0}$ on both macro- and microscale responses. Specifically, p₀ influences the response until a certain strain level, while ε_v∞ emerges as the primary decisive factor affecting particulate assemblies’ response under high strain levels (beyond ε_q > 20 [%]).

Fig. 10

Simulations for the influence of initial mean stress on the behaviors of three medium-loose particulate assemblies under linear (left side) and exponential (right side) coupling with the same initial void ratio (e₀ ≈ 0.678) under p_c = 50, 100, and 200 [kPa]: a and e q versus ε_q; b and f q versus p′; c and g CN versus ε_q; d and h ||dev F|| versus ε_q

In Eq. (10), R controls the pace of change in ε_v with ε₁ and the greater R, the faster ε_v varies to reach ε_v∞. The mechanical response of six initially identical medium-loose assemblies (e₀ = 0.706, $p_{{\text{c}}}^{\prime }$ = 100 [kPa], and $\psi_{0}$ = − 0.025) under R = 0.75, 0.45, 0.35, 0.25, 0.15, and 0.05 with ε_v∞ = − 0.60 [%] are simulated in Fig. 11. The transient ε_v–ε₁ coupling paths and the corresponding e versus p′, q versus ε_q, q versus p′ responses are, respectively, depicted in Fig. 11a–d. Even though the target asymptotic void ratio was slightly (change in void ratio is $\Delta e = - (1 + e_{0} )\,\varepsilon_{v\infty } \approx + 0.0102$) greater than e₀ [see Fig. 11b], Fig. 11c, d indicates that R played a profound role in the overall macroscale response of the assemblies in as much as its increase intensified post-peak softening remarkably.

Fig. 11

Simulations for the influence of exponential dilative transient coupling under ε_v∞ = − 0.60 [%] and R = 0.75, 0.45, 0.35, 0.25, 0.15, and 0.05 on the behaviors of seven medium-loose (e₀ = 0.706) particulate assemblies with $p_{{\text{c}}}^{\prime }$ = 100 [kPa]: a ε_v versus ε₁; b e versus ε_q; c q versus ε_q; d q versus p′; e CN versus ε_q; and f ||dev F|| versus ε_q

However, a unique ultimate shear strength is expected to reach by all specimens at large shear strains owing to gradual nullification of ε_v–ε₁ coupling at identical target void ratio [say $e_{0} - (1 + e_{0} )\,\varepsilon_{v\infty }$]. Figure 11e, f signify that the transient minimums in CN decrease with R, while the peak ||dev F|| values increased with R. Nonetheless, the asymptotic CN and ||dev F|| values are practically identical irrespective of R. Similar numerical simulations for three initially identical assemblies (e₀ = 0.706 [-], $p_{{\text{c}}}^{\prime }$ = 100 [kPa], and $\psi_{0}$ = − 0.025 [-]) sheared under R = 0.75, 0.45, and 0.15 with contractive ε_v∞ =  + 0.60 [%] are depicted in Fig. 12 [see Fig. 12a, b for the strain paths and e vs. ε_q curves]. A side-by-side comparison of parts “c” to “f” of Fig. 12 with the corresponding ones in Fig. 11 indicates that change in R under contractive ε_v, at least for the range of values studied here, does not change q versus ε_q, q versus p′, CN versus ε_q, and ||dev F|| versus ε_q responses, profoundly.

Fig. 12

Simulations for the influence of exponential contractive transient coupling under ε_v∞ =  + 0.60 [%] and R = 0.75, 0.45, and 0.15 on the behaviors of three medium-loose (e₀ = 0.706) particulate assemblies with $p_{{\text{c}}}^{\prime }$ = 100 [kPa]: a ε_v versus ε₁; b e versus ε_q; c q versus ε_q; d q versus p′; e CN versus ε_q; and f ||dev F|| versus ε_q

Soon after a seismic event, pore-water pressure redistribution may occur following migration of pore water from high-pressure regions to the low-pressure ones. Figure 6, 7, 8, and 9 imply that pore-water influx-induced expansion of soil elements (as caused by $\zeta > 0$ and $\varepsilon_{v\infty } > 0$) weakens sand strength against flow liquefaction instability significantly. Moreover, the increase in rate of pore-water influx intensifies post-peak softening after triggering of flow liquefaction instability. In contrast, pore-water outflux causes volumetric contraction and mitigates flow liquefaction vulnerability depending on the amount and rate of pore-water outflux.

4.3 Critical state of specimens sheared under transient ε _v–ε ₁ coupling

A condition of nil volume change while soil element continues shearing without any further change in effective stress tensor is a prerequisite for the critical state (e.g., [5, 13, 20, 47, 70]). Nullification of ε_v never fulfills in strain paths with linear ε_v–ε₁ coupling, and accordingly, such strain paths are unable to direct soil state toward the CSL [see Sect. 3.2]. However, ε_v is gradually terminated in strain paths with transient ε_v–ε₁ coupling. The ultimate states of specimens sheared under ε_v∞ = − 2.50, − 1.50, − 0.90, − 0.60, − 0.30, + 0.30, + 0.60, and + 0.90 [%] are depicted in the q versus p′ [see Fig. 13a] and e versus p′ [see Fig. 13b] planes. For the sake of comparison, the critical state data achieved from uncoupled tests [i.e., constant volume, conventional drained and p′-constant drained tests in Sect. 4.1] are also superimposed on the figures. Figure 13a, b indicates that a single CSL describes the ultimate states of the assemblies sheared under uncoupled stress paths and the ones sheared with transient ε_v–ε₁ coupling.

Fig. 13

Critical state characteristics of the assemblies subjected to transient coupling between the volumetric and axial strains: a CSL in the q versus p′ plane; b CSL in the e versus p′ plane; c CN_cs versus $p_{{{\text{cs}}}}^{\prime }$; d ||dev F||_cs versus CN_cs; and e ||dev F ||_cs versus $p_{{{\text{cs}}}}^{\prime }$

Similarly, the data for CN_cs versus $p_{{{\text{cs}}}}^{\prime }$, ||dev F||_cs versus CN_cs and ||dev F||_cs versus $p_{{{\text{cs}}}}^{\prime }$, respectively, illustrated in Fig. 13c–e are uniquely interrelated with each other in the numerical simulations of the uncoupled stress paths as well as strain paths with transient ε_v–ε₁ coupling. Using the data of CN_cs for a total of 34 uncoupled tests in Fig. 13c leads to the following nonlinear empirical correlation between CN_cs and $p_{{{\text{cs}}}}^{\prime }$:

$${\text{CN}}_{{{\text{cs}}}} = 8.79 - 6.69\,\exp \,\left[ { - 0.414\,\left( {\frac{{p_{{{\text{cs}}}}^{\prime } }}{{p_{{{\text{ref}}}} }}} \right)^{0.26} } \right]$$

(15)

It is observed that Eq. (15) can reasonably predict the CN_cs versus $p_{{{\text{cs}}}}^{\prime }$ data for the uncoupled and transient coupling tests in Fig. 13c. The data in Fig. 13d suggest a linear correlation between ||dev F||_cs and CN_cs:

$$\left\| {{\text{dev}}\,{\mathbf{F}}} \right\|_{{{\text{cs}}}} = 0.242 - 0.0275\;{\text{CN}}_{{{\text{cs}}}}$$

(16)

Equations (15) and (16) imply that an increase in $p_{{{\text{cs}}}}^{\prime }$ [or a decrease in e_cs from Eq. (14)_b] results in an increase in CN_cs and subsequently a decrease in ||dev F||_cs. Figure 13e corroborates that ||dev F||_cs versus $p_{{{\text{cs}}}}^{\prime }$ data vary in a narrow range in a way that the increase in $p_{{{\text{cs}}}}^{\prime }$ causes a decrease in ||dev F||_cs that can be expressed through the following empirical relationship:

$$\left\| {{\text{dev}}\,{\mathbf{F}}} \right\|_{{{\text{cs}}}} = 0.184\,\exp \;\left[ { - 0.414\,\left( {\frac{{p_{{{\text{cs}}}}^{\prime } }}{{p_{{{\text{ref}}}} }}} \right)^{0.26} } \right]$$

(17)

5 Anisotropy of contact force networks

Inter-particle contact forces in granular systems are transmitted by means of contact force networks. Azéma and Radjaï [4], Guo and Zhao [14], Zhou et al. [71], Shi and Guo [55], and Liu et al. [37] have addressed the bimodal nature of the force network and partitioned it into the strong and weak networks. In the strong network, the transmitted forces are greater than a certain threshold and distribution of force chains may demonstrate a considerable anisotropy leading to anisotropic macroscale response of the assembly. On the other hand, contact forces less than the threshold value form the weak network and their spatial distribution remains almost unchanged during loading. Inspired by Azéma and Radjaï [4], Guo and Zhao [14], and Shi and Guo [55] among others, $f_{{{\text{thr}}}} = \left\langle f \right\rangle$ is adopted here as the dividing threshold whereby $\left\langle f \right\rangle$ is the average value of the transmitted forces in the force network.

To understand the macro-mechanical behavior of assemblies when subjected to linear coupling, three dense assemblies were studied. These assemblies were subjected to linear ε_v–ε₁ coupling with $\zeta$ values of − 0.25, 0.0, and + 0.25 as shown in Fig. 6. The evolution of the strong force networks at ε_q values of 0, 2, 10, and 30 [%] is shown in Fig. 14a–c. After isotropic compression and prior to shear, the average contact force in strong network of the identical specimens was uniformly distributed and amounted 0.00757 [N], which corresponds to an initial isotropic stress condition. For the three specimens, a slight increase in the strong force networks is observed at ε_q = 2 [%] which may be attributed to the increase in shear stress at the early stage of shearing [see Fig. 6a]. However, at this stage, the distribution of the strong network force remains generally isotropic. In Fig. 14a, loosening of the strong force networks emerges for higher shear strains until the end of the experiment with $\zeta$ = − 0.25 in a way that the average force magnitudes in the strong networks amounted 0.0110, 0.0173, and 0.00006 [N] corresponding to ε_q = 2, 10, and 30 [%], respectively. Clearly, up to εq = 10 [%], the shear stress shown in Fig. 6a for ζ = − 0.25 increases, which results from the increase in contact forces as illustrated in Fig. 14a. Furthermore, as the shear stress increases, the vertical stress σ₁ also increases, a result of the development of anisotropic contact forces, particularly in the vertical direction at this strain level [see Fig. 14a]. Finally, at high strains, the shear stress decreases, which is attributed to the diminishing contact force network. For the assembly sheared under the constant volume condition [see Fig. 14b], the strong force networks improve gently and the average force magnitudes of 0.0139, 0.03458, and 0.05949 [N] were obtained at ε_q = 2, 10, and 30 [%], respectively. For the assembly sheared under contractive ε_v–ε₁ coupling with $\zeta$ =  + 0.25 [see Fig. 14c], the strong networks fortified progressively in view of the fact that the average force magnitudes of 0.01718, 0.05230, and 0.1147 [N] were obtained at ε_q = 2, 10, and 30 [%], respectively. The increase in strong force networks, along with the progressive alignment of strong contact forces in the vertical direction, leads to an increase in vertical stress and consequently a progress in the shear stress as shown in Fig. 6a for ζ = 0.0, and especially for ζ = − 0.25.

Fig. 14

Evolution of strong force network with shear strain in tests with the linear coupling between the volumetric and shear strains on three specimens with e₀ = 0.629 and $p_{{\text{c}}}^{\prime }$ = 100 [kPa]: a ζ = − 0.25, b ζ = 0, and c ζ = + 0.25

For three medium-loose assemblies sheared under transient ε_v–ε₁ coupling with ε_v∞ = − 0.90, 0.0, and + 0.90 [%] in Fig. 9, evolution of strong force networks is demonstrated in Fig. 15. The average contact force in the strong networks of identical assemblies was 0.01175 [N] just prior to shear. At this stage, due to isotropic stress conditions, the contact forces are also distributed isotropically. It is important to note that the higher value of the void ratio accounts for the higher average contact force in the strong networks under the same confining pressure (lower value of coordination number). For the assembly sheared under ε_v∞ = − 0.90 [%], the average contact force in the strong networks dropped steadily to reach a minimum of 0.00034 [N] around ε_q = 10 [%], and then, it improved increasingly and reached 0.00966 [N] at ε_q = 30 [%], which is reflected in the development of the shear stress in Fig. 9a. A similar trend is observed in the build-up of excess or negative pore-water pressure, resulting from the decrease or increase in the mean effective pressure, respectively, as depicted in the e versus p’ space in Fig. 9c. Since the variation of shear stress compared to the mean effective stress is not significant, a nearly isotropic distribution of the strong force network is observed in Fig. 15a during almost all stages.

Fig. 15

Evolution of strong force network with shear strain in tests with transient coupling between the volumetric and shear strains on three specimens with e₀ = 0.706 and $p_{{\text{c}}}^{\prime }$ = 100 [kPa]: a ε_v∞ = − 0.9 [%], b ε_v∞ = 0, and c ε_v∞ =  + 0.9 [%]

In the specimen sheared under nil volume change (ε_v∞ = 0), the average contact force in the strong networks decreased slightly to a minimum of 0.0090 [N] around ε_q = 10 [%], and thereafter, it increased until the average contact force of 0.01981 [N] was achieved at ε_q = 30 [%]. The assembly sheared under ε_v∞ = + 0.90 shows no transient weakening of the strong force networks, and the average contact force increase until 0.03039 [N] was reached at q = 30 [%]. These observations align with the macro-mechanical illustrations in Fig. 9a, c, where a decrease in the strong force network correlates with a decrease in mean effective stress and shear stress, while an increase in the strong force network leads to an increase in mean effective stress at a constant or decreasing void ratio. Furthermore, the development of anisotropy in the direction of the strong force network, coupled with the simultaneous growth of the strong force network, results in an increase in shear stress.

The random distribution of strong force networks in Figs. 14 and 15 at ε_q = 0 highlights nearly initial isotropy of the load carrying microstructure prior to the shear. However, horizontal force chains were gradually diminished and a majority of the force chains were aligned vertically (parallel with the major principal stress axis) with increasing ε_q visualizing gradual formation of strong contact anisotropy in the assemblies. It is worth noting that assemblies undergoing contractive coupled paths, both linear and transient coupling, exhibit evident force chain pathways through particles, which facilitate the transmission of contact forces significantly larger than the average. Conversely, under expansive coupled paths, a discernible shift occurs in the pattern of load transfer within the assemblies. In such cases, grain loops or clusters surrounding the force chains form what is referred to as the weak force chain, playing a crucial role in stabilizing these linear patterns.

3D histograms of the contact force networks for three dense specimens subjected to linear coupling with $\zeta$ = − 0.25, 0, and + 0.25 (see Fig. 6) at ε_q = 0, 2, 10, and 30 [%] are illustrated in Fig. 16a–c. At ε_q = 0, the specimens are nearly isotropic with almost no preferential orientation of contact forces. For the dilative linear coupling with $\zeta$ = − 0.25 [see Fig. 16a], the contact force networks became highly anisotropic at ε_q = 2 [%] and thereafter became shrunk and more isotropic steadily with further straining. On the other hand, for the constant volume case with $\zeta$ = 0 as well as contractive linear coupling with $\zeta$ = + 0.25, applying shear strain caused continuous strengthening of the contact force networks and made their distribution progressively anisotropic until the end of simulation in Figs. 16b, c. 3D histograms of the contact force networks for three medium-loose assemblies sheared under transient coupling with ε_v∞ = − 0.90, 0, and + 0.90 (see Fig. 11) computed at ε_q = 0, 2, 10, and 30 [%] are demonstrated in Fig. 17a–c. For the case with transient ε_v–ε₁ coupling under ε_v∞ = − 0.90 [see Fig. 16a], 3D histograms got smaller steadily until ε_q = 10 [%] was reached, and then, the assembly became progressively anisotropic with further straining. However, for the constant volume (say ε_v∞ = 0) and contractive transient ε_v–ε₁ coupling under ε_v∞ = − 0.90, 3D histograms of the contact forces became increasingly anisotropic in Fig. 17b, c.

Fig. 16

3D histograms for evolution of contact fabric with shear strain in tests with the linear ε_v–ε₁ coupling with e₀ = 0.629 and $p_{{\text{c}}}^{\prime }$ = 100 [kPa] under: a ζ = − 0.25, b ζ = 0, and c ζ = + 0.25

Fig. 17

3D histograms for evolution of contact fabric with shear strain in tests with transient ε_v–ε₁ coupling with e₀ = 0.706 and $p_{{\text{c}}}^{\prime }$ = 100 [kPa] under: a ε_v∞ = − 0.9 [%], b ε_v∞ = 0, and c ε_v∞ = + 0.9 [%]

6 Investigation of instability

6.1 Redundancy index

In Fig. 18a, the sliding contact fraction f_s [see Sect. 2.1 for the definitions of f_s and I_R] of the dense specimens sheared under linear ε_v–ε₁ coupling with $\zeta$ = − 1.0, − 0.50, and − 0.25 [see Fig. 6 for the tests] is almost doubled following abrupt jumps at ε_q ≈ 3.4, 9.7, and 28.3 [%], respectively, and thereafter, the redundancy index I_R drops below 1.0 and the specimens become hypostatic [as marked by red asterisks in Fig. 18b]. The latter observation and sudden decrease in CN in Fig. 6e are definite signs of flow instability in the aforementioned tests. For the tests sheared under linear ε_v–ε₁ coupling with $\zeta$ = 0.0, + 0.25, and + 0.50, I_R initially drops from 1.45 to 1.12, 1.16, and 1.21, respectively, which are all sufficiently greater than 1. Subsequently, I_R increases steadily with ε_q. Therefore, the assemblies sheared under $\zeta$ = 0.0, + 0.25, and + 0.50 always remain hyperstatic and resistant to flow instability. For the loose assemblies in Fig. 18c [see Fig. 7 for the tests], the abrupt jumps in the f_s versus ε_q curves diminish gradually with increasing $\zeta$ in so far as no jump is observed for the tests with $\zeta$ =  + 0.10 and + 0.25. At ε_q = 0 in Fig. 7e, CN is slightly above 4.0 for identical loose specimens, and accordingly, I_R ≈ 1.055 is obtained prior to shear. I_R drops below 1.0 in the specimens sheared with $\zeta$ = − 0.50, − 0.25, − 0.05, 0.0, and + 0.10 and these assemblies become hypostatic in shear strains not greater than 0.4 [%]. However, for the specimens sheared under $\zeta$ = 0.0 and + 0.10, I_R becomes greater than 1.0 at ε_q ≈3.7, and 16.3 [%] following a tangible improvement of CN in Fig. 7e [see blue asterisks in Fig. 18d]. Consequently, the latter specimens become hyperstatic and demonstrate strain hardening subsequent to transient flow instability [see Fig. 7a]. Finally, CN and I_R improve progressively in specimen sheared under $\zeta$ =  + 0.50 and therefore, this specimen always remains hyperstatic and no sign of flow instability is observed in its macroscale behavior.

Fig. 18

Sliding contacts fraction, and redundancy index for: a and b seven dense (e₀ = 0.629) particulate assemblies under linear coupling with ζ = − 1.0, − 0.50, − 0.25, − 0.05, 0, + 0.25, and + 0.50; c and d six loose (e₀ = 0.718) particulate assemblies under linear coupling with ζ = − 0.50, − 0.25, − 0.05, 0, + 0.10, and + 0.25 [%] sheared from $p_{{\text{c}}}^{\prime }$ = 100 [kPa]

All I_R versus ε_q curves for nine dense assemblies subjected to ε_v∞ = − 2.50, − 1.50, − 0.90, − 0.60, − 0.30, 0, + 0.30, + 0.60, and + 0.90 [%] in Fig. 19a drop to reach their minimums around ε_q = 2.3–3.1 [%] and then begin to improve with ε_q until the critical state. Except for the specimen sheared under ε_v∞ = − 2.50 [%], I_R for the rest of the experiments always remains greater than 1 explaining strong strain hardening in the tests with ε_v∞ ≥ − 1.50 [%]. The specimen sheared under ε_v∞ = − 2.50 [%] becomes hypostatic around ε_q = 1.3 [%]; however, further shearing results in an increase in CN following a transient minimum and the specimen becomes hyperstatic again around ε_q = 5.7 [%] {onset of hypo- and hyperstatic regimes are superimposed on stress paths in Fig. 19b}. The mentioned scenario clarifies limited flow instability under transient expansive ε_v–ε₁ coupling with ε_v∞ = − 2.50 [%] in Fig. 19a. Similar I_R versus ε_q curves in Fig. 19c for the seven medium-loose particulate assemblies sheared under ε_v∞ = − 0.90, − 0.60, − 0.30, 0, + 0.30, + 0.60, and + 0.90 [%] (Fig. 9 for the tests) indicate that the increase in the expansive ε_v∞ accelerates transition from hyperstatic regime to hypostatic regime and causes triggering of the post-peak flow instability at lower shear strengths in the tests sheared under ε_v∞ = − 0.90, − 0.60, − 0.30, and 0 [%].

Fig. 19

Redundancy index and onset of the mechanical instability for: a and b nine dense (e₀ = 0.629) assemblies under R = 0.75 and ε_v∞ = − 2.50, − 1.50, -0.90, − 0.60, − 0.30, 0, + 0.30, + 0.60, and + 0.90 [%]; c and d seven medium-loose (e₀ = 0.706) assemblies under R = 0.75 and ε_v∞ =− 0.90, − 0.60, − 0.30, 0, + 0.30, + 0.60, and + 0.90 [%]; e and f six medium-loose (e₀ = 0.706) assemblies under R = 0.05, 0.15, 0.25, 0.35, 0.45, and 0.75, and ε_v∞ = − 0.60 [%] from $p_{{\text{c}}}^{\prime }$ = 100 [kPa]

On the other hand, the increase in the expansive ε_v∞ postpones the reverse transition from the hypostatic regime to the hyperstatic one and widens the limited flow instability domain. Figure 19d also signifies the certain states for the transition from hyperstatic to hypostatic regime stand slightly after the beginning of the post-peak softening, and the increase in the expansive ε_v∞ causes a concrete decrease in the stress ratio (i.e., $q/p^{\prime }$) at the onset of phase transition from hyperstatic to hypostatic regime. Of note, $I_{R} > 0$ holds in the entire shear strain range studied here for the initially medium-loose specimens sheared with ε_v∞ = + 0.30, + 0.60, and + 0.90 [%] and such systems remain hyperstatic until the end of simulations.

An increase in the pace of expansive volume change (i.e., R) in Fig. 19e, f for six tests sheared under ε_v–ε₁ coupling with ε_v∞ = − 0.60 [%] speeds up the decrease in I_R and p' leading to faster transition from hyperstatic to hypostatic regime, while delaying reverse transition from hypostatic to hyperstatic regime following phase transformation.

A side-by-side comparison of Figs. 6a, b and 18b indicates that I_R = 1 for the dense (e₀ = 0.629) assemblies with ζ = − 0.25, − 0.50, and − 1.0 fulfilled at ζ = 0.734, 0.917, and 0.968, respectively. Accordingly, the mobilized friction angles at onset of I_R = 1, $\varphi_{{I_{R} = 1}}^{\prime } = \sin^{ - 1} \left[ {\frac{3\eta }{{6 + \eta }}} \right]_{{\,I_{R} = 1}}$, under ζ = − 0.25, − 0.50, and − 1.0 become 19.1°, 23.4°, and 24.6°, respectively. A similar practice for the loose (e₀ = 0.718) assemblies in Figs. 7 and 18d sheared under ζ = 0, − 0.05, − 0.25, and − 0.50 results in $\eta$ = 0.412, 0.398, 0.301, and 0.225 corresponding to $\varphi_{{I_{R} = 1}}^{\prime } =$ 11.1°, 10.7°, 8.2°, and 6.2° at the onset of I_R = 1, respectively. Accordingly, the onset of mechanical instability (i.e., I_R = 1) of the initially dense assemblies sheared under expansive linear ε_v–ε_a coupling may be triggered at mobilized friction angles more or less greater than the critical state friction angle [i.e., $\varphi_{{{\text{cs}}}}^{\prime } = 20.6^{ \circ }$ (see Sect. 4)] of the assemblies. On the other hand, the mobilized friction angle of the initially loose assemblies at the onset of mechanical instability is significantly lower than $\varphi_{{{\text{cs}}}}^{\prime }$. CN at the onset of I_R = 1 varied in a narrow range from 4.20 to 4.22 for the loose assemblies, whereas it varied 4.13 to 4.29 for the dense ones. In the dense assemblies in Fig. 18b, ||dev F|| at I_R = 1 increased from 0.118 to 0.138 owing to the increase in ζ from − 0.25 to − 1.0; however, only a negligible change in ||dev F|| from 0.035 to 0.015 occurred in the loose assemblies following the increase in ζ from zero to − 0.5.

Intensifying expansive ε_v–ε_a coupling from ε_v∞ = 0 (i.e., constant volume condition) to − 0.9 [%] in Figs. 9 and 19d caused $\varphi_{{I_{R} = 1}}^{\prime }$ and ||dev F|| to decrease from 15.9° to 12.9°, and 0.062 to 0.045, respectively, at the onset of I_R = 1 for the medium-loose assemblies. For the medium-loose assemblies subjected to transient ε_v–ε_a coupling with ε_v∞ = − 0.6 [%] in Figs. 11 and 19f, an increase in R from 0.05 to 0.75 led to a decrease in $\varphi_{{I_{R} = 1}}^{\prime }$ from 15.7° to 14.0°, and decrease in ||dev F|| from 0.066 to 0.049 at the onset of I_R = 1. However, CN ≈ 4.35 remained insensitive to change in ε_v∞ and R for the experiments in Fig. 19c–f. Analysis of the data mentioned above indicates that expansive transient ε_v–ε_a coupling may cause mechanical instability earlier than the complete mobilization of $\varphi_{{{\text{cs}}}}^{\prime }$ in the loose and medium-loose assemblies.

6.2 Instability line

Yang [67] has suggested that $\eta$ at the onset of post-peak softening (i.e., $\eta$_PPS) in undrained TXC tests can be interrelated to $\psi_{0}$ (i.e., initial value of $\psi$) through:

$$\eta_{{{\text{PPS}}}} = A\,M\exp (B\,\,\psi_{0} )$$

(18)

wherein $A(\; < 1)$ and B are positive state-dependent sand parameters, and M is the slope of CSL in the q–p′ plane. Recently, Lashkari et al. [32] showed that Eq. (18) is valid for sands sheared under the constant volume condition in the direct simple shear apparatus. More recently, analysis of the $\eta$_PPS data obtained from direct simple shear tests with linear coupling between the volumetric and shear strains in Lashkari et al. [30] has revealed that A in Eq. (18) is not fixed and decreases with ζ. In other words, dilative ε_v–ε₁ coupling necessitates lower “A” values.

For a total of 51 DEM tests suffering from flow instability under linear ε_v–ε₁ coupling, data of stress ratio at the onset of post-peak strain softening (i.e., $\eta_{{{\text{PPS}}}} = q_{{{\text{PPS}}}} {/}p_{{{\text{PPS}}}}^{\prime }$ wherein $q_{{{\text{PPS}}}}$ and $p_{{{\text{PPS}}}}^{\prime }$ are, respectively, q and p′ at the onset of post-peak strain softening) in Fig. 20a indicate that $\eta$ _PSS decreases with increasing $p_{c}^{\prime }$ and e₀; however, the increase in ζ and transition from dilative to contractive coupling yields a tangible rise in $\eta_{{{\text{PPS}}}}$. Figure 20a corroborates that, when the initial state is looser at the beginning (e.g., $e_{0} \ge 0.677$), the greater expansive ε_v–ε₁ coupling, achieved through the decrease in ζ, or higher confining pressure, results in a lower η_PSS (compared to than M). This implies that the onset of post-peak softening in loose state is expected to be triggered before the mobilization of $\varphi_{{{\text{cs}}}}^{\prime }$. However, for the initially medium and dense (e.g., $e_{0} \le 0.646$) assemblies with relatively low values (e.g., $p_{{\text{c}}}^{\prime } \le 200\,{\text{[kPa]}}$), post-peak softening at stress ratios greater than M_c arises due to linear ε_v–ε₁ coupling-induced progressive loosening of the assemblies.

Fig. 20

Stress ratio at the onset of post-peak strain softening in tests with linear ε_v–ε₁ coupling: a DEM data, b empirical equation prediction versus DEM data

Using the data shown in Fig. 20a, a calibration of Eq. (18) in the following form can link η_PSS to ζ, e₀, and ψ₀ for the virtual DEM experiments with linear ε_v–ε₁ coupling:

$$\eta_{{{\text{PPS}}}} \approx 0.948\;\left[ {1 - \left( {\frac{{e_{0} }}{0.741}} \right)^{25.5} + 0.20\,{\text{sign}}(\zeta )\,\left| {\,\zeta \,} \right|^{\,0.5} } \right]\;M\;\exp ( - 5.18\,\psi_{0} )$$

(19)

wherein $B = - 5.18$ and $A = 0.948\;\left[ {1 - \left( {\frac{{e_{0} }}{0.741}} \right)^{25.5} + 0.20\,{\text{sign}}(\zeta )\,\left| {\,\zeta \,} \right|^{\,0.5} } \right]$ is a function of e₀ and ζ. In Fig. 20b, the predicted η_PPS values from Eq. (19) demonstrate a reasonable agreement with the corresponding data from DEM simulations. Similar studies for a total of 20 DEM tests suffering from flow instability under transient ε_v–ε₁ coupling with R = 0.75 are presented in Fig. 21a wherein $\eta_{PPS}$ increases with decreasing ε_v∞, e₀, and ψ₀. Compiling the data shown in Fig. 20a together with data of the specimens suffering from flow stability under different R values, the following calibration of Eq. (18) is suggested linking η_PSS to ε_v∞, R, e₀, and ψ₀ through:

$$\eta_{{{\text{PPS}}}} \approx 0.948\;\left[ {1 - \left( {\frac{{e_{0} }}{0.741}} \right)^{25.5} + 8.41\,\left( {\frac{{\varepsilon_{v\infty } \;[\% ]}}{100\;[\% ]}} \right)\;\left( {1 + 1.184\,R} \right)} \right]\;M\;\exp ( - 5.18\,\psi_{0} )$$

(20)

wherein $B = - 5.18$ and $A = 0.948\;\left[ {1 - \left( {\frac{{e_{0} }}{0.741}} \right)^{25.5} + 8.41\,\left( {\frac{{\varepsilon_{v\infty } \;[\% ]}}{100\;[\% ]}} \right)\;\left( {1 + 1.184\,R} \right)} \right]$ is a function of e₀, ε_v∞, and R. Of note, Eqs. (19) and (20) predict identical post-peak stress ratio [i.e., $\eta_{{{\text{PPS}}}} \approx 0.948\;\left[ {1 - \left( {\frac{{e_{0} }}{0.741}} \right)^{25.5} } \right]\;M\;\exp ( - 5.18\,\psi_{0} )$] under the constant volume condition wherein ζ = 0 or ε_v∞ = 0 holds. Predictions obtained from Eq. (20) agree with the DEM data in Fig. 21b.

Fig. 21

Stress ratio at the onset of post-peak strain softening in tests with transient ε_v–ε₁ coupling: a DEM, b empirical equation prediction versus DEM data

Figure 22 illustrates a 3D visualization of post-peak softening properties in e_PPS-η_PPS-CN space under transient coupling conditions. This analysis takes into consideration a wide range of initial void ratios and confining pressures. To enhance the visual clarity of this multidimensional dataset, a color bar is utilized to represent the distance from the x-axis within the 3D space. In Fig. 22, a direct correlation is observed between e_PPS and η_PPS, where an increase in e_PPS corresponds to a decrease in η_PPS. This observation highlights that higher initial void ratios are associated with lower stress ratios at the onset of instability. Additionally, post-peak softening is prominently evident for CN values within the range of 4.1 to 5.8.

Fig. 22.

3D visualization of post-peak softening properties in e_PPS-η_PPS-CN space under transient coupling conditions

7 Conclusions

The influence of coupling between the volumetric and axial strains on flow instability of the cubic particulate assemblies was investigated in this paper. To this end, a total of 213 DEM simulations (i.e., 24 constant volume, 28 drained, 68 tests with linear ε_v–ε₁ coupling, and 93 tests with transient nonlinear ε_v–ε₁ coupling) were carried out. The main findings from this study are:

• Loosening of load bearing microstructure during expansive ε_v–ε₁ coupling may lead to flow instability of particulate assemblies. It was observed that the decrease in $\zeta < 0$ (under linear ε_v–ε₁ coupling), and decrease in $\varepsilon_{v\infty } < 0$ and increase in R (under transient ε_v–ε₁ coupling) result in the decrease in peak shear strength, faster post-peak loss of shear strength, rapid accumulation of positive pore-water pressure, quick decrease in CN, and easier evolution of anisotropy in contact force networks.

• Densification of load bearing microstructure under contractive ε_v–ε₁ coupling mitigates (or even prevents) the potential of flow instability in particulate assemblies. An increase in $\zeta > 0$ (under linear coupling) and increases in $\varepsilon_{{{\text{v}}\infty }} > 0$ and R (under transient coupling) cause the increase in peak shear strength, slower post-peak loss of shear strength (if any), slow accumulation of positive pore-water pressure, faster increase in CN, and later evolution of anisotropy.

• Linear ε_v–ε₁ coupling prevents particulate specimens from reaching the critical state. No asymptotic values were obtained for CN and $\left\| {{\text{dev }}{\mathbf{F}}} \right\|$ at large shear strains in particulate assemblies shared under linear ε_v–ε₁ coupling.

• State of the particulate assemblies sheared under transient ε_v–ε₁ coupling approaches the CSL progressively owing to gradual nullification of coupling. For such strain paths, CSL and CN versus p′, and $\left\| {{\text{dev }}{\mathbf{F}}} \right\|$ versus CN data at the critical state are identical with those of the specimens sheared under the uncoupled paths.

• Flow instability can be investigated by means of sudden jumps in sliding contacts fraction f_s and redundancy index $I_{{\text{R}}} < 1$. It was observed that $I_{{\text{R}}} < 1$ is fulfilled faster with the decrease in $\zeta < 0$ (in case if dilative linear ε_v–ε₁ coupling), decrease in $\varepsilon_{v\infty } < 0$ and increase in R (in case if transient dilative ε_v–ε₁ coupling). On the other hand, increases in $\zeta > 0$, $\varepsilon_{{{\text{v}}\infty }} > 0$ [both leading to contractive ε_v–ε₁ coupling] and decrease in R alleviate flow instability susceptibility through a progressive improvement of I_R in medium and large shear strains.