Experimental and Constitutive Modeling Investigations of the Mechanical Behaviors of a Gravelly Soil Material Under Large-Size Triaxial Cyclic Tests

Abstract

This study investigates the mechanical behavior of gravelly soil under various confining pressures using large-size triaxial cyclic tests and a novel constitutive model. Key properties analyzed include stress-dependent dilatation, nonlinear strength, cumulative plastic strain, cyclic hysteresis, hardening, and particle breakage. Experimental results show that confining pressure significantly affects volume deformation, strength, and failure modes. Specifically, volume deformation shifts from dilatation to contraction with increasing pressure, and failure modes transition from drum-shaped to compressive shear. The developed model integrates stress-dilatancy equations, plastic flow directions, and plastic moduli within the critical state soil mechanics framework, effectively capturing cyclic loading and unloading behaviors. A particle breakage index and a differential equation for void ratio evolution are included to reflect relative density changes. The material constants of this constitutive model are derived from large-size triaxial cyclic tests. The model's material constants are derived from large-size triaxial cyclic tests. Comparison with experimental data confirms the model's accuracy and potential applications in stress path analysis and complex engineering projects, demonstrating its adaptability to varying mechanical stress conditions.

1 Introduction

Gravelly soil, a blend of gravel and finer soil particles, is extensively utilized in various geotechnical applications such as roadbeds, dams, and mountainous slopes due to its material composition. Over the past few decades, numerous large-size triaxial monotonic and cyclic experiments have been conducted on this material [1,2,3]. These studies have revealed several critical mechanical properties of gravelly soil, including particle breakage [4, 5], confinement-dependent volume changes, whether contraction or dilatation [3, 6], nonlinear strength [7], and the hysteresis effects of unloading and reloading stresses [2]. When developing constitutive models for gravelly soil materials, it is essential to accurately incorporate these significant mechanical properties [1, 8, 9].

Experimental observations indicated that particle breakage is a critical factor having an undeniable impact on soil behavior. In theoretical field, the continuous breakage mechanics theory [10] reveals the similarity between the breakage mechanisms of granular materials and damage. This theory explores the coupling effect between particle breakage and plasticity, demonstrating the impact of active and passive breakage on material hardening and plastic behavior under compression and shear conditions [11]. These studies [10, 11] provide a comprehensive framework for understanding the effects of particle breakage on granular materials.

In the field of experimental research, several tests have explored the effects of experimental conditions [12] and particle size distribution [13] on the breakage of sand particles, emphasizing the importance of input energy per unit volume and axial strain [12] and revealing the relationship between particle size and breakability [13]. Additionally, mechanical experiments on calcareous and siliceous sands have established the correlation between particle shape and breakage at the dilation-contraction transition point of volumetric strain, as well as the impact on volumetric shrinkage trends [14]. In a one-dimensional compression test, it was even observed that particle breakage significantly affects the compressibility and settlement of sands [15]. These studies highlight the intricate relationship between particle breakage and other mechanical properties, offering crucial insights for developing more precise constitutive models. Consequently, particle breakage is routinely considered in the constitutive models of gravelly soils [7, 16,17,18].

Developing a constitutive model is crucial for predicting the engineering and mechanical responses of gravelly soil materials under cyclic loading conditions. The transition between loading and unloading phases results in the formation of stress–strain hysteresis curves, highlighting significant differences in plastic moduli between these phases, particularly when initial loading is considered [8, 19]. This complexity poses a challenge for classical elastoplastic models [20], which rely on yield surface definitions, making it difficult for them to comprehensively describe the intricate mechanical behavior of gravelly soil materials under cyclic loading conditions.

To overcome this limitation, many researchers [9, 17, 21] have adopted a generalized plastic framework combined with critical state soil mechanics [22]. Generalized plasticity offers a straightforward approach for modeling stress–strain behavior without explicitly defining yield and plastic potential functions [22], allowing greater flexibility in defining loading and unloading plastic moduli [8, 19, 22, 23]. Moreover, within these frameworks, state-dependent variables such as particle breakage and void ratio can be incorporated into the equation for the critical state line (CSL), capturing the effects of particle breakage on the CSL [7, 17, 18] and the relationship between void ratio evolution and volume change [18, 24, 25]. Due to these advantages, generalized plasticity models and critical state soil mechanics are ideal for developing constitutive models for gravelly soils under cyclic loading conditions.

In this study, large-size triaxial cyclic tests were conducted on a specific gravelly soil material to investigate its mechanical behavior under varying confining pressures. Key mechanical properties of the gravelly soil, including stress-dependent dilatation, nonlinear strength, cumulative plastic strain, cyclic hysteresis effects, and particle breakage, were analyzed. Novel mechanical formulations, such as stress-dilatancy equations, plastic flow directions, and plastic moduli, were developed for initial loading, unloading, and reloading. Then, these mechanical formulations were combined to construct a generalized plastic model within the critical state soil mechanics framework, incorporating void ratio and particle breakage index as state-dependent variables. Methods for determining material constants using large-size triaxial cyclic tests were provided. The model's effectiveness was validated by its high consistency with experimental results, suggesting its potential for advanced stress path analysis and engineering modeling projects in the future.

2 Experiment Programs

2.1 Gravelly Soil Material and Specimen Preparation

The gravelly soil used in this study was sourced from a slope adjacent to a large reservoir in Southwest China. This soil consists of angular gravels, sands, silts, and clays. After sampling, the moisture content of this gravelly soil was measured to be approximately 3.45%. The primary lithology of the gravels included strongly and moderately weathered sandstone and limestone. For experimental purposes, the soil collected from the engineering site was sieved and categorized into six particle size groups: less than 2, 2–5, 5–10, 10–20, 20–40 and 40-60 mm, as depicted in Fig. 1. When the aggregates of these six grain groups are mixed to make a specimen, its sectional image is shown in Fig. 2.

Fig. 1

The gravelly soil material, which was sieved into 6 grain groups

Fig. 2

A prepared specimen of the gravelly soil (its top surface was photographed vertically)

To ensure consistency among all gravelly soil specimens used in the experiment, aggregates from the six particle groups were meticulously mixed in the ratios specified in Table 1. The resulting gradation curve of the specimens, as illustrated in Fig. 3, closely matches that of the actual gravelly soil collected from the engineering site. Basic physical parameters were also determined for the soil prepared according to these proportions. Specifically, the plastic limit and liquid limit of the gravelly soil's moisture content were found to be 4.08% and 11.92%, respectively.

Table 1 Soil mixing scheme for a cylindrical gravelly soil specimen with a diameter of 300 mm and a height of 600 mm

Fig. 3

Particle size distribution of the gravelly soil specimen

The gravelly soil aggregate was placed inside a 3 mm-thick cylindrical rubber film and housed in a split cylindrical steel mold. The dimensions of the constructed gravelly soil specimen were 300 mm in diameter and 600 mm in height. To create a specimen of this size, 91.20 kg of dry aggregates and 3.15 kg of water were used, resulting in a moisture content of 3.45% (Table 1). This moisture content was close to the in-situ condition at the engineering site.

To ensure uniform distribution of water within the dry aggregates, the water was first uniformly mixed with the dry aggregates, and then the mixture was divided into five equal parts. Each portion of the gravelly soil was sequentially added to the rubber film and compacted into a cylindrical shape using a hammer equipped with a rubber pad. After compacting each portion, the top surface was roughened using a soil knife before adding the next portion. This method ensured that the different layers of the gravelly soil form a cohesive unit without layering. During this process, the hammering force was kept as consistent as possible to control particle breakage and to prevent significant deviations in the aggregate's particle size distribution from its initial state.

The resulting specimen had a density of 2.23 g/cm³, a dry density of 2.15 g/cm³, and the specific gravity $d_{\text{s}}$ of the gravelly soil material had been determined to be 2.78. The void ratio $e$ was subsequently calculated using the following formula:

$$e = \frac{{d_{\text{s}} \rho_{\text{w}} }}{{\rho_{\text{d}} }} - 1$$

(1)

where, $\rho_{\text{w}}$ is the density of water, with a value of 1.00 g/cm³. Using Eq. (1), the initial void ratio of the gravelly soil specimen was estimated to be 0.29.

Experimental determination showed that the gravelly soil had a minimum dry density $\rho_{\text{d min}}$ of 2.04 g/cm³, a maximum dry density $\rho_{\text{d max}}$ of 2.28 g/cm³, and an optimum moisture content of 5.71%. Thus, the relative density $D_{\text{r}}$ of the prepared specimen was about 0.49, estimated using the following formula:

$$D_{\text{r}} = \frac{{\left( {\rho_{\text{d}} - \rho_{\text{d min}} } \right)\rho_{\text{d max}} }}{{\left( {\rho_{\text{d max}} - \rho_{\text{d min}} } \right)\rho_{\text{d}} }}$$

(2)

2.2 Large-Size Triaxial Test Apparatus

A large-size triaxial testing machine at Kunming University (Fig. 4) was employed to analyze the mechanical properties of the gravelly soil. This sophisticated geotechnical test equipment consists of six components: the main unit, a dynamic axial pressure automatic excitation system, a static axial pressure system, a triaxial pressure chamber, a confining pressure control system, and an integrated automatic control program with data acquisition and processing capabilities. Designed to accommodate a broad spectrum of geotechnical tests, the machine not only facilitates conventional triaxial tests but also supports various stress path tests, enhancing its versatility and utility in geotechnical research.

Fig. 4

A prepared specimen of the gravelly soil that would be put into large-size triaxial test apparatus

This large-size triaxial test apparatus accommodates cylindrical soil specimens up to 300 mm in diameter and 600 mm in height. It is capable of automatically measuring mechanical properties including axial stress, strain, confining pressure, pore pressure, specimen volume change, saturated water inflow, and consolidation discharge. The machine offers a maximum confining pressure of 3 MPa and a maximum axial static load of 1500 kN, with an axial force resolution of 0.1 kN. Axial displacement measurements range from 0 to 300 mm with a high resolution of 0.001 mm, and pore water pressure can be measured between − 0.1 and 3.0 MPa, with reverse pressure capabilities up to 1.0 MPa.

The testing machine is equipped with an LVDT extensometer, which is used to accurately measure the axial strain of soil specimens. The loading process of confining pressure and axial force is controlled by computer servo, ensuring the accuracy and efficiency of the experiment. The apparatus allows for either force or displacement methods to control axial load applications, typically employing displacement methods with an adjustable strain gauge axial loading rate from 0.01 to 3.0 mm/min, suitable for the testing requirements of most coarse-grained soil materials.

2.3 Experimental Procedures

Following the preparation of the gravelly soil specimen, it is completely sealed within a rubber film as depicted in Fig. 4 and then placed inside the triaxial chamber. The chamber is sealed, and water is injected from its bottom plate until full. An initial confining pressure of about 30 kPa is applied to stabilize the specimen. The specimen takes approximately 2.5 h to become fully saturated. Subsequently, the drainage valve is opened, and confining pressure is increased at a steady rate of 20 kPa/min until the target pressure is reached. This pressure is maintained for one hour to allow for isobaric consolidation with a consolidation ratio of 1.0. Once consolidation is complete, the triaxial drainage shear test can can be conducted.

In this study, six gravelly soil specimens underwent large-size triaxial drained shear cyclic tests with confining pressures ranging from 100 to 600 kPa, increasing in increments of 100 kPa. Axial compression loading was controlled by displacement at a rate of 1 mm/min, while unloading was performed in constant pressure mode at a rate of 2.5 kN/min, stopping when axial compression reached approximately zero. Following this, the mode was switched to displacement control for reloading. The target axial displacement values for unloading points were set at 10, 20, 30, 40, 50, 60 and 70 mm, meaning unloading commenced once axial pressure reached these predetermined values. The process continued as previously described. After the final unloading from 70 mm, displacement loading resumed to 90 mm, at which point the axial strain reached approximately 15%, concluding the experiment. Therefore, the large-size triaxial cyclic tests were uniformly conducted with 7 cycles for each of the six confining pressure conditions.

3 Experimental Observations and Analyses

3.1 Failure Behaviors

The photos of the failed gravelly soil specimens after the large-size triaxial cyclic tests are displayed in Fig. 5. Observations reveal that under confining pressures of 100 and 200 kPa (Fig. 5a, b), the specimens underwent 7 cycles of progressively increasing load and exhibited drum-shaped failure. This suggests that under lower confining pressures, the failure mode of the dense gravelly soil specimens is characterized by volume expansion.

Fig. 5

Photos of the gravelly soil specimens after the large-size triaxial cyclic tests

As illustrated in Fig. 5c, d, under confining pressures of 300 kPa and 400 kPa, the middle part of the gravelly soil specimens exhibits a bulging effect, with radial deformation significantly more pronounced than at the ends. At higher confining pressures of 500 kPa and 600 kPa, no significant bulging is observed in the middle sections of the specimens, as depicted in Fig. 5e, f. This progression indicates that as confining pressure increases, the failure mode of the gravelly soil transitions from volume expansion to compression shear, demonstrating that higher confining pressures suppress lateral deformation of the specimen.

3.2 Particle Breakage

This study focused on particle crushing in the gravelly soil material. During specimen preparation, using a hammer with a rubber pad to compact the materials inevitably led to particle breakage. To clearly understand the impact of specimen preparation on particle breakage, a specimen was dismantled and subjected to sieving experiments immediately after its preparation. The obtained particle grading distribution data were labeled as "After specimen preparation" and recorded in Table 2.

Table 2 Particle grading distributions and particle breakage indices of the gravelly soil under different confining pressures

Subsequently, each failed gravelly soil specimen that underwent large-size triaxial cyclic tests was also subjected to a sieving experiment. The obtained particle grading distribution data are presented in Table 2. When using the particle grading distribution data labeled as "After specimen preparation" as a reference, the influence of large-size triaxial cyclic tests on the particle size distribution of the gravelly soil can be analyzed and quantitatively assessed.

The failed soil specimens were utilized to carry out the sieving tests. The particle grading data, detailed in Table 2, show that large-size triaxial cyclic loading significantly altered the grading distribution of the gravelly soil. As confining pressure increased, the particle grading curve diverged further from the initial one, indicating that large-size triaxial tests not only caused gravelly soil particles to crush but also increased the proportion of fine grains.

When the stresses applied to particles exceed their strength, particle breakage occurs, impacting the soil's mechanical properties. This study incorporates the effects of particle breakage into the constitutive descriptions using the Grain Size Distribution (GSD) method, initially proposed by Hardin [26] and subsequently refined by Einav [10] and Liu et al. [27].

According to GSD method [10, 27], the grain size distribution curve is represented by a percentage function $P\left( {x^{\text{p}} } \right)$ in relation to the grain size $x^{\text{p}}$, where the superscript "p" denotes "particle." The area $A$ bounded by the current GSD curve and the grain size axis can be calculated using the following integration:

$$A = \int_{x_{{\text{max}}}^{\text{p}} }^{x_{{\text{max}}}^{\text{p}} } {P\left( {x^{\text{p}} } \right){\text{d}}x^{\text{p}} } ,$$

(3)

where $x^{\text{p}}_{\max }$ and $x^{\text{p}}_{{\text{m}} {\text{in}}}$ represent the maximum and minimum particle sizes, respectively. The relative breakage index $B_{\text{r}}$ is then defined by Einav [10]:

$$B_{\text{r}} = \frac{A - A_0 }{{A_{\text{u}} - A_0 }},$$

(4)

where $A_0$ and $A_{\text{u}}$ are the areas bounded by the initial and ultimate GSD curves, respectively.

In practice, the values of $A_0$ and $A$ can be directly estimated from the GSD curves before and after the experiment. Figure 6 illustrates the relationship between the area bounded by the GSD curve and the effective mean stress $p^{\prime}$ ($p^{\prime} = {{\sigma^{\prime}_{{\text{kk}}} } / 3}$, the subscript "kk" represents the summary operation of tension) under the failure state of gravelly soil material. While the area $A$ bounded by the GSD curve increases with rising effective mean stress, the growth rate diminishes over time and eventually stabilizes at $A_{\text{u}}$. This pattern of change can be mathematically represented by the following formula:

$$A = A_{\text{u}} - k_0 \exp \left( { - k_1 \frac{{p^{\prime}}}{{p_{\text{a}} }}} \right)$$

(5)

Fig. 6

Relationship between the area $A$ bounded by GSD curves and normalized effective mean stress ${{p^{\prime}} / {p_{\text{a}} }}$ at failure state of the gravelly soil material

Using Eq. (5) to fit the data points in Fig. 6, the ultimate area $A_{\text{u}}$ boundaried by the GSD curve can be estimated. Following the triaxial cyclic tests, the particle breakage indices of the failed specimens were calculated using Eq. (4) and are listed in Table 2.

Given the strong correlation between particle breakage and plastic deformation, Lade et al. [28] and Liu et al. [7] respectively proposed hyperbolic and exponential functions to model the relationship between the particle breakage index and plastic properties. The function proposed by Lade et al. [28] has been extensively used to analyze particle crushing behavior in coarse-grained geological materials [17, 18, 29]. In this study, we have made subtle modifications to the exponential function originally proposed by Liu et al. [7] to better understand the relationship between the particle fragmentation index and plastic dissipation energy in gravelly soil. The modified exponential function is outlined as follows:

$$B_{\text{r}} = 1 - \exp \left( { - b_{\text{r}} \frac{{w^{\text{p}} }}{{p_{\text{a}} }}} \right)$$

(6)

where, $b_{\text{r}}$ is a material constant and $p_{\text{a}}$ is the atmospheric pressure, which usually takes the value of engineering atmospheric pressure, i.e. $p_{\text{a}}$ = 98 kPa. $w^{\text{p}}$ denotes the plastic dissipation energy which is calculated using the following equation:

$$w^{\text{p}} = \int {\sigma_{{\text{ij}}} {\text{d}}\varepsilon_{{\text{ij}}}^{\text{p}} }$$

(7)

By using Eq. (7), the plastic dissipated energy ($w^{\text{p}}$) can be calculated. The test data pairs of ($w^{\text{p}}$, $B_{\text{r}}$) for the six gravelly soil specimens under different confining pressures are displayed in Fig. 7. Subsequently, Eq. (6) was used to fit the data pairs of ($w^{\text{p}}$, $B_{\text{r}}$) and obtain the value of the material constant $b_{\text{r}}$, also as shown in Fig. 7.

Fig. 7

Determination of the material constant $b_{\text{r}}$ describing particle breakage behavior

3.3 Strength Behavior

This study employs a power function strength criterion [7, 30] to describe the nonlinear strength behavior of the gravelly soil material, as detailed below:

$$q = m_{\text{f}} p_{\text{a}} \left( {\frac{{p^{\prime}}}{{p_{\text{a}} }}} \right)^{n_{\text{f}} }$$

(8)

where $m_{\text{f}}$ and $n_{\text{f}}$ are the strength constants. $q = \sqrt {{1.5s_{{\text{ij}}} s_{{\text{ij}}} }}$ are mean effective stress and deviatoric shear stress, respectively. $s_{{\text{ij}}} = \sigma^{\prime}_{{\text{ij}}} - p^{\prime}\delta_{{\text{ij}}}$ represents the deviatoric stress tensor, in which $\delta_{{\text{ij}}}$ is the Kronecker tensor.

In conventional large-size triaxial compression experiments for coarse-grained geomaterials exhibiting strain-hardening behaviors, failure is typically defined when the axial strain reaches 15%, at which point the stress state is accepted as the strength. In this study, the strength data points $\left( {p^{\prime}_{\text{f}} ,\ \ \ \ \ q_{\text{f}} } \right)$ for each gravelly soil specimen were identified and illustrated in Fig. 8. Subsequently, Eq. (8) was applied to fit these data to derive the strength envelope (see Fig. 8) and determine the values of the strength constants $m_{\text{f}}$ and $n_{\text{f}}$ for the gravelly soil material.

Fig. 8

Strength data of the tested gravelly soil material and strength envelope obtained by data-fitting

4 Formulations of the Constitutive Model

4.1 Generalized Plasticity Framework

In a elastoplastic constitutive model, it is assumed that the total strain $e$ is decomposed into the elastic part ${{\upvarepsilon }}^{\text{e}}$ and the plastic part ${{\upvarepsilon }}^{\text{p}}$, as follows:

$${{\upvarepsilon }} = {{\upvarepsilon }}^{\text{e}} + {{\upvarepsilon }}^{\text{p}} \quad {\text{and}}\quad {\text{d}}{{\upvarepsilon }} = {\text{d}}{{\upvarepsilon }}^{\text{e}} + {\text{d}}{{\upvarepsilon }}^{\text{p}}$$

(9)

In this research, the generalized plasticity framework initially proposed by Pastor et al. [22] was used to build the constitutive model for gravelly soils. In a generalized plasticity model, the incremental stress–strain relationship is expressed as follows:

$${\text{d}}{{\upsigma }} = {{\mathbf{D}}}^{{\text{ep}}} {\text{d}}{{\upvarepsilon }} = {{\mathbf{D}}}^{\text{e}} {\text{d}}{{\upvarepsilon }}^{\text{e}} = {{\mathbf{D}}}^{\text{e}} \left( {{\text{d}}{{\upvarepsilon }} - {\text{d}}{{\upvarepsilon }}^{\text{p}} } \right)$$

(10)

where ${\text{d}}{{\upsigma }}$ and ${\text{d}}{{\upvarepsilon }}$ represent the increments of stress ${{\upsigma }}$ and strain ${{\upvarepsilon }}$, respectively. ${{\mathbf{D}}}^{{\text{ep}}}$ is the elastoplastic stiffness matrix, explicitly expressed as follows:

$${{\mathbf{D}}}^{{\text{ep}}} = {{\mathbf{D}}}^{\text{e}} - \frac{{{{\mathbf{D}}}^{\text{e}} {{\mathbf{n}}}_{{\text{g}}\ {\text{L/U}}} {{\mathbf{n}}}_{\text{f}}^{\text{T}} {{\mathbf{D}}}^{\text{e}} }}{{{{\mathbf{n}}}_{\text{f}}^{\text{T}} {{\mathbf{D}}}^{\text{e}} {{\mathbf{n}}}_{{\text{g}}\ {\text{L/U}}} + H_{\text{L/U}}^{\text{p}} }}$$

(11)

where ${{\mathbf{D}}}^{\text{e}}$ is the elastic stiffness matrix, ${{\mathbf{n}}}_{{\text{g}}\ {\text{L/U}}}$ and ${{\mathbf{n}}}_{\text{f}}$ are the plastic flow and plastic loading direction, respectively, and $H_{\text{L/U}}^{\text{p}}$ denotes the loading/unloading plastic modulus $H^{\text{p}}$. The subscripts “${\text{L}}$” and “${\text{U}}$” represent loading and unloading, respectively. The upper subscript “${\text{T}}$” of ${{\mathbf{n}}}_{\text{f}}$ represents the transpose operation of a matrix or vector.

Additionally, using the compliance matrices, the incremental strain–stress relationship of the generalized plasticity model can be expressed as follows:

$${\text{d}}{{\upvarepsilon }} = {{\mathbf{C}}}^{{\text{ep}}} {\text{d}}{{\upsigma }} = \left( {{{\mathbf{C}}}^{\text{e}} + \frac{{{{\mathbf{n}}}_{{\text{g}}\ {\text{L/U}}} \ \ \ \ {{\mathbf{n}}}_{\text{f}}^{\text{T}} }}{{H^{\text{p}} }}} \right){\text{d}}{{\upsigma }}$$

(12)

where, ${{\mathbf{C}}}^{\text{e}} = \left( {{{\mathbf{D}}}^{\text{e}} } \right)^{ - 1}$ and ${{\mathbf{C}}}^{{\text{ep}}} = \left( {{{\mathbf{D}}}^{{\text{ep}}} } \right)^{ - 1}$ are the compliance matrix of elasticity and elastoplasticity, respectively.

4.2 Elastic Behavior

The empirical formulation suggested by Hardin and Richart [31] has been broadly applied to describe the elastic shear modulus $G$ and bulk modulus $K$ of gravelly soils. They are defined as follows:

$$G = G_0 F\left( {p^{\prime}} \right)F\left( e \right) = G_0 p_{\text{a}} \sqrt {{\frac{{p^{\prime}}}{{p_{\text{a}} }}}} \frac{{\left( {2.97 - e} \right)^2 }}{1 + e}$$

(13)

$$K = K_0 F\left( {p^{\prime}} \right)F\left( e \right) = K_0 p_{\text{a}} \sqrt {{\frac{{p^{\prime}}}{{p_{\text{a}} }}}} \frac{{\left( {2.97 - e} \right)^2 }}{1 + e}$$

(14)

where $G_0$ and $K_0$ is the material constants which can be named as shear-modulus and bulk-modulus coefficients. According to [31], $F\left( {p^{\prime}} \right) = p_{\text{a}} \sqrt {{{{p^{\prime}} / {p_{\text{a}} }}}}$ and $F\left( e \right) = {{\left( {2.97 - e} \right)^2 } / {\left( {1 + e} \right)}}$ characterize the effects of effective stress and compaction state on shear and bulk moduli, respectively.

4.3 Critical State Line With Considering Particle Breakage

In soil mechanics, the critical state is reached during monotonic compression loading when the shear-compression stress ratio (${q / p}$) remains constant, shear strain continuously increases, and volumetric strain stabilizes. This ultimate condition is represented by the Critical State Line (CSL) [32], extensively used in constitutive modeling of gravelly soils [18, 27, 33].

Unlike finer soils such as silts, clays, and sandy soils [34], experiments indicate that particle breakage significantly affects the mechanical behavior of gravelly soils by altering the critical state void ratio [29, 35,36,37]. As particle breakage increases, the actual CSL notably deviates from the ideal CSL [38]. To quantify this effect, researchers have incorporated the particle breakage index into the CSL equation. For example, Konrad [35] introduced a bi-linear CSL to account for the reduction in critical state void ratio due to particle breakage, while Liu and Zou [17], and Saberi et al. [18] developed movable CSL equations to address particle breakage impacts on gravelly soils.

In this study, we adopted a sigmoid function proposed by Liu et al. [7] to define the CSL affected by particle breakage. This function effectively describes the nonlinear and continuous shifts in the critical state void ratio due to particle breakage and is expressed as follows:

$$e_{\text{c}} = e_{\min } + \left( {e_{\max } - e_{\min } } \right)\sqrt {{1 - B_{\text{r}} }} \exp \left( { - \lambda ({{p^{\prime}} / {p_{\text{a}} }})^\xi } \right)$$

(15)

where $e_{\text{c}}$ stands for the critical state void ratio, $e_{\max }$ and $e_{{\text{m}} {\text{in}}}$ are the maximum and minimum critical state void ratios of a gravelly soil mass, respectively, $\lambda$ and $\xi$ are material constants, and $B_{\text{r}}$ is the particle breakage index, which has already been discussed in the previous section.

4.4 Volumetric Strain-Void Ratio Evolution Formula

In soil mechanics, it is conventionally assumed that the volume of the solid phase ($V_{\text{s}}$) remains constant during loading, with any volume changes attributed solely to the void phase ($V_{\text{v}}$). Assuming $V_{\text{s}} = 1$, the volume of the void phase ($V_{\text{v}}$) equates to the void ratio ($e$). Consequently, any change in the void ratio results in a corresponding change in the volumetric strain. This relationship is captured by the following equation:

$${\text{d}}\varepsilon_{\text{v}} = \frac{{{\text{d}}V_{\text{v}} }}{{V_{\text{s}} { + }V_{\text{v}} }} = \frac{{{\text{d}}e}}{1 + e}\ \to {\text{d}}e = {\text{d}}\varepsilon_{\text{v}} \left( {1 + e} \right)$$

(16)

The second formula in Eq. (16) is a differential equation. It has a solution expressed as follows:

$$e = A\exp \left( {\varepsilon_{\text{v}} } \right) - 1$$

(17)

In Eq. (17), the parameter $A$ can be determined using the initial condition. That is, $\varepsilon_{\text{v}} = 0$, $e = e_0$$\Rightarrow A = 1 + e_0$.

The current void ratio ($e$), however, does not always lie in the CSL, but usually lies above the CSL (i.e. wet/loose side of the CSL) or below the CSL (i.e. dry/dense side of the CSL). To identify the density state of soils compared to the critical state, Been and Jefferies [39] proposed the following state parameter $\psi$:

$$\psi = e - e_{\text{c}}$$

(18)

The evolution of the volumetric behavior from the contraction state towards the dilation state and then towards the critical state is directly controlled by variations of the state parameter $\psi$. As displayed in Fig. 9, $\psi < 0$ indicates that the relative density state of gravelly soil is denser than the critical state. Hence, the gravelly soil may show a dilative response. Also, $\psi > 0$ means the gravelly soil is in the looser state than the critical state. Therefore, the gravelly soil displays a contractive response.

Fig. 9

Diagrammatic presentation of the critical state line and state variable $\psi$, showing a loose state or dense state relative to the critical state line (after [9, 18])

4.5 Stress Ratio and Dilatancy Equation

Accoding to Eq. (8), the failure stress ratio $M_{\text{f}}$ of the gravelly soil material under the conventional triaxial compression condition can be expressed as follows:

$$M_{\text{f}} = m_{\text{f}} \left( {\frac{{p^{\prime}}}{{p_{\text{a}} }}} \right)^{n_{\text{f}} - 1}$$

(19)

By introducing a function $F_\pi \left( \theta \right)$ that describes the shape of the strength criterion on the deviatoric plane,the failure stress ratio $M_{\text{f}}$ can be extended to three-dimensional stress space, with

$$M_{{\text{f}}\theta } = M_{\text{f}} F_\pi \left( \theta \right) = m_{\text{f}} \left( {{{p^{\prime}} / {p_{\text{a}} }}} \right)^{n_{\text{f}} - 1} \cdot F_\pi \left( \theta \right)$$

(20)

The formula for $F_\pi \left( \theta \right)$ [7] was adopted in this study, which is expressed as follows:

$$\begin{aligned} & F_\pi \left( \theta \right) = \frac{{\sin \left( {{\pi / 3} - {{\sin^{ - 1} \left( \chi \right)} / 3}} \right)}}{{\sin \left( {{\pi / 3} - {{\sin^{ - 1} \left( {\chi \sin 3\theta } \right)} / 3}} \right)}}, \\ & \chi = \frac{{3\sqrt {3} R(1 - R)}}{{2(R^2 - R + 1)^{3 / 2} }} \\ \end{aligned}$$

(21)

where, $\theta = {{\sin^{ - 1} \left( {{{ - 27J_3 } / {\left( {2q^3 } \right)}}} \right)} / 3}$ is Lode’s angle, in which $J_3$ is the third invariant of the deviatoric stress tensor, namely, $J_3 = \det s_{{\text{ij}}}$. In Eq. (21), $R$ is the ratio of triaxial tensile strength (TE, $\theta = - 30^\circ$) to triaxial compressive strength (TC, $\theta = 30^\circ$).

Gravelly soils usually exhibit complicated volumetric behavior closely related to stress ratio ($\eta = {q / p}$). In soil mechanics, the dilation/phase transformation stress ratio ($M_{{\text{d}}\theta } = {{q_{\text{d}} } / {p_{\text{d}} }}$) is used to represent the stress state of the phase transformation point. Figure 10 schematically shows the nonlinear strength envelope and the dilatation/dilation line, which distinguishes both contraction and dilatation regions. Any stress state with a stress ratio ($\eta = {q / p}$) greater than $M_{{\text{d}}\theta }$ results in shear-dilatation during plastic volumetric deformation, and a stress ratio ($\eta = {q / p}$) less than $M_{{\text{d}}\theta }$ leads to contraction. To predict such behavior numerically, the stress-dilatancy relationship is usually defined as the difference between the current phase transformation stress ratio and the current stress ratio multiplied by a constant.

Fig. 10

Schematic illustration for showing volume contraction/dilatation regions of a gravelly soil material

Following the studies of Dafalias and Manzari [40], the dilation/phase transformation stress ratio ($M_{{\text{d}}\theta }$) in this study is defined as a function of the state parameter ($\psi$) as follows:

$$M_{{\text{d}}\theta } = M_{{\text{f}}\theta } \cdot \exp \left( {\kappa \psi } \right) = m_{\text{f}} \left( {\frac{p}{{p_{\text{a}} }}} \right)^{n_{\text{f}} - 1} \cdot F_\pi \left( \theta \right) \cdot \exp \left( {\kappa \psi } \right)$$

(22)

where $\kappa$ is a material constant for describing the dilative behavior.

As already mentioned, the state parameter ($\psi$) controls the evolution of the dilation/phase transformation line towards the critical state line. Also, we know that the relative magnitude of the current $M_{{\text{d}}\theta }$ compared to that of the current stress ratio $\eta = {q / p}$ determines the sign of the dilatancy rate. Since the magnitude of the current $M_{{\text{d}}\theta }$ itself is controlled by $\psi$, it implies that the change of behavior from contractive to dilative is directly controlled by $\psi$.

The dilatation coefficient (i.e. dilatancy rate) $d_{\text{g}}$ is defined as the ratio of the plastic volumetric-strain increment to the plastic shear-strain increment. That is, $d_{\text{g}} = {{{\text{d}}\varepsilon_{\text{v}}^{\text{p}} } / {{\text{d}}\varepsilon_{\text{s}}^{\text{p}} }}$. By relating $d_{\text{g}}$ to the dilatation/dilation stress ratio (i.e. phase transformation stress ratio $M_{{\text{d}}\theta }$, the stress-dilatancy equation can be described using the following equation [21]:

$$d_{\text{g}} = \frac{{{\text{d}}\varepsilon_{\text{v}}^{\text{p}} }}{{{\text{d}}\varepsilon_{\text{s}}^{\text{p}} }} = d_0 \left( {M_{{\text{d}}\theta } - \eta } \right)\exp \left( {{\eta / {M_{{\text{d}}\theta } }}} \right)$$

(23)

In Eq. (23), $\varepsilon_{\text{v}}^{\text{p}}$ denotes the plastic component of the volumetric strain $\ \varepsilon_{\text{v}} = \varepsilon_{{\text{kk}}}$, and $\varepsilon_{\text{s}}^{\text{p}}$ is the plastic component of the equivalent shear strain $\ \varepsilon_{\text{s}} = \sqrt {{{{2\left( {\varepsilon_{{\text{ij}}} - {{\varepsilon_{\text{v}} \delta_{{\text{ij}}} } / 3}} \right)\left( {\varepsilon_{{\text{ij}}} - {{\varepsilon_{\text{v}} \delta_{{\text{ij}}} } / 3}} \right)} / 3}}}$.

In Eq. (23), $d_0$ is a material constant representing dilative behavior. When $\eta \to 0$, $d_{\text{g}} = d_0 M_{{\text{d}}\theta }$, the stress-dilatancy equation described in Eq. (23) cannot satisfy the isotropic compression condition ($d_{\text{g}} \to { + }\infty$ since ${\text{d}}\varepsilon_{\text{s}}^{\text{p}} = 0$). However, considering the engineering practice and numerical calculation, this study employed Eq. (23) as the stress-dilatancy equation. Equation (23) uses a finite number to avoid the infinitely large number and to not cause too much calculation error.

4.6 Plastic Flow and Loading Directions

Within the framework of generalized plasticity, the plastic flow direction ${{\mathbf{n}}}_{{\text{gL}}}$ is proposed by Pastor et al. [22], that is:

$${{\mathbf{n}}}_{{\text{gL}}} = \frac{{d_{\text{g}} }}{{\sqrt {{d_{\text{g}}^2 { + }1}} }}\frac{\partial p}{{\partial \sigma_{{\text{ij}}} }} + \frac{1}{{\sqrt {{d_{\text{g}}^2 { + }1}} }}\frac{\partial q}{{\partial \sigma_{{\text{ij}}} }}$$

(24)

As the nonassociated flow rule is assumed, the plastic loading direction ${{\mathbf{n}}}_{\text{f}}$ is different from the plastic flow direction ${{\mathbf{n}}}_{{\text{gL}}}$. For the sake of simplicity, ${{\mathbf{n}}}_{\text{f}}$ can be expressed using a formula similar to ${{\mathbf{n}}}_{\text{g}}$ as follows:

$${{\mathbf{n}}}_{\text{f}} = \frac{{d_{\text{f}} }}{{\sqrt {{d_{\text{f}}^2 { + }1}} }}\frac{\partial p}{{\partial \sigma_{{\text{ij}}} }} + \frac{1}{{\sqrt {{d_{\text{f}}^2 { + }1}} }}\frac{\partial q}{{\partial \sigma_{{\text{ij}}} }}$$

(25)

where $d_{\text{f}}$ is defined as the loading direction factor, which is a state variable relevant to the critical state. Similar to $d_{\text{g}}$, $d_{\text{f}}$ is defined as follows:

$$d_{\text{f}} = d_0 \left( {M_{{\text{f}}\theta } - \eta } \right)\exp \left( {{\eta / {M_{{\text{f}}\theta } }}} \right)$$

(26)

4.7 Plastic Modulus

Figure 11 provides a schematic diagram illustrating the typical hysteresis effect of coarse-grained geomaterials during unloading–reloading cycles. In the generalized plastic model, these hysteresis effects are captured by separately defining the plastic moduli $H^{\text{p}}$ for the initial loading, unloading, and reloading phases. The upcoming subsections will use the schematic in Fig. 11 to detail how the plastic moduli $H^{\text{p}}$ are defined.

Fig. 11

The evolution of stress ratio $\eta$ during a monotonic or cyclic loading process, modified after [19]

4.7.1 Initial Loading Plastic Modulus

In classical plastic models, the plastic modulus is defined based on the yield function, whereas in the generalized plastic model, its definition allows for greater flexibility. Researchers [8, 33, 41] have developed various mathematical expressions for the plastic modulus $H^{\text{p}}$ that demonstrate strong applicability. Key factors in defining the plastic modulus $H^{\text{p}}$ in generalized plastic models include effective confining stress, stress path, evolution of void ratio state, and stress ratio.

Figure 11 illustrates the stress ratio's development along the monotonic loading curve ABH to point H, which represents the failure state (corresponding to the stress ratio $M_{{\text{f}}\theta }$). During initial loading, the plastic modulus decreases as the stress ratio $\eta$ increases. This study employs a mathematical expression for the initial loading plastic modulus $H_{\text{L}}^{\text{p}}$ inspired by the research mentioned, which is formulated as follows:

$$H_{\text{L}}^{\text{p}} = H_{0} \cdot F\left( {p^{\prime}} \right) \cdot F\left( e \right) \cdot \left( {M_{{\text{f}}\theta } - \eta } \right)\exp \left( {{\eta / {M_{{\text{f}}\theta } }}} \right)$$

(27)

where $H_{0}$ signifies the plastic modulus coefficient. It is a material constant.

The plastic modulus $H_{\text{L}}^{\text{p}}$ expressed in Eq. (27) is only applicable to describe the mechanical behavior of soil with strain-hardening behavior, just like the gravelly soil material in this study. The plastic modulus in some studies can simultaneously describe the strain-hardening and strain-softening behavior of dense coarse-grained geomaterials. Generally, these expressions for plastic modulus $H^{\text{p}}$ introduce the concept of "virtual" peak stress ratio, such as [9, 17].

4.7.2 Unloading Plastic Modulus

Figure 11 shows that if unloading occurs at point B, the stress ratio $\eta$ will decline along the curve BCD to endpoint E (corresponding to stress ratio $\eta_{\text{r}}$). Point D represents the lowest unloading point, where the stress ratio $\eta_{\text{r}}$ = 0. During unloading, volume deformation appears as compression, a phenomenon observed in previous large-size triaxial tests [1, 2] and in this study. This behavior challenges the classical plastic theory's assumption that unloading results in purely elastic deformation. The conventional triaxial compression experiment provided here serves as an example to demonstrate the volume compression behavior during unloading.

In conventional triaxial compression experiments, when the confining pressure $\sigma_3$ remains constant and the axial compressive stress $\sigma_1$ ($\sigma_1 > \sigma_3$) is unloaded, the resulting elastic volume deformation typically expands. Simultaneously, the effective mean stress $p^{\prime}$ decreases. Thus, the volume contraction observed in numerous triaxial tests is attributable to plastic behavior [8, 19]. This contraction during unloading, which contradicts the predictions of classical plastic theory, presents challenges in both explanation and simulation.

In generalized plastic models, the stress-dilatancy equation, plastic flow direction ${{\mathbf{n}}}_{\text{g}}$, and plastic modulus $H^{\text{p}}$ for loading and unloading can be distinctly defined, allowing for accurate mechanical description of plastic volume dilatation during unloading. Consequently, the plastic flow direction ${{\mathbf{n}}}_{{\text{gU}}}$ during unloading in these models is specified as follows [8, 22]:

$${{\mathbf{n}}}_{{\text{gU}}} = \frac{{ - {\text{abs}}(d_{\text{g}} )}}{{\sqrt {{d_{\text{g}}^2 { + }1}} }}\frac{\partial p}{{\partial \sigma_{{\text{ij}}} }} + \frac{1}{{\sqrt {{d_{\text{g}}^2 { + }1}} }}\frac{\partial q}{{\partial \sigma_{{\text{ij}}} }}$$

(28)

Although the stress ratio $\eta = {q / p}$ gradually decreases during the unloading process, there is not much change in the slope of the $q - \varepsilon_1$ and $\varepsilon_{\text{v}} - \varepsilon_1$ curves. For the sake of simplification, the formula for the unloading plastic modulus $H_{\text{U}}^{\text{p}}$ in this study can be expressed as follows:

$$H_{\text{U}}^{\text{p}} = H_{0} \cdot F\left( {p^{\prime}} \right) \cdot F\left( e \right) \cdot \exp (s_{\text{U}} {\eta / {\eta_{\text{m}}^{\text{r}} }})$$

(29)

where, $s_{\text{U}}$ is a material constant that characterizes the shape of the unloading curve and can be determined through large-size triaxial cyclic tests.

In the generalized plastic model, the values (positive or negative) of the stress-dilatancy equation in Eq. (23) dictate whether the plastic volume deformation of geomaterials manifests as contraction or dilatation. During large-size triaxial compression tests at lower confining pressures (such as $\sigma_3$ = 100 or 200 kPa in this study), the volume deformation of dense gravelly soil specimens transitions from contraction to dilatation as deviatoric stress increases. The introduction of the unloading plastic flow direction ${{\mathbf{n}}}_{{\text{gU}}}$ defined by Eq. (28) allows for the description of volume contraction during unloading, irrespective of the values being positive or negative. Thus, in the generalized plastic model, the stress-dilatancy equation during unloading generally mirrors that of the loading stage.

However, in actual experiments, it was observed that the magnitude of volume contraction strain caused by unloading remained consistent, regardless of whether the deviatoric stress level $q$ (stress ratio $\eta = {q / p}$) was high or low. This indicates that even when unloading at a stress ratio $\eta$ very close to $M_{{\text{d}}\theta }$, where $d_{\text{g}}$ is a positive or negative value very close to zero, the volume contraction strain caused by unloading cannot be ignored. Given this observation, it becomes necessary to propose a stress-dilatancy equation suitable for unloading. For simplicity, this study proposes that the unloading stress-dilatancy equation can remain constant and need not be directly related to the stress ratio $\eta$. Thus, the following formula in Eq. (30) is adopted as the unloading stress-dilatancy equation.

$$d_{\text{g}} = d_0 M_{{\text{d}}\theta }$$

(30)

4.7.3 Reloading Plastic Modulus

As depicted in Fig. 11, the reloading process begins from point E, and the stress ratio $\eta$ evolves along the curve BGH to point H, representing the failure state. In large-size triaxial compression experiments, the reloading curve of coarse-grained geomaterials typically mirrors the unloading curve, albeit with a hysteresis loop between them. This implies that during the reloading stage $\eta \le \eta_{\text{m}}^{\text{r}}$, stress history and densification significantly influence the plastic modulus $H^{\text{p}}$.

Furthermore, continuous reloading from point E to point G brings the stress ratio $\eta$ close to the recorded maximum historical value $\eta_{\text{m}}^{\text{r}}$ at point B during the last unloading. Throughout this process, the reloading plastic modulus $H_{{\text{RL}}}^{\text{p}}$ gradually decreases from a higher value and approaches the value anticipated at point G. While the exact value of plastic modulus $H^{\text{p}}$ at point G is unknown, the experimental curve's shape suggests it should be slightly higher than the plastic modulus at point B during initial loading.

It's important to note that the reloading curve GH after point G smoothly transitions and gradually coincides with the initial loading curve ABH. Consequently, the plastic modulus $H_{{\text{RL}}}^{\text{p}}$ under reloading at point G can be considered the same as the plastic modulus $H^{\text{p}}$ under initial loading at point B. Considering these factors, a new expression for plastic modulus can be formulated as follows:

$$H_{{\text{RL}}}^{\text{p}} = H_{0} \cdot F\left( {p^{\prime}} \right) \cdot F\left( e \right) \cdot \left( {M_{{\text{f}}\theta } - \eta } \right)\exp \left( {{\eta / {M_{{\text{f}}\theta } }}} \right) \cdot \left( {1 + \beta_1 \beta_2 H_{{\text{den}}} } \right)$$

(31)

where, in previous studies, $H_{{\text{den}}}$ was defined as a hardening function, such as $H_{{\text{den}}} = \exp \left( { - \gamma_{\text{d}} \varepsilon_{\text{v}} } \right)$ [8], which characterizes the densification and hardening effects caused by cyclic loading. However, in this study, it was considered $H_{{\text{den}}}$ as a material constant that characterizes the improvement of reloading plastic modulus $H_{{\text{RL}}}^{\text{p}}$ by densification. This processing is for simplification and does not cause significant simulation errors.

In Eq. (31), $\beta_1$ and $\beta_2$ are two functions that characterize the shape of the stress–strain curve under reloading. They are combined to characterize the evolution of reloading plastic modulus $H_{{\text{RL}}}^{\text{p}}$ as the stress ratio $\eta$ increases. Hereby, $\beta_1$ adopts the following expression:

$$\beta_1 = h_{\text{n}} + \left( {h_{\text{m}} - h_{\text{n}} } \right)\frac{{{\eta / {\eta_{\text{m}}^{\text{r}} }}}}{{c + {\eta / {\eta_{\text{m}}^{\text{r}} }}}}$$

(32)

where, $h_{\text{n}}$ and $h_{\text{m}}$ are the lower and upper bounds of the function $\beta_1$, respectively. $c$ is a parameter that characterizes the growth rate of function $\beta_1$. After verification, when the value of parameter $c$ is set to 0.05, function $\beta_1$ can better simulate the evolution law of the plastic modulus of the gravelly soil material under reloading.

In Eq. (31), $\beta_2$ adopts the following piecewise function form:

$$\beta_2 = \left\{ {\begin{array}{*{20}l} {1 - \exp \left[ {s_{\text{R}} \left( {{\eta / {\eta_{\text{m}}^{\text{r}} - 1}}} \right)} \right],} \hfill & {\eta < \eta_{\text{m}}^{\text{r}} } \hfill \\ {0,} \hfill & {\eta \ge \eta_{\text{m}}^{\text{r}} } \hfill \\ \end{array} } \right.$$

(33)

where, $s_{\text{R}}$ is a parameter that characterizes the shape of the stress–strain curve during reloading. Therefore, in Eq. (31), term $\left( {1 + \beta_1 \beta_2 H_{{\text{den}}} } \right)$ can describe the entire process of the evolution of the plastic modulus $H_{{\text{RL}}}^{\text{p}}$ from point D along the curve DGH during reloading.

At unloading point B, with a stress ratio $\eta_{\text{m}}^{\text{r}} > M_{{\text{d}}\theta }$, the soil specimen exhibits a dilative state. If the stress-dilatancy equation remains unchanged, following the expression in Eq. (26), the stress ratio $\eta$ will transition from volume contraction $d_{\text{g}} = d_0 M_{{\text{d}}\theta } > 0$ to volume dilation $d_{\text{g}} = d_0 \left( {M_{{\text{d}}\theta } - \eta_{\text{m}}^{\text{r}} } \right)\exp \left( {{{\eta_{\text{m}}^{\text{r}} } / {M_{{\text{d}}\theta } }}} \right) < 0$ during the reloading process from point D to point G. Consequently, even a minor increase in axial strain during reloading will lead to a significant shift in the volume strain curve. Numerical experiments conducted by the authors support this observation. Hence, it becomes imperative to appropriately redefine the stress-dilatancy equation to suit the reloading process. In this study, the reloading stress-dilatancy equation adopts the following modified form:

$$d_{\text{g}} = \left\{ {\begin{array}{*{20}l} {d_0 \left( {M_{{\text{d}}\theta } - \eta_{\text{m}}^{\text{r}} } \right)\exp \left( {{{\eta_{\text{m}}^{\text{r}} } / {M_{{\text{d}}\theta } }}} \right),} \hfill & {\eta < \eta_{\text{m}}^{\text{r}} } \hfill \\ {d_0 \left( {M_{{\text{d}}\theta } - \eta } \right)\exp \left( {{\eta / {M_{{\text{d}}\theta } }}} \right)\ ,} \hfill & {\eta \ge \eta_{\text{m}}^{\text{r}} } \hfill \\ \end{array} } \right.$$

(34)

From Eq. (34), it can be seen that in the reloading stage at $\eta < \eta_{\text{m}}^{\text{r}}$, the dilatation coefficient $d_{\text{g}}$ is constant, which can avoid the above problems. In addition, at point G, the dilatation coefficient $d_{\text{g}}$ achieves a smooth transition. The stress-dilatancy equation described in Eq. (34) is consistent with the plastic modulus $H^{\text{p}}$ under reloading.

4.8 Loading–Unloading Criterion

In the generalized plastic models [17], determining whether a point's stress is in a loading or unloading state is usually defined using the loading–unloading criteria consistent with classical elastic–plastic theory, namely

$$\begin{aligned} & {{\mathbf{n}}}_{\text{f}}^{\text{T}} {\text{d}}\sigma > 0\;({\text{loading}}), \\ & {{\mathbf{n}}}_{\text{f}}^{\text{T}} {\text{d}}\sigma = 0\;({\text{neutral}}\;{\text{loading}})\;{\text{and}} \\ & {{\mathbf{n}}}_{\text{f}}^{\text{T}} {\text{d}}\sigma < 0\;\left( {{\text{unloading}}} \right) \\ \end{aligned}$$

(35)

5 Numerical Modeling and Comparisons with Experimental Results

5.1 Determination of the Model Constants

This proposed model involves 18 material constants, which can be established through large-size triaxial cyclic loading tests. The process of determining 2 strength constants and 1 particle breakage constant has been detailed in Sect. 3. The determination of the remaining 15 material constants is outlined in the following section.

5.1.1 Initial Void Ratio

As previously mentioned, the void ratio of the gravelly soil specimens after formation is approximately 0.30. Drainage consolidation experiments were conducted under various confining pressure conditions. By incorporating the measured volumetric strain values into Eq. (17), the void ratio of the gravelly soil specimens after drainage consolidation under different confining pressure conditions can be calculated within a range of 0.279–0.297, averaging 0.29. This average void ratio value serves as the initial state for subsequent drainage shear experiments and is utilized as a material constant (listed in Table 3) in the proposed model.

Table 3 Material constants of the gravelly soils

5.1.2 Elastic Constants

The model includes two elastic constants ($G_0$ and $K_0$) to characterize the elastic behavior of the gravelly soil. Typically, for geomaterials exhibiting significant plastic deformation, their elastic constants can be determined from unloading curves. However, it has been noted that during the unloading stage, the volume deformation of gravelly soil manifests as contraction, which is essentially plastic. Therefore, separating the elastic part from the total volumetric strain is challenging. It should be noted that the axial strain rebound caused by unloading is relatively small, indicating it is primarily an elastic strain. Thus, under triaxial unloading conditions with constant confining pressures, the elastic modulus $E$ can be estimated from the unloading curve using the following formula:

$$E = \frac{{\sigma_1^{\text{B}} - \sigma_1^{\text{E}} }}{{\varepsilon_1^{\text{B}} - \varepsilon_1^{\text{E}} }}$$

(36)

where $\sigma_1^{\text{B}}$ and $\varepsilon_1^{\text{B}}$ are the axial stress and axial strain at unloading point B (see Fig. 11), respectively. $\sigma_1^{\text{E}}$ and $\varepsilon_1^{\text{E}}$ are the axial stress and axial strain at the unloading endpoint E (see Fig. 11), respectively.

Due to the challenge of separating elastic volume strain from unloaded volume strain, the radial elastic strain of the gravelly soil specimen remains unknown. Consequently, accurately determining Poisson's ratio $\nu$ through unloading is difficult. However, in the early loading stage (e.g., $\varepsilon_1 \le 0.5\%$), it's assumed that the deformation of the gravelly soil is elastic. Hence, strain test data in this range can be utilized to estimate the value of Poisson's ratio $\nu$. Once the values of elastic modulus $E$ and Poisson's ratio $\nu$ are obtained, they can be used to calculate the values of shear modulus and bulk modulus using formulas $G = {E / {\left( {2 + 2\nu } \right)}}$ and $K = {E / {\left( {3 - 6\nu } \right)}}$. Subsequently, the elastic constants $G_0$ and $K_0$ of the constitutive model developed in this study can be further determined. Their values are presented in Table 3.

5.1.3 Critical State Constants

The triaxial testing curves demonstrate that the differential/deviatoric stress and volumetric strain tend to be constant when the axial strain approaches 15%. This means that the soil specimens were loaded to the critical state. The void ratio at this time can be considered to be approximately equal to the critical state void ratio ($e_{\text{c}}$). Equation (15) is employed to fit the data pairs ($B_{\text{r}}$, ${{p^{\prime}} / {p_{\text{a}} }}$, $e_{\text{c}}$) and a 3D-data fitting surface for the critical state void ratio is obtained, as displayed in Fig. 12a. Then, the values of $e_{\max }$, $e_{{\text{m}} {\text{in}}}$, $\lambda$, and $\xi$ can be determined accordingly. Their values are presented in Table 3. As an additional display, the values of particle breakage index $B_{\text{r}}$ ($B_{\text{r}}$ = 22.76% and $B_{\text{r}}$ = 54.56%) obtained from these tests conducted under the confining pressures of 100 kPa and 600 kPa, respectively, are brought into Eq. (15). The upper and lower bounds of CSL corresponding to these tests can be plotted on $e - {{p^{\prime}} / {p_{\text{a}} }}$ plane, as shown in Fig. 12b.

Fig. 12

Determination of the critical state constants: $e_{\max }$, $e_{{\text{m}} {\text{in}}}$, $\lambda$ and $\xi$

5.1.4 Dilatancy Constants

The determination of the value of $d_0$ only requires data from the curve ABH (see Fig. 11). By determining the elastic constants (elastic modulus $E$ and Poisson's ratio $\nu$, or shear modulus $G$ and bulk modulus $K$), elastic strain and plastic strain can be separated from the total strain. Furthermore, using the experimental data on plastic volumetric and shear strains, the plastic expansion coefficient can be estimated by the difference in measured data with ${{\Delta \varepsilon_{\text{v}}^{\text{p}} } / {\Delta \varepsilon_{\text{s}}^{\text{p}} }}$. Thus, the discrete data points $\left( {\eta \ \ ,\ \ \ d_{\text{g}} } \right)$ composed of the dilatation coefficient $d_{\text{g}}$ and the stress ratio $\eta \ \ { = }{q / p}$ are shown in Fig. 13. Subsequently, Eq. (23) is used to fit the relationship between $d_{\text{g}}$ and $\eta ,$ the value of material constant $d_0$ can be determined, as shown in Fig. 13. Finally, the average value of $d_0$ is adopted and listed in Table 3.

Fig. 13

Determination of the dilatation constant: $d_0$

Through the test data pairs at the dilatation point, the dilatation stress ratio ($M_{\text{d}}$) can be calculated. Using the value of the initial void ratio ($e_0$), Eq. (17) is used to compute the current void ratio ($e$). Then, using Eq. (22), the state parameter ($\psi$) can be determined. In this way, the dilatation constant ($\kappa$) can be calculated by the formula $\ln \left( {{{M_{\text{d}} } / {M_{\text{c}} }}} \right) = \kappa \psi$ to fit the test data pairs $\left[ {\ln \left( {{{M_{\text{d}} } / {M_{\text{c}} }}} \right),\ \ \ \ \psi } \right]$ as shown in Fig. 14. The value of $\kappa$ is listed in Table 3.

Fig. 14

Determination of the dilatation constant: $\kappa$

5.1.5 Plastic Constant

The determination of the values of plastic constants $H_{0}$, $s_{\text{U}}$, $H_{{\text{den}}}$, $h_{\text{n}}$, $h_{\text{m}}$ and $s_{\text{R}}$ requires the use of the relationship between equivalent plastic shear strain $\varepsilon_{\text{s}}^{\text{p}}$ and deviatoric stress $q$, as follows:

$$H^{\text{p}} = \frac{{{\text{d}}q}}{{{\text{d}}\varepsilon_{\text{s}}^{\text{p}} }}$$

(37)

Based on the test data on the curve ABH (see Fig. 11), the difference formula ${{\Delta p^{\prime}} / {\Delta \varepsilon_{\text{v}}^{\text{p}} }}$ is used to estimate the value of the initial loading plastic modulus $H_{\text{L}}^{\text{p}}$. Correspondingly, the values of $F\left( {p^{\prime}} \right)$, $F\left( e \right)$, and the term $\left( {M_{{\text{f}}\theta } - \eta } \right)\exp \left( {{\eta / {M_{{\text{f}}\theta } }}} \right)$ are calculated. Subsequently, a series of data pairs [$F\left( {p^{\prime}} \right) \cdot F\left( e \right) \cdot \left( {M_{{\text{f}}\theta } - \eta } \right)\exp \left( {{\eta / {M_{{\text{f}}\theta } }}} \right)$, $H_{\text{L}}^{\text{p}}$] were obtained and plotted in Fig. 15. Using Eq. (27) to fit these data pairs, the value of plastic constant $H_{0}$ can be determined, as shown in Fig. 15, which is listed in Table 3.

Fig. 15

Determination of the critical state constants: $H_{{\text{L0}}}$

Similarly, the difference formula ${{\Delta p^{\prime}} / {\Delta \varepsilon_{\text{v}}^{\text{p}} }}$ is used to estimate the value of the unloading plastic modulus $H_{\text{U}}^{\text{p}}$. Correspondingly, the values of $F\left( {p^{\prime}} \right)$ and $F\left( e \right)$ are also calculated. Figure 16 shows a series of data pairs ($\tfrac{{H_{\text{U}}^{\text{p}} }}{{H_{0} F\left( {p^{\prime}} \right)F\left( e \right)}}$, ${\eta / {\eta_{\text{m}}^{\text{r}} }}$), which are fitted by the function in Eq. (29) to obtain the value of the plasticity constant $s_{\text{U}}$. Similarly, the average value of $s_{\text{U}}$ is the final choice, as listed in Table 3.

Fig. 16

Determination of the plastic constant $s_{\text{U}}$

Similarly, the test data on the curve DG during the reloading stage (as shown in Fig. 11) were selected. Then, the values of $F\left( {p^{\prime}} \right)$, $F\left( e \right)$ and $\left( {M_{{\text{f}}\theta } - \eta } \right)\exp \left( {{\eta / {M_{{\text{f}}\theta } }}} \right)$, as well as the value of the reloaded plastic modulus $H_{{\text{RL}}}^{\text{p}}$, were obtained using the same method. Based on Eq. (31), it can be derived that

$$\beta_1 \beta_2 H_{{\text{den}}} = \frac{{H_{{\text{RL}}}^{\text{p}} }}{{H_{0} \cdot F\left( {p^{\prime}} \right) \cdot F\left( e \right) \cdot \left( {M_{{\text{f}}\theta } - \eta } \right)\exp \left( {{\eta / {M_{{\text{f}}\theta } }}} \right)}} - 1.$$

(38)

Subsequently, the value of $\beta_1 \beta_2 H_{{\text{den}}}$ is calculated using Eq. (38). The data points (${\eta / {\eta_{\text{m}}^{\text{r}} }}$, $\beta_1 \beta_2 H_{{\text{den}}}$) are plotted in Fig. 17. Using Eq. (32) and the first term in Eq. (33) to fit these data points, the values of plastic constants $H_{{\text{den}}}$, $h_{\text{n}}$, $h_{\text{m}}$ and $s_{\text{R}}$ are determined. Their average values are listed in Table 3 as the final choices.

Fig. 17

Determination of the plastic constant: $H_{{\text{den}}}$, $h_{\text{n}}$, $h_{\text{m}}$ and $s_{\text{R}}$

5.2 Modeling Results and Analyses

By using the material constants given in, the proposed constitutive model was applied to simulate the triaxial cyclic tests of the gravelly soil. The simulation curves provided by the proposed constitutive model are compared with the experimental observations in Fig. 18. The green curves represent the experimental results for deviatoric stress vs. axial strain ($q - \varepsilon_1$) curves and volumetric strain vs. axial strain ($\varepsilon_{\text{v}} - \varepsilon_1$) curves, respectively. The red curves display the corresponding simulation results from the constitutive model.

Fig. 18

Model predations versus experimental results of the gravelly soil material

The left subfigures in each row in Fig. 18 shows that the gravelly soil material specimens exhibits significant strain-hardening behavior within the range of confining pressure from 100 to 600 kPa in the experiments. During the stage where the axial strain $\varepsilon_1$ is less than 0.03, the deviatoric stress $q$ increases at a relatively fast rate. This indicates that the elasticity plays a dominant role in the early deformation of gravelly soil material. Then, the growth rate gradually slowed down and the deviatoric stress $q$ gradually stabilized. But it still grows very slowly with the increase of axial strain $\varepsilon_1$. The strain-hardening effect is further reduced, and gravelly soil exhibits characteristics approaching ideal plasticity. From these subplots, it can be seen that the constitutive model developed in this study is able to simulate the stress–strain relationship of gravelly soil material as a whole and capture their strain-hardening behavior appropriately.

From these subplots, it's evident that the unloading curves exhibit steep slopes and large tangents, indicating relatively small rebound deformations of the gravelly soil specimens upon unloading. Even a slight unloading of axial displacement leads to a sharp decrease in deviatoric stress $q$. The reloading curve closely follows the unloading curve, with a narrow hysteresis loop between them. As reloading progresses, the deviatoric stress $q$ approaches and surpasses the deviatoric stress at the previous unloading point, causing the reloading curve of the triaxial cyclic experiment to gradually flatten. Under confining pressures of $\sigma_3$ = 100 and 200 kPa, the reloading curves transition from high slopes to gentle curves earlier, while this characteristic weakens with increasing confining pressure (e.g. $\sigma_3$ = 500 and 600 kPa). This may be attributed to the stress conditions of stable loading formed by higher confining pressure constraints. Comparisons between simulation and experimental results indicate that the proposed model effectively reproduces the hysteresis effects of the gravelly soil material under unloading–reloading cycles. The numerical predictions for these cycles exhibit good consistency with the experimental results.

The right subfigurea in each row in Fig. 18 shows the volumetric strain versus axial strain ($\varepsilon_{\text{v}} - \varepsilon_1$) curves under different confining pressures. It is evident that the confining pressure significantly influences the volume deformation behavior of the gravelly soil material. Under lower confining pressures $\sigma_3$ = 100 and 200 kPa, the volume deformation of gravelly soil material exhibits a significant transition from contraction to dilatation. Corresponding to $\sigma_3$ = 100 and 200 kPa, the volumetric strain of the gravelly soil specimen is compressed to extreme values $\varepsilon_{\text{v}} \approx$ 0.01 and 0.017, respectively, at axial strains $\varepsilon_1$ = 0.02 and 0.04 approiximately. Subsequently, as the axial strain $\varepsilon_1$ increases, the volumetric strain changes from contraction to dilatation. Under moderate confining pressure $\sigma_3$ = 300 and 400 kPa, the gravelly soil material specimen exhibited a less obvious transition from volume contraction to dilatation. However, when the confining pressure $\sigma_3$ increases to 500 and 600 kPa, the gravelly soil material predominantly experiences volume contraction.

In the right subfigures of each row in Fig. 18, the phenomenon of volume contraction due to unloading is evident under all confining pressures. This contradicts the expectation that elastic volume deformation caused by unloading should result in dilation. Thus, the volume deformation behavior induced by unloading in the gravelly soil material is inherently plastic. However, the stress reduction resulting from unloading must manifest as expansion in the elastic volume. To illustrate the disparities between the simulated results of total volume and plastic strain and the experimental results, red and blue curves are utilized to represent the numerical predictions of plastic and total volume strains, respectively. Upon comparison, it is apparent that the model generally captures the volume deformation behavior of the gravelly soil material, transitioning gradually from dominance by dilation under lower confining pressures to dominance by contraction under higher confining pressures. Moreover, because the stress-dilatancy equation, plastic flow direction ${{\mathbf{n}}}_{\text{g}}$, and plastic modulus $H^{\text{p}}$ for loading/unloading are appropriately and separately defined, this model can more accurately replicate the plastic and total volume deformations of the gravelly soil material during the unloading and reloading stages.

6 Conclusions

In this study, we conducted a comprehensive investigation into the mechanical behavior of gravelly soil materials subjected to varying confining pressures through large-size triaxial cyclic experiments and developed a novel constitutive model. The primary contributions and innovations of this research are summarized as follows:

• (1) The large-size triaxial tests revealed that confining pressure significantly impacts the volume deformation, strength, and failure modes of gravelly soil material. The gravelly soil demonstrated a strain-hardening stress–strain relationship under varying confining pressures. Notably, an increase in confining pressure induced a transition in volume change behavior from dilatation to contraction.

• (2) Under confining pressures ranging from 100 to 400 kPa, the gravelly soil specimens exhibited a drum-shaped failure mode with predominant volume expansion. As the confining pressure increased to 500 and 600 kPa, the failure mode shifted to predominantly compression shear. This shift indicates that higher confining pressures can effectively suppress lateral deformation of the specimens.

• (3) Post-experiment analysis showed significant changes in the particle size distribution of the gravelly soil material. As the confining pressure increased from 100 to 600 kPa, the particle breakage index rose from 22.76 to 54.56%, demonstrating a positive correlation between particle breakage rate and confining pressure. Higher confining pressures resulted in a greater proportion of fine particles, leading to a particle size distribution curve that deviated increasingly from its initial state.

• (4) With increasing confining pressure, particle breakage in the gravelly soil material led to strength attenuation and displayed nonlinear characteristics. By employing a power function strength criterion, we successfully described the nonlinear strength behavior of the gravelly soil material and determined the strength constant.

• (5) A generalized plastic model appropriate for gravelly soil material was developed by incorporating the shift in the critical state line due to particle breakage, in conjunction with the volume strain porosity evolution equation. This constitutive model necessitates a total of 18 material constants, each with a distinct physical meaning. These constants can be calibrated through large-size triaxial cyclic tests.

• (6) The generalized plastic model defines specific stress-dilatancy equations, plastic flow directions, and plastic moduli for the initial loading, unloading, and reloading phases. This differentiation enables the model to accurately replicate the complex mechanical behavior of gravelly soil material under triaxial cyclic loading conditions.

• (7) Comparisons between numerical predictions and experimental data demonstrated that the model effectively simulates the volume dilatation of gravelly soil material under lower confining pressures and volume contraction under higher confining pressures. Additionally, the model captures the stress–strain hysteresis effects observed during unloading–reloading cycles and the volume contraction behavior resulting from unloading.