Triaxial Response and Elastoplastic Constitutive Model for Artificially Cemented Granular Materials

Abstract

Because artificially cemented granular (ACG) materials employ diverse combinations of aggregates and binders—including cemented soil, low-cement-content cemented sand and gravel (LCSG), and concrete—their stress–strain responses vary widely. In LCSG, the binder dosage is typically limited to 40–80 kg/m³ and the sand–gravel skeleton is often obtained directly from on-site or nearby excavation spoil, endowing the material with a markedly lower embodied carbon footprint and strong alignment with current low-carbon, green-construction objectives. Yet, such heterogeneity makes a single material-specific constitutive model inadequate for predicting the mechanical behavior of other ACG variants, thereby constraining broader applications in dam construction and foundation reinforcement. This study systematically summarizes and analyzes the stress–strain and volumetric strain–axial strain characteristics of ACG materials under conventional triaxial conditions. Generalized hyperbolic and parabolic equations are employed to describe these two families of curves, and closed-form expressions are proposed for key mechanical indices—peak strength, elastic modulus, and shear dilation behavior. Building on generalized plasticity theory, we derive the plastic flow direction vector, loading direction vector, and plastic modulus, and develop a concise, transferable elastoplastic model suitable for the full spectrum of ACG materials. Validation against triaxial data for rock-fill materials, LCSG, and cemented coal–gangue backfill shows that the model reproduces the stress and deformation paths of each material class with high accuracy. Quantitative evaluation of the peak values indicates that the proposed constitutive model predicts peak deviatoric stress with an error of 1.36% and peak volumetric strain with an error of 3.78%. The corresponding coefficients of determination R² between the predicted and measured values are 0.997 for peak stress and 0.987 for peak volumetric strain, demonstrating the excellent engineering accuracy of the proposed model. The results provide a unified theoretical basis for deploying ACG—particularly its low-cement, locally sourced variants—in low-carbon dam construction, foundation rehabilitation, and other sustainable civil engineering projects.

1. Introduction

Artificially cemented granular (ACG) materials consist of loose particles of soil, sand, and gravel bonded by hydraulic binders in amounts that can vary from almost negligible to levels comparable to conventional concrete. A notable subclass is low-cement-content cemented sand and gravel (LCSG), in which the binder dosage is typically limited to 40–80 kg/m³, while the aggregates are frequently sourced from on-site excavation spoil or nearby natural deposits [1]. By combining a minimal quantity of cement with locally available soil and sand–gravel blends, LCSG preserves the constructability associated with concrete yet substantially lowers embodied carbon and primary resource consumption.

To investigate the mechanical behavior of ACG systems, researchers initially used uniaxial compression tests; however, such tests cannot fully reproduce the three-dimensional stress paths encountered in foundation and dam structures. Consequently, triaxial shear testing has become the preferred laboratory method for studying cemented soils, LCSG, and related concrete-based composites, providing a more realistic assessment of their strength, stiffness, and volumetric responses under engineered loading conditions.

For cemented soils and fine-grained mixtures, Chen et al. [2] demonstrated that cement modification enhances compressive strength and elastic modulus while reducing volumetric strain and altering shear dilation. Zeng et al. [3] investigated soil–rock mixtures and observed how rock block content and shape affect strength and deformation. Lepakshi et al. [4] reported increased shear strength in cement-stabilized rammed earth (CSRE) with higher cement content and confining pressure. Kong et al. [5] revealed how fiber content affects the strength and dilatancy of polypropylene fiber-reinforced sand. Rabbi et al. [6] examined the effects of confining pressure and curing time on the mechanical behavior of cemented sand.

The materials of the second type in

Table 1. Classification of ACG materials.

Number of Category	Material Name	Cementing Content	Loose Granular Material
1	Cemented soil, cemented sand	-	Soil
			Sand
2	Cemented sand and gravel	20 kg/m3–120 kg/m3	Sand and gravel Coal gangue
3	Concrete or roller-compacted concrete	>120~140 kg/m3	Sand and gravel

are cemented coarse-grained soil and cemented gravel with a little cement or other cementing agent added to the aggregate. Wu et al. [7] analyzed peak strength and initial modulus under varying curing ages and confining pressures. Chen et al. [8], Zhang et al. [9], and Cai et al. [10] studied the influence of cement content on the mechanical parameters of cemented sand and gravel. Yang et al. [11] investigated resilient behavior in polyurethane foam-reinforced rockfill. Xu et al. [12] found that cement content, curing age, and confining pressure significantly affect the strength and deformation of cemented tailing backfill. Ding et al. [13] established a constitutive model for cemented coarse-grained soils, capturing post-peak stress variation. Amini et al. [14] employed acoustic emission and X-CT to study fracture mechanisms in cemented sand gravel. Liu et al. [15] analyzed the effect of curing temperature on cemented rock fill strength. Lohani et al. [16] showed that cement content, water content, and dry density influence the deformation characteristics of cement–gravel mixtures.

Concrete and similar high-cement-content ACG materials have also been studied. Chen et al. [17] reported that confining pressure alters the failure mode of recycled aggregate concrete. Further, Chen et al. [18] developed a triaxial constitutive model for coral coarse aggregate–sea sand–seawater concrete. Vu et al. [19] found that reducing aggregate size or paste volume enhances stiffness under high stress. Dong et al. [20] showed gravel addition improves the drainage strength of cement-treated soils. Luan et al. [21] found that a 25% sand content maximizes the strength and cohesion of sand concrete. Bai et al. [22] compared different concrete types and found that full-solid waste concrete has a faster strength growth rate than that of natural aggregate concrete.

In addition to experimental studies, various constitutive models have been developed to simulate the mechanical behavior of ACG materials. Luo [23] introduced a chemically degradable Cam-clay-based model using porosity and cement content as parameters. Rossi [24] formulated a micromechanical model for weakly cemented materials by extending the Tengattini theory, incorporating porosity and shear dilation prediction. Liu [25] proposed a multi-scale hydrogen bond (H-bond) model to link local bond failure with macro behavior. Wu [26] constructed a double-yield surface model for cemented sand and gravel using the modified Cam-clay theory. Yang [27] applied statistical damage mechanics with Weibull-distributed micro-defect strength to improve simulations of softening and dilation. Qian [28] developed a subloading plasticity model accounting for particle breakage. Pirmoradi [29] built a multi-scale model for fiber-reinforced concrete using homogenization and rate-dependent formulations to simulate boundary value problems.

Although these models are backed by an advanced understanding, most are tailored to specific materials and are either overly complex or suffer from poor generalizability. The diversity in both aggregate types and, critically, binder content makes it difficult for existing models to effectively simulate the stress–strain behavior across all ACG material categories, limiting their application in practical engineering design and predictive modeling.

This study proposes a novel unified constitutive model that employs the generalized hyperbolic–parabolic equation set described in Equations (3) and (4) to generate a full-spectrum representation of the stress–volumetric response, covering uncemented rockfill where the cement content c equals 0 kg/m³ and extending to highly cemented concrete where c surpasses 120 kg/m³. Relative to the segmental linearization inherent in Shen’s bilinear model and the empirical parameter dependence characteristic of Wei’s hump formula, this new model introduces a nonlinear strength criterion together with an explicit parabolic dilatancy equation, thereby establishing for the first time a quantitative linkage among cement content, brittleness, and dilatancy evolution. Engineering validation indicates that the proposed model predicts the peak stress and volumetric strain responses of rockfill, cemented sand and gravel, and coal gangue backfill with errors of 1.36% and 3.78%, respectively. The corresponding R² values reach 0.997 and 0.987, demonstrating an 8% improvement in accuracy compared to traditional models. The core innovations include a mathematical structure that spans the entire material spectrum, parameters that are fully calibrated through physical testing, and explicit quantification of strength attenuation under high confining pressure, thus delivering the first integrated theoretical and computational tool for deformation control of low-carbon cemented granular materials with cement contents between 40 and 80 kg/m³ in dam engineering. Building upon the unified formulation introduced above, we now consolidate it into a streamlined constitutive framework that faithfully reproduces the full mechanical spectrum of artificially cemented granular (ACG) materials. By integrating a generalized hyperbolic stress–strain equation with a parabolic volumetric strain equation, we derive closed-form expressions for peak strength, initial elastic modulus, and dilatancy. These expressions exhibit applicability across a wide range of materials, from weakly bonded rockfill to highly cemented concrete, thereby overcoming the necessity for multiple empirical relations tailored individually to different material classes. On this basis, we further establish a nonlinear strength criterion that accurately captures the experimentally observed deceleration in strength gain when confining pressures exceed 1.5 MPa, an effect typically overestimated by conventional linear criteria. Employing generalized plasticity theory, our model incorporates only ten physically interpretable parameters, effectively predicting both axial and volumetric responses of rockfill, low-cement-content sand and gravel (LCSG), and coal gangue backfill with errors consistently below 5%. The proposed elastoplastic model thus provides an optimal balance between computational simplicity and predictive precision, demonstrating significant potential for practical engineering analysis and design.

2. Macro–Microscopic Analysis Results

2.1. Experimental Materials

2.1.1. Design of Materials and Mixture Proportions

This research focused on low-cement-content cemented sand and gravel (LCSG) as the primary testing material, which was selected as a representative type of artificially cemented granular (ACG) material, owing to its well-controlled and relatively low levels of cementation. This deliberately weak bonding environment isolates the fundamental role of binder content in granular skeleton behavior, distinguishing it from conventional concrete while maintaining typical ACG characteristics: particulate composition, cement-dependent cohesion, and strain-driven failure mechanisms. In practical engineering applications, the maximum particle size of crushed stones used in CSG typically reaches up to 150 mm. However, due to the constraints of laboratory testing conditions, the particle size was limited to 40 mm. The specific physical characteristics and component details of both the crushed stones (Nantong Sanjian Quarry Co., Ltd., Nantong, China) and the medium sand (Nantong Xinghua Building Materials Co., Ltd., Nantong, China)—characterized by a fineness modulus of 2.48—are presented in

Table 2. Physical properties and composition of crushed stones and sand.

Aggregate Type	Specific Gravity	Bulk Density (kg/m3)	Water Content	Clay Content
Crushed stone	2.71	1650	0.01%	0.01%
Sand	2.62	1450	0.01%	0.01%

. The aggregate framework was compacted to achieve a target dry density of 2.29 g/cm³. Ordinary Portland cement (OPC) of grade P.C. 32.5 (Anhui Digang Hailuo Cement Co., Ltd., Wuhu, China) served as the cementitious binder, and its detailed properties are provided in

Table 3. Physical properties and chemical composition of the cement.

The Fineness	The Content of SO3	The Content of MgO
2.26%	2.56%	1.78%

. To systematically evaluate how variations in cement content influence the deformation behavior of CSG materials, four mix designs were formulated with cement dosages of 0, 40, 80, and 100 kg/m³, respectively, as shown in

Table 4. Details of the test specimens.

Group ID	Cement (kg/m3)	Sand (kg/m3)	Gravel (kg/m3)
			5–10 mm	10–20 mm	20–40 mm
1	0	477	340.8	596.4	715.7
2	40	477	340.8	596.4	715.7
3	80	477	340.8	596.4	715.7
4	100	477	340.8	596.4	715.7

. The water-to-cement ratio was uniformly maintained at 1.0 across all mixes.

2.1.2. Specimen Production

Large cylindrical specimens of cemented sand and gravel (LCSG) with a diameter of 300 mm and a height of 700 mm were fabricated, and their dimensional accuracy was maintained within ±1%. According to the mix design, the required amounts of cement, sand, gravel, and water were calculated, weighed, and thoroughly blended. The fresh mixture was placed into a cylindrical mold (Figure 1) in five equal lifts; each lift was successively vibrated, troweled, and compacted. After compaction of the fifth lift, an integral LCSG specimen was obtained. Two identical specimens were prepared for each test condition to ensure result reliability. All specimens were cured at 20 ± 2 °C for 28 days, a curing period chosen on the basis of Wu et al. [7], who reported that LCSG reaches sufficient strength for engineering applications (e.g., slope stabilization and dam construction) by 28 days, with only marginal gains thereafter. Finally, the specimens were jacketed with rubber sleeves to prevent damage during handling and testing.

2.2. Experimental Apparatus and Procedure

Testing Protocol for LCSG Device (Highly Permeable ACG Materials)

The experiments strictly complied with the geotechnical testing standard SL237-1999 [30]. Consolidated-drained (CD) triaxial shear tests were conducted on the highly permeable cemented sand and gravel (LCSG). Specimens were allowed to stand for 2–3 h prior to testing and were then vacuum-saturated [31] to achieve a water saturation level greater than 95%. After consolidation under the prescribed confining pressure, axial loading was applied under drained conditions at a constant displacement rate of 2 mm/min. Four confining pressures, σ₃ = 300, 600, 900, and 1200 kPa, were selected to represent stress states typical of LCSG applications such as slope stabilization and dam construction; loading proceeded until the axial strain reached 15%. Axial load, displacement, and drainage volume were continuously recorded to obtain stress–strain curves. When a distinct peak emerged, the corresponding major principal stress difference (deviator stress) was taken as the failure strength; otherwise, the strength at 15% axial strain was adopted. Tests were repeated under each confining pressure to establish the shear strength envelope and stress–strain behavior. Using a displacement rate of 2 mm/min during compaction of low-permeability mixtures where the permeability coefficient k is less than 10⁻⁸ m/s is rational because, through a three-fold mechanism, it ensures that Skempton’s B value exceeds 0.95 before shearing. First, characteristic time analysis shows that expelling air from such soils requires about 10–100 s, whereas compacting a 10 mm displacement at 2 mm/min takes 300 s, thereby preventing air entrapment and guaranteeing an initial saturation S_r > 98%. In addition, this rate strikes an optimal balance between efficiency and reliability: it avoids the prolonged duration and boundary effects associated with 1 mm/min and the air entrapment risk of 5 mm/min that can reduce B to 0.80–0.93. Moreover, experimental evidence, for example in Wroth [32], confirms that for clay where the permeability coefficient k is 5 × 10⁻⁹ m/s, the rate yields B values of 0.96–0.98, satisfying ASTM D4767 [33] and BS 1377 [34] requirements. Furthermore, this method is applicable with permeability coefficients ranging from 10⁻⁹ to 10⁻⁷ m/s, provided t compaction exceeds t₉₀. Collectively, the theoretical grounding, efficiency advantages, experimental reliability, and standard compliance make 2 mm/min a sound choice for preparing specimens of low-permeability mixtures. For low-permeability but high-strength ACG materials (e.g., cemented soil and concrete), loading was performed with a dedicated rock-mechanics triaxial apparatus that offered both stress-controlled and displacement-controlled modes: loading typically started under stress control to a predetermined level, then switched to displacement control until failure. Two displacement transducers and a strain ring synchronously monitored axial displacement and lateral deformation, ensuring precise capture of the complete stress–deformation response [18,20,22,35,36,37].

A static–dynamic triaxial testing system (Figure 2) was employed to perform large-scale consolidated-drained (CD) shear tests. The system consists of a hydraulic power unit with digital signal control, load sensors, a triaxial pressure chamber, pressure/volume controllers, and dedicated software (TriaxPro v2.3.1, GCTS Instruments, Tempe, AZ, USA); it offers a maximum confining pressure of 4 MPa, a maximum axial load capacity of 1500 kN, and an overall measurement accuracy better than 1%. Axial displacement was recorded with an optical grating transducer accurate to 0.01 mm and converted to axial strain, while volumetric strain was obtained from the ratio of the drained water volume to the specimen’s initial volume. The static–dynamic triaxial testing system is employed to impose precisely controlled confining pressure and axial loading on large-scale ACG specimens under drained conditions, allowing effective stress paths, axial and volumetric deformations, and shear strength parameters to be measured with better than 1% accuracy for subsequent constitutive calibration and cross-material comparison. In the present study, the apparatus is operated chiefly in static-loading mode to obtain monotonic stress–strain responses. To ensure consistency and comparability of the mechanical parameters for all artificially cemented granular (ACG) materials considered—namely LCSG, rockfill, and cemented coal gangue backfill—all data were derived from conventional large-scale CD triaxial tests. The CD path was chosen because it allows complete dissipation of pore water pressure, enabling the measurement of effective stresses and long-term shear strength, both essential for calibrating constitutive model parameters. Testing procedures for rockfill and cemented gravel strictly followed SL237-1999 [30], and previous studies [13,26,27,38] confirmed drained conditions during shearing; investigations of cemented coal gangue backfill likewise used CD triaxial loading with full prior consolidation [15,39]. This unified database of drained shear tests underpins the applicability and reliability of the constitutive model parameters obtained in the present study.

2.3. Quantitative Description of Stress–Strain Relationship

2.3.1. Analysis of Experimental Curves and Their Inherent Mechanisms

Figure 3, Figure 4, Figure 5, Figure 6 and Figure 7 present the stress–strain and volumetric–axial strain responses of artificially cemented granular (ACG) materials as the cement content increases from 0 to 100 kg/m³. The specimen containing 0 kg/m³ cement, shown in Figure 3, hardens continuously: deviatoric stress rises nonlinearly with axial strain, and volumetric strain evolves from an initial contraction of about 1.5% to a dilation of roughly 4% once shear localization begins. Introducing 40 kg/m³ of the binder, as shown in Figure 4, raises the initial tangent modulus and shortens the early nonlinear segment, yet a distinct peak remains absent; volumetric contraction is milder and the onset of dilation is delayed, indicating that nascent cement bridges partially restrain particle rotation. A dosage of 80 kg/m³, depicted in Figure 5, marks the entry into the low-cement-content CSG regime: an almost linear elastic branch extends to an approximately 0.3% axial strain, a clear peak emerges before 0.5%, and post-peak softening accelerates; volumetric change is governed by a slight compression of about 0.5%, with dilation is largely suppressed. With 100 kg/m³ of the binder, Figure 6 shows that the specimen achieves lean concrete-like stiffness, develops a peak deviator stress nearly twice that of the 0 kg/m³ benchmark under identical confinement, and undergoes an abrupt stress drop once the continuous cement matrix fractures; volumetric strain remains marginally negative, confirming the shift from friction- to cementation-controlled behavior. Figure 7 overlays all the curves, clearly illustrating the monotonic increase in stiffness and strength and the progressive attenuation of dilation with rising cement content, thereby providing the experimental foundation for the unified constitutive model developed in the following section.

As the cementitious content gradually increases, the nonlinear characteristics become significantly less pronounced, and the curve exhibits a more rigid growth trend, resembling the linear peak-softening behavior typical of concrete. During axial loading, the internal cemented structure of the material gradually contributes to load bearing, reflected in the continuous increase in stress until a peak is reached at a certain strain level. At this stage, the resistance provided by the cemented structure becomes dominant, and the overall shear strength is significantly enhanced. Once the peak is exceeded, the stress begins to decline, indicating the initiation of damage in the cemented structure and failure of the bonded interfaces between particles. Thereafter, the load-bearing mechanism of the material gradually shifts from cementation-dominated to friction-dominated, with inter-particle sliding becoming the primary mechanism, leading to an overall stress-softening behavior [40].

Moreover, the experiments also show that under the same confining pressure, the higher the cementitious content, the greater the principal stress difference at the same strain level, indicating that the cementitious agent plays a significant role in enhancing strength and improving structural stability.

Triaxial shear tests on cemented sand and gravel (CSG) demonstrated that its mechanical behavior changes markedly as binder content increases. When the binder content is low, such as 0 or 40 kg/m³, the stress–strain response displays continuous strain hardening without a distinct peak. Under a confining pressure of 300 kPa, the material strength does not reach a stable value even at an axial strain of 15%. The initial tangent modulus remains within a low range of 100 to 200 MPa, and a prolonged nonlinear segment is observed, indicating that the cementation is weak and particle sliding governs the deformation mechanism. The volumetric strain response at this stage is characterized by an initial marked compression, with volume strain approaching −1.5% at axial strains less than 2%, followed by pronounced dilatancy of approximately +4%, which initiates near the peak stress.

In contrast, at higher binder contents, such as 80 or 100 kg/m³, the material behavior becomes significantly more brittle. The peak strength increases to approximately 3.5 MPa, followed by a sharp drop in residual strength to around 1.0 MPa. The initial tangent modulus exceeds 300 MPa, and the pre-peak stress–strain behavior becomes nearly linearly elastic, with peak strength occurring at axial strains below 0.5%. The corresponding volumetric strain response shows only minor compression, generally around −0.5%, and dilatancy is largely suppressed. This suggests that a continuous and stable cemented skeleton is formed, effectively restricting particle rearrangement and deformation.

In summary, a binder content of approximately 80 kg/m³ represents a critical threshold for the macroscopic transition of CSG from soil-like behavior dominated by friction and particle rearrangement to concrete-like behavior characterized by stiffness, brittleness, and cementation-dominated load bearing.

Confining pressure plays a significant role in modulating the strength, deformation, and failure characteristics of cemented sand and gravel (CSG). When the confining pressure increases from 300 kPa to 600 kPa, the peak strength exhibits an enhancement of approximately 40% to 60%, while the initial tangent modulus rises by about 50% to 80%. However, as the confining pressure continues to increase beyond 1.5 MPa, the rate of strength enhancement diminishes in a nonlinear manner. This attenuation trend can be described by a power–law relationship, wherein the peak strength is expressed as a constant term A plus a coefficient B multiplied by the normalized confining pressure raised to the exponent 0.91, with normalization conducted using atmospheric pressure as the reference. Meanwhile, the increment in the initial modulus becomes more moderate under high confining pressures, with gains reduced to approximately 20% to 30%.

Binder content interacts strongly with confining pressure. At an axial strain of 10%, a binder content of 40 kg/m³ under a 300 kPa confining pressure produced a deviator stress of 1.2 MPa and a volumetric strain of +3.5%, indicating dominant particle sliding and pronounced dilatancy. Increasing the pressure to 1200 kPa raised the deviator stress to 2.8 MPa and reduced the volumetric strain to +1.0%, demonstrating marked suppression of dilatancy. With a binder content of 100 kg/m³ and a 300 kPa confining pressure, the deviator stress fell from a peak of 3.5 MPa to a residual of 1.0 MPa, and the volumetric strain was −0.3%, revealing failure governed by cement bond softening. Under the same binder content but at 1200 kPa, the deviator stress decreased only from 6.2 MPa to 4.5 MPa and the volumetric strain reached −0.7%, showing that a high confining pressure significantly slows cement bond failure.

Previous studies have thoroughly validated the sensitivity of cemented sand and gravel (CSG) to variations in binder content and confining pressure. Large-scale triaxial tests by Cai et al. [41] demonstrated that CSG shear strength obeys the Mohr–Coulomb criterion; increasing confining pressure markedly enhances the secant modulus, whereas binder content exerts only a minor influence. Wei et al. [42] reported pronounced dilatancy and strain softening in the 100–3000 kPa range and observed that strength gains attenuate nonlinearly (m = 0.91) once the confining pressure exceeds 1.5 MPa. Huang et al. [9] corroborated the strengthening effect of confinement under dynamic loading and noted that strength increments level off when the binder content exceeds 80 kg/m³. Fu et al. [8] showed that specimens with low binder contents (20–40 kg/m³) are governed by strain hardening, whereas those with high contents (≥80 kg/m³) display a combination of linear elasticity and brittle softening; Chai et al. [43] found that each additional 20 kg/m³ of binder raises compressive strength by 40–50%, yet the modulus increment diminishes as content increases. Collectively, these findings highlight the complementary roles of confining pressure and binder content in controlling the strength, modulus, and deformation mechanisms of CSG.

2.3.2. Micromechanical Interpretation

The aforementioned stress–strain response characteristics can be further interpreted from a microscopic perspective. At the initial stage of loading, the interfacial bond between the particles and the cementitious agent remains intact, resulting in high overall structural stiffness, with the cemented matrix bearing the majority of the stress transmission. As strain increases, microcracks gradually develop within the cemented structure, and with continued loading, they propagate and coalesce, ultimately leading to structural discontinuity. This process marks the peak of the stress–strain curve and the onset of the softening stage. Thereafter, the material primarily relies on inter-particle contact friction for load transfer, exhibiting a pronounced decline in residual strength.

2.3.3. Evolution of Volumetric Strain Behavior

The volumetric strain curve further reveals the deformation characteristics of ACG materials during the triaxial shearing process. At the early stage of shearing, the specimen exhibits compaction behavior, with volumetric strain increasing as axial strain increases, indicating a contraction response. With continued loading, some specimens exhibit dilation, as evidenced by a slight increase in volume. This phenomenon indicates that the material undergoes a transition from particle compaction to particle sliding and rearrangement during the loading process.

With increasing cementitious content, the initial slope of the volumetric strain curve becomes gentler, the dilation magnitude decreases, and the axial strain required to reach the peak volumetric strain is reduced, indicating that the cemented structure enhances the stability of inter-particle connections and suppresses the tendency toward volumetric expansion. In addition, under different confining pressures, the volumetric strain–axial strain curves exhibit a crossover behavior, suggesting that the contact patterns and force transmission paths among particles are reorganized during the failure process.

At the micro-scale, this contraction–dilation process is closely related to the arrangement of particles. In the initial stage, high porosity allows particles to slide and compact along the void spaces; after entering the dilation stage, bond breakage induces local particle rearrangements such as rolling, arching, and sliding, leading to localized void expansion and volumetric rebound. With increasing cementitious content, particle motion becomes more constrained, resulting in smaller volumetric changes and a weaker dilation tendency. After failure, the particle arrangement within the shear band becomes more disordered, leading to the crossover trend observed in the volumetric strain–axial strain relationship under different confining pressures.

2.3.4. Numerical Simulation Validation and Failure Mode Observation

To further validate the experimental results, Chen [44] conducted triaxial shear simulations of ACG materials using the Lattice Discrete Particle Model (LDPM). The simulation results showed that under certain confining pressure conditions, ACG materials initially experienced localized failure of the cemented structure, followed by particle compaction and inter-particle friction, eventually leading to the formation of a localized shear band, which was highly consistent with the experimental observations.

The simulation process shown in Figure 8 indicates that material failure generally proceeds through three main stages: bond breakage, particle compaction, frictional transfer, and shear band development, leading to structural instability. This process clearly illustrates the evolutionary path of ACG materials from a cementation-dominated to a friction-dominated mechanism under varying cementitious contents.

In the Lattice Discrete Particle Model (LDPM-CSIC v4.2), the surface mesh size h_s is determined as the product of a scaling factor ξ and the minimum particle diameter d₀. The scaling factor ξ is 1.5, and the minimum particle diameter d₀ is one quarter of the maximum aggregate size d_a of 40 mm, namely 10 mm; consequently, the surface mesh size equals h_s 15 mm. The model assigns mechanical parameters for three different cement contents. When the cement content c is 60 kg/m³, the normal elastic modulus E₀ is 610 MPa, the interfacial tensile strength σ_t is 0.30 MPa, and the compressive yield strength is σ_c0 10 MPa. When the cement content c is 80 kg/m³, the normal elastic modulus E₀ is 800 MPa, the interfacial tensile strength is σ_t 0.55 MPa, and the compressive yield strength σ_c0 is 12 MPa. When the cement content c is 100 kg/m³, the normal elastic modulus E₀ is 1500 MPa, the interfacial tensile strength σ_t is 0.70 MPa, and the compressive yield strength σ_c0 is 20 MPa. Additional parameters include an initial hardening modulus ratio H_c0/E₀ of 0.6, an initial friction coefficient μ₀ of 0.2, and a softening index n_t of 0.2. With a fixed cement content, an increase in confining pressure enlarges the initial elastic modulus, raises the peak deviatoric stress, and correspondingly increases the axial strain. Moreover, the post-peak stress–strain curve becomes more gradual, and the maximum crack width at failure decreases. Conversely, under constant confining pressure, increasing cement content markedly elevates the peak deviatoric stress while reducing the associated strain and further narrowing the crack width at failure. Regarding failure mode, all specimens develop shear fracture planes oriented approximately forty-five degrees to the loading direction, exhibiting typical shear failure characteristics. These observations demonstrate that the LDPM accurately reproduces the meso-scale deformation processes and shear failure mechanisms of cemented sand and gravel under triaxial loading, indicating strong applicability and predictive capability for engineering purposes.

2.3.5. Influence of Cementitious Content on Failure Mode

The experiments also revealed that cement content has a significant influence on the failure mode of the material. When the cement content is low, the material exhibits high internal porosity and weak bonding, with load transfer primarily relying on inter-particle friction. During shearing, it shows contraction with slight dilation, low residual strength, and a failure pattern similar to granular soils; in contrast, when the cement content is high, the structure becomes denser, bonding strength improves, contraction is reduced, dilation becomes more pronounced, stress softening is evident, and post-peak strength drops rapidly.

Under conditions of high cementitious content, pronounced “rolling” and “arching” phenomena are observed within the shear band, with particle behavior becoming more complex and irregular, further contributing to the crossover observed in the volumetric strain curve. These observations suggest that cementation enhances the initial structural stability; however, once bonding support is lost after failure, inter-particle sliding becomes more unconstrained and chaotic, thereby intensifying structural softening.

However, under conditions of high cementitious content, the stress–strain response of ACG materials tends to resemble that of concrete-like materials, and continued use of Equation (1) for fitting may result in significant prediction errors. To address this issue, He et al. [45] introduced the Ottosen equation to model the stress–strain relationship of cemented sand and gravel materials, but the physical interpretation of its parameters remains unclear, and its predictive accuracy still needs improvement.

2.4. Stress–Strain Equation

Artificially cemented granular (ACG) materials exhibit significant differences in their mechanical response under varying cementitious contents. When the cementitious content is low, the stress–strain curve exhibits a typical nonlinear behavior prior to reaching the peak strength, which can be described by the following equation [45]:

$$q={\displaystyle \frac{{\epsilon }_1}{a_0+b_0{\epsilon }_1}}$$

(1)

In this equation, a₀ and b₀ are parameters characterizing the nonlinear behavior of the material. This model effectively captures the deformation characteristics of ACG materials with low cementitious content.

To further optimize the representation of the stress–strain curve, Wei et al. [42] proposed describing the mechanical behavior in the softening stage using a “hump-shaped curve” formulation, for which the mathematical expression is given as follows:

$$q={\displaystyle \frac{\left(a_1+c_1{\epsilon }_1\right){\epsilon }_1}{{\left(a_1+b_1{\epsilon }_1\right)}^2}}$$

(2)

In this equation, q denotes the shear stress; a₁, b₁, and c₁ are fitting coefficients; and ε_a represents the axial strain. Although the model offers certain advantages in capturing the stress-softening trend, the actual fitting results indicate that discrepancies still exist between the model and the experimental data.

To develop a stress–strain relationship model with a simpler structure and broader applicability, the following expression is proposed:

$$q={\displaystyle \frac{{\epsilon }_1}{c{\epsilon }_1^2+b{\epsilon }_1+a}}$$

(3)

In this equation, a, b, and c are characteristic parameters reflecting the nonlinear response of the material. The formulation is structurally similar to the Saenz equation and is capable of describing the nonlinear stress–strain behavior of various materials, including concrete, rockfill, and granular soils. When c = 0, Equation (3) is reduced to Equation (1), thereby enhancing its generality.

Given that most ACG materials are applied in engineering contexts involving low-to-moderate stress levels, such as dam embankment construction and foundation reinforcement, the potential softening segment in their stress–strain response can be neglected in the analysis. Therefore, Equation (3) demonstrates good applicability in such practical scenarios.

In terms of representing volumetric deformation behavior, Liu et al. [15] did not account for the nonlinear relationship between axial strain and volumetric strain at low cementitious contents when developing their nonlinear elastic model, which limits its applicability. In contrast, the “Nanshui” bilinear constitutive model [46] employs the following expression to characterize the nonlinear evolution between the two variables:

$${\epsilon }_{\mathrm{v}}=d{\left({\epsilon }_1-{\epsilon }_{\mathrm{n}}\right)}^2+{\epsilon }_{\mathrm{vd}}$$

(4)

In this expression, ${\epsilon }_n$ denotes the axial strain corresponding to the peak volumetric strain, and d is a fitting coefficient. This expression is not only structurally concise but also effectively captures the typical deformation behavior of “initial contraction followed by dilation,” providing a feasible approach for describing the volumetric response of ACG materials.

2.5. Study on Strength Characteristics

2.5.1. Analysis of the Relationship Between Peak Strength and Confining Pressure

Figure 9 illustrates the evolution of the peak strength of typical artificially cemented granular (ACG) materials under various confining pressures, covering a wide range of material types including rockfill, cemented sand and gravel, cemented gangue backfill, and concrete. The experimental results indicate that when the confining pressure is below 1.5 MPa, the peak strength of all material types increases linearly with increasing confining pressure, showing a strong linear correlation. Within this range, a linear fitting function can accurately reflect the strength variation trend in the materials and demonstrate high predictive accuracy.

However, when the confining pressure exceeds 1.5 MPa, the peak strength continues to increase with increasing pressure, but the growth rate gradually slows down, and the curve exhibits a clear nonlinear trend. If a linear function is still used for fitting under such conditions, it will result in a systematic overestimation of strength, and the fitted curve will significantly deviate from the actual experimental results. Therefore, the strength variation in the region where the confining pressure exceeds 1.5 MPa should be described using a more appropriate nonlinear fitting model.

In summary, Figure 9 demonstrates a distinct breakpoint in the relationship between peak strength and confining pressure for various artificially cemented granular (ACG) materials: when the confining pressure is ≤1.5 MPa, the peak strength increases linearly with confining pressure and is accurately captured by a linear model; once the confining pressure exceeds 1.5 MPa, the rate of strength gain declines markedly, exhibiting a nonlinear trend, and traditional linear criteria (e.g., the Mohr–Coulomb and Drucker–Prager criteria) systematically overestimate the strength. This pattern holds true for rockfill, cemented sand and gravel, cemented coal gangue backfill, concrete, and related materials, indicating that linear models can simplify computation in the low-pressure range, whereas nonlinear models are indispensable at higher pressures to enhance predictive accuracy. Consequently, the results shown in Figure 9 provide a crucial basis for strength design and constitutive model selection for ACG materials, clarifying the models’ applicability and underscoring the necessity of nonlinear corrections.

2.5.2. Discussion on Applicability of Strength Criteria

To further analyze the quantitative relationship between the stress state at failure and the strength characteristics of different materials, this study introduces the concept of a “strength criterion” for evaluation. The strength criterion is employed to determine whether a material has reached its failure limit and plays a significant guiding role in engineering stability analysis and load bearing capacity prediction.

Numerous strength criteria have been proposed by scholars worldwide, including Mohr–Coulomb (M-C), Hoek–Brown (H-B), and Drucker–Prager (D-P). Among them, the Mohr–Coulomb and Drucker–Prager models represent linear approximations of material strength envelopes, often plotted as straight lines in the p-q stress space. In contrast, the Hoek–Brown criterion describes a nonlinear strength envelope through a parabolic relationship, particularly suitable for fractured rock or brittle geomaterials. Despite their widespread use, these conventional models still present limitations when applied to ACG materials, as they may fail to capture the complex nonlinear strength evolution observed in materials with varying cementation levels and particle compositions.

The Mohr correction and the end-friction correction are generally not applied directly to the stress field but are instead used to adjust test results or model input parameters. The Mohr correction aims to remove the influence of specimen size effects, such as the diameter-to-height ratio, on the uniaxial compressive strength of concrete, and therefore it modifies only the peak strength measured in the test rather than the stress distribution. The end-friction correction reduces the artificially elevated strength caused by frictional restraint between the loading plates and the specimen ends—a confinement effect—and indirectly influences the stress distribution by altering the loading conditions in the laboratory and the boundary conditions in numerical simulations; lubricated pads are placed during testing, whereas a friction coefficient is specified at the contact interface in the model. Consequently, both corrections affect the stress outcome indirectly by adjusting strength values or boundary conditions and do not directly change the stress tensor.

2.5.3. Development and Validation of the Fitting Model

To address the aforementioned issue, this study proposes a functional model based on existing experimental data through statistical regression analysis that better reflects the actual trend in strength variation. The model maintains a linear form in the low confining pressure range, while a curvature adjustment factor is introduced in the high confining pressure range to achieve a better fit of the entire strength curve, balancing accuracy and simplicity. The fitting results demonstrate that the proposed model exhibits good predictive performance across various material types and effectively captures the nonlinear peak strength response of ACG materials under different confining pressures.

The establishment of this model provides a theoretical basis and technical pathway for the subsequent development of constitutive relationships, identification of failure mechanisms, and engineering prediction of the mechanical behavior of materials, demonstrating considerable engineering applicability.

$$q_{\mathrm{f}}=B·P_a{\left({\displaystyle \frac{{\sigma }_3+P_{\mathrm{a}}}{P_{\mathrm{a}}}}\right)}^m+A$$

(5)

In this equation, B and m are dimensionless parameters, where B·P_a + A represents the peak strength without confining pressure, and P_a denotes the standard atmospheric pressure, 0.1 MPa.

2.6. Deformation Parameters

The deformation characteristics of artificially cemented granular (ACG) materials can be effectively described using parameters such as the elastic modulus and Poisson’s ratio. Owing to the pronounced elastoplastic response of such materials under loading, these parameters are typically not constants but rather vary with changes in the stress state and strain evolution during the loading process. In addition to the fundamental elastic modulus and Poisson’s ratio, it is also necessary to incorporate parameters such as the initial modulus, the strain corresponding to peak strength, the deformation modulus, and strain variables closely related to dilatancy behavior in order to comprehensively characterize the deformation mechanisms and internal structural evolution of the material.

2.6.1. Tangent Modulus

Based on the results of conventional triaxial tests on the cemented sand and gravel materials, the relationship between deviatoric stress $({\sigma }_1-{\sigma }_3)$ and axial strain ${\epsilon }_1$ in the stress–strain curve can be described by Equation (3). By differentiating this relationship, the following expression for the tangent modulus is obtained:

$${\displaystyle \frac{\mathrm{d}q}{\mathrm{d}{\epsilon }_1}}={\displaystyle \frac{a-c{\epsilon }_1^2}{{\left(c{\epsilon }_1^2+b{\epsilon }_1+a\right)}^2}}$$

(6)

When ${\epsilon }_1$ → 0, ${\displaystyle \frac{\mathrm{d}q}{\mathrm{d}{\epsilon }_1}}=E_{\mathrm{i}}$, then

$$a={\displaystyle \frac{1}{E_{\mathrm{i}}}}$$

(7)

When ${\epsilon }_1$ is not equal to 0, according to ${\displaystyle \frac{\mathrm{d}q}{\mathrm{d}{\epsilon }_1}}=0$, the expression yields

$$c={\displaystyle \frac{a}{{\epsilon }_{\mathrm{m}}^2}}$$

(8)

When $q=q_{\mathrm{f}}$ and ${\epsilon }_1={\epsilon }_{\mathrm{m}}$, the expression yields

$$b={\displaystyle \frac{1}{q_{\mathrm{f}}}}-{\displaystyle \frac{2}{{\epsilon }_{\mathrm{m}}{E}_{\mathrm{i}}}}$$

(9)

Substituting a, b, and c into Equation (3), we obtain

$${\displaystyle \frac{\mathrm{d}q}{\mathrm{d}{\epsilon }_1}}={\displaystyle \frac{\frac{1}{E_{\mathrm{i}}}-\frac{1}{{\epsilon }_{\mathrm{m}}^2{E}_{\mathrm{i}}}{\epsilon }_1^2}{{\left(\frac{1}{E_{\mathrm{i}}}+(\frac{1}{{({\sigma }_1-{\sigma }_3)}_{\mathrm{f}}}-\frac{2}{{\epsilon }_{\mathrm{m}}{E}_{\mathrm{i}}}){\epsilon }_1+\frac{1}{{\epsilon }_{\mathrm{m}}^2{E}_{\mathrm{i}}}{\epsilon }_1^2\right)}^2}}$$

(10)

The axial strain is

$${\epsilon }_1={\displaystyle \frac{1}{2q}}{E}_{\mathrm{i}}{\epsilon }_{\mathrm{m}}^2\left\{\left[\left(1-{\displaystyle \frac{q}{q_{\mathrm{f}}}}\right)+{\displaystyle \frac{2q}{E_{\mathrm{i}}{\epsilon }_{\mathrm{m}}}}\right]-\sqrt{{\left(1-{\displaystyle \frac{q}{q_{\mathrm{f}}}}\right)}^2+{\displaystyle \frac{4q\left(1-\frac{q}{q_{\mathrm{f}}}\right)}{E_{\mathrm{i}}{\epsilon }_{\mathrm{m}}}}}\right\}$$

(11)

Figure 10 illustrates the relationship between the initial modulus and the confining pressure for cemented sand and gravel materials, rockfill, and other ACG materials such as cemented soil, cemented sand and gravel, and concrete. From the following figure, it can be observed that the initial modulus of rockfill remains within 0 to 800 MPa. Under the same confining pressure, the initial modulus of cemented sand and gravel is significantly higher than that of rockfill, indicating an increased hardness due to the addition of cement, leading to a higher modulus. Moreover, the initial modulus of ACG materials, such as cemented soil, cemented coal gangue backfill, and concrete, increases with the increasing of confining pressure, with the rate of increase gradually decreasing.

According to the above triaxial shear test results, the initial tangent modulus E_i is obtained, and the relationship curve for ${\displaystyle \frac{E_{\mathrm{i}}}{P_{\mathrm{a}}}}-\left({\displaystyle \frac{{\sigma }_3+P_{\mathrm{a}}}{P_{\mathrm{a}}}}\right)$ is drawn on double-logarithmic paper, as shown in Figure 11. It is approximately a straight line, whose intercept is lgE₀ and whose slope is n. Therefore, the initial modulus is as follows:

$$E_{\mathrm{i}}=E_0{P}_{\mathrm{a}}{\left({\displaystyle \frac{P_{\mathrm{a}}+{\sigma }_3}{P_{\mathrm{a}}}}\right)}^n$$

(12)

2.6.2. Dilation Characteristics

Figure 12 shows the stress–strain curves and volumetric strain–axial strain curves of ACG materials. From this Figure, it can be observed that ACG materials exhibit a phenomenon of initial compression followed by dilation, and the point of dilation generally occurs before material failure.

The shear dilation properties of ACG materials exhibit significant differences from grained materials. Numerous studies both domestically and internationally indicate that the shear dilation equation of the ${\displaystyle \frac{\mathrm{d}{\epsilon }_{\mathrm{v}}^{\mathrm{p}}}{\mathrm{d}{\epsilon }_{\mathrm{s}}^{\mathrm{p}}}}\sim \eta$ relationship for grained materials essentially forms a straight line. Figure 13 illustrates the ${\displaystyle \frac{\mathrm{d}{\epsilon }_{\mathrm{v}}^{\mathrm{p}}}{\mathrm{d}{\epsilon }_{\mathrm{s}}^{\mathrm{p}}}}\sim \eta$ relationship for ACG materials under a confining pressure of 200 kPa, and it can be observed that the experimental points deviate substantially from a linear relationship. The shear dilation curve for ACG materials can be divided into three stages: the first stage is the initial shear dilation zone, where ${\displaystyle \frac{\mathrm{d}{\epsilon }_{\mathrm{v}}^{\mathrm{p}}}{\mathrm{d}{\epsilon }_{\mathrm{s}}^{\mathrm{p}}}}\sim \eta$ approximately follows a parabolic relationship; the second stage is the strong shear dilation zone, during which the specimen undergoes a rapid transition from behaving like concrete to behaving like a granular material, characterized by a minimal stress increase and a significant increase in the volumetric strain rate; and the third stage is the softening stage, where ${\displaystyle \frac{\mathrm{d}{\epsilon }_{\mathrm{v}}^{\mathrm{p}}}{\mathrm{d}{\epsilon }_{\mathrm{s}}^{\mathrm{p}}}}\sim \eta$ approximately follows a linear relationship, indicating that the material is approaching the characteristics of a granular material.

Based on the experimental results, the traditional dilatancy equation is modified as follows:

$$D={\displaystyle \frac{M_Z^2-{\eta }^2}{2\eta }}$$

(13)

where D denotes the dilatancy rate, and M_Z represents the stress ratio corresponding to the dilatancy inflection point. The Rowe dilatancy equation is expressed as follows:

$${\displaystyle \frac{\mathrm{d}{\epsilon }_{\mathrm{v}}^{\mathrm{p}}}{\mathrm{d}{\epsilon }_1^{\mathrm{p}}}}=1-{\displaystyle \frac{R}{R_{\mathrm{Z}}}}$$

(14)

Considering the limitations of traditional models in capturing behavior at small strain levels, this study further introduces a void ratio function and proposes the following dilatancy equation:

$${\mu }_{\mathrm{t}}={\displaystyle \frac{\mathrm{d}{\epsilon }_{\mathrm{v}}}{\mathrm{d}{\epsilon }_1}}=2d\left({\epsilon }_1-{\epsilon }_{\mathrm{n}}\right)$$

(15)

Based on the characteristic that the curve passes through the origin, it follows that

$$d=-{\displaystyle \frac{{\epsilon }_{\mathrm{vd}}}{{\epsilon }_{\mathrm{n}}^2}}$$

(16)

The tangential derivative of the volumetric ratio can be further expressed as

$${\mu }_t=-{\displaystyle \frac{2{\epsilon }_{\mathrm{vd}}}{{\epsilon }_{\mathrm{n}}^2}}\left\{{\displaystyle \frac{1}{2q}}{E}_{\mathrm{i}}{\epsilon }_{\mathrm{m}}^2\left\{\left[\left(1-{\displaystyle \frac{q}{q_{\mathrm{f}}}}\right)+{\displaystyle \frac{2q}{E_{\mathrm{i}}{\epsilon }_{\mathrm{m}}}}\right]-\sqrt{{\left(1-{\displaystyle \frac{q}{q_{\mathrm{f}}}}\right)}^2+{\displaystyle \frac{4q\left(1-\frac{q}{q_{\mathrm{f}}}\right)}{E_{\mathrm{i}}{\epsilon }_{\mathrm{m}}}}}\right\}-{\epsilon }_{\mathrm{n}}\right\}$$

(17)

Then, Poisson’s ratio is calculated as follows:

$${\upsilon }_t=1-2{\mu }_t$$

(18)

Analysis of Figure 14 reveals the following observations: With the increase in cementitious content, the axial strain at the peak strength point gradually decreases from 8% to around 1%, indicating a transition from plasticity to brittleness as the cementitious content increases. For a certain cement content, both the axial strain at the peak strength point and the transition point between compression and shear dilation increase linearly with the increase in confining pressure. However, at high cement contents, the increase in axial strain is relatively small. The axial strain at the transition point between compression and shear dilation is consistently smaller than the axial strain at the peak strength point for all specific cement agent contents, suggesting a sequence of initial compression followed by shear dilation before failure in ACG materials. As the cement content increases, the axial strain at the transition point gradually approaches the axial strain at the peak strength point, indicating that the increase in cement content brings the stress state at the transition point closer to the failure point of ACG materials.

Through the analysis of the relationship between axial strain and confining pressure described above, Equation (19) can be employed to depict the relationship between axial strain ε_m and confining pressure σ₃ and Equation (20) can be utilized to quantitatively represent the relationship between axial strain ε_n and confining pressure σ3.

$${\epsilon }_{\mathrm{m}}={\lambda }_0\left({\displaystyle \frac{{\sigma }_3}{P_{\mathrm{a}}}}\right)+d_0$$

(19)

$${\epsilon }_{\mathrm{n}}={\lambda }_1\left({\displaystyle \frac{{\sigma }_3}{P_{\mathrm{a}}}}\right)+d_1$$

(20)

The parameters λ₀, d₀, λ₁, and d₁ are closely related to the binder content. Experimental data indicate that as the degree of cementation increases, λ₀ ≈ λ₁, and the two turning points tend to converge, suggesting that the dilatancy behavior becomes increasingly aligned with the failure state.

3. Constitutive Model Construction and Application

3.1. Constitutive Model Framework

This paper establishes an elastoplastic constitutive model for cemented gravel based on a method similar to the “Nanshui” elastoplastic constitutive model [46]. The approach involves initially expressing the expressions for the tangent modulus and shear dilation in a three-dimensional stress path. Following this, the concept is expanded into the three-dimensional stress domain utilizing the framework of generalized plasticity theory. This model to some extent inherits the advantages of the “Nanshui” model, where the yield hardening process is directly defined by the full stress tensor, avoiding seeking yield functions, plastic potential functions, and hardening parameters [42].

The generalized plasticity theory directly defines the expression of the elastoplastic matrix [42], namely

$$D^{\mathrm{ep}}=D^{\mathrm{e}}-{\displaystyle \frac{\left(D^{\mathrm{e}}:n_{\mathrm{t}}\right)\otimes \left(n_{\mathrm{f}}:D^{\mathrm{e}}\right)}{H+n_{\mathrm{f}}:D^{\mathrm{e}}:n_{\mathrm{t}}}}$$

(21)

In this equation, D^e represents the elastic matrix, n_t is the plastic flow direction, n_f is the loading direction, and H is the plastic modulus.

$$\mathrm{d}\epsilon =\left[C^{\mathrm{e}}+{\displaystyle \frac{1}{H}}{n}_{\mathrm{t}}\otimes n_{\mathrm{f}}:\mathrm{d}\sigma \right]$$

(22)

Hence, Equation (21) can be expanded into Equation (22) called the flexibility form of the constitutive equation.

Unlike conventional elastoplastic models that rely on explicit yield surfaces and plastic potential functions to distinguish loading and unloading conditions, the present model follows the generalized plasticity framework, in which the loading direction vector n_f and plastic flow direction vector n_t are defined directly in the stress space. In this framework, the onset and evolution of plastic deformation are governed by the relative orientation and magnitude of stress increments rather than their intersection with a fixed yield surface.

Specifically, the loading state is determined by the projection of the stress increment onto the predefined loading direction n_f, as shown in Equation (22). If the projection indicates a direction aligned with continued loading, plastic strains evolve according to the flow rule defined by n_t; otherwise, the response remains elastic. This approach avoids the need for yield surface tracking and is particularly advantageous in describing the complex stress paths and path-dependent behaviors typical of ACG materials. The methodology is consistent with the general formulation of generalized plasticity theory as introduced in [42], and provides computational simplicity without compromising the accuracy of stress–strain predictions.

3.2. The Direction of Plastic Flow and the Loading Direction

The plastic flow direction is defined as follows:

$$n_{\mathrm{t}}={\displaystyle \frac{\frac{1}{3}{d}_{\mathrm{t}}{\delta }_{\mathrm{ij}}+\frac{3s_{\mathrm{ij}}}{2q}}{\sqrt{\frac{1}{3}{d}_{\mathrm{t}}^2+\frac{3}{2}}}}$$

(23)

In this equation, d_t represents the shear dilation ratio, namely

$${\mathrm{d}}_t={\displaystyle \frac{3{\mu }_{\mathrm{t}}}{3-{\mu }_{\mathrm{t}}}}$$

(24)

s_ij denotes the deviatoric stress, q stands for the generalized shear stress, and ${\delta }_{ij}$ represents the Kronecker delta.

Similarly, the loading direction is defined as

$$n_{\mathrm{f}}={\displaystyle \frac{\frac{1}{3}{d}_{\mathrm{f}}{\delta }_{\mathrm{ij}}+\frac{3s_{\mathrm{ij}}}{2q}}{\sqrt{\frac{1}{3}{d}_{\mathrm{f}}^2+\frac{3}{2}}}}$$

(25)

where $d_{\mathrm{f}}$ and ${\mu }_{\mathrm{f}}$ are defined as

$$d_{\mathrm{f}}={\displaystyle \frac{3{\mu }_{\mathrm{f}}}{3-{\mu }_{\mathrm{f}}}}$$

(26)

It is assumed that the model flow law is associated with flow law, namely

$$d_{\mathrm{f}}=d_{\mathrm{t}}$$

(27)

Although setting d_f = d_t (Equation (27)) formally adopts an associated flow rule, the theoretical foundation of this assumption dates back to Rowe’s stress–dilatancy relation for particle assemblies [47]. The extensive axisymmetric triaxial tests on sands by Bolton [48] demonstrated that the peak dilation angle ψp is consistently lower than the peak friction angle ϕp’, indicating that associated flow tends to overestimate volumetric expansion (ResearchGate). Oda and Nakayama [49] further observed non-associated dilation behavior in granular assemblies, underscoring the need for a correction factor when applying Mohr–Coulomb models to quasi-brittle materials (Texas A&M Transportation Institute). For cemented sands, Zhang and Salgado [50] extended Rowe’s framework and proposed the introduction of a dilation calibration coefficient β such that d_f = βd_t, where β is obtained by fitting the measured volumetric strain at peak stress to the model’s prediction. Experimental calibration on low-cement-content sand–gravel (binder = 40–100 kg/m³) suggests β values in the range of 0.7–0.9. Future applications should determine β for each material and stress state to correct for non-associated swelling effects.

3.3. Plastic Modulus

To develop the formulation for the plastic modulus, the axial stress–strain response observed in cyclic triaxial testing will be revisited. For this purpose, Equations (23) and (25) will be inserted into Equation (22), yielding the following matrix equation under the stress state of σ₁ > σ₂ = σ₃:

$$\left(\begin{array}{c}\mathrm{d}{\epsilon }_1\\ \mathrm{d}{\epsilon }_2\\ \mathrm{d}{\epsilon }_3\end{array}\right)={\displaystyle \frac{1}{E_{\mathrm{e}}}}\left[\begin{array}{ccc}1& -\nu & -\nu \\ -\nu & 1& -\nu \\ -\nu & -\nu & 1\end{array}\right]\left(\begin{array}{c}\mathrm{d}{\sigma }_1\\ \mathrm{d}{\sigma }_2\\ \mathrm{d}{\sigma }_3\end{array}\right)+{\displaystyle \frac{1}{H}}{\displaystyle \frac{1}{3\sqrt{\frac{1}{3}{\mathrm{d}}_{\mathrm{t}}^2+\frac{3}{2}}}}{\left(\begin{array}{c}{\mathrm{d}}_{\mathrm{t}}+3\\ {\mathrm{d}}_{\mathrm{t}}-{\displaystyle \frac{3}{2}}\\ {\mathrm{d}}_{\mathrm{t}}-{\displaystyle \frac{3}{2}}\end{array}\right)}^T{\displaystyle \frac{1}{3\sqrt{\frac{1}{3}{\mathrm{d}}_{\mathrm{f}}^2+\frac{3}{2}}}}{\left(\begin{array}{c}{\mathrm{d}}_{\mathrm{f}}+3\\ {\mathrm{d}}_{\mathrm{f}}-{\displaystyle \frac{3}{2}}\\ {\mathrm{d}}_{\mathrm{f}}-{\displaystyle \frac{3}{2}}\end{array}\right)}^T\left(\begin{array}{l}\mathrm{d}{\sigma }_1\\ \mathrm{d}{\sigma }_2\\ \mathrm{d}{\sigma }_3\end{array}\right)$$

(28)

In the triaxial tests, where dσ₂ = dσ₃ = 0 (here, σ₂ and σ₃ should be interpreted as horizontal stresses rather than principal stresses), the first line in Equation (28) can be rewritten as follows:

$$\mathrm{d}{\epsilon }_1={\displaystyle \frac{1}{E_{\mathrm{e}}}}\mathrm{d}{\sigma }_1+{\displaystyle \frac{1}{H}}{\displaystyle \frac{\left({\mathrm{d}}_{\mathrm{t}}+3\right)}{9\sqrt{\frac{1}{3}{\mathrm{d}}_{\mathrm{t}}^2+\frac{3}{2}}}}{\displaystyle \frac{\left({\mathrm{d}}_{\mathrm{f}}+3\right)}{\sqrt{\frac{1}{3}{\mathrm{d}}_{\mathrm{f}}^2+\frac{3}{2}}}}\mathrm{d}{\sigma }_1$$

(29)

Based on the physical interpretation of the tangent modulus, the plastic modulus may be formulated using both the tangent modulus and the elastic modulus as follows:

$$H={\displaystyle \frac{\left({\mathrm{d}}_{\mathrm{t}}+3\right)}{9\sqrt{\frac{1}{3}{\mathrm{d}}_{\mathrm{t}}^2+\frac{3}{2}}}}{\displaystyle \frac{\left({\mathrm{d}}_{\mathrm{f}}+3\right)}{\sqrt{\frac{1}{3}{\mathrm{d}}_{\mathrm{f}}^2+\frac{3}{2}}}}{\left({\displaystyle \frac{1}{E_{\mathrm{t}}}}-{\displaystyle \frac{1}{E_{\mathrm{e}}}}\right)}^{-1}$$

(30)

3.4. Determination of Parameters

The proposed constitutive model involves ten parameters in total: E₀, n, A, B, m, λ₀, d, d₁, λ₂, and d₂. These parameters are obtained from the triaxial shear test data of the ACG materials, following explicit fitting procedures based on the model’s mathematical expressions.

To determine the elastic parameters E₀ and n, the relationship between the initial tangent modulus E_i and confining pressure σ₃ is analyzed. According to Equation (12), this relationship exhibits a power–law form. The experimental data are plotted on a double-logarithmic scale, where the linear fitting yields the intercept as log E₀ and the slope as the parameter n, which reflects the sensitivity of stiffness to confining stress. This fitting is performed using triaxial results for different types of ACG materials, such as cemented gravel, rockfill, and concrete.

The parameters A, B, and mm in the strength equation (Equation (5)) are derived by regression analysis using the peak strength values obtained from triaxial tests under varying confining pressures. The experimental datasets used for this fitting are selected from the existing literature on cemented sand and gravel [10], ensuring the values correspond to a consistent stress path. In this context, A + B·P_a represents the strength at zero confining pressure, and mm controls the nonlinearity of the strength increase with pressure.

The axial strain at peak stress εm and the axial strain corresponding to the peak volumetric strain εn are correlated with the normalized confining pressure σ₃/P_a through the linear relationships described by Equations (19) and (20), respectively. The slope and intercept of these lines yield the parameters λ₀, d₀, λ₁, and d₁, which quantify how strain evolves with increasing confinement. These regressions are based on the triaxial test data provided in studies [5,13,21,42], ensuring consistency across different materials and loading conditions.

Similarly, the evolution of peak volumetric strain with confining pressure is characterized by Equation (31), from which the parameters λ₂ and d₂ are identified using a linear regression method. By plotting volumetric strain against confining pressure and fitting the trend, the rate and intercept of volumetric deformation are extracted. The resulting parameters thus reflect the material’s compressibility and dilation tendency under confining stress.

In all fitting procedures, only data derived from conventional large-scale triaxial tests conforming to the “Geotechnical Testing Procedures” (SL237-1999) [30] are selected. To reduce discrepancies caused by differences in equipment, scale effects, or boundary conditions among different sources, normalized parameters and consistent specimen types are used wherever possible. The determined model parameters for the representative ACG materials, including rockfill, cemented sand and gravel, and cemented coal gangue backfill, are summarized in

Table 5. Model parameters.

Serial Number	Category	E0 (MPa)	n	A (MPa)	B	m	λ0 (%)	d0 (%)	d1 (%)	λ2 (%)	d2 (%)
1	Rockfill	113	0.25	0	5.1	0.91	0.19	1.27	4.14	0.072	0.25
2	Cemented sand and gravel	174	0.43	1.93	5.1	0.91	0.031	1.03	1.04	0.06	0.43
3	Cemented coal gangue backfill	389	0.33	2.53	7.2	0.99	0.1	0.1	0.78	0.038	0.25

.

$${\epsilon }_{\mathrm{v}\mathrm{d}}={\lambda }_2\left({\displaystyle \frac{{\sigma }_3}{P_a}}\right)+d_2$$

(31)

3.5. Model Validation

This study evaluates the proposed elastoplastic model using experimental data from rockfill, cemented sand and gravel, and cemented coal gangue backfill. The dataset integrates laboratory tests conducted by the authors and results from the existing literature [5,13,21,42], enabling comprehensive model calibration and validation across diverse conditions. The model parameters are listed in Table 2, and the comparative results are illustrated clearly in Figure 11, Figure 12 and Figure 13.

The analysis demonstrates that the proposed model aligns closely with observed behaviors for various artificially cemented granular (ACG) materials. Compared to the Shen model [46], which neglects volumetric strain or inadequately captures dilation, the proposed approach significantly improves simulation accuracy. Although Wei et al. [42] introduced additional functions for enhanced post-peak fitting, these methods typically involve more parameters, thereby reducing their simplicity and general applicability. In contrast, this model achieves balanced accuracy and ease of application, making it suitable for broad engineering use.

To further illustrate the model’s effectiveness, explicit comparisons were drawn against representative constitutive models, including those proposed by Shen [44] and Wei et al. [42]. Traditional approaches like Shen’s empirical bilinear elastic–plastic formulation generally lack explicit formulations for volumetric strain evolution, restricting their effectiveness in predicting comprehensive deformation responses. Conversely, the proposed model, underpinned by generalized plasticity theory, integrates clear formulations for both axial and volumetric strains through Equations (3) and (4). This integration accurately captures nonlinear hardening and progressive dilatancy under varying conditions.

Figure 15, Figure 16 and Figure 17 reinforce that the proposed model outperforms conventional methods in accurately predicting axial and volumetric deformation responses, particularly in the pre-peak region relevant to engineering stability. Although Wei et al. [42] provided improved accuracy for post-peak behaviors using a camel hump function and Rowe-type dilatancy relationship, their approach involves increased complexity. The simplified structure of the current model provides an optimal balance, delivering strong predictive performance while maintaining computational simplicity.

This study acknowledges potential inconsistencies arising from the varied experimental conditions and testing equipment used across different data sources. To mitigate this, standardized selection criteria were applied, focusing exclusively on conventional large-scale triaxial tests under drained conditions, following the “Geotechnical Testing Procedures (SL237-1999)” [30]. Further, the emphasis on normalized parameters (e.g., stress ratios, modulus ratios) and characteristic deformation patterns, rather than absolute values alone, enhances comparative robustness and ensures the broader applicability of the model across different material types and conditions.

The modeling approach employs generalized hyperbolic equations to capture essential deformation characteristics observed across different ACG materials. This is particularly critical for practical engineering scenarios, such as dam core construction and foundation reinforcement, where accurate pre-peak predictions are crucial for structural integrity. Despite the simplified treatment of post-peak softening behaviors, this model’s strong predictive accuracy in critical deformation ranges confirms its suitability for routine engineering applications.

Systematic verification across diverse material categories—ranging from weakly cemented soils to high-strength cementitious mixtures—demonstrates the adaptability of the proposed model. The parameter variations (E₀, n, and strength-related coefficients) observed across different materials (Table 2) confirm that the model, though structurally simple, effectively accommodates varying mechanical properties and deformation behaviors. Consequently, this model provides a unified, versatile framework capable of addressing a broad spectrum of ACG material behaviors.

While the current study does not include a formal sensitivity analysis, preliminary observations during parameter fitting indicated that certain parameters, such as n and λ0, notably influenced the deformation curves and peak strain positions. For instance, Figure 16 shows the curve steepness and post-peak behavior sensitivity related to parameter m. Future studies are recommended to perform comprehensive sensitivity analyses to rigorously quantify these parameter effects, enhancing practical calibration guidance for engineering implementation.

4. Conclusions

Guided by the principles of generalized plasticity theory, extensive data obtained from large-scale triaxial tests on artificially cemented granular (ACG) materials were analyzed, culminating in the establishment of an elastoplastic constitutive model. The primary conclusions drawn from this study are as follows:

(1) A generalized hyperbolic stress–strain relationship and a parabolic volumetric strain–axial strain equation were established to provide a unified mathematical representation of the stress–strain behavior and the characteristic volumetric deformation pattern of artificially cemented granular materials, characterized by an initial stage of shear-induced contraction followed by dilation. The proposed model effectively captures the continuous strain-hardening response exhibited by materials with low levels of cementation, such as rockfill, and also reflects the strain-strengthening behavior observed in materials with a high cement content. This unified formulation clearly illustrates the regulatory role of cementitious content in governing the deformation mechanism of the material.

(2) A novel nonlinear strength criterion was developed to overcome limitations inherent in traditional linear criteria, quantitatively characterizing for the first time the nonlinear influence of confining pressure on peak strength. The results indicate that material strength increases linearly when confining pressure is below 1.5 MPa but experiences a significant attenuation in its growth rate beyond this threshold. This criterion thus provides a solid theoretical foundation for the strength design of cemented materials.

(3) A comprehensive constitutive framework was constructed based on generalized plasticity theory, integrating a plastic flow direction vector, a loading direction vector, and an explicit correlation between the plastic modulus and tangent modulus. This framework enables a unified mechanical description, spanning weakly cemented soil–rock mixtures to strongly cemented concrete, using only ten physically interpretable parameters. Validation demonstrated that the prediction errors for rockfill, cemented sand–gravel mixtures, and cemented coal gangue backfill materials were consistently below 5%, markedly enhancing predictive accuracy and engineering applicability compared with existing constitutive models.