Cyclic Dynamic Behaviour of Reconstructed Soil–Rock Mixtures: Hysteresis Response, Normalized Shear Modulus, and Damping Evolution

Highlights

What are the main findings?

• SRMs show strain-dependent nonlinear cyclic behaviour.
• Shear modulus degrades nonlinearly with increasing strain.
• Rock content and confinement control normalized evolution.

What are the implications of the main findings?

• Dynamic stability should include modulus and damping evolution.
• Deformation compatibility is critical for complex SRM foundations.
• Findings provide dynamic-parameter evidence for SRM foundation assessment.

Abstract

Protective coatings and surface-protection systems improve structural durability, but the long-term performance of durability-sensitive infrastructure also depends on the cyclic stability of supporting soil–rock mixture (SRM) foundations. In this study, undrained multistage strain-controlled cyclic triaxial tests were conducted on reconstructed SRMs with rock block contents of 0%, 10%, 20%, 40%, and 60% under confining pressures of 100, 200, and 400 kPa. Hysteresis-loop morphology, secant shear modulus, normalized shear modulus ratio, damping ratio, normalized damping ratio, and fitting parameters were evaluated. The results show that hysteresis loops evolved from narrow and steep to wider and fuller forms as strain amplitude increased, indicating stiffness degradation and enhanced hysteretic dissipation. The secant shear modulus decreased from 35.835 to 158.871 MPa to 3.296–12.854 MPa, corresponding to an overall reduction of approximately 85%–94%, while the damping ratio increased from 0.036 to 0.063 to 0.195–0.268. Higher rock block content and stronger confinement increased absolute stiffness, but rock block content advanced normalized degradation and damping development, whereas confinement delayed these normalized responses. These findings provide experimental evidence for dynamic-parameter selection, deformation-compatibility evaluation, and cyclic stability assessment of complex SRM foundations.

1. Introduction

1.1. Engineering Background

Protective coatings and surface-protection systems are widely used to improve the durability of engineering structures exposed to corrosion, erosion, wetting–drying cycles, and other aggressive service environments. However, the long-term performance of such surface systems is not governed by material durability alone. When protected structures are founded on heterogeneous geomaterials, cyclic foundation deformation, stiffness degradation, and interface incompatibility may change the mechanical boundary conditions imposed on structural components and protected surfaces. Therefore, for durability-sensitive infrastructure, the stability of the supporting ground and the deformation compatibility at structural interfaces should be considered together with surface-protection measures.

This issue is particularly relevant to durability-sensitive and safety-critical infrastructure constructed on complex ground, including coastal nuclear power plants, liquefied natural gas terminals, large dams, transportation hubs, bridges, and tunnels [1,2]. In these systems, even limited foundation deformation or local stiffness discontinuity may affect load transfer, serviceability, and safety margins. Therefore, foundation problems should not be evaluated only in terms of static bearing capacity or settlement, but should also consider cyclic stiffness retention, damping evolution, and deformation compatibility under repeated environmental, operational, or seismic disturbances.

The foundation–ground interface is a critical zone for load transfer, deformation compatibility, and dynamic interaction [2,3]. When the supporting ground consists of heterogeneous soil–rock mixtures, cyclic stiffness degradation and damping evolution may alter the stress-transfer path and local deformation compatibility of the foundation system. This is especially important for safety-critical structures, because foundation response may be amplified through the structural system under repeated vibration, seismic excitation, or long-term operational disturbance. Therefore, the cyclic behaviour of SRM foundations should be characterized not only by initial stiffness, but also by stiffness retention, energy dissipation, and interface stability under repeated loading [1,2,3].

Soil–rock mixtures are widely encountered in slope deposits, embankment fills, colluvial accumulations, cut-and-fill transition zones, and complex foundation formations [4,5,6]. Owing to the coexistence of a deformable soil matrix and relatively rigid rock blocks, their cyclic response is governed by matrix deformation, rock block interaction, contact rearrangement, and soil–rock interfacial sliding [1,2,3,7,8,9]. Under repeated loading, these internal processes may lead to nonlinear stiffness degradation, hysteresis-loop expansion, and increasing energy dissipation [7,9,10]. As a result, SRMs should be regarded as heterogeneous block-in-matrix geomaterials whose dynamic behaviour cannot be fully represented by a single equivalent homogeneous parameter.

For SRMs, the macroscopic cyclic response is affected by matrix deformation, rock block interaction, contact rearrangement, and soil–rock interfacial sliding [3,8]. These internal processes are reflected externally through hysteresis-loop morphology, shear modulus degradation, damping-ratio evolution, and normalized dynamic-parameter changes. For safety-critical infrastructure such as coastal nuclear power plants, the relevance of SRM behaviour lies in the possible influence of cyclic foundation response on deformation compatibility, stress redistribution, and interface stability [7,8]. Therefore, material-scale cyclic triaxial evidence can provide useful geo-mechanical information for evaluating complex SRM foundations.

Despite the engineering importance of SRMs, most existing foundation dynamic analyses and site-response evaluations still rely on relatively homogeneous soil assumptions or idealized layered ground. Previous studies have improved the understanding of SRM strength, modulus, damping, and seismic response, but the unified experimental characterization of hysteresis-loop morphology, secant shear modulus degradation, normalized modulus reduction, damping-ratio evolution, and normalized damping development remains insufficient [6,9,10]. Therefore, the central contribution of this study is to provide a systematic cyclic triaxial characterization of reconstructed SRMs under controlled rock block contents and confining pressures, and to clarify how stiffness degradation and energy dissipation evolve within a consistent strain-controlled framework [3,9]. This provides a clearer parameter basis for dynamic assessment of complex SRM foundations, including those supporting coastal nuclear power plants and other safety-critical infrastructure [3,7,8].

1.2. Previous Studies

Previous studies on soil–rock mixtures (SRMs) under cyclic and dynamic loading have primarily concentrated on the evolution of stiffness, deformation, and energy dissipation. Wang et al. conducted large-scale cyclic triaxial tests on SRMs with different rock block contents and confining pressures, and showed that the maximum shear modulus increased with rock content and confining pressure, while the dynamic shear modulus ratio decreased and the damping ratio increased as strain developed [11]. Their work established an important basis for describing SRM dynamics using normalized shear strain and normalized dynamic parameters [11]. Building on this line, Du et al. further investigated the effects of rock shape and rock volume fraction on the dynamic properties of SRMs under cyclic loading, and reported that threshold cyclic stress ratio, pore-pressure evolution, cumulative strain, and stiffness response were all sensitive to block geometry and coarse-phase proportion [12]. More recently, Chen et al. examined the dynamic behaviour of clay–gravel mixtures under varying gravel contents and mean effective stresses, and confirmed that the coupled evolution of shear modulus and damping is strongly governed by coarse fraction and confinement [13]. Although clay–gravel mixtures are not identical to SRMs, these results provide valuable adjacent evidence for interpreting the dynamic response of block-in-matrix geomaterials [13]. Collectively, these studies indicate that the cyclic behaviour of SRMs is controlled not only by loading amplitude, but also by rock content, confining pressure, and block geometry [11,12,13]. However, most existing work still emphasizes global dynamic indices or cumulative response variables, whereas the direct linkage between hysteretic morphology, secant shear modulus degradation, normalized modulus reduction, damping evolution, and fitting parameters within a unified strain-controlled framework remains insufficiently clarified [11,12].

Recent studies have also extended SRM dynamics from the material scale to the engineering scale. Zhao et al. used large-scale shaking-table tests to investigate SRM slopes with different rock contents, and demonstrated that rock content significantly affected acceleration amplification, deformation development, and seismic failure characteristics [10]. Li et al. further explored the effect of underlying tunnels on seismic response in soil–rock strata through shaking-table tests, and showed that tunnels could modify the surface acceleration response in such heterogeneous strata [14]. Chen et al. recently examined the dynamic response of SRM embankments under the coupled effects of cyclic loading and seepage using a coupled discrete-element and finite-difference framework, and reported that hydraulic conditions further altered amplification characteristics and dynamic stability [13]. These engineering-scale studies clearly show that SRM-related dynamic problems are not confined to laboratory specimens, but are expressed through slope response, embankment performance, and underground structure–ground interaction [10,14]. Nevertheless, most of these studies still rely on prescribed or indirectly inferred material parameters, and therefore do not fully resolve how the intrinsic cyclic response of SRMs should be characterized in a concise, transferable, and experimentally grounded parameter system [10,11,14].

Another important branch of recent research concerns interface behaviour, which is highly relevant to the present study. Liu et al. performed cyclic shear tests on soil–rock mixture–concrete interfaces and found that concrete surface roughness, rock content, and cyclic shear amplitude all had significant effects on cyclic shear characteristics [3]. In a related study on concrete–soil interfaces, Liu et al. showed that interface roughness strongly affected load transfer and deformation between pile and soil [15]. Zhang et al. further investigated the cyclic shear behaviour of rock–concrete interfaces and demonstrated that repeated shearing led to systematic degradation of shear strength, shear stiffness, and friction coefficient, accompanied by nonlinear evolution of interfacial damage and energy release [16]. Although these studies do not directly address the dynamic parameter system of SRMs themselves, they indicate that interface conditions can influence shear stiffness, frictional dissipation, and load-transfer behaviour under cyclic loading [3,15,16]. This is especially relevant for SRMs, in which internal soil–rock interfaces and external engineering interfaces may both participate in stiffness loss and dissipation development. However, current interface studies and SRM dynamic studies are still largely separated, and the connection between internal interfacial activity and bulk dynamic parameter evolution remains underdeveloped [3,16].

Parallel to laboratory testing, recent numerical and mesoscopic studies have substantially improved the understanding of SRM mechanics by introducing more realistic block geometry. Wu et al. incorporated realistic particle shapes into DEM-based mini-triaxial simulations and showed that particle shape, size, and irregular-particle proportion had strong influences on deformation and failure characteristics [7]. Liu et al. developed a high-precision computational modelling method for SRMs considering the realistic shape of rock blocks, highlighting the importance of preserving actual morphology in mechanical simulation [8]. He et al. further demonstrated that the spatial distribution orientation of blocks could introduce significant bias in DEM calculations, and proposed a three-dimensional modelling strategy to reduce this effect [17]. Liu et al. subsequently proposed a mesoscopic numerical modelling method for SRMs capable of efficiently generating actual block shapes and analyzing their influence on stress–strain response and failure mode [18]. Li et al. used computer-vision-aided DEM modelling to study the compaction characteristics of subgrade fillers with real coarse-particle shapes, again confirming that realistic geometry substantially affects macro- and micro-mechanical behaviour [19]. Tu et al. later advanced refined DEM modelling of SRMs by emphasizing the structural effects of rock blocks and the role of realistic block geometry under different loading paths [20]. In addition, Zhang et al. investigated the influence of gravel shape on the triaxial shear behaviour of SRMs, while Sun et al. examined the shear behaviour of breakable SRMs and highlighted the roles of fragmentation and shape-related meso-mechanics [21]. Together, these studies show that realistic geometry, internal arrangement, and mesoscopic structure are increasingly recognized as essential to SRM analysis [7,8,17,18,19,20,21]. Yet most of this work has focused on strength, deformation, or failure, whereas the extraction and interpretation of cyclic dynamic parameters from hysteresis-based experimental evidence remain relatively limited [7,18,20].

At the same time, several recent studies have attempted to bridge macro-scale behaviour and advanced constitutive or coupled numerical descriptions. Li et al. proposed a coupled MPM-DEM framework for modelling SRMs with large particle-size ratios, providing a useful route for handling multiscale heterogeneity [22]. Li et al. later applied a coupled MPM-DEM method to the failure analysis of SRM slopes, further demonstrating the value of hybrid numerical approaches for complex geomaterials [5]. Qian et al. developed a thermodynamically consistent constitutive model for SRMs and explicitly considered the effects of fine content and particle crushing, thereby improving the theoretical representation of block-in-matrix mechanical behaviour [6]. These developments are important because they suggest that the next stage of SRM research should move beyond isolated strength descriptions and toward multiscale, parameter-transferable formulations [5,6,22]. However, even in these advanced studies, the emphasis remains mainly on strength, stability, seepage, or constitutive enhancement, while the unified experimental characterization of cyclic hysteresis, shear modulus degradation, normalized modulus reduction, damping ratio, and normalized damping evolution is still comparatively scarce [5,6,11,22].

Overall, recent studies have advanced the understanding of SRMs from several complementary perspectives, including cyclic dynamic parameters, engineering-scale seismic response, interface degradation, and realistic geometry-based mesoscopic modelling. Existing cyclic triaxial studies have provided important evidence on modulus reduction, damping evolution, pore-pressure response, cumulative deformation, and the effects of rock content, confining pressure, and block geometry [11,12,13]. Engineering-scale investigations have further shown that SRM-related dynamic problems may be expressed through slope response, embankment performance, tunnel–ground interaction, and heterogeneous site response [3,10,14,15,16]. In parallel, interface studies have demonstrated that cyclic degradation at soil–rock, soil–concrete, and rock–concrete interfaces can substantially affect shear stiffness, frictional dissipation, and load-transfer behaviour [3,10,14,15,16]. Recent mesoscopic and numerical studies have also highlighted the influence of realistic block geometry, spatial arrangement, and multiscale modelling on the mechanical response of SRMs [5,6,7,8,17,18,19,20,21,22]. However, these aspects are still not fully connected within a single experimental parameter system. In particular, the transition from hysteresis-loop morphology to secant shear modulus degradation, normalized shear modulus reduction, damping-ratio evolution, normalized damping development, and empirical fitting-parameter variation remains insufficiently organized under controlled rock-content and confining-pressure conditions [11,12].

Building on these previous advances, the present study connects the above response quantities within one strain-controlled cyclic triaxial framework. The emphasis is placed on how rock block content and confining pressure affect both the absolute dynamic response and the relative evolution path of reconstructed SRMs. This integrated treatment allows stiffness degradation and energy dissipation to be interpreted not only through separate modulus or damping indices, but also through their coupled evolution in the normalized parameter space. Therefore, the present work provides a more coherent experimental basis for subsequent constitutive fitting, mesoscopic interpretation, and interface-related dynamic assessment of complex SRM foundation materials [11,12].

1.3. Objective of This Study

Against the above background, the present study develops an integrated experimental characterization of the cyclic dynamic behaviour of reconstructed soil–rock mixtures under a systematic strain-controlled testing framework. The analysis is organized around a continuous parameter chain derived from the hysteresis loop, including secant shear modulus degradation, normalized shear modulus evolution, damping-ratio development, normalized damping evolution, and fitting-parameter variation. Special attention is given to how rock block content and confining pressure influence the absolute stiffness and damping levels, as well as the relative degradation and dissipation paths of reconstructed SRMs.

Specifically, this study seeks to:

• Characterize the evolution of hysteresis-loop morphology of reconstructed SRMs under multistage strain-controlled cyclic loading and identify the strain-dependent changes in cyclic response;
• Quantify the effects of rock block content and confining pressure on secant shear modulus degradation and stiffness-retention behaviour over the investigated strain range;
• Evaluate the damping-ratio response and normalized damping development to characterize the strain-dependent mobilization of hysteretic energy dissipation under different material compositions and stress conditions;
• Apply previously reported empirical relationships to describe normalized shear modulus and normalized damping evolution, and examine the variation in the corresponding fitting parameters under different test conditions;
• Provide a parameter-based basis for foundation treatment, construction control, and interface-related stability assessment of safety-critical infrastructures, such as coastal nuclear power plants, founded on complex SRM ground.

2. Materials and Test Programme

2.1. Test Materials

The basic physical properties of the two components were determined before specimen preparation. For the rock blocks, the material was collected from a slope near a quarry site in Ipoh, Malaysia, and was subsequently processed into crushed rock particles for specimen preparation. The natural density and dry density of the rock particles were 2.61 g/cm³ and 2.55 g/cm³, respectively. The rock particles used in this study were granitic crushed rock particles [23]. They were predominantly light grey to grey-white in colour, with scattered dark mineral speckles, rough surfaces, and angular to sub-angular outlines. These particles were dense, hard, and crystalline, with low porosity, relatively high surface roughness, and strong interlocking potential. In engineering terms, the rock phase therefore provided a rigid coarse skeleton with high interface-friction capacity within the reconstructed soil–rock mixture. A water absorption of 1.2% was adopted for the rock particles.

For the soil matrix, the material was obtained from the deep part of a slope near the Engineering Campus of Universiti Sains Malaysia, Malaysia. Based on field information and laboratory identification, the soil matrix was classified as a fines-dominated silty clay. The soil matrix had a natural density of 2.04 g/cm³, a specific gravity of 2.64, a natural water content of 23.12%, a dry unit weight of 16.28 kN/m³, an optimum water content of approximately 16%, and an initial void ratio of 0.522. Index characterization showed that the matrix soil had a liquid limit of 43%, a plastic limit of 25%, and a plasticity index of 18%. The fines content (<0.075 mm) was 62%, and the clay fraction (<0.002 mm) was 19%. According to the Unified Soil Classification System, the soil matrix was classified as CL and is therefore described herein as a low- to medium-plasticity silty clay. These results indicate that the tested geomaterial was composed of two components with clearly different mechanical characteristics, namely a relatively deformable soil matrix and a comparatively rigid granitic rock block phase. The test materials used for specimen preparation are presented in Figure 1, including the rock block particles, the soil matrix, and the particle-size distribution curve of the soil matrix.

The division between soil matrix and rock block particles was defined according to particle size. In the natural soil–rock mixture, the original rock block size mainly ranged from 2 mm to 60 mm. For the present experimental study, the rock particles used in the reconstructed mixture were controlled within the range of 6–10 mm. This particle-size range was selected by considering both the original gradation characteristics of the natural soil–rock mixture and the geometric limitation of the cyclic triaxial specimen [24]. In the present study, the specimen diameter and height were 50 mm and 100 mm, respectively. Accordingly, the adopted rock-particle range corresponds to a maximum particle-size-to-specimen-diameter ratio ($d_{\mathit{max}}/D$) of 0.20 and a specimen-height-to-maximum-particle-size ratio ($H/d_{\mathit{max}}$) of 10 [25]. These values are close to the upper practical boundary for small-scale triaxial testing with coarse inclusions. Therefore, the selected 6–10 mm range should be regarded as a controlled dimensional compromise rather than a complete reproduction of the in situ particle-size distribution. This choice allowed the rock phase to retain a clear block-in-matrix structural role while keeping the specimen geometry compatible with the cyclic triaxial apparatus. The reconstructed specimens were therefore designed to capture the essential coarse-particle effect and comparative influence of rock block content, while possible specimen-scale and boundary effects associated with the relatively large particle-size ratio are acknowledged as an inherent limitation of the small-scale testing programme [26].

To investigate the role of rock content in the cyclic response of soil–rock mixtures, five groups of materials were prepared with rock block contents of 0%, 10%, 20%, 40%, and 60%, respectively. In this study, the rock block content is denoted by $R_c$ and is defined as the volumetric proportion of rock blocks in the reconstructed soil–rock mixture specimen, namely

$$R_c={\displaystyle \frac{V_r}{V_r+{ V}_s}} \times 100\%$$

(1)

where $V_r$ is the volume of rock particles and $V_s$ is the volume of the soil matrix in the specimen. Accordingly, $R_c$ in this study refers to volumetric rock block content rather than mass percentage. This definition was adopted because the volumetric proportion of rock blocks is more directly related to the internal block-in-matrix structure, contact state, and skeleton development of the reconstructed soil–rock mixture. The selected $R_c$ levels were arranged to establish a gradual transition from pure soil to block-rich mixtures, so that the influence of rock block content on the basic dynamic behaviour of the material could be evaluated more clearly.

2.2. Specimen Preparation

Cylindrical specimens were prepared for the cyclic triaxial tests, with a diameter of 50 mm and a height of 100 mm. Based on the particle-size selection principle described in Section 2.1, the reconstructed soil–rock mixture specimens were prepared using rock particles in the range of 6–10 mm. This particle-size selection was used to balance two requirements: maintaining a recognizable block-in-matrix structure of the coarse phase and satisfying the dimensional constraints of the 50 mm × 100 mm cyclic triaxial specimen. Five rock block content levels were considered, namely 0%, 10%, 20%, 40%, and 60%. In addition, three confining pressures, namely 100 kPa, 200 kPa, and 400 kPa, were adopted in the testing programme.

The specimens were prepared using a compaction-based reconstruction method [27]. In the present study, the target state of the reconstructed specimens was defined as a moisture-conditioned and uniformly compacted state prior to saturation, rather than as a direct replication of the in situ density of a natural deposit. For each target rock block content, the required quantities of soil matrix and rock particles were calculated according to the prescribed volumetric rock block content $R_c$, the specimen volume, and the physical properties of the two constituent materials. Considering that the optimum water content of the soil matrix was approximately 16%, the target water content of the matrix soil during specimen preparation was set at 18% to compensate for minor moisture loss during mixing and placement. The granitic rock particles were pre-conditioned at a moisture content of 1.2% before mixing so as to reduce uncontrolled water transfer from the soil matrix to the rock phase. After water addition and mixing, the reconstructed mixture was sealed and left to stand for 24 h to allow a more uniform redistribution of moisture. Prior to the formal testing programme, a pilot compaction calibration was conducted, and the adopted layered compaction procedure was fixed such that specimens prepared under the same $R_c$ reached a stable and repeatable compacted state. Therefore, the target state of all specimens in this study corresponds to the same controlled compaction condition and pre-mixing moisture condition, thereby ensuring the comparability of cyclic test results obtained under different rock block contents and confining pressures.

According to the dry unit weight of the soil matrix and the dry density of the rock particles, the target dry unit weights of the reconstructed specimens after moulding were controlled at approximately 16.28, 17.17, 18.13, 19.82, and 21.57 kN/m³ for R_c = 0%, 10%, 20%, 40%, and 60%, respectively. These values were used together with the fixed six-layer compaction procedure, target water content, and dimensional acceptance criteria to control the initial moulding state of the reconstructed specimens. Therefore, the specimen preparation was controlled not only by the prescribed volumetric rock block content, but also by an R_c-specific target dry unit weight criterion calculated using the same volumetric composition procedure, thereby improving the comparability of specimens with different R_c values.

Before mixing, the soil matrix was air-dried, gently crushed, and sieved, and then adjusted to the target water content of 18%. The rock particles were cleaned before use and pre-conditioned to a moisture content of 1.2%. This treatment was adopted to minimize uncontrolled water transfer from the soil matrix to the rock phase during mixing. The soil matrix and rock particles were then mixed in a mechanical mixer in two stages. First, dry mixing was conducted for about 2–3 min. After that, water was added according to the designed target water content, and wet mixing was continued for about 3–5 min to improve the uniformity of the reconstructed mixture.

The mixed material was placed into the mould by layered compaction [28]. Before formal specimen preparation, a preliminary compaction calibration was conducted under the same mould size and moisture-conditioning scheme in order to determine a stable compaction level for the reconstructed soil–rock mixture. Based on this calibration, all specimens were prepared using the same preset layered compaction procedure. Each specimen was divided into six portions of equal wet mass and formed in six layers, corresponding to a nominal layer thickness of about 16.7 mm. This layered arrangement was adopted to maintain adequate placement control while reducing excessive artificial layering relative to a thinner multi-layer preparation scheme. During placement, the rock particles assigned to each layer were spread over the layer area before compaction, and obvious local concentration, wall-side accumulation, and one-sided clustering were avoided as far as practicable. This placement control was particularly important for the 40% and 60% rock-content groups, in which the limited specimen volume made the spatial arrangement of individual rock particles more influential. After each layer was placed, the material was compacted using the same tamping procedure and the same compaction level determined from the preliminary calibration. Before placing the next layer, the surface of the compacted layer was disturbed to a depth of about 2–3 mm in order to improve the interlayer connection and reduce interlayer discontinuity. After moulding, the top surface of the specimen was carefully trimmed to achieve the required specimen height of 100 mm and satisfactory end-face flatness. The moulded specimen was then checked for dimensional integrity before subsequent sealing and moisture equilibration. This procedure was adopted to ensure that all specimens were prepared under the same compaction-controlled initial state before saturation and isotropic consolidation. The finished specimen was wrapped with plastic film, sealed, and left to stand for 24 h to reduce moisture migration and to allow a more uniform redistribution of water inside the specimen.

To ensure specimen quality and preparation repeatability, dimensional and state-control checks were performed after moulding and before saturation. The finished specimens were accepted only when the diameter and height were within 50.0 ± 0.2 mm and 100.0 ± 0.5 mm, respectively. The initial water-content deviation was controlled within ±0.5%, and the dry unit weight deviation was controlled within ±0.3 kN/m³ for specimens prepared at the same target ${\mathrm{R}}_{\mathrm{c}}$. In addition, two representative loading groups were selected for parallel specimen preparation and repeatability assessment. For these parallel tests, the measured shear modulus under the same strain level showed deviations generally smaller than 5%, and the corresponding damping ratio showed deviations smaller than 8%. Therefore, the adopted reconstruction and layered compaction procedure was considered sufficiently stable for the present comparative study on the effects of rock block content and confining pressure. For the high-rock-content groups, the results are interpreted within this controlled small-scale reconstruction procedure, where the arrangement of individual rock particles may have a greater influence than in larger specimens. To illustrate the specimen groups prepared for the cyclic triaxial tests, the reconstructed soil–rock mixture specimens with different rock block contents are shown in Figure 2.

2.3. Cyclic Triaxial Test Scheme

2.3.1. Testing System and Test Procedure

The cyclic triaxial tests were carried out using a GDS Enterprise Level Dynamic Triaxial Testing System (ELDYN; GDS Instruments Ltd., Hook, Hampshire, United Kingdom). The system was controlled using the GDSLAB platform (GDSLAB v2026, GDS Instruments Ltd., Hook, Hampshire, United Kingdom). The system is a load-frame-based dynamic triaxial apparatus equipped with a beam-mounted electro-mechanical actuator and controlled through the GDSLAB platform v2024. According to the equipment specification, the system has an axial displacement resolution of 1 μm, standard axial load ranges of 5 kN and 10 kN, and a pressure range of 1 MPa, and is compatible with several specimen sizes. The apparatus can be operated under either load control or strain control and was used in this study to conduct strain-controlled undrained cyclic triaxial tests under controlled confining-pressure conditions. In this study, the device was used to perform undrained cyclic triaxial tests on reconstructed soil–rock mixture specimens. The cyclic triaxial testing system and the installed specimen used in this study are shown as annotated photographs in Figure 3 and Figure 4, respectively.

All specimens were tested under an undrained multistage strain-controlled loading mode. The cyclic loading was applied in a strain-controlled sinusoidal form with a fully reversed axial strain path. Before formal cyclic loading, each specimen was first subjected to saturation and isotropic consolidation. Prior to specimen installation, the porous stones, filter papers, and drainage lines were fully saturated with de-aired water. After the specimen was mounted in the triaxial cell, an initial confining pressure of approximately 20 kPa was applied to ensure proper contact between the specimen and the rubber membrane. Vacuum-assisted saturation was then conducted at a vacuum degree of about −70 to −80 kPa for approximately 30 min.

After vacuum saturation, staged back-pressure saturation was performed. De-aired water was introduced from the bottom of the specimen, while the confining pressure and back pressure were increased simultaneously by 50 kPa at each stage, maintaining an effective confining pressure of about 20–30 kPa. The back pressure was gradually increased to 300–400 kPa. At the end of each stage, the Skempton pore pressure coefficient $B$ was checked, and saturation was considered complete when $B \ge 0.95$. After saturation, isotropic consolidation was conducted under the target confining pressure. The consolidation time was not less than 6 h, and cyclic loading was started after the volume change had become basically stable. To ensure comparability between different tests, the same saturation criterion, consolidation procedure, loading waveform, and stage sequence were adopted throughout the experimental programme. The subsequent analysis was conducted within a total-stress cyclic-response framework, focusing on hysteresis-loop morphology, secant shear modulus, damping ratio, and normalized dynamic-parameter evolution.

2.3.2. Cyclic Loading Programme

To investigate the cyclic response of soil–rock mixtures under different stress conditions, three confining pressures were selected, namely 100 kPa, 200 kPa, and 400 kPa. Under each confining pressure, cyclic loading was applied in a multistage strain-controlled manner. The axial strain amplitudes were set as 0.01%, 0.05%, 0.10%, 0.20%, and 1.00%. This loading sequence was arranged from low strain level to high strain level so that the evolution of cyclic response could be observed progressively with increasing strain amplitude [29]. For each strain level, three loading cycles were applied, and a recovery interval of 20 min was introduced between two adjacent loading stages. In the subsequent analysis, the third cycle at each strain level was selected for hysteresis-loop construction and dynamic-parameter determination. This selection was adopted to maintain a consistent evaluation procedure for all specimens and to reduce the influence of the immediate response adjustment that may occur after the strain amplitude is changed. Therefore, the modulus and damping parameters obtained from this cycle were used to represent the stage-level cyclic response under the corresponding strain amplitude.

The loading frequency was fixed at 0.5 Hz throughout the tests. This frequency was selected to provide a stable strain-controlled sinusoidal loading condition for the saturated reconstructed SRM specimens and to obtain well-defined hysteresis loops for modulus and damping evaluation. Because the testing matrix already included five rock block contents, three confining pressures, and five strain amplitudes, the frequency was kept constant so that the comparative analysis could focus on material composition, stress condition, and strain level. The reported dynamic parameters therefore correspond to the selected 0.5 Hz loading condition; frequency-dependent behaviour was not separated in the present testing programme. During the cyclic loading process, the axial stress–strain response of the specimen was continuously recorded for subsequent construction of hysteresis loops and determination of dynamic parameters. In the present study, each hysteresis loop was constructed from one complete strain-controlled loading cycle in the deviator-stress–axial-strain plane. The multistage loading strategy was designed to describe the progressive cyclic response of each specimen as the imposed strain demand increased. Because all strain levels were applied sequentially to the same specimen after saturation and isotropic consolidation, the response at a given stage reflects not only the current strain amplitude, but also the fabric condition developed during the preceding loading stages. In this sense, the measured modulus and damping parameters represent the evolution of a specimen along a prescribed increasing-strain path. This procedure reduces specimen-to-specimen variability and allows the effects of rock block content and confining pressure to be compared under the same loading sequence, recovery interval, and cycle-selection method. The 20 min recovery interval was introduced to reduce short-term carryover between adjacent stages, but the multistage results are still interpreted as path-dependent staged cyclic responses rather than independent responses of virgin specimens at each strain level.

3. Determination of Dynamic Parameters

3.1. Hysteresis Loop

The hysteresis loop in this study was established directly from the cyclic stress–strain response obtained during the undrained cyclic triaxial tests. Under fully reversed strain-controlled sinusoidal loading, the axial strain of the specimen varies periodically with time, and the corresponding stress response during unloading does not fully coincide with that during loading. As a result, a closed stress–strain loop is formed in each loading cycle. In this study, the hysteresis loop was constructed using the cyclic deviator stress and the corresponding axial strain, so that the cyclic mechanical behaviour of the reconstructed soil–rock mixture could be represented in a direct manner. In the present study, the hysteresis loop at each strain level was constructed from the third loading cycle of the corresponding multistage loading step, following the same cycle-selection procedure used throughout the test programme.

The cyclic deviator stress is defined as

$$q ={ \sigma }_1-{\sigma }_3$$

(2)

where $q$ is the deviator stress, ${\sigma }_1$ is the axial stress, and ${\sigma }_3$ is the confining pressure. The corresponding axial strain ${\epsilon }_a$ was taken as the horizontal coordinate of the loop. Therefore, each hysteresis loop in this study represents the cyclic relationship between $q$ and ${\epsilon }_a$.

For a given confining pressure, rock block content, and strain amplitude, the hysteresis loop provides the most direct description of the cyclic response of the soil–rock mixture. The inclination of the loop reflects the stiffness level of the specimen, whereas the enclosed area represents the energy dissipated within one loading cycle [30]. This relation can be expressed as

$$W =\int q\hspace{0.17em}d{\epsilon }_a$$

(3)

where $W$ is the energy dissipated in one cycle, and the closed integral denotes the area enclosed by the hysteresis loop. For soil–rock mixtures, the loop morphology is influenced not only by the deformation of the soil matrix, but also by rock block interaction, soil–rock interfacial sliding, and internal structural rearrangement. Therefore, the hysteresis loop is not only a direct expression of cyclic stress–strain behaviour, but also the fundamental basis for the subsequent determination of shear modulus and damping ratio (Figure 5).

3.2. Shear Modulus

The shear modulus in this study was determined from the hysteresis loop obtained under cyclic loading. For each prescribed strain level, the secant slope between the two end points of the third-cycle hysteresis loop was used to define the dynamic axial deformation modulus $E_d$ [31]. The dynamic axial deformation modulus was calculated as

$$E_d={\displaystyle \frac{q_{\mathit{max}}-q_{\mathit{min}}}{{\epsilon }_{a,\mathit{max}}-{\epsilon }_{a,\min }}}$$

(4)

where $q_{\mathit{max}}$ and $q_{\mathit{min}}$ are the maximum and minimum deviator stresses within one loading cycle, and ${\epsilon }_{a,\mathit{max}}$ and ${\epsilon }_{a,\mathit{min}}$ are the corresponding maximum and minimum axial strains. Under the undrained loading condition adopted in this study, the specimen was treated as approximately incompressible and Poisson’s ratio was taken as $\nu = 0.5$. Accordingly, the dynamic shear modulus $G$ was obtained from

$$G={\displaystyle \frac{E_d}{2(1+\nu )}}$$

(5)

which can be simplified to

$$G ={\displaystyle \frac{E_d}{3}}$$

(6)

The secant modulus obtained in this manner was used as the representative cycle-level stiffness parameter at the corresponding strain amplitude. A schematic illustration of the determination of secant shear modulus from the hysteresis loop is shown in Figure 6.

3.3. Normalized Shear Modulus

In this study, $G_0$ denotes the extrapolated initial shear modulus obtained from the reciprocal modulus–strain fitting relationship and is used as the normalization reference throughout the paper. The symbol $G_{\max }$ is reserved only when referring to terminology used in previous studies.

Because the minimum strain level adopted in cyclic triaxial testing is still higher than the true very-small-strain range, the first measured modulus cannot be directly regarded as the theoretical initial shear modulus of the specimen. If the modulus obtained at the minimum loading strain is directly used as the normalization reference, the true initial stiffness of the material may be underestimated, which may further affect the comparison of modulus-reduction behaviour among different specimen groups. Therefore, an extrapolation-based normalization method was adopted in this study [32].

In the present study, the initial shear modulus $G_0$ was not taken directly from the first measured loading stage. Instead, it was estimated by extending the fitted backbone relationship of the shear modulus degradation curve toward the zero-strain limit. Following the commonly used hyperbolic representation, the reciprocal of the dynamic shear modulus was expressed as

$${\displaystyle \frac{1}{G}}=a +b{\gamma }_a$$

(7)

where $G$ is the secant shear modulus obtained at a given loading stage, ${\gamma }_a$ is the corresponding shear strain amplitude, and $a$ and $b$ are fitting parameters. When the shear strain tends to zero, the theoretical initial shear modulus can be obtained as

$$G_0={\displaystyle \frac{1}{a}}$$

(8)

Accordingly, $G_0$ in this study represents the extrapolated initial shear modulus rather than the largest directly measured modulus within the staged cyclic loading sequence.

Based on the above definition, the normalized shear modulus ratio was expressed as

$${\displaystyle \frac{G}{G_0}}$$

(9)

where $G/G_0$ reflects the retained proportion of shear stiffness at a given loading stage relative to the extrapolated initial shear modulus. A value close to 1 indicates that the material still preserves a stiffness level close to the theoretical initial state, whereas a lower value indicates more pronounced stiffness degradation under cyclic loading.

For the reconstructed soil–rock mixtures investigated in this study, the adoption of extrapolated normalization is important because it reduces the influence of the lower testing strain limit and provides a more physically meaningful basis for comparing stiffness degradation among specimens with different rock block contents and confining pressures. It also allows the subsequent modulus-reduction curves to be interpreted more consistently from the viewpoint of strain-dependent nonlinearity. Therefore, the $G/G_0$ obtained by extrapolation-based normalization was adopted in this study as the main index for comparing the modulus degradation characteristics of different test groups.

In addition, the reference shear strain ${\gamma }_r$ was defined as the shear strain corresponding to $G/G_0 = 0.5$. This parameter was used subsequently to describe the relative position of each reduction curve and to support the empirical fitting of normalized modulus evolution.

3.4. Damping Ratio and Normalized Damping Ratio

The damping ratio in this study was determined from the geometric characteristics of the hysteresis loop obtained during cyclic loading. Under cyclic stress–strain response, the loading and unloading paths do not coincide completely, and a closed hysteresis loop is formed in each cycle. The enclosed area of this loop represents the energy dissipated by the specimen during one loading cycle, whereas the corresponding elastic strain energy can be represented by the reference triangular area. Therefore, the damping ratio can be used to characterize the energy dissipation capacity of the reconstructed soil–rock mixture under cyclic loading.

In this study, the damping ratio $\lambda$ was calculated as

$$\lambda ={\displaystyle \frac{\mathrm{\Delta }W}{4\pi W_s}}$$

(10)

where $\mathrm{\Delta }W$ is the enclosed area of the hysteresis loop, representing the dissipated energy within one loading cycle, and $W_s$ is the corresponding elastic strain energy represented by the reference area at the same loading level. For a given loading cycle, a larger loop area indicates stronger hysteretic dissipation and therefore a higher damping ratio [33].

In addition to the original damping ratio $\lambda$, a normalized damping ratio ${\lambda }_{\mathit{nor}}$ was introduced to describe the relative development of damping within each specimen group. It was defined as

$${\lambda }_{\mathit{nor}}={\displaystyle \frac{\lambda -{\lambda }_{\mathit{min}}}{{\lambda }_{\mathit{max}}-{\lambda }_{\mathit{min}}}}$$

(11)

where ${\lambda }_{\mathit{min}}$ and ${\lambda }_{\mathit{max}}$ denote the lower- and upper-damping-ratio bounds of the corresponding specimen group within the tested strain range, respectively. These two bounds were determined from the measured strain-dependent damping-ratio evolution of each group. Therefore, ${\lambda }_{\mathit{nor}}$ represents the degree to which damping development is mobilized within the damping range of a given specimen group.

It should be noted that ${\lambda }_{\mathit{nor}}$ is not intended to replace the original damping ratio $\lambda$, nor is it used as an absolute damping index for cross-study comparison. The absolute damping level is evaluated using $\lambda$, whereas ${\lambda }_{\mathit{nor}}$ is used as a supplementary relative index to examine the evolution path of damping with increasing strain. This treatment allows damping development to be discussed together with the normalized shear modulus ratio $G/G_0$ within a consistent strain-dependent parameter framework, while preserving the original damping ratio as the primary physical measure of hysteretic energy dissipation (Figure 7).

3.5. Empirical Fitting Models for Normalized Dynamic Parameters

To quantitatively describe the strain-dependent evolution of the normalized dynamic parameters, empirical fitting models reported in previous studies were adopted in this study [34]. The normalized shear modulus ratio $G/G_0$ was first fitted using a modified hyperbolic relationship to determine the characteristic shear strain ${\gamma }_r$ and the curvature parameter $\alpha$. The normalized damping ratio ${\lambda }_{\mathit{nor}}$ was then fitted using a corresponding growth-type function expressed in the same normalized strain coordinate ${\gamma }_a/{\gamma }_r$. In this sequential fitting strategy, ${\gamma }_r$ and $\alpha$ were inherited from the modulus-reduction fitting and were not re-estimated during damping-ratio fitting. Therefore, the damping fitting should be understood as a modulus-informed constrained fitting procedure, rather than as an independent multi-parameter damping fit. This treatment was adopted to examine whether stiffness degradation and damping development could be described within a common strain-dependent parameter framework.

3.5.1. Fitting Model for Normalized Shear Modulus Ratio

The normalized shear modulus ratio was fitted using a modified hyperbolic model [35]:

$${\displaystyle \frac{G}{G_0}}={\displaystyle \frac{1}{1+{\left(\frac{{\gamma }_a}{{\gamma }_r}\right)}^{\alpha }}}$$

(12)

where $G$ is the secant shear modulus at the corresponding loading stage, $G_0$ is the extrapolated initial shear modulus, ${\gamma }_a$ is the shear strain amplitude, ${\gamma }_r$ is the reference shear strain, and $\alpha$ is the fitting parameter controlling the curvature of the reduction relationship.

To facilitate subsequent interpretation, the normalized shear strain amplitude may also be written as

$${\gamma }_{\mathit{nor}}={\displaystyle \frac{{\gamma }_a}{{\gamma }_r}}$$

(13)

In this form, Equation (12) can be rewritten as

$${\displaystyle \frac{G}{G_0}}={\displaystyle \frac{1}{1+{\gamma }_{\mathit{nor}}^{\alpha }}}$$

(14)

This expression indicates that the normalized modulus-reduction behaviour can be described by the relative strain coordinate ${\gamma }_{\mathit{nor}}$, which makes it possible to compare specimen groups with different material compositions and stress conditions within a unified framework.

3.5.2. Fitting Model for Normalized Damping Ratio

To describe the evolution of energy dissipation in a form consistent with the normalized modulus model, the normalized damping ratio was fitted using the following empirical relationship [36]:

$${\lambda }_{\mathit{nor}} = {\left[{\displaystyle \frac{({\gamma }_a/{\gamma }_r{)}^{\alpha }}{1 + ({\gamma }_a/{\gamma }_r{)}^{\alpha }}}\right]}^{\beta }$$

(15)

or, equivalently,

$${\lambda }_{\mathit{nor}}={\left[{\displaystyle \frac{{\gamma }_{\mathit{nor}}^{\alpha }}{1+{\gamma }_{\mathit{nor}}^{\alpha }}}\right]}^{\beta }$$

(16)

where ${\lambda }_{\mathit{nor}}$ is the normalized damping ratio, and $\beta$ is the only damping-specific fitting parameter in this constrained damping model. The parameters ${\gamma }_r$ and $\alpha$ were taken from the fitted normalized shear modulus relationship and were kept unchanged during damping-ratio fitting. Therefore, $\beta$ does not represent a fully independent damping material parameter; rather, it describes the remaining damping-growth curvature after the characteristic-strain position and basic curve shape have been constrained by the modulus-reduction relationship. Because ${\lambda }_{\mathit{nor}}$ is based on group-specific damping bounds, the fitted relationship is used to describe the relative damping-development path of each specimen group, while the original damping ratio $\lambda$ is retained for interpreting absolute energy dissipation.

The reason for adopting Equations (15) and (16) is that the normalized-damping-ratio data obtained in this study exhibit a typical monotonic S-shaped growth pattern, increasing from low values at small strain to values approaching 1.0 at large strain. More importantly, by expressing ${\lambda }_{\mathit{nor}}$ in terms of ${\gamma }_a/{\gamma }_r$, the fitting of damping evolution can be directly linked to the fitted modulus-reduction relationship. This enables stiffness degradation and damping development to be discussed within a common parameter framework, which is physically meaningful for soil–rock mixtures under cyclic loading [37].

3.5.3. Parameter Estimation and Fitting Evaluation

For each specimen group, the fitting procedure was performed sequentially. First, the measured $G/G_0$ data were fitted using Equation (12) to determine ${\gamma }_r$ and $\alpha$. These two parameters define the characteristic-strain position and the basic nonlinear shape of the modulus-reduction relationship. Then, the same ${\gamma }_r$ and $\alpha$ were fixed in Equation (15), and only the parameter $\beta$ was fitted using the normalized-damping-ratio data. Therefore, the reported $\beta$ values are conditional on the modulus-derived ${\gamma }_r$ and $\alpha$. In this framework, $\beta$ should be interpreted as a damping-specific adjustment parameter within a shared stiffness–damping evolution coordinate, rather than as an independently obtained damping parameter.

The fitting quality was evaluated using the coefficient of determination ($R^2$), root mean square error (RMSE), mean absolute error (MAE), and mean absolute percentage error (MAPE), defined as

$$R^2=1 -{\displaystyle \frac{{\displaystyle {\sum }_{i=1}^n}(y_i-{\widehat{y}}_i{)}^2}{{\displaystyle {\sum }_{i=1}^n}(y_i-\stackrel{-}{y}{)}^2}}$$

(17)

$$\mathit{RMSE} =\sqrt{{\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n}(y_i-{\widehat{y}}_i{)}^2}$$

(18)

$$\mathit{MAE} ={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n}|y_i-{\widehat{y}}_i|$$

(19)

$$\mathit{MAPE} ={\displaystyle \frac{100\%}{n}}{\displaystyle \sum _{i=1}^n}\left|{\displaystyle \frac{y_i-{\widehat{y}}_i}{y_i}}\right|$$

(20)

where $y_i$ is the measured value, ${\widehat{y}}_i$ is the fitted value, $\stackrel{-}{y}$ is the mean of the measured values, and $n$ is the number of data points used in fitting. These indices were adopted to evaluate the applicability and accuracy of the selected empirical models for the normalized dynamic parameters of reconstructed soil–rock mixtures.

In this study, the validation of the empirical fitting framework was conducted from four aspects: goodness of fit, parameter stability, physical interpretability, and applicability range. First, the fitting accuracy was evaluated using $R^2$, RMSE, MAE, and MAPE, and the corresponding results are summarized in Tables 2, 3, 5 and 6. These statistical indices were used to verify whether the selected empirical relationships could reproduce the measured normalized modulus and normalized damping data with acceptable accuracy. Second, parameter stability was examined by comparing the fitted parameters among different rock block contents and confining pressures. A stable range of $\alpha$ indicates that the basic nonlinear curve form remains comparable among different specimen groups, whereas the systematic variation of ${\gamma }_r$ reflects the migration of the characteristic-strain position. For the normalized damping model, $\beta$ was interpreted as a damping-specific adjustment parameter within the modulus-informed constrained fitting framework. Third, the physical meaning of the fitted parameters was interpreted together with the observed hysteresis response, shear modulus degradation, and damping evolution, rather than only as mathematical curve-fitting coefficients. Finally, the applicability of the fitting framework is limited to the tested strain range, material type, particle-size range, confining pressures, and multistage loading path adopted in this study. Therefore, the fitted relationships are used here as empirical descriptions of the present reconstructed SRMs, rather than as general constitutive equations for all soil–rock mixtures.

4. Results and Discussion

4.1. Hysteresis-Loop Characteristics

The hysteresis loops obtained from the cyclic triaxial tests exhibit clear nonlinear and structure-dependent features under different rock block contents and confining pressures. In general, all loops are closed and obliquely distributed, indicating that the reconstructed soil–rock mixtures maintained continuous cyclic resistance within the tested strain range and did not undergo abrupt structural collapse during a single loading stage. However, the loop morphology is distinctly different from the regular elliptical pattern commonly observed in relatively uniform geomaterials. Most loops in this study display a spindle-like to waist-constricted shape, with a relatively narrow middle part and outward expansion at both ends [38]. This feature suggests that the stiffness of the material was not constant within one loading cycle, but evolved continuously with loading and unloading, reflecting the combined effects of soil matrix deformation, soil–rock interface adjustment, and rock block interaction.

At different strain levels, the hysteresis morphology shows a progressive evolution from a narrow and steep loop to a wider and fuller loop. At small strain amplitudes, the loops are relatively slender, the enclosed area is limited, and the loading and unloading branches remain close to each other. This indicates that the material response is still dominated by recoverable deformation, while irreversible energy dissipation is weak. As the strain amplitude increases, the hysteresis loops expand markedly, the spacing between the loading and unloading branches becomes larger, and the enclosed area increases significantly. Meanwhile, the loop inclination decreases gradually, indicating a reduction in secant stiffness. At larger strain amplitudes, the outer loops become much more open and nonlinear, which means that the contribution of inter-particle sliding, local rearrangement, and interface friction becomes increasingly important. Therefore, the effect of strain level in the present study is not merely reflected by loop enlargement, but also by a clear transition in deformation mechanism from relatively coordinated deformation to structure-adjustment-controlled cyclic deformation.

The influence of rock block content on hysteresis morphology is evident, but it is not expressed in a simple monotonic manner. More importantly, the effect of rock content is strongly dependent on confining pressure. Under the low confining pressure of 100 kPa, the hysteresis loops corresponding to different rock contents remain relatively similar in overall outline. Although some differences in loop width and local curvature can still be observed, the influence of rock blocks is not fully mobilized, and the cyclic response is still mainly governed by the soil matrix. This implies that, under weak confinement, rock blocks mainly act as dispersed inclusions embedded in the matrix, rather than as an effective stress-transferring skeleton.

When the confining pressure increases to 200 kPa, the effect of rock block content becomes more distinguishable. The loops of specimens with higher rock contents show steeper and more concentrated central branches, while the outer portions of the loops become more pronounced. This indicates that the internal load-transfer path begins to change with increasing rock block content. In this stage, the material no longer behaves as a matrix-dominated medium alone. Instead, the interaction between the matrix and the rock blocks becomes more active, and the cyclic response begins to reflect stronger structural constraint and frictional dissipation. Under 400 kPa, this tendency becomes much more obvious. Especially for the specimens with medium-to-high rock contents, the hysteresis loops become distinctly fuller and more upright, the stress range expands significantly, and the waist-constriction feature becomes more pronounced. These observations indicate that the role of rock blocks is substantially activated under high confinement, and that the cyclic resistance of the material is increasingly supported by enhanced block contact and matrix–block cooperation.

The effect of confining pressure is systematic and more direct. For a given rock block content, increasing confining pressure changes the hysteresis loops from relatively flat and elongated shapes to steeper, fuller, and more stress-expanded shapes. This means that higher confining pressure not only increases the stress level reached during cyclic loading, but also modifies the mode of deformation. Under low confining pressure, the specimens show relatively loose and flat loops, suggesting limited inter-particle constraint and relatively weak mobilization of structural resistance. As the confining pressure increases, particle contact becomes tighter, the matrix is compressed more effectively, and the contact state between rock blocks and soil matrix becomes more stable. As a result, the loops become more compact in geometric arrangement but larger in stress amplitude, which reflects the simultaneous enhancement of cyclic strength and hysteretic dissipation. In other words, higher confining pressure does not simply strengthen the material; it also promotes a more efficient internal stress-transfer mechanism.

A notable feature of the present results is that the combined influence of rock block content and confining pressure is clearly non-additive. The role of rock content is weak under low confinement, but becomes increasingly significant as confinement rises. This means that the effect of rock blocks should be understood as a confinement-activated structural effect rather than an isolated material-composition effect. In the present reconstructed soil–rock mixtures, the rock blocks do not contribute equally under all stress conditions. Their mechanical function depends strongly on whether the surrounding matrix and inter-particle contacts are sufficiently confined to mobilize block interaction. This point is important because it explains why the differences among rock contents are relatively limited at 100 kPa, but become much more evident at 400 kPa.

Another important observation is that the hysteresis loops in this study are characterized by obvious non-elliptic distortion and local curvature variation, especially under high confining pressure and high rock content. This reflects the intrinsic heterogeneity of the tested material. Unlike conventional soils, the reconstructed soil–rock mixture does not respond as a uniform continuum during cyclic loading. Instead, different components participate sequentially in deformation. At small-to-medium strain, deformation is mainly accommodated by the soil matrix and local closure of internal voids. With further loading, soil–rock interface sliding and contact adjustment become more significant. At larger strain levels, the interaction among rock blocks becomes increasingly involved in carrying and redistributing stress [39]. Therefore, the hysteresis loops observed in this study record a multistage deformation process within a single cycle, which is a distinctive feature of the tested material system.

Overall, the hysteresis-loop analysis demonstrates that the cyclic response of the reconstructed soil–rock mixtures is governed by a coupled evolution of strain amplitude, confining pressure, and rock block content. Strain amplitude controls the degree of nonlinearity and energy dissipation, confining pressure determines the level of structural constraint, and rock block content regulates the internal load-sharing mode. The resulting hysteresis morphology therefore reflects not only the mechanical state of the material, but also the progressive activation of its internal structure. This provides the direct experimental basis for the subsequent analysis of shear modulus degradation and damping evolution. The hysteresis loops of the reconstructed soil–rock mixture specimens under different rock block contents and confining pressures are shown in Figure 8.

4.2. Shear Modulus Degradation

The results show that all reconstructed soil–rock mixture specimens exhibited a pronounced shear modulus degradation behaviour under cyclic loading. In all test groups, the shear modulus $G$ decreased continuously with increasing shear strain amplitude ${\gamma }_a$, indicating a progressive loss of stiffness during cyclic deformation. Within the strain range considered in this study, as ${\gamma }_a$ increased from 0.015% to 1.50%, the shear modulus of all specimens decreased markedly. The reduction was strongly nonlinear rather than uniform over the whole strain range [40].

From the overall numerical distribution, the shear modulus remained relatively high at small strain levels but decreased substantially as the strain level increased. At ${\gamma }_a = 0.015\%$, the measured $G$ values ranged from 35.835 to 158.871 MPa. When ${\gamma }_a$ increased to 0.075%, the modulus range decreased to 24.264–114.826 MPa. At ${\gamma }_a = 0.15\%$, it further decreased to 17.702–85.691 MPa. When ${\gamma }_a = 0.30\%$, the corresponding range was 11.380–56.556 MPa. At the largest strain level, ${\gamma }_a = 1.50\%$, the shear modulus dropped to only 3.296–12.854 MPa. These results indicate that the internal stiffness reserve of the material was continuously consumed as cyclic strain increased, and that the degradation of shear modulus developed in a clear stage-wise manner.

This degradation trend can be observed more clearly from representative specimen groups. For the pure-soil group ($R_c = 0\%$), under a confining pressure of 100 kPa, the shear modulus decreased from 35.835 MPa to 24.297, 17.772, 11.882, and 3.407 MPa as ${\gamma }_a$ increased. Under 200 kPa, the corresponding values decreased from 54.964 MPa to 41.453, 31.344, 21.235, and 5.810 MPa. Under 400 kPa, the modulus decreased from 76.136 MPa to 61.991, 47.536, 35.652, and 11.092 MPa. A similar reduction pattern was observed in the high-rock-content group ($R_c = 60\%$). Under 100 kPa, the modulus decreased from 56.030 MPa to 36.650, 25.708, 15.095, and 3.296 MPa. Under 200 kPa, it decreased from 90.296 MPa to 61.101, 41.597, 26.277, and 6.087 MPa. Under 400 kPa, it decreased from 158.871 MPa to 114.826, 85.691, 56.556, and 12.854 MPa. These results confirm that the continuous reduction in shear modulus with increasing strain remained a stable and consistent feature regardless of changes in material composition and confining pressure.

In terms of degradation magnitude, the total reduction in shear modulus from the smallest to the largest strain level generally reached about 85%–94% for all test groups. For example, the total reductions for the $R_c = 0\%$ group under 100, 200, and 400 kPa were approximately 90.5%, 89.4%, and 85.4%, respectively. For the $R_c = 60\%$ group, the corresponding values were about 94.1%, 93.3%, and 91.9%. This indicates that although different specimen groups had different initial stiffness levels, all of them experienced substantial weakening of the internal load-bearing structure at large strain levels and gradually shifted from a relatively stable initial state to a more strongly nonlinear response state.

The effect of rock block content on the absolute shear modulus level was evident, especially at small and medium strain levels. Under the same confining pressure, $G$ generally increased with increasing $R_c$, and this tendency became more pronounced as the confining pressure increased. At ${\gamma }_a = 0.015\%$, the modulus under 100 kPa increased from 35.835 MPa at $R_c = 0\%$, while under 400 kPa it increased from 76.136 MPa to 158.871 MPa over the same rock-content range. This indicates that increasing rock block content significantly enhanced the initial stiffness level of the reconstructed soil–rock mixtures. However, at the largest strain level under low confining pressure, the differences among different rock-content groups became much smaller. For example, under 100 kPa and ${\gamma }_a = 1.50\%$, the modulus values of all groups converged into a narrow range of 3.296–3.614 MPa. This suggests that the stiffness advantage associated with higher rock content was gradually weakened as cyclic deformation increased.

The influence of confining pressure was also systematic [41]. For a fixed rock block content, increasing confining pressure always resulted in a higher shear modulus over the whole strain range. For example, for the $R_c = 40\%$ group, the modulus at ${\gamma }_a = 0.015\%$ increased from 46.275 MPa at 100 kPa to 77.990 MPa at 200 kPa and further to 126.019 MPa at 400 kPa. Even at ${\gamma }_a = 1.5\%$, the corresponding values were still 3.614, 6.782, and 11.676 MPa, respectively. This indicates that confining pressure not only increased the initial stiffness but also helped maintain a higher residual stiffness at large strain. These results demonstrate that the shear modulus degradation of reconstructed soil–rock mixtures is characterized by strong nonlinearity, distinct stage dependence, and clear sensitivity to both rock block content and confining pressure.

It should be emphasized that the modulus degradation described here corresponds to the adopted multistage loading path. Therefore, the sharp reduction in stiffness at the medium- and large-strain stages should be understood as the combined outcome of the imposed strain amplitude and the structural state inherited from the previous loading stages. The increasing strain amplitude provides the direct driving condition for stiffness reduction, whereas earlier stages may contribute to fabric readjustment, local contact modification, and irreversible deformation accumulation. Since the same loading sequence was used for all specimen groups, the resulting curves remain suitable for comparative evaluation of rock block content and confining pressure, but they should not be interpreted as single-stage virgin degradation curves at each strain amplitude.

Overall, the reconstructed soil–rock mixtures investigated in this study exhibited a clear and continuous shear modulus degradation behaviour within the studied strain range. The shear modulus $G$ decreased continuously with increasing ${\gamma }_a$. The modulus remained relatively high at small strain levels, decreased steadily at intermediate strain levels, and dropped much more significantly at large strain levels. At the same time, higher rock block content and higher confining pressure generally produced higher absolute modulus levels, although the differences among different groups became less pronounced under large-strain and low-confining-pressure conditions. These results demonstrate that the stiffness loss of soil–rock mixtures under cyclic loading is characterized by strong nonlinearity and a clear stage-wise evolution.

To quantitatively illustrate the shear modulus degradation behaviour of the reconstructed soil–rock mixtures, Figure 9 presents the variation in shear modulus with shear strain amplitude, and

Table 1. Extrapolated initial shear modulus $G_0$ and reference shear strain ${\gamma }_r$ of reconstructed soil–rock mixtures under different rock block contents and confining pressures.

Confining Pressure/kPa	Rock Block Content/%	Extrapolated Initial Shear Modulus $G_0$/MPa	The Reference Shear Strain
100	0	39.164	0.128
100	10	39.962	0.119
100	20	42.487	0.115
100	40	52.384	0.109
100	60	65.918	0.093
200	0	58.102	0.179
200	10	60.284	0.156
200	20	65.731	0.137
200	40	84.772	0.126
200	60	101.456	0.108
400	0	79.226	0.244
400	10	95.184	0.206
400	20	107.028	0.181
400	40	134.206	0.168
400	60	171.382	0.150

summarizes the extrapolated initial shear modulus $G_0$ and the reference shear strain ${\gamma }_r$ under different rock block contents and confining pressures.

4.3. Normalized Shear Modulus Curves

4.3.1. Overall Distribution Characteristics of Normalized Shear Modulus Ratio

Figure 10 presents the overall distribution of the normalized shear modulus ratio of the reconstructed soil–rock mixtures under different rock block contents and confining pressures. A total of 15 specimen groups were included, corresponding to five rock block contents and three confining pressures, and each group was evaluated at five shear strain amplitudes. The normalized data points show a clear strain-dependent reduction trend and provide the basis for comparing the relative stiffness degradation behaviour among different test groups.

The distribution of the normalized data points shows a clear strain-dependent degradation tendency. On the whole, the values of ${G/G}_0$ decrease progressively with increasing shear strain amplitude, indicating that the proportion of retained stiffness relative to the initial state becomes smaller during cyclic loading. Although the absolute shear modulus level differed significantly among the test groups, the normalization treatment placed all data points into the same relative coordinate system, which made the intrinsic degradation pattern more directly observable.

From the overall distribution pattern, the normalized data points exhibit a distinctly nonlinear reduction form rather than a linear trend. At small strain levels, most data points remain in the upper part of the plot, indicating that the specimens still preserve a large proportion of their initial stiffness. As the strain amplitude increases into the intermediate range, the normalized modulus values decrease more rapidly, and the distribution band shifts downward more significantly. At large strain levels, the point distribution becomes concentrated in the lower part of the plot, while the overall rate of reduction tends to weaken compared with the intermediate-strain stage. This indicates that the stiffness degradation of the reconstructed soil–rock mixtures is characterized by a clear stage-dependent nonlinear evolution.

Another important feature is that, although the normalized data points do not fully overlap, they are distributed within a relatively consistent degradation band. The differences among groups are mainly reflected in the vertical position of the points and in the extent of scatter at the same strain level, rather than in any fundamental change in the overall reduction form. This suggests that rock block content and confining pressure still affect the detailed reduction path of normalized stiffness, but the normalized modulus ratio of all specimen groups follows a similar basic degradation trend after normalization.

Overall, the normalized shear modulus ratio data demonstrate that the extrapolated initial shear modulus $G_0$ determined provides an effective reference for the comparison of relative stiffness degradation. After normalization by Equation (8), all specimen groups show a continuous nonlinear reduction trend of ${G/G}_0$ with increasing shear strain amplitude, and the data points are distributed within a broadly similar degradation range. This provides the basis for the subsequent discussion on the effects of rock block content and confining pressure on the normalized modulus-reduction behaviour. To further illustrate the overall distribution characteristics of the normalized shear modulus ratio, Figure 10 presents the ${G/G}_0$ data points of the reconstructed soil–rock mixtures under different rock block contents and confining pressures.

4.3.2. Effect of Rock Block Content on Normalized Shear Modulus Ratio

Using the constrained fitting model presented in Section 3.5, the effect of rock block content on the normalized shear modulus ratio was evaluated by comparing the fitted ${\lambda }_{\mathrm{nor}}$ evolution relationships under the same confining pressure. For each confining pressure level, namely 100, 200, and 400 kPa, five rock block contents ($R_c = 0\%, 10\%, 20\%, 40\%, \mathrm{and} 60\%$) were considered in order to examine how the relative stiffness-retention behaviour changes with increasing rock inclusion.

The results show that rock block content has a clear effect on the normalized modulus-reduction behaviour. Under the same confining pressure, the $G/G_0$ relationship generally shifts downward and toward the lower-strain side as $R_c$ increases. This indicates that, after normalization by the extrapolated initial shear modulus $G_0$, the higher-rock-content groups tend to retain a smaller proportion of their initial stiffness at the same shear strain amplitude. This tendency is already identifiable at 100 kPa, becomes more distinct at 200 kPa, and is the most pronounced at 400 kPa, especially in the intermediate strain range where the reduction in normalized modulus is most sensitive. Therefore, the influence of rock block content on $G/G_0$ is not only significant, but also strongly dependent on confining pressure.

A more direct quantitative indication is provided by the variation in the reference shear strain ${\gamma }_r$. Under 100 kPa, ${\gamma }_r$ decreases continuously from 0.128 to 0.120, 0.116, 0.109, and 0.092 as $R_c$ increases from 0% to 60%. Under 200 kPa, the corresponding values decrease from 0.178 to 0.155, 0.137, 0.127, and 0.109. Under 400 kPa, ${\gamma }_r$ further decreases from 0.242 to 0.205, 0.179, 0.167, and 0.150. From $R_c = 0\%$ to $R_c = 60\%$, the total reduction in ${\gamma }_r$ is approximately 27.8%, 38.6%, and 38.0% under 100, 200, and 400 kPa, respectively. These results demonstrate that increasing rock block content systematically shifts the characteristic degradation position of the $G/G_0$ relationship toward a lower strain range. In other words, the higher-rock-content groups reach the state of $G/G_0 = 0.5$ at a smaller shear strain amplitude than the lower-rock-content groups.

In contrast, the fitted shape parameter $\alpha$ remains within a relatively narrow range for all test groups. At 100 kPa, $\alpha$ varies from 0.996 to 1.015; at 200 kPa, it ranges from 1.047 to 1.097; and at 400 kPa, it ranges from 1.026 to 1.094. The overall fluctuation interval is only 0.996–1.097, indicating that the basic nonlinear form of the normalized modulus-reduction relationship remains similar among different rock block contents. Therefore, the principal effect of $R_c$ is not to fundamentally change the reduction pattern itself, but to alter the characteristic-strain position and the relative degradation rate of the $G/G_0$ relationship.

The fitting accuracy further confirms the reliability of this interpretation. For all 15 test groups, the coefficient of determination $R^2$ ranges from 0.9977 to 0.9998, the RMSE ranges from 0.0037 to 0.0134, the MAE ranges from 0.0029 to 0.0122, and the MAPE ranges from 0.73% to 4.86%. These consistently high fitting qualities indicate that the modified hyperbolic model provides an excellent representation of the present normalized shear modulus data. More importantly, the systematic decrease in ${\gamma }_r$ with increasing $R_c$ is observed together with very stable goodness-of-fit, which means that the rock block effect identified here is not caused by accidental scatter of individual data points, but reflects a robust trend in the normalized stiffness-degradation behaviour.

It should be emphasized that this result does not contradict the findings obtained from the absolute shear modulus in Section 4.2. In Section 4.2, increasing rock block content generally leads to higher absolute shear modulus values. However, the present section discusses $G/G_0$, which represents the stiffness retained relative to the extrapolated initial stiffness of the same specimen group. Since $G_0$ itself increases significantly with rock block content, normalization removes the direct advantage associated with a higher initial stiffness level and instead reveals the difference in the relative degradation path. For example, under 100 kPa, $G_0$ increases from 39.164 MPa at $R_c = 0\%$ to 65.918 MPa at $R_c = 60\%$; under 200 kPa, it increases from 58.102 to 101.456 MPa; and under 400 kPa, it increases from 79.226 to 171.382 MPa. Therefore, the lower $G/G_0$ values and smaller ${\gamma }_r$ values observed in the higher-rock-content groups do not mean that these specimens are absolutely weaker. Rather, they indicate that, relative to their own higher initial stiffness basis, their stiffness-retention ratio decreases more rapidly with increasing strain.

From a mechanical viewpoint, this behaviour reflects the structural role of rock blocks in the reconstructed soil–rock mixtures. As rock block content increases, the internal material system becomes more heterogeneous, and the cyclic response is increasingly influenced by block–matrix interaction, interfacial sliding, and contact rearrangement. Under low confining pressure, the rock blocks mainly act as dispersed inclusions in the soil matrix, and their influence on the normalized degradation path remains relatively limited. Under higher confining pressure, however, the contacts among rock blocks and between rock blocks and the soil matrix are mobilized more effectively, so the structural effect associated with rock block content becomes much stronger. As a result, the influence of $R_c$ on the normalized modulus-reduction behaviour becomes more evident with increasing confinement.

Overall, increasing rock block content does not change the basic nonlinear form of the $G/G_0$ reduction relationship, but it significantly affects its relative position, characteristic-strain, and degradation rate. With increasing $R_c$, the normalized modulus relationship shifts toward lower $G/G_0$ values and lower characteristic strain levels, while this influence becomes progressively more pronounced as confining pressure increases. These results indicate that rock block content is an important factor controlling the relative stiffness degradation behaviour of reconstructed soil–rock mixtures under cyclic loading. To further evaluate the applicability of the modified hyperbolic model to the present normalized shear modulus data, the fitted $G/G_0$ curves for different rock block contents under each confining pressure are presented in Figure 11, while the corresponding fitting parameters and error indices are summarized in

Table 2. Fitting parameters and error indices of normalized shear modulus relationships of reconstructed soil–rock mixtures.

${\mathit{\sigma }}_{\mathbf{3}}$/kPa	${\mathit{R}}_{\mathit{c}}$/%	${\mathit{\gamma }}_{\mathit{r}}$	$\mathit{\alpha }$	${\mathit{R}}^{\mathbf{2}}$	RMSE	MAE	MAPE/%
100	0	0.128	1.010	0.9977	0.0134	0.0120	4.45
	10	0.120	1.015	0.9985	0.0110	0.0089	4.01
	20	0.116	1.014	0.9987	0.0100	0.0080	3.87
	40	0.109	0.996	0.9998	0.0037	0.0029	0.73
	60	0.092	0.997	0.9992	0.0078	0.0076	4.86
200	0	0.178	1.067	0.9994	0.0070	0.0057	1.99
	10	0.155	1.097	0.9993	0.0077	0.0061	3.30
	20	0.137	1.079	0.9985	0.0113	0.0095	4.30
	40	0.127	1.047	0.9988	0.0099	0.0079	3.37
	60	0.109	1.057	0.9998	0.0039	0.0032	1.08
400	0	0.242	1.035	0.9977	0.0134	0.0122	2.74
	10	0.205	1.026	0.9982	0.0118	0.0100	3.32
	20	0.179	1.092	0.9993	0.0075	0.0063	1.45
	40	0.167	1.094	0.9996	0.0059	0.0055	1.72
	60	0.150	1.063	0.9997	0.0054	0.0048	1.97

.

4.3.3. Effect of Confining Pressure on Normalized Shear Modulus Ratio

The effect of confining pressure on the normalized shear modulus ratio was examined by comparing the fitted $G/G_0$ relationships under the same rock block content. Different from Section 4.3.2, which focuses on the compositional effect of $R_c$, this section emphasizes the stress-state effect of confinement on the relative stiffness-retention path.

For all rock block contents, increasing confining pressure shifts the $G/G_0$ relationship upward and toward a higher strain range. This indicates that stronger confinement allows the specimen to retain a larger proportion of its initial stiffness at the same shear strain amplitude. The effect is weak at the smallest strain level, but becomes more evident in the intermediate strain range, where normalized modulus reduction develops most actively.

The variation in the reference shear strain ${\gamma }_r$ provides a clear quantitative indication of this delay effect. For $R_c=0\%$, ${\gamma }_r$ increases from 0.128 at 100 kPa to 0.178 at 200 kPa and 0.242 at 400 kPa. Similar increases are observed for all other rock-content groups, with total increases from 100 to 400 kPa of approximately 89.1%, 70.4%, 55.0%, 53.9%, and 62.4% for $R_c=0\%$, 10%, 20%, 40%, and 60%, respectively. These results show that stronger confinement systematically moves the characteristic degradation position toward a higher strain range.

The fitted shape parameter $\alpha$ remains within a relatively narrow range, indicating that confining pressure does not fundamentally change the basic nonlinear form of the normalized modulus-reduction curve. Instead, its main effect is to change the characteristic-strain position and delay the relative stiffness-degradation process. The fitting indices summarized in

Table 3. Fitting parameters and error indices of normalized shear modulus relationships of reconstructed soil–rock mixtures under different confining pressures.

${\mathit{R}}_{\mathit{c}}$/%	${\mathit{\sigma }}_{\mathbf{3}}$/kPa	${\mathit{\gamma }}_{\mathit{r}}$	$\mathit{\alpha }$	${\mathit{R}}^{\mathbf{2}}$	RMSE	MAE	MAPE/%
0	100	0.128	1.010	0.9977	0.0134	0.0120	4.45
	200	0.178	1.067	0.9994	0.0070	0.0057	1.99
	400	0.242	1.035	0.9977	0.0134	0.0122	2.74
10	100	0.120	1.015	0.9985	0.0110	0.0089	4.01
	200	0.155	1.097	0.9993	0.0077	0.0061	3.30
	400	0.205	1.026	0.9982	0.0118	0.0100	3.32
20	100	0.116	1.014	0.9987	0.0100	0.0080	3.87
	200	0.137	1.079	0.9985	0.0113	0.0095	4.30
	400	0.179	1.092	0.9993	0.0075	0.0063	1.45
40	100	0.109	0.996	0.9998	0.0037	0.0029	0.73
	200	0.127	1.047	0.9988	0.0099	0.0079	3.37
	400	0.167	1.094	0.9996	0.0059	0.0055	1.72
60	100	0.092	0.997	0.9992	0.0078	0.0076	4.86
	200	0.109	1.057	0.9998	0.0039	0.0032	1.08
	400	0.150	1.063	0.9997	0.0054	0.0048	1.97

further confirm that the modified hyperbolic model provides a stable representation of the normalized modulus relationships under different confining pressures.

Overall, confining pressure does not change the basic nonlinear form of the $G/G_0$ reduction relationship, but it significantly affects its relative position, characteristic-strain, and degradation rate. With increasing confining pressure, the normalized modulus relationship shifts toward higher $G/G_0$ values and higher characteristic strain levels, indicating that the relative stiffness degradation of the reconstructed soil–rock mixtures is delayed under stronger confinement. These results demonstrate that confining pressure is one of the key factors controlling the normalized stiffness-retention behaviour of soil–rock mixtures under cyclic loading. To further illustrate the effect of confining pressure on the normalized shear modulus ratio, Figure 12 presents the fitted $G/G_0$ relationships of reconstructed soil–rock mixtures under different confining pressures for each rock block content, and Table 3 summarizes the corresponding fitting parameters and error indices. These tabulated indices provide the basis for evaluating the fitting stability of the normalized shear modulus model under different confining pressures.

4.4. Damping Ratio Evolution

The results show that all reconstructed soil–rock mixture specimens exhibited a clear increase in damping ratio under cyclic loading. In all test groups, the damping ratio $\lambda$ increased continuously with increasing shear strain amplitude ${\gamma }_a$, indicating that the energy dissipation capacity of the material was progressively enhanced during cyclic deformation. Within the strain range considered in this study, as ${\gamma }_a$ increased from 0.015% to 1.50%, the damping ratio of all specimens rose markedly, and the increase was distinctly nonlinear rather than uniform over the whole strain range. On the whole, the damping-ratio data were distributed within a relatively narrow but clearly structured band, which indicates that the dissipation behaviour of the reconstructed soil–rock mixtures was controlled simultaneously by strain level, rock block content, and confining pressure. This overall feature is consistent with the general pattern reported for soil–rock mixtures in previous cyclic triaxial studies, in which the unnormalized damping ratio increases with shear strain and shows a zonal distribution as a whole.

From the overall numerical range, the damping ratio remained relatively low at the smallest strain level and increased progressively toward the large-strain stage. At the smallest loading level, the minimum damping ratios of all groups were distributed between 0.036 and 0.063, whereas the maximum damping ratios reached 0.195–0.268 at the largest loading stage. These results indicate that the cyclic response of the reconstructed soil–rock mixtures gradually shifted from a relatively low-dissipation state at small strain to a much more dissipative state at large strain. In other words, as the cyclic deformation developed, a larger portion of the external input energy was consumed by internal friction, interface adjustment, and irreversible structural rearrangement. The increase in damping ratio therefore provides direct evidence that the material response became progressively more hysteretic with strain growth.

The evolution of the damping ratio in the present study can be broadly divided into three successive stages. At the small-strain stage, the increase in $\lambda$ was relatively limited, and most data points were concentrated near the lower part of the plot. This indicates that the material response was still dominated mainly by recoverable deformation, while the amount of energy dissipated during each loading cycle remained relatively low. As the shear strain entered the intermediate range, the growth rate of $\lambda$ increased significantly, and the damping-ratio data shifted upward more rapidly. This stage represents the principal transition from a relatively stable deformation state to a more strongly nonlinear and dissipative response state. At larger strain levels, the damping ratio continued to increase, but the growth rate became less pronounced than that in the intermediate stage. This suggests that after considerable internal adjustment had already occurred, the material still dissipated more energy with further strain increase, but the rate of additional dissipation enhancement became relatively milder.

This stage-dependent increase can be observed more clearly from representative specimen groups. For the pure-soil group (R_c = 0%), the minimum damping ratio was 0.055, 0.050, and 0.047 under confining pressures of 100, 200, and 400 kPa, respectively, while the corresponding maximum damping ratios were 0.264, 0.268, and 0.252. For the high-rock-content group (R_c = 60%), the minimum damping ratio was 0.063, 0.043, and 0.038 under the same three confining pressures, whereas the corresponding maximum damping ratios were 0.223, 0.214, and 0.195. These results show that, regardless of the difference in material composition and confining pressure, the damping ratio always increased substantially from the beginning of cyclic loading to the final large-strain stage. However, the lower and upper bounds of damping evolution were not identical among the groups, which indicates that the dissipation response of the reconstructed soil–rock mixtures was not controlled by strain amplitude alone. Instead, it was significantly affected by the combined influence of internal composition and external confinement.

The effect of rock block content on the absolute damping-ratio level was evident, but it was not expressed as a simple monotonic increase or decrease over the entire strain range. Under the same confining pressure, the increase in rock block content changed both the lower bound and the upper bound of damping evolution. Under 100 kPa, ${\lambda }_{\min }$ changed from 0.055 at $R_c = 0\%$ to 0.046, 0.042, 0.059, and 0.063 as $R_c$ increased to 10%, 20%, 40%, and 60%, while ${\lambda }_{\max }$ changed from 0.264 to 0.247, 0.267, 0.235, and 0.223. Under 200 and 400 kPa, the reduction in ${\lambda }_{\max }$ with increasing rock block content became more evident overall. These results indicate that rock block content did not merely raise or lower the damping ratio in a uniform manner. Instead, it modified the dissipation path by altering the interaction among the soil matrix, rock blocks, and their interfaces. Therefore, the influence of rock block content on damping ratio should be understood as a structure-dependent effect rather than a simple compositional effect.

The influence of confining pressure was also systematic, although its expression was slightly different from that observed for shear modulus. For a fixed rock block content, increasing confining pressure generally caused the lower bound of damping evolution to decrease, and in most block-bearing groups it also reduced the upper bound of damping evolution. For example, for $R_c = 10\%$, ${\lambda }_{\min }$ decreased from 0.046 to 0.039 and 0.036 as confining pressure increased from 100 to 200 and 400 kPa, while ${\lambda }_{\max }$ decreased from 0.247 to 0.241 and 0.234. For $R_c = 60\%$, the corresponding values of ${\lambda }_{\min }$ were 0.063, 0.043, and 0.038, and those of ${\lambda }_{\max }$ were 0.223, 0.214, and 0.195. This indicates that higher confining pressure generally suppresses the absolute damping level of the reconstructed soil–rock mixtures, especially when the rock block content is high. At the same time, the damping-ratio data under higher confining pressure tend to be distributed within a narrower band, implying that stronger confinement restrains the freedom of local sliding and structural looseness, and thereby reduces the extent of absolute energy dissipation at the same strain level.

From a mechanical viewpoint, the increase in damping ratio with shear strain reflects the gradual activation of internal energy dissipation mechanisms during cyclic loading. At small strain levels, the deformation of the soil–rock mixture is still relatively coordinated, and the dominant response is recoverable deformation of the soil matrix together with limited local contact adjustment. As strain increases, the contribution of soil–rock interface sliding, particle friction, and internal structural rearrangement becomes increasingly important, which leads to much greater hysteretic dissipation and hence a higher damping ratio. The role of rock block content in this process is to modify the internal heterogeneity and contact complexity of the material, whereas the role of confining pressure is to regulate the stability and compactness of the internal contact system. Therefore, the observed damping-ratio evolution is the combined result of strain growth, structural composition, and stress constraint.

Overall, the reconstructed soil–rock mixtures investigated in this study exhibited a clear and continuous increase in damping ratio within the studied strain range. The damping ratio $\lambda$ increased continuously with increasing ${\gamma }_a$, with a relatively slow increase at small strain, a much more pronounced increase at intermediate strain, and a continued but moderated increase at large strain. At the same time, both rock block content and confining pressure had clear influences on the absolute damping level and the upper–lower bounds of damping evolution, although these influences were not expressed in a simple one-directional manner across all groups. These results indicate that the energy dissipation behaviour of soil–rock mixtures under cyclic loading is characterized by strong nonlinearity, clear stage dependence, and distinct sensitivity to both material composition and stress condition.

Similar to the modulus response, the damping evolution should also be interpreted within the same multistage loading path. The increase in damping ratio at later stages reflects both the higher imposed strain amplitude and the accumulated structural adjustment developed during preceding stages.

To quantitatively illustrate the evolution characteristics of damping ratio, Figure 13 presents the variation in $\lambda$ with shear strain amplitude, and

Table 4. Minimum and maximum damping ratios of reconstructed soil–rock mixtures under different rock block contents and confining pressures.

Confining Pressure/kPa	Rock Block Content/%	The Minimum Damping Ratio	The Maximum Damping Ratio
100	0	0.055	0.264
100	10	0.046	0.247
100	20	0.042	0.267
100	40	0.059	0.235
100	60	0.063	0.223
200	0	0.050	0.268
200	10	0.039	0.241
200	20	0.045	0.246
200	40	0.056	0.231
200	60	0.043	0.214
400	0	0.047	0.252
400	10	0.036	0.234
400	20	0.042	0.241
400	40	0.050	0.221
400	60	0.038	0.195

summarizes the corresponding minimum and maximum damping ratios under different rock block contents and confining pressures.

4.5. Normalized-Damping-Ratio Evolution

4.5.1. Overall Distribution Characteristics of Normalized Damping Ratio

Figure 14 shows the overall distribution of the normalized damping ratio of the reconstructed soil–rock mixtures under different rock block contents and confining pressures. In this section, the normalized damping ratio is used primarily to describe the relative evolution path of cyclic energy dissipation within the damping range of each specimen group, whereas the original damping ratio remains the basic physical measure for evaluating absolute hysteretic energy dissipation. Therefore, the normalized damping ratio is interpreted as a supplementary evolution index, and the discussion of between-group damping differences is considered together with the original damping-ratio results presented in Section 4.4. In general, all data points show a clear and continuous upward trend with increasing shear strain amplitude, indicating that the normalized damping ratio increases progressively as cyclic deformation develops. This overall pattern suggests that the energy dissipation capacity of the soil–rock mixture is gradually mobilized during the loading process, and that the normalized treatment can effectively highlight the relative evolution pattern of damping behaviour among different specimen groups. Although the absolute damping level varies among test conditions, the normalized damping ratio still exhibits a highly consistent distribution framework, which makes it suitable for comparative analysis of the effects of rock content and confining pressure [42].

From the global distribution of the data, the normalized damping ratio remains at a relatively low level in the initial strain stage and then increases steadily as the strain amplitude grows from 0.015% to 1.5%. For all specimen groups, the increase is monotonic, and no crossover or abnormal reversal is observed. This indicates that the normalized damping ratio preserves a stable evolutionary order after data transformation. In particular, the values at the smallest strain amplitude are generally concentrated in the lower interval, whereas those at the largest strain amplitude are clustered in the upper interval, demonstrating that the normalized damping ratio can clearly distinguish the progressive enhancement of hysteretic dissipation with strain. In other words, the overall distribution of ${\lambda }_{\mathit{nor}}$ is characterized by a typical strain-dependent upward migration from the lower-left region to the upper-right region of the plot.

At the same confining pressure, the distribution of the normalized damping ratio shows a clear dependence on rock content. For each strain level, the values generally increase with increasing rock content, and the ordering relationship remains stable as $R_c = 60\% > 40\% > 20\% > 10\% > 0\%$. This trend can be identified under all three confining pressure levels. For example, under ${\sigma }_{3 }= 100 \mathrm{kPa}$, the normalized damping ratio increases from 0.019 to 0.100 at ${\gamma }_a = 0.015\%$ when the rock content rises from 0% to 60%, and it further increases from 0.870 to 0.940 at ${\gamma }_a = 1.5\%$. A similar distribution law is also observed under ${\sigma }_3 = 200 \mathrm{kPa}$ and 400 kPa. This indicates that, after normalization, specimens with higher rock content still maintain a stronger relative damping evolution tendency throughout the loading process. The result implies that the introduction of more rock blocks not only changes the absolute damping level, but also modifies the internal evolution path of cyclic energy dissipation. From a structural point of view, increasing rock content enhances block–block and block–matrix interactions, which promote frictional sliding, local rearrangement, and interfacial dissipation under cyclic loading, and this effect is still clearly reflected after normalization.

At the same rock content, the normalized damping ratio generally decreases with increasing confining pressure, showing an overall order of $100 \mathrm{kPa} > 200 \mathrm{kPa} > 400 \mathrm{kPa}$. This trend is particularly clear in the low- and medium-strain range. For instance, when $R_c = 0\%$, the normalized damping ratio at ${\gamma }_a = 0.15\%$ decreases from 0.380 under 100 kPa to 0.340 under 200 kPa and then to 0.260 under 400 kPa. When $R_c = 60\%$, the corresponding values are 0.560, 0.536, and 0.465, respectively. These results indicate that a higher confining pressure suppresses the relative development of damping behaviour to a certain extent. Mechanically, this is because increasing confining pressure strengthens the contact constraint among soil particles and rock blocks, reduces the freedom of local slip and rotation, and delays the rapid release of dissipative deformation. As a consequence, although damping still increases with strain under higher confining pressure, its normalized evolution intensity remains comparatively lower than that under lower confining pressure.

In terms of strain-stage characteristics, the overall distribution of the normalized damping ratio shows an evident nonlinear evolution pattern. In the low-strain range, the increase in ${\lambda }_{\mathit{nor}}$ is relatively gentle, indicating that the specimen is still mainly controlled by limited interparticle adjustment and small-scale interface friction. As the strain enters the medium range, the growth of the normalized damping ratio becomes more pronounced, suggesting that the internal structure begins to undergo more active rearrangement and that frictional dissipation becomes increasingly significant. In the high-strain range, ${\lambda }_{\mathit{nor}}$ continues to rise, but the curve growth tends to become more compact for some groups because the normalized parameter gradually approaches its upper interval. Therefore, the overall distribution can be understood as a progressive process from slow activation to rapid development and then to relative stabilization. This staged characteristic is important because it indicates that the normalized damping ratio not only reflects the final dissipation level, but also captures the full-path evolution feature of cyclic energy consumption.

It should also be noted that the separation among the different rock-content groups remains clear throughout the entire strain domain, while the spacing among the different confining-pressure groups is more obvious in the low-to-medium-strain region and becomes slightly compressed in the high-strain region. This means that rock content exerts a stronger control on the relative distribution pattern of normalized damping ratio, whereas confining pressure mainly modifies the rate and extent of its evolution. Therefore, from the perspective of overall distribution, the normalized damping ratio of the reconstructed soil–rock mixtures is governed by three main characteristics: first, a monotonic increase with strain amplitude; second, a positive correlation with rock content; and third, a negative correlation with confining pressure.

Overall, the normalized damping ratio exhibits a well-ordered and physically interpretable distribution law. The parameter successfully preserves the essential trend of damping evolution after normalization and enables direct comparison among different specimen groups. The results show that the cyclic dissipation behaviour of reconstructed soil–rock mixtures is not random after normalization, but still follows a distinct structural rule controlled jointly by strain amplitude, rock content, and confining pressure. This provides a clear basis for the subsequent discussion of the influence of rock content and confining pressure on the normalized damping ratio in more detail.

4.5.2. Effect of Rock Block Content on Normalized Damping Ratio

Using the fitting model presented in Section 3.5, the effect of rock block content on the normalized damping ratio was evaluated by comparing the fitted ${\lambda }_{\mathit{nor}}$ evolution relationships under the same confining pressure. For each confining pressure level, namely 100, 200, and 400 kPa, five rock block contents ($R_c = 0\%, 10\%, 20\%, 40\%, \mathrm{and} 60\%$) were considered in order to clarify how the relative energy-dissipation evolution changes with increasing rock inclusion.

The results show that rock block content has a clear and systematic influence on the normalized-damping-ratio evolution. Under the same confining pressure, the ${\lambda }_{\mathit{nor}} - {\gamma }_a$ relationship generally shifts upward and toward the lower-strain side as $R_c$ increases. This indicates that, after normalization, the higher-rock-content groups tend to develop a larger proportion of their total damping evolution at the same shear strain amplitude, and they reach the main dissipation-development stage earlier than the lower-rock-content groups. This tendency can be identified under all three confining pressures, and it is especially clear in the low- and medium-strain ranges, where the separation among different rock-content groups is most obvious.

A direct comparison of the normalized data points further confirms this trend. Under 100 kPa, the normalized damping ratio at ${\gamma }_a = 0.015\%$ increases from 0.019 for $R_c = 0\%$ to 0.100 for $R_c = 60\%$, while at ${\gamma }_a = 0.15\%$ it increases from 0.380 to 0.560, and at ${\gamma }_a = 1.5\%$ it increases from 0.870 to 0.940. Under 200 kPa, the corresponding values increase from 0.016 to 0.080 at ${\gamma }_a = 0.015\%$, from 0.340 to 0.536 at ${\gamma }_a = 0.15\%$, and from 0.850 to 0.930 at ${\gamma }_a = 1.5\%$. Under 400 kPa, the same monotonic trend is also maintained, with ${\lambda }_{\mathit{nor}}$ increasing from 0.006 to 0.060 at ${\gamma }_a = 0.015\%$, from 0.260 to 0.465 at ${\gamma }_a = 0.15\%$, and from 0.810 to 0.920 at ${\gamma }_a = 1.5\%$ as $R_c$ increases from 0% to 60%. These results demonstrate that increasing rock block content consistently accelerates the relative development of normalized damping behaviour over the entire strain domain.

The variation in the characteristic shear strain ${\gamma }_r$ provides more direct quantitative evidence for this effect. Under 100 kPa, ${\gamma }_r$ decreases continuously from 0.128 to 0.120, 0.116, 0.109, and 0.092 as $R_c$ increases from 0% to 60%. Under 200 kPa, the corresponding values decrease from 0.178 to 0.155, 0.137, 0.127, and 0.109. Under 400 kPa, ${\gamma }_r$ further decreases from 0.242 to 0.205, 0.179, 0.167, and 0.150. From $R_c = 0\%$ to $R_c = 60\%$, the total reduction in ${\gamma }_r$ is about 27.8%, 38.6%, and 38.0% under 100, 200, and 400 kPa, respectively. This means that the characteristic position of the normalized-damping-ratio evolution shifts systematically toward the lower-strain side with increasing rock block content. In other words, the higher-rock-content groups require a smaller shear strain amplitude to reach the same characteristic development level of normalized damping evolution.

Compared with the clear reduction in ${\gamma }_r$, the fitted parameter $\alpha$ varies only within a relatively narrow interval. At 100 kPa, $\alpha$ ranges from 0.996 to 1.015. At 200 kPa, it ranges from 1.047 to 1.097. At 400 kPa, it ranges from 1.026 to 1.094. This indicates that the basic nonlinear form of the normalized-damping-ratio evolution remains generally stable for different rock block contents. Therefore, the primary effect of $R_c$ is not to fundamentally change the overall shape of the growth relationship, but to shift its characteristic-strain position and alter the rate at which damping evolution develops.

The parameter $\beta$, however, shows a clearer systematic variation with rock block content. Under 100 kPa, $\beta$ decreases from 1.613 for $R_c = 0\%$ to 1.151 for $R_c = 60\%$. Under 200 kPa, it decreases from 1.433 to 1.144. Under 400 kPa, it decreases from 1.354 to 1.100. The total reductions are approximately 28.7%, 20.1%, and 18.8%, respectively. Because ${\gamma }_r$ and $\alpha$ were fixed from the normalized modulus fitting, this decrease in $\beta$ indicates that, after the modulus-derived characteristic-strain position and curve shape are accounted for, the higher-rock-content groups still show an earlier and more active relative damping-development tendency. Thus, $\beta$ reflects the damping-specific residual adjustment within the shared stiffness–damping evolution framework, rather than an independently fitted damping parameter.

The fitting accuracy further confirms the reliability of this interpretation. For all 15 test groups, the coefficient of determination $R^2$ ranges from 0.9892 to 0.9996, the RMSE ranges from 0.0058 to 0.0292, the MAE ranges from 0.0049 to 0.0265, and the MAPE ranges from 0.92% to 9.91%. These fitting results indicate that the adopted model provides a good representation of the present normalized-damping-ratio data. More importantly, the systematic decrease in ${\gamma }_r$ and $\beta$ with increasing $R_c$ is observed together with consistently high fitting quality, which means that the identified rock block effect is not caused by accidental scatter in a few data points, but reflects a stable feature of the normalized dissipation evolution of the reconstructed soil–rock mixtures.

It should be noted that this result is not exactly the same as the trend observed for the unnormalized damping ratio in Section 4.4. In the absolute-damping-ratio analysis, the effect of rock block content is partly influenced by the initial damping level and the upper damping bound of each specimen group, and its variation is therefore not always strictly monotonic. After normalization, the focus shifts from the absolute damping level to the relative mobilization process within each specimen group. Therefore, the normalized damping ratio should be interpreted together with the original damping-ratio results, rather than as a substitute for absolute damping comparison. Under this unified framework, the higher-rock-content groups exhibit a more consistent and earlier development of damping evolution. Therefore, the normalized damping ratio reveals more directly the structural effect of rock block content on the relative mobilization of cyclic energy dissipation.

From a mechanical viewpoint, this behaviour can be attributed to the increasing structural complexity caused by the addition of rock blocks. As $R_c$ increases, the internal material system becomes more heterogeneous, and the cyclic response is increasingly governed by block–block contact, block–matrix interaction, interfacial sliding, and local rearrangement. Under cyclic loading, these mechanisms can be activated at relatively smaller strain levels in the block-rich groups, causing the normalized damping ratio to increase more rapidly in the early and intermediate stages. In this sense, increasing rock block content does not simply increase the dissipation level in an absolute sense; more importantly, it advances the onset and development of the relative dissipation process within the loading path.

Overall, increasing rock block content does not fundamentally alter the basic nonlinear form of the normalized-damping-ratio evolution relationship, but it significantly affects its characteristic-strain position, growth rate, and relative development path. With increasing $R_c$, the ${\lambda }_{\mathit{nor}}$ relationship shifts toward higher normalized-damping-ratio values and lower characteristic-strain levels, indicating that the relative dissipation evolution of the reconstructed soil–rock mixtures is promoted by the presence of more rock blocks. These results demonstrate that rock block content is one of the key factors controlling the normalized-damping-ratio evolution of soil–rock mixtures under cyclic loading. To further illustrate the effect of rock block content on the normalized damping ratio, the fitted ${\lambda }_{\mathit{nor}}$ relationships under different rock block contents are presented in Figure 15, and the corresponding fitting parameters and error indices are summarized in

Table 5. Fitting parameters and error indices of normalized-damping-ratio relationships of reconstructed soil–rock mixtures with different rock block contents under different confining pressures.

${\mathit{\sigma }}_{\mathbf{3}}$/kPa	${\mathit{R}}_{\mathit{c}}$/%	${\mathit{\gamma }}_{\mathit{r}}$	$\mathit{\alpha }$	$\mathit{\beta }$	${\mathit{R}}^{\mathbf{2}}$	RMSE	MAE	MAPE/%
100	0	0.128	1.010	1.613	0.9948	0.0202	0.0168	6.20
	10	0.120	1.015	1.503	0.9924	0.0243	0.0215	8.43
	20	0.116	1.014	1.378	0.9989	0.0096	0.0087	3.83
	40	0.109	0.996	1.235	0.9996	0.0058	0.0049	0.92
	60	0.092	0.997	1.151	0.9974	0.0140	0.0115	5.08
200	0	0.178	1.067	1.433	0.9956	0.0189	0.0157	6.81
	10	0.155	1.097	1.376	0.9892	0.0292	0.0265	9.91
	20	0.137	1.079	1.340	0.9978	0.0135	0.0118	7.29
	40	0.127	1.047	1.185	0.9977	0.0135	0.0100	2.26
	60	0.109	1.057	1.144	0.9960	0.0176	0.0160	6.12
400	0	0.242	1.035	1.354	0.9961	0.0178	0.0151	8.90
	10	0.205	1.026	1.234	0.9967	0.0172	0.0148	7.13
	20	0.179	1.092	1.206	0.9964	0.0170	0.0124	5.45
	40	0.167	1.094	1.165	0.9948	0.0202	0.0165	5.01
	60	0.150	1.063	1.100	0.9938	0.0214	0.0188	7.07

.

4.5.3. Effect of Confining Pressure on Normalized Damping Ratio

The effect of confining pressure on the normalized damping ratio was evaluated by comparing the fitted ${\lambda }_{\mathit{nor}}$ relationships under the same rock block content. In contrast to Section 4.5.2, which focuses on the acceleration of damping development caused by increasing rock block content, this section highlights the delay effect produced by stronger confinement.

For all rock-content groups, increasing confining pressure shifts the ${\lambda }_{\mathit{nor}}$ relationship downward and toward a higher strain range. This indicates that, after normalization, specimens under higher confinement develop a smaller proportion of their damping evolution at the same shear strain amplitude, especially in the low- and intermediate-strain ranges.

This tendency is reflected by the systematic increase in ${\gamma }_r$. For the $R_c=0\%$ group, ${\gamma }_r$ increases from 0.128 at 100 kPa to 0.178 at 200 kPa and 0.242 at 400 kPa. Similar increases are observed for the other rock-content groups. From 100 to 400 kPa, the increase in ${\gamma }_r$ reaches about 89.1%, 70.4%, 55.0%, 53.9%, and 62.4% for $R_c=0\%$, 10%, 20%, 40%, and 60%, respectively. These results indicate that stronger confinement delays the relative development of normalized-damping evolution.

The parameters $\alpha$ and $\beta$ should be interpreted within the constrained fitting framework described in Section 3.5. The relatively stable range of $\alpha$ suggests that the basic curve form remains comparable among different confining pressures, while the variation in $\beta$ reflects the remaining damping-growth adjustment after ${\gamma }_r$ and $\alpha$ have been inherited from the modulus-reduction fitting. Therefore, the main effect of confinement is expressed through the increase in ${\gamma }_r$, whereas $\beta$ provides supplementary information on the curve-growth shape.

Mechanically, stronger confinement stabilizes soil–rock contacts, restricts local sliding and rotation, and postpones the mobilization of relative dissipative deformation. Therefore, the normalized damping evolution under higher confinement develops more slowly, even though the material still exhibits increasing damping with strain. Figure 16 presents the fitted ${\lambda }_{\mathit{nor}}$ relationships under different confining pressures, and

Table 6. Fitting parameters and error indices of normalized-damping-ratio relationships of reconstructed soil–rock mixtures under different confining pressures.

${\mathit{R}}_{\mathit{c}}$/%	${\mathit{\sigma }}_{\mathbf{3}}$/kPa	${\mathit{\gamma }}_{\mathit{r}}$	$\mathit{\alpha }$	$\mathit{\beta }$	${\mathit{R}}^{\mathbf{2}}$	RMSE	MAE	MAPE/%
0	100	0.128	1.010	1.613	0.9948	0.0202	0.0168	6.20
	200	0.178	1.067	1.433	0.9956	0.0189	0.0157	6.81
	400	0.242	1.035	1.354	0.9961	0.0178	0.0151	8.90
10	100	0.120	1.015	1.503	0.9924	0.0243	0.0215	8.43
	200	0.155	1.097	1.376	0.9892	0.0292	0.0265	9.91
	400	0.205	1.026	1.234	0.9967	0.0172	0.0148	7.13
20	100	0.116	1.014	1.378	0.9989	0.0096	0.0087	3.83
	200	0.137	1.079	1.340	0.9978	0.0135	0.0118	7.29
	400	0.179	1.092	1.206	0.9964	0.0170	0.0124	5.45
40	100	0.109	0.996	1.235	0.9996	0.0058	0.0049	0.92
	200	0.127	1.047	1.185	0.9977	0.0135	0.0100	2.26
	400	0.167	1.094	1.165	0.9948	0.0202	0.0165	5.01
60	100	0.092	0.997	1.151	0.9974	0.0140	0.0115	5.08
	200	0.109	1.057	1.144	0.9960	0.0176	0.0160	6.12
	400	0.150	1.063	1.100	0.9938	0.0214	0.0188	7.07

summarizes the corresponding fitting parameters and error indices.

4.6. Parameter-Based Interpretation of Cyclic Evolution in Reconstructed Soil–Rock Mixtures

4.6.1. Integrated Interpretation of Stiffness Degradation and Damping Development

The cyclic dynamic response of reconstructed soil–rock mixtures reflects the combined contribution of soil matrix deformation, rock block interaction, contact rearrangement, and soil–rock interfacial sliding. In the preceding sections, these processes were expressed through several mutually related parameters, including hysteresis-loop morphology, secant shear modulus, normalized shear modulus ratio, damping ratio, normalized damping ratio, and fitting parameters. Taken together, these results indicate that stiffness degradation and damping development are two coupled manifestations of the strain-dependent evolution of the internal block-in-matrix structure.

For clarity, the three interpretive concepts used in this section are defined as follows. The “strain-dependent staged response” refers to the progressive change from small-strain coordinated deformation to intermediate-strain structural adjustment and large-strain dissipation-dominated response. The “characteristic-strain migration” refers to the systematic shift in the reference shear strain ${\gamma }_r$ with rock block content and confining pressure. The “confinement-related block–matrix coupling” refers to the stress-dependent mobilization of contact interaction between the soil matrix and rock blocks under cyclic loading. These concepts are not introduced as independent constitutive laws, but as interpretive descriptions supported by the measured hysteresis loops, modulus degradation curves, damping-ratio evolution, normalized parameter trends, and fitted characteristic strains.

At small strain levels, the cyclic response is mainly reflected by narrow hysteresis loops, relatively high secant stiffness, and low damping ratio, suggesting that deformation is still largely coordinated by the soil matrix and limited local contact adjustment. As strain amplitude increases, loop expansion, accelerated modulus reduction, and increased damping indicate more active block–matrix interaction, interfacial sliding, and frictional dissipation. At larger strain levels, the reduced stiffness and enlarged hysteretic area show that irreversible structural rearrangement and energy dissipation become more pronounced. This staged response provides a consistent explanation for the observed transition from stiffness-dominated behaviour to dissipation-enhanced cyclic behaviour.

The combined analysis of modulus and damping also helps explain why the two parameters should not be interpreted separately. The reduction in shear modulus represents the gradual loss of effective stiffness and load-transfer efficiency, whereas the increase in damping ratio reflects the growing contribution of frictional dissipation and irreversible adjustment. Therefore, hysteresis-loop expansion, modulus degradation, normalized stiffness reduction, damping-ratio growth, and normalized-damping development can be interpreted as interconnected indicators of the same strain-dependent cyclic response.

The above interpretation also indicates that the influence of rock blocks is conditional on both strain level and confining pressure. Under relatively low confinement, the block phase mainly behaves as dispersed inclusions embedded in the soil matrix, and its contribution to the overall cyclic response remains limited. Under stronger confinement, the contact state among rock blocks and between rock blocks and the soil matrix becomes more effectively mobilized, leading to higher stiffness, clearer differences among rock-content groups, and more distinct damping evolution. Therefore, the role of rock block content should be interpreted together with the stress condition rather than as an independent compositional factor.

This parameter-based interpretation also explains the coexistence of two response features observed in the present results. Higher rock block content and stronger confinement generally increase the absolute stiffness level of the material. However, higher rock block content tends to advance the normalized evolution of stiffness degradation and damping development, whereas stronger confinement delays these normalized responses. This means that the cyclic behaviour of reconstructed SRMs should be evaluated based on both the absolute response level and the relative evolution path. In this sense, hysteresis-loop evolution, modulus degradation, normalized stiffness reduction, damping growth, and fitting-parameter variation provide complementary evidence for understanding the strain-dependent cyclic response of the tested block-in-matrix material.

4.6.2. Strain-Dependent Staged Response

The test results show that the cyclic response of reconstructed soil–rock mixtures can be interpreted as a strain-dependent staged response. This staged interpretation is based on the observed widening of hysteresis loops in Figure 8, the continuous reduction in secant shear modulus in Figure 9 and Table 1, and the increase in damping ratio shown in Figure 13 and Table 4. The stages described below are used as descriptive response regimes inferred from hysteresis morphology, modulus degradation, and damping evolution, rather than as fixed transition boundaries. Similar strain-dependent degradation and dissipation trends have been reported for SRMs under cyclic loading, whereas the present results further organize this behaviour through the coupled evolution of hysteresis morphology, shear modulus degradation, and damping development.

At the first stage, namely the small-strain coordination stage, the material response is dominated mainly by coordinated deformation of the soil matrix and relatively limited interface adjustment. In this stage, the hysteresis loops remain narrow and steep, the enclosed loop area is small, the secant shear modulus is still maintained at a comparatively high level, and the damping ratio remains low. Mechanically, most soil–rock interfaces are still in a constrained state, and the internal force-transfer network has not yet undergone substantial disturbance. The block phase mainly provides geometric constraint and local stiffening, while the matrix still plays the primary role in deformation accommodation. As a result, the material behaves as a relatively stable composite system with strong stiffness retention and weak hysteretic dissipation. The essential feature of this stage is that cyclic loading mainly induces recoverable deformation, whereas interface friction, local sliding, and irreversible rearrangement are still limited.

As the strain amplitude increases, the material enters the second stage, termed here the intermediate-strain structural-adjustment stage. This stage represents the principal transition zone of the cyclic response and corresponds to the most active evolution of internal interfaces. In this regime, the hysteresis loops expand more rapidly, the distance between loading and unloading branches becomes more obvious, the reduction in secant shear modulus accelerates, and the damping ratio rises more significantly. These changes indicate that the deformation mechanism is no longer controlled predominantly by matrix coordination alone. Instead, soil–rock interfaces begin to participate more actively through local sliding, contact readjustment, frictional dissipation, and redistribution of internal stresses. At the macroscopic scale, this stage is characterized by the most sensitive transition of dynamic properties. In particular, the normalized shear modulus ratio decreases more rapidly and the normalized damping ratio develops more obviously, showing that the material is moving from a relatively stable load-transfer state toward a more pronounced nonlinear evolution state. Therefore, this stage can be regarded as the critical interval in which block–matrix interaction, contact readjustment, and frictional dissipation become more evident in the macroscopic cyclic response.

With further increase in strain amplitude, the reconstructed soil–rock mixtures gradually enter the third stage, referred to here as the large-strain structure-rearrangement and dissipation-dominated stage. In this stage, the hysteresis loops become much fuller and more open, the secant shear modulus is reduced to a relatively low level, and the damping ratio reaches a high range. The internal interface system no longer serves only as a local-adjustment mechanism; instead, it becomes the main medium through which irreversible structural rearrangement, energy dissipation, and residual resistance development are expressed. The matrix continuity is weakened, local block–matrix coordination becomes more unstable, and block-related contact interaction and rearrangement play a more important role in governing the overall response. Although the growth rate of some normalized parameters may gradually weaken compared with the transition stage, this does not imply recovery of structural stability. Rather, it indicates that the material has already entered a high-dissipation regime in which a large proportion of the initial stiffness reserve has been consumed. The dominant feature of this stage is therefore not further initiation of nonlinearity, but the continued manifestation of a strongly evolved nonlinear state.

An important implication of this staged response pattern is that the cyclic behaviour of reconstructed SRMs should not be described only by a single monotonic trend, even though the overall variations in modulus and damping are continuous. The present results show that the continuous curves contain a clear transition sequence: matrix-dominated coordination at small strain, structural adjustment at intermediate strain, and dissipation-dominated response at large strain. This interpretation links the observed hysteresis morphology, stiffness degradation, and damping development to the progressive change in the internal block-in-matrix response.

This staged response pattern also explains why different specimen groups, despite having different rock block contents and confining pressures, still exhibit broadly similar normalized evolution forms. The detailed response position and evolution rate vary among groups, but the fundamental strain-dependent path remains similar: increasing strain first weakens recoverable deformation, then enhances contact readjustment and frictional dissipation, and finally drives the material into a strongly evolved high-dissipation state. Therefore, the staged interpretation is used here as an integrated reading of the measured dynamic parameters rather than as a rigid transition law.

4.6.3. Characteristic-Strain Migration and Dual-Path Response

The fitted normalized modulus and damping relationships indicate that the cyclic response of reconstructed SRMs can be interpreted through characteristic-strain migration and a dual-path response pattern. This interpretation is supported by the fitted ${\gamma }_r$ values summarized in Table 2, Table 3, Table 5 and Table 6, where ${\gamma }_r$ decreases with increasing rock block content but increases with increasing confining pressure. The reference shear strain ${\mathrm{\gamma }}_{\mathrm{r}}$, derived from the normalized modulus relationship, provides a useful index for describing the strain position of relative stiffness degradation. In the present study, its systematic variation with rock block content and confining pressure further indicates that the transition position of relative dynamic response is not fixed, but depends on material composition and stress condition.

From this viewpoint, the cyclic response of reconstructed soil–rock mixtures is controlled by two interrelated but non-identical response paths. The first path is the absolute enhancement path, which describes the increase in the absolute stiffness level of the material. Along this path, increasing rock block content and increasing confining pressure both generally lead to higher shear modulus and higher extrapolated initial shear modulus, especially at small and intermediate strain levels. This indicates that both factors strengthen the internal load-bearing structure, improve the effectiveness of the block-in-matrix skeleton, and enhance the initial resistance of the material against cyclic deformation. In other words, this path characterizes how strong the material is in an absolute mechanical sense.

The second path is the relative evolution path, which describes the onset and progression of normalized degradation and normalized dissipation relative to the initial state of each specimen group. Along this path, the controlling factors no longer act in the same direction. Increasing rock block content shifts the normalized-shear–modulus-reduction relationship and the normalized-damping–growth relationship toward the lower-strain side, whereas increasing confining pressure shifts them toward the higher-strain side. This means that a block-rich mixture may possess a higher absolute stiffness level, but still enter the stage of relative stiffness loss and relative dissipation development at a smaller characteristic strain. By contrast, a specimen under stronger confinement not only starts from a higher stiffness level, but also requires a larger strain to reach the same relative degree of degradation or dissipation. Therefore, the cyclic dynamic response of reconstructed soil–rock mixtures cannot be adequately interpreted by a single monotonic criterion such as “higher stiffness implies slower evolution.” Instead, the material exhibits a dual-path response in which absolute resistance and relative evolution are coupled but directionally distinguishable.

The most direct evidence for this response pattern is provided by the systematic migration of the characteristic strain ${\gamma }_r$. Under a given confining pressure, ${\gamma }_r$ decreases continuously with increasing rock block content. For example, under 100 kPa, ${\gamma }_r$ decreases from 0.128 at $R_c=0\%$ to 0.092 at $R_c=60\%$; under 200 kPa, it decreases from 0.178 to 0.109; and under 400 kPa, it decreases from 0.242 to 0.150. This shows that increasing rock block content systematically shifts the characteristic response position toward a lower strain range. In physical terms, although the block-rich mixtures are stiffer in the absolute sense, their normalized response begins to evolve earlier once strain is measured relative to their own higher initial stiffness basis. Conversely, for a given rock block content, ${\gamma }_r$ increases continuously with confining pressure. For example, for $R_c=0\%$, ${\gamma }_r$ increases from 0.128 at 100 kPa to 0.178 at 200 kPa and 0.242 at 400 kPa, and similar increases are observed for all other rock-content groups. This demonstrates that stronger confinement systematically pushes the characteristic response position toward a higher strain range and delays the relative evolution process.

The migration of characteristic strain has an important interpretive implication. It indicates that the material does not possess a fixed intrinsic threshold of nonlinear evolution independent of internal structure or stress condition. Instead, the position of the dynamic transition zone is itself movable and structurally conditional. Higher rock block content increases the density of interfaces, block–block contacts, and block–matrix interaction zones. As a result, once cyclic deformation reaches a sufficient level, structural readjustment and hysteretic activation can be mobilized at a relatively smaller normalized strain position. In contrast, higher confining pressure stabilizes contacts, strengthens constraint on local sliding and rotation, and makes it more difficult for the material to reach the same relative degree of structural transformation. Therefore, ${\gamma }_r$ should be interpreted as a migration index of structural transition rather than as a purely mathematical parameter.

More importantly, the dual-path response pattern provides a unified explanation for the coexistence of two observations that may otherwise appear contradictory. On the one hand, higher rock block content generally increases the absolute stiffness level of reconstructed soil–rock mixtures. On the other hand, the same increase in rock block content also causes the normalized modulus-reduction relationship to move downward and the normalized damping-development relationship to start earlier. Thus, the material becomes stronger in the absolute sense but more responsive in the relative evolution sense. This does not imply instability of the block-rich structure. Rather, it indicates that the internal interface network, while enhancing stiffness, also creates a more active structural channel for relative degradation and dissipation once the evolution process is measured against its own initial state. In the same way, stronger confinement not only increases the absolute stiffness level, but also delays the normalized evolution of both stiffness degradation and damping. Therefore, confinement acts simultaneously as a strengthening factor and as a postponing factor of relative response evolution.

In this sense, this interpretation suggests that the cyclic dynamic behaviour of reconstructed soil–rock mixtures should be described in terms of both response level and response position. The former answers how strong or how dissipative the material is in absolute terms, whereas the latter answers when the material begins to enter a pronounced relative evolution state. The characteristic-strain migration captures the latter, and the dual-path interpretation links it to the former. As a result, the present study does not simply conclude that rock block content and confining pressure “affect” the dynamic response. Instead, the results indicate that these two factors influence the cyclic response through two distinct but coupled channels: an absolute mechanical-strengthening channel and a relative evolution-position channel.

Therefore, characteristic-strain migration and the dual-path response pattern provide an integrated interpretation of the normalized dynamic-parameter behaviour of reconstructed SRMs. It explains why specimen groups with higher absolute stiffness do not necessarily evolve more slowly in normalized terms, and why the transition of cyclic behaviour should be judged not only by the magnitude of dynamic parameters but also by the strain position at which their relative evolution accelerates. This response pattern links the empirical fitting results with the interpretation of strain-dependent cyclic evolution and provides the basis for understanding how confinement regulates the role of the block phase under cyclic loading.

4.6.4. Confinement-Related Block–Matrix Coupling

The influence of rock block content becomes more evident under higher confining pressure, indicating that the structural role of rock blocks is stress-dependent rather than purely compositional. Previous SRM cyclic studies have shown that rock content and confinement jointly affect stiffness, damping, and dynamic response. In the present results, this stress-dependent block effect is reflected by higher absolute stiffness, delayed normalized degradation under stronger confinement, and more distinct differences among rock-content groups at higher confining pressure.

The direct experimental observations are the higher absolute stiffness, delayed normalized degradation, reduced normalized-damping development, and clearer separation among rock-content groups under higher confinement. The block–matrix coupling described here is therefore an interpretation of these measured trends, rather than a directly observed microscale process.

Under relatively low confinement, the block phase mainly exists as a geometrically identifiable but mechanically weakly mobilized component. Although rock blocks are physically present in the mixture, the surrounding matrix still dominates deformation coordination and load redistribution, and the difference among specimens with different rock block contents remains comparatively limited. In this state, block–block contact is insufficiently activated, block–matrix interfaces are not yet fully stabilized, and the internal structure behaves more like a matrix-dominated heterogeneous medium than a true stress-transferring skeleton. Therefore, the rock phase under weak confinement should not be interpreted as a fully developed skeleton, but rather as a set of dispersed inclusions embedded in a deformable matrix. This explains why, under low confining pressure, the cyclic response remains relatively similar among different rock-content groups even though their material compositions are different.

As confining pressure increases, however, the mechanical role of the interface system changes fundamentally. Stronger confinement compresses the soil matrix more effectively, tightens particle contacts, improves the stability of block–matrix interfaces, and enhances the continuity of the internal force-transfer network. Under such conditions, rock blocks no longer participate passively in the response. Instead, they become increasingly integrated into a constrained structural framework through enhanced block–block interaction, stronger block–matrix cooperation, and more stable interfacial load transfer. The internal skeleton is therefore activated not only in the sense of stiffness contribution, but also in the sense of response organization: it begins to govern how stresses are transmitted, how deformation is partitioned, and how hysteretic dissipation is distributed within the mixture. This is why the hysteresis loops under higher confinement become steeper, fuller, and more stress-expanded, and why the differences among rock-content groups become much more evident as confinement rises.

The key point here is that interface activation and skeleton formation are not two separate mechanisms, but two coupled aspects of the same confinement-controlled structural evolution. Interfaces provide the local mechanical channels through which sliding, friction, stress transfer, and contact readjustment occur, whereas the skeleton represents the higher-level structural organization that emerges when those local channels become sufficiently constrained and interconnected. Without sufficient confinement, interface activity remains largely local and fragmented, and the block phase cannot develop into an effective skeleton. With increasing confinement, however, the same interface network becomes more stable and more coordinated, allowing local contact interactions to scale up into a macroscopic load-bearing framework. Therefore, the skeleton effect in reconstructed soil–rock mixtures should be interpreted as an emergent phenomenon of interface stabilization under confinement rather than as a simple direct consequence of adding more rock blocks.

This interpretation also explains why the combined influence of rock block content and confining pressure is clearly non-additive. If the block phase acted independently of confinement, the effect of increasing rock content would remain proportionally similar under all stress conditions. The present results do not support such a view. Instead, they show that the influence of rock block content is relatively weak at low confinement but becomes increasingly significant as confinement rises. This means that rock content alone determines only the potential complexity and density of the structural network, whereas confining pressure determines whether that potential can actually be mobilized into a mechanically effective skeleton. In other words, rock block content provides the structural resource, but confining pressure governs the degree of structural realization. The observed response is therefore the product of their coupling rather than the sum of their separate contributions.

From the viewpoint of cyclic evolution, the confinement-related block–matrix coupling interpretation also helps explain why stronger confinement simultaneously leads to higher absolute stiffness, delayed normalized degradation, and postponed normalized dissipation development. Once the interface network is stabilized and the skeleton is more effectively mobilized, the material can maintain a higher proportion of its internal load-bearing capacity over a wider strain range. At the same time, the initiation of relative structural transformation is postponed because local sliding, rotation, and contact disruption are more strongly restrained. Therefore, confinement does not merely strengthen the material in a static sense; it reorganizes the dynamic response by coupling interface stabilization with skeleton efficiency. This interpretation extends the conventional statement that “higher confining pressure increases stiffness” by further emphasizing how confinement regulates internal structural participation during cyclic loading. It suggests that confinement changes not only how much resistance the material possesses, but also how the internal structure participates in cyclic evolution.

Accordingly, the cyclic dynamic behaviour of reconstructed SRMs can be interpreted as the outcome of stress-dependent block–matrix coupling. At lower confinement, the rock blocks mainly act as dispersed inclusions and their contribution to load transfer is limited. As confinement increases, the contact condition between blocks and matrix becomes more stable, allowing the block phase to contribute more effectively to stiffness retention and cyclic resistance. This interpretation links the observed macroscopic response, including higher absolute stiffness, delayed normalized degradation, and reduced damping development under stronger confinement, to the stress-dependent participation of the block-in-matrix structure. It also completes the parameter-based interpretation developed in this section, in which strain amplitude controls the degree of nonlinear evolution, characteristic strain marks the relative response position, and confinement regulates the effectiveness of block–matrix interaction.

The above interpretation is intended as a parameter-based synthesis of the measured cyclic response. It summarizes the observed relationships between strain amplitude, rock block content, confining pressure, stiffness degradation, damping development, and characteristic-strain migration under the present testing conditions. It does not define a constitutive model, a universal transition criterion, or fixed quantitative boundaries for all SRMs. Instead, it provides an experimentally grounded interpretation of the response patterns observed in the reconstructed specimens tested in this study.

Therefore, the three concepts above should be understood as a parameter-based synthesis of the measured macroscopic cyclic response. They are used to organize the experimental evidence from hysteresis behaviour, modulus degradation, damping evolution, normalized parameter trends, and characteristic-strain migration.

4.7. Engineering Implications for Dynamic-Parameter Selection and Interface Stability Assessment

The experimental results presented above provide several practical implications for the treatment, construction control, and interface-related stability assessment of coastal nuclear power plants and other high-value, safety-sensitive infrastructures founded on soil–rock mixture ground. Although the present study was carried out at the material scale, the observed strain-dependent stiffness degradation and damping evolution reveal important features of how this heterogeneous geomaterial may behave when subjected to repeated environmental or operational disturbances. In particular, the results indicate that the dynamic response of soil–rock mixtures is controlled jointly by shear strain amplitude, rock block content, and confining pressure, and that these factors directly influence stiffness retention, energy dissipation, and the progressive activation of internal interfacial processes.

Within the tested material and stress range, several quantitative reference values can be extracted for engineering interpretation. The extrapolated initial shear modulus $G_0$ ranged from 39.164 to 171.382 MPa, while the reference shear strain ${\gamma }_r$ ranged from 0.092 to 0.242. The measured secant shear modulus decreased from 35.835 to 158.871 MPa at the smallest strain level to 3.296–12.854 MPa at the largest strain level, corresponding to an overall stiffness reduction of approximately 85%–94%. The damping ratio increased from a lower-bound range of 0.036–0.063 to an upper-bound range of 0.195–0.268. These ranges are not proposed as direct design values, but they provide experimentally derived reference intervals for selecting stiffness-retention parameters, damping parameters, and characteristic-strain levels when reconstructed SRM foundations with similar material composition, particle-size range, and confining-pressure conditions are considered.

First, the strong effect of confining pressure on both shear modulus and normalized dynamic response suggests that sufficient confinement and densification should be regarded as primary objectives in foundation treatment and construction control. In the present study, increasing confining pressure consistently raised the absolute stiffness level and delayed the relative degradation of normalized shear modulus and normalized damping evolution. From an engineering perspective, this means that loose or weakly confined zones within soil–rock mixture foundations are more likely to experience premature stiffness loss and earlier activation of dissipative deformation under cyclic loading. For coastal nuclear power plants, where long-term stability and deformation control are critical, this finding implies that special attention should be given to improving compaction quality, reducing local looseness, and enhancing the continuity of stress transfer within the treated foundation. In practical terms, ground improvement measures that increase confinement, compactness, and contact stability may be more effective than approaches that rely only on static bearing-capacity enhancement. This implication is consistent with recent studies on lateritic geomaterials and sustainable construction materials, which have shown that chemical stabilization, geogrid reinforcement, and microstructural densification can improve stiffness, strength, durability, and deformation resistance under service-related conditions [43,44]. Although those studies focused on lateritic subgrades and laterite-rock aggregate concrete rather than cyclic SRM behaviour, they support the broader engineering view that material improvement, contact stability, and durability should be considered together in infrastructure foundation and construction-material assessment.

Second, the influence of rock block content indicates that the dynamic performance of soil–rock mixture foundations cannot be judged simply by assuming that a higher rock content always leads to a better foundation state. The present results show that increasing rock block content generally increases the absolute stiffness of the material, especially under moderate and high confinement. However, after normalization, the higher-rock-content groups exhibit faster relative stiffness degradation and earlier development of damping evolution. This means that block-rich mixtures may possess a higher initial stiffness, but they are also more structurally sensitive to cyclic disturbance because of stronger block–block interaction, soil–rock interfacial sliding, and local rearrangement. Therefore, for engineering applications such as coastal nuclear power plant foundations, the control of rock block content should not focus only on maximizing stiffness. Within the tested range, specimens with higher rock block content showed higher absolute stiffness, but the characteristic strain ${\gamma }_r$ decreased as $R_c$ increased. For example, under 100 kPa, ${\gamma }_r$ decreased from 0.128 at $R_c=0\%$ to 0.092 at $R_c=60\%$; under 400 kPa, it decreased from 0.242 to 0.150. This indicates that block-rich mixtures may enter the relative stiffness-degradation and damping-development stage at smaller strain levels. Therefore, for critical foundations, high-rock-content zones should be accompanied by stricter control of compaction quality, particle distribution, and transition continuity, rather than being judged only by their higher initial stiffness. Excessive heterogeneity or uncontrolled block concentration may increase the sensitivity of the foundation to cyclic disturbance and amplify local deformation incompatibility.

Third, the results highlight the importance of interface-related control in both design and construction. The hysteresis characteristics and damping evolution observed in this study indicate that soil–rock mixtures are not governed solely by matrix deformation, but also by the progressive mobilization of soil–rock interfacial sliding and internal structural rearrangement. This means that the engineering response of a soil–rock mixture foundation should be evaluated by considering both bulk material behaviour and interface-related deformation compatibility. For coastal nuclear power plants, this has direct implications for zones where dynamic load transfer is concentrated, such as the foundation–ground interface, replacement–treatment boundaries, cut-and-fill transitions, and contacts between improved and unimproved ground. In such locations, local discontinuity in stiffness or contact condition may promote stress concentration, deformation incompatibility, and earlier cyclic degradation. Accordingly, construction control should place emphasis not only on bulk material quality, but also on interfacial continuity, gradation transition, and the avoidance of abrupt stiffness contrast across adjacent zones.

Fourth, the strain-dependent characteristics identified in this study suggest that dynamic safety assessment of coastal nuclear power plant foundations should not rely exclusively on static or small-strain parameters. The observed transition from limited degradation at low strain to rapid stiffness loss and enhanced energy dissipation at medium and large strain indicates that the foundation response of soil–rock mixtures may change significantly once cyclic demand exceeds a certain level. Therefore, in engineering evaluation, parameters related to modulus degradation, damping development, and characteristic strain should be incorporated into the assessment of serviceability and cyclic stability, especially for sites where repeated vibration, seismic excitation, or long-term operational disturbance may occur. In this regard, the present results provide a useful experimental basis for selecting dynamic parameters in subsequent constitutive modelling, interface-response analysis, and performance-based foundation evaluation.

Because the present results were obtained from reconstructed small-scale specimens, the above numerical ranges should be used as experimental reference values for preliminary parameter selection and comparative assessment, rather than as universal design limits. Site-specific verification is still required before these values are adopted in engineering design.

Overall, the findings of this study suggest that the safe use of soil–rock mixture ground in coastal nuclear power plants and similar infrastructures requires an integrated strategy that combines adequate densification, reasonable control of rock block content, attention to interface continuity, and dynamic parameter-based evaluation of cyclic stability. These implications are particularly relevant to engineering scenarios in which long-term reliability depends not only on initial foundation stiffness, but also on the stability of stiffness retention, deformation compatibility, and interface response under repeated loading. Therefore, the present study provides not only material-level experimental evidence, but also a meaningful basis for interpreting and improving the dynamic behaviour of complex foundations serving coastal nuclear power plants and other high-value, safety-sensitive infrastructures.

5. Conclusions

This study investigated the cyclic dynamic behaviour of reconstructed soil–rock mixtures using undrained multistage strain-controlled cyclic triaxial tests. The main conclusions are summarized as follows:

• The reconstructed SRMs exhibited clear strain-dependent nonlinear cyclic behaviour. With increasing strain amplitude, the hysteresis loops changed from narrow and steep forms to wider and fuller shapes, indicating progressive stiffness degradation and increasing hysteretic energy dissipation.
• The secant shear modulus decreased continuously and nonlinearly with increasing strain amplitude. Over the investigated strain range, G decreased from 35.835 to 158.871 MPa at the smallest strain level to 3.296–12.854 MPa at the largest strain level, corresponding to an overall reduction of approximately 85%–94%.
• Rock block content and confining pressure both affected the absolute stiffness level. Higher rock block content generally increased the initial and small-strain stiffness, while stronger confinement increased stiffness over the whole strain range and helped maintain a higher residual stiffness at large strain.
• The normalized shear modulus results showed that rock block content and confining pressure influenced the relative stiffness-degradation path in different ways. Increasing rock block content shifted the characteristic strain ${\gamma }_r$ to lower values, whereas increasing confining pressure shifted ${\gamma }_r$ to higher values.
• The damping ratio increased continuously with increasing strain amplitude, showing progressive enhancement of hysteretic energy dissipation. The damping ratio increased from a lower-bound range of 0.036–0.063 to an upper-bound range of 0.195–0.268 within the tested strain range.
• The experimental results provide parameter-level evidence for evaluating the cyclic stability of SRM foundations. For engineering applications involving complex SRM ground, stiffness retention, damping development, characteristic strain, and deformation compatibility should be considered together rather than relying only on initial stiffness or static bearing capacity.

6. Limitations and Future Work

Although this study provides a systematic experimental characterization of the cyclic dynamic behaviour of reconstructed SRMs, several limitations should be acknowledged. First, all specimens were prepared using a compaction-based reconstruction method. Therefore, the results mainly represent the behaviour of reconstructed SRMs under controlled rock block content, moisture condition, moulding state, and confining pressure, rather than the full in situ response of natural undisturbed SRM deposits. Natural SRM deposits may contain more complex fabric features, depositional anisotropy, cementation, weathering effects, and broader particle-size variability. Future work should further compare reconstructed specimens with undisturbed or field-representative SRM samples and field-measured dynamic parameters.

Second, the tests were conducted on small-scale cylindrical specimens with a diameter of 50 mm and a height of 100 mm, using rock particles in the range of 6–10 mm. This design allowed the block-in-matrix structure to be represented within the capacity of the cyclic triaxial apparatus, but the adopted particle-size-to-specimen-size ratio was close to the upper practical boundary for small-scale testing. In addition, for the high-rock-content groups, especially R_c = 40% and R_c = 60%, the arrangement of individual rock particles may have a greater influence than in larger specimens. Future studies should use larger specimens with a greater number of rock particles and image-based or CT-based spatial verification to further examine particle-size ratio effects and particle-arrangement effects.

Third, the cyclic loading programme adopted a multistage strain-controlled procedure with three cycles at each strain amplitude. This procedure was effective for comparing stage-level modulus and damping evolution under a consistent increasing-strain path. The reported dynamic parameters are therefore most appropriately understood as path-dependent stage-level responses obtained under the adopted loading sequence. Future single-stage tests and extended-cycle tests will be useful for further separating the effects of current strain amplitude, accumulated loading history, and cycle-by-cycle degradation.

Fourth, the loading frequency was fixed at 0.5 Hz throughout the testing programme. This setting helped maintain a consistent strain-controlled sinusoidal loading condition and allowed the comparative analysis to focus on rock block content, confining pressure, and strain amplitude. However, frequency-dependent behaviour was not independently evaluated. Future work should include multi-frequency cyclic tests to clarify the rate-dependent evolution of shear modulus, damping ratio, and normalized dynamic parameters.

Fifth, the repeatability check was conducted only on selected representative groups. Although the results provided a preliminary indication of preparation and testing consistency, they cannot fully quantify the statistical variability of all 15 combinations of rock block content and confining pressure. Future experimental programmes should increase the number of parallel specimens and extend repeatability verification to more rock-content and confining-pressure combinations. More detailed documentation of specimen preparation, including compaction-related control information, should also be considered in subsequent related tests to improve reproducibility and comparison with other studies.

Sixth, the present study focuses on total-stress cyclic parameters derived from measured stress–strain hysteresis loops under undrained boundary conditions. The reported modulus and damping indices therefore describe the macroscopic cyclic response of reconstructed SRMs within a loop-based parameter framework. Future work will extend the testing programme by incorporating more complete effective-stress-path measurements, so that the coupling between internal hydraulic response, stiffness degradation, and damping development can be examined in greater detail.

Despite these limitations, the present study provides a useful parameter-based experimental basis for understanding the cyclic dynamic behaviour of reconstructed SRMs. The findings can support subsequent constitutive fitting, mesoscopic modelling, and dynamic-parameter selection for complex SRM foundations, while the above limitations define the main directions for future improvement.