Three-Dimensional Mesoscopic DEM Modeling and Compressive Behavior of Macroporous Recycled Concrete

Abstract

The mesoscopic-scale discrete element method (DEM) modeling approach demonstrated high compatibility with macroporous recycled concrete (MRC). However, existing DEM models failed to adequately balance modeling accuracy and computational efficiency for recycled aggregate (RA), replicate the three distinct interfacial transition zone (ITZ) types and pore structure of MRC, or establish a systematic calibration methodology. In this study, PFC 3D was employed to establish a randomly polyhedral RA composite model and an MRC model. A systematic methodology for parameter testing and calibration was proposed, and compressive test simulations were conducted on the MRC model. The model incorporated all components of MRC, including three types of ITZs, achieving an aggregate volume fraction of 57.7%. Errors in simulating compressive strength and elastic modulus were 3.8% and 18.2%, respectively. Compared to conventional concrete, MRC exhibits larger strain and a steeper post-peak descending portion in stress–strain curves. At peak stress, stress is concentrated in the central region and the surrounding arc-shaped zones. After peak stress, significant localized residual stress persists within specimens; both toughness and toughness retention capacity increase with rising porosity and declining compressive strength. Failure of MRC is dominated by tension rather than shear, with critical bonds determining strength accounting for only 1.4% of the total. The influence ranking of components on compressive strength is as follows: ITZ (new paste–old paste) > ITZ (new paste–natural aggregates) > new paste > old paste > ITZ (old paste–natural aggregates). The Poisson’s ratio of MRC (0.12–0.17) demonstrates a negative correlation with porosity. Predictive formulas for peak strain and elastic modulus of MRC were established, with errors of 2.6% and 3.9%, respectively.

1. Introduction

The recycled concrete studied in this work, characterized by 20–30% porosity, is conventionally termed pervious recycled concrete. However, this designation may be inadequate since the material exhibits multifunctional capabilities beyond water permeability, including acoustic absorption [1,2], energy dissipation [3], and water purification [4]. Herein, we define it as macroporous recycled concrete (MRC), thereby describing it by its objective structural feature (macroporosity) rather than a singular functional perspective. MRC is a sustainable construction material typically employed in permeable pavements, which effectively mitigates the urban heat island effect while valorizing construction waste [5,6,7]. The current design process for macroporous concrete is complex, requiring multiple experimental tests such as permeability and mechanical tests [8,9,10]. Furthermore, high dispersion in test results [11,12] necessitates larger test sample sizes. To address this, numerous researchers have conducted numerical simulations of concrete to reduce experimental efforts while obtaining more accurate performance parameters [13,14,15,16].

As a composite material, concrete possesses an interfacial transition zone (ITZ) of relatively low strength, which is widely regarded as the weakest region [17,18,19]. Consequently, the ITZ must be accurately replicated at the mesoscopic level in numerical simulations [20]. Mesoscopic modeling of concrete commonly employs the finite element method (FEM), discrete element method (DEM), lattice method, finite difference method (FDM), and their coupled approaches [20,21,22,23,24], among which FEM is the most prevalent. Key aspects of FEM mesoscopic modeling involve ITZ configuration and aggregate placement. To enhance computational efficiency, the ITZ thickness is frequently amplified to 0.1–1.0 mm through mesh size adjustment [14,25,26], or represented by zero-thickness elements [27]. Placed aggregates must remain non-overlapping, typically achieved via overlap detection and random walk algorithms [20,22,28]. Overlap detection involves post-placement verification of aggregate interference through edge–face relationships [21] or vertex–face relationships [29]. Additionally, the method of separating planes [30] and method of separating axes [22] constitute classical techniques. The random walk approach enables aggregates to “walk” within the target domain with real-time collision checks [28]. These methods achieve aggregate volume fractions of 30–55% [20,22,29], yet realistic replication of contact behavior between adjacent aggregates and spatial distribution of aggregate centroids remains challenging. Owing to methodological complexity, minimum aggregate spacing exceeding 5 mm persists in recent studies [14,31]. For macroporous concrete, strength is significantly influenced by the skeleton structure [32], necessitating high packing density of aggregates.

The DEM serves as another effective approach for mesoscopic concrete modeling, wherein the parallel bond model between particles accurately simulates bonding behavior in cementitious materials [33,34]. DEM-based mesoscale models have been successfully applied to investigate fundamental mechanical properties [24,35,36,37], aggregate and ITZ effects [36,38,39,40], size effects [41], creep [42], cracking [43,44,45], rebar bond-slip [46], old–new interface degradation [47], diffusion [48], and surface spalling [49]. Simulation errors for splitting tensile strength, compressive strength, and elastic modulus are now controllable within 0.49%, 0.96%, and 1.56%, respectively [36]. DEM models utilize discrete entities as basic elements, enabling active compression of aggregate models to achieve maximum packing density. Random polyhedral aggregates can be modeled via clump or cluster approaches [35,36]: Clump aggregates represent rigid, non-breakable aggregates with higher computational efficiency (fewer particles); cluster aggregates simulate deformable, and crushable aggregates with enhanced accuracy. Current research predominantly employs single aggregate models [50,51,52]. In MRC, aggregates interact through point contacts, with compressive failures primarily occurring in the paste or ITZ [9,53,54], while natural aggregates (NAs) rarely fracture. Consequently, adopting cluster models for old paste (OP) on recycled aggregate (RA) surfaces and clump models for NA cores offers a balanced strategy to optimize computational accuracy and efficiency—though this hybrid approach remains underexplored. While DEM mesoscale models have been well-established for conventional concrete [55,56], their application to MRC remains limited due to structural complexities, including three distinct ITZ types and macroporosity. Mesoscopic parameters required for DEM modeling—primarily interparticle interaction properties—cannot be experimentally determined and are typically calibrated through trial-and-error [33,34,36], reflecting the absence of systematic calibration methodologies. Furthermore, bond gap between particles (governing bond quantity) constitutes another critical parameter requiring strict control, as cross-particle bonds induce significant fluctuations in DEM mesoscale simulation results. The strength of the specimen demonstrates a strong correlation with the number of bonds. Without strict control of bond quantity, the calibration of bond parameters becomes invalid—an oversight pervasive in prior research.

To address the aforementioned DEM modeling challenges for MRC, this study utilized PFC 3D (Version 5.0) to integrate clump and cluster models for developing a random polyhedral RA composite model, then constructed the MRC model. Bond gap was controlled to restrict bonds exclusively to adjacent particles. Based on mesoscale tests of paste, ITZ, and RA, a systematic parameter calibration methodology was proposed. Uniaxial compression simulations were performed on the MRC model. Mechanical properties—including stress–strain response, toughness, compressive strength, elastic modulus, and Poisson’s ratio—were analyzed alongside test data; concurrently, internal force distribution and failure states of all components (encompassing three distinct ITZ types) were investigated.

2. Mesoscopic DEM Modeling of RA

The modeling of random polyhedral RA was implemented in two stages: (1) geometry and entity design, followed by (2) OP and ITZ configuration on RA surfaces. All modeling procedures were executed through custom scripting in the FISH language.

2.1. RA Shape and Entity Design

The geometry of RA was designed according to the following methodology. Constraints on the ranges of OM and PE were implemented to prevent the formation of elongated and flaky aggregates, as such aggregates diminish concrete performance. GB/T 25177-2010 [57] stipulates that the content of elongated and flaky aggregates in concrete shall not exceed 10%. Consequently, elongated and flaky aggregates were excluded from this study. Random seeds were derived from a function of time (i.e., time as the independent variable) to guarantee distinct shapes for each RA.

(1)

A unit circle O with diameter 1 was constructed, intersecting the x-axis at points A and C. A random point M was selected on the positive y-axis such that OM ∈ (0.25, 0.50). A line parallel to the x-axis passing through M intersected circle O at D₁ and D₂, from which a random point D was chosen on segment D₁D₂. Using an identical method, a random point B was determined in the third and fourth quadrants, forming quadrilateral ABCD as shown in Figure 1a.

(2)

Within quadrilateral ABCD, a random point P was selected. A perpendicular line segment EF to the xy-plane was drawn through P, with PE ∈ (0.30, 0.70) and PF = 1 − PE, resulting in octahedron ABCDEF illustrated in Figure 1b.

(3)

The surface areas of all triangular facets constituting octahedron ABCDEF were computed individually. For triangles exceeding 50% of the quadrilateral ABCD area (e.g., △ABF), expansion points were added: point G was defined by position vector OG = 0.5(OA + OB + OF) as depicted in Figure 1c. Newly formed surfaces underwent identical area verification until geometric stability was achieved.

During failure, MRC exhibited predominant damage within ITZ and paste regions, while NA components rarely fractured. Consequently, NA cores in RA were modeled using clump models with a density of 2695 kg/m³ (measured NA density), whereas OP on RA surfaces was represented by cluster models assigned a density of 2037 kg/m³ (measured OP density). Key parameters of the clump modeling approach and representative image are summarized in

Table 1. Parameters and representative image for clump model generation.

Parameters	Value	Representative Image
Distance/°	150
Ratio	0.3
Radfactor	1.05
Refinenum	10,000

.

2.2. Configuration of OP and ITZ on RA Surfaces

RAs were classified into Types I, II, and III based on OP content, as illustrated in Figure 2. Water-saturated RA specimens underwent 300 freeze–thaw cycles in air, followed by Los Angeles abrasion testing without steel balls. The mass of detached OP was recorded every 10 revolutions, with the OP detachment rate stabilizing at 23.1%, as shown in Figure 3. This stabilized value served as an approximation for the OP content. Consequently, the RA model in this study was designed as Type I.

Attachment of OP to the clump model surface modified clump size. Randomizing OP location and thickness at this stage would complicate final aggregate size statistics. Thus, a novel OP distribution method was developed, as illustrated in Figure 4. A random pebble element on the clump surface was replaced by a geometrically identical cluster model with cubic-packed balls. Balls overlapping the clump core were deleted; residual balls were defined as surface OP. This process iterated until 23.0% OP content was attained. The resultant RA model maintained identical size to the original clump model, simplifying gradation control.

OP volume was calculated as the sum of circumscribed cube volumes of ball elements; this approach was adopted because cubic-packed balls occupy the space of their circumscribed cubes, as illustrated in Figure 4. Overlapping balls between cluster models were adjusted by radius scaling via Equations (1) and (2), eliminating all physical overlaps while almost preserving OP volume. Initial radii of ball elements were linearly determined based on RA sizes, as shown in

Table 2. OP particle radii under different RA sizes.

RA Sizes/mm	9.5–13.2	13.2–16	16–19	19–26.5	26.5–37.5
OP Particle Radii/mm	0.3	0.4	0.5	0.7	1.0

, so that each RA contained approximately equal numbers of balls at fixed OP content.

$$L_{\mathrm{overlap}}=2R_{\mathrm{original}}-D$$

(1)

$$R_{\mathrm{update}}=R_{\mathrm{original}}-0.5L_{\mathrm{overlap}}$$

(2)

where $D$ is the centroid distance between overlapping ball elements (mm); $R_{\mathrm{original}}$ is the initial radius of ball elements (mm); $L_{\mathrm{overlap}}$ is the overlap distance of overlapping ball elements (mm); $R_{\mathrm{update}}$ is the scaled radius of overlapping ball elements (mm).

Parallel bond models were adopted for both ball–ball bonds within cluster models and ball–pebble bonds between cluster models and clump models. The bond gap was set to the diameter d (mm) of ball elements. This configuration was necessitated by the RA modeling approach: distinct distances (0–d) existed between adjacent balls, and between adjacent balls and pebbles. A bond gap < d might prevent bond formation, while > d could induce bonds spanning a third particle. Crucially, cross-particle bonds constitute a primary source of significant fluctuations in DEM mesoscale simulation results, as it prevents stable and controllable regulation of bond quantities. Since the strength of the specimen is directly correlated with the number of bonds, an unconstrained bond quantity undermines the effectiveness of parameter calibration. This aspect has received minimal attention in previous studies. To resolve this, bond gap was strictly controlled to ensure bonds existed exclusively between adjacent particles. Per Table 2, d varied with RA size; thus, bond gap was dynamically determined for each RA.

Structural analysis of the parallel bond model revealed its close resemblance to RA–RA bonding in MRC, as shown in Figure 5. Bonds activate when the surface gap g_s < 0; thus, bond gap (i.e., ITZ thickness) was controlled by modifying reference gap g_r. Theoretically, adjusting g_r to 0 μm–100 μm to simulate realistic ITZ thickness would not significantly reduce computational efficiency, but would pose significant modeling challenges while failing to account for ITZ steric hindrance. A novel ITZ design method was implemented: the first layer of ball elements surrounding pebble elements was designated as ITZ entity elements, as illustrated in Figure 4. All bonds adjacent to these elements were assigned ITZ parameters, serving as a mechanical ITZ model. This approach simplified modeling while simulating realistic ITZ steric hindrance. The ITZ thickness in this model ranged from 0.6 to 2.0 mm, constituting a reasonable simplification within this field. Subsequent ITZ representations adopted this modeling approach.

3. Mesoscopic Parameter Assignment and Calibration for Matrix

Six groups of new paste (NP) were designed, with mix proportions detailed in

Table 3. Mix proportions of the cementitious matrix (kg/m³).

Groups	W/B	Cement	Water	Silica Fume	Superplasticizer
M1	0.25	1665	434	69 (4%)	7
M2	0.35	1427	520	59 (4%)	–
M3	0.45	1242	582	52 (4%)	–
M4	0.25	1566	426	136 (8%)	15
M5	0.35	1356	516	118 (8%)	–
M6	0.45	1182	578	103 (8%)	–

. The chemical compositions of P.O. 42.5 ordinary Portland cement and silica fume are presented in

Table 4. Chemical compositions of the P.O. 42.5 cement and silica fume (%).

Materials	SiO2	CaO	Al2O3	SO3	Fe2O3	MgO	Na2O	K2O
Cement	22.75	52.56	7.38	0.79	6.37	1.02	0.73	1.31
Silica fume	93.58	0.47	0.22	1.21	0.11	0.56	0.22	0.84

. The average mesh size of the silica fume is 10 μm. Parallel bond models governed all interparticle bonds within NP and RA systems. In this section, mesoscopic parameters for these bonds were calibrated separately for NP and RA. Random seeds were generated as functions of time. Each set of calibration tests was performed in triplicate, and resultant parameters were averaged.

To enforce bonds exclusively between adjacent particles, ball elements in NP were arranged in cubic-packed configuration. In concrete DEM modeling, diameters of paste particles typically range from 0.15 mm to 2.00 mm [36,41,43]; herein, diameter of NP particles was fixed at 2.00 mm. Particle arrangement and sizing in calibration models were maintained identical to those in MRC models. Since parallel bond models do not fail under compression, compressive thresholds were enforced for regularly arranged particles. Uniaxial compression tests and simulations were performed on 10 mm × 10 mm × 15 mm specimens, as shown in Figure 6a. Loading rate was set at 1.5 MPa/s per GB/T 17671-2021 [58]. At peak load, the load was primarily borne by 25 vertical force chains in the compression zone. Taking M1 as an example: at peak load (6250 N), force chain loads fluctuated between 248.2 N and 253.1 N, as illustrated in Figure 6b. Consequently, 1/25 of peak load was assigned as the normal force threshold for all groups, as shown in

Table 5. Normal force thresholds of contact and peak load for NP in groups M1–M6.

Items	M1	M2	M3	M4	M5	M6
Peak Load/N	6250.0	5660.0	4620.0	7240.0	6040.0	5080.0
Normal Force Thresholds/N	250.0	226.4	184.8	289.6	241.6	203.2

. In subsequent models, when normal contact force reached threshold, the parallel bond model was actively switched to a linear contact model, with elastic modulus reduced to 1% of its original value.

Linear contact model was applied to the interactions between walls and NP/ RA (including pebble and ball elements). The stiffness ratio was approximated at 1.00. The friction coefficient was assigned a value of 0.463, derived from spring dynamometer measurements between loading platens and NP specimens. The effective elastic modulus for wall–NP interactions was determined by contact quantity: where n denotes the number of contacts between wall and NP, the effective modulus for ball-facet was set to 2/n of the ball–ball effective modulus. This controlled ball–facet overlap at 1/2 of ball–ball overlap. For wall–RA contacts, the effective elastic modulus was approximated at 1.0 GPa. Unspecified parameters retained default values.

3.1. Parameter Assignment and Calibration for NP

Parameter list, their initial values, and calibration status for NP are summarized in

Table 6. Parameter list, initial values, and calibration status for NP.

Parameters		M1	M2	M3	M4	M5	M6
Linear Group	Effective Modulus $E^{\ast }$/GPa	40.5	31.1	26.5	44.6	37.8	28.6
	Stiffness Ratio $k^{\ast }$	0.25	0.25	0.25	0.25	0.25	0.25
	Friction Coefficient $\mu$	0.795	0.776	0.790	0.807	0.773	0.781
	Reference Gap $g_r$/mm	−0.1	−0.1	−0.1	−0.1	−0.1	−0.1
Parallel-Bond Group	Bond Effective Modulus ${\overline{E}}^{\ast }$/GPa	40.5	31.1	26.5	44.6	37.8	28.6
	Bond Stiffness Ratio ${\overline{k}}^{\ast }$	0.25	0.25	0.25	0.25	0.25	0.25
	Tensile Strength ${\overline{\sigma }}_c$/MPa	6.28	5.49	4.16	7.48	5.86	4.94
	Cohesion $\overline{c}$/MPa	16.8	14.1	11.0	19.4	14.7	11.9
	Friction Angle $\overline{\varphi }$/°	38.5	37.8	38.3	38.9	37.7	38.0
Other Parameters		Default Values

Note: Bolded values are parameters to calibrate; others are fixed.

. Cohesion (tangential bond strength) was derived from shear strength of 10 mm × 10 mm × 15 mm paste specimens. Tensile strength (normal bond strength) corresponded to splitting tensile strength of 10 mm × 10 mm × 30 mm paste specimens. (Bond) Effective elastic modulus was mesoscopic modulus obtained via nanoindentation tests. Friction angle $\overline{\varphi }$ was approximated by the repose angle of paste powder. Friction coefficient μ was assigned as μ = tan$\overline{\varphi }$, ensuring continuous frictional transition after bond failure. Given cubic-packed arrangement of balls, the stiffness ratio could be equated to Poisson’s ratio. Poisson’s ratio for cement paste ranges from 0.24 to 0.27 [59]; herein, 0.25 was adopted.

Based on the initial values of the parameters in Table 6, the parameter calibration was carried out as follows.

(1)

Calibration of $E^{\ast }$ and ${\overline{E}}^{\ast }$

Specimen deformation alters internal stress distribution, exerting influence on strength. Consequently, $E^{\ast }$ and ${\overline{E}}^{\ast }$ were calibrated first. Elastic modulus was estimated via Equation (3) as calibration reference. Per ACI 318-19 [60] and CSA A23.3-19 [61], the coefficient $\beta$ is specified as 4700 and 4500, respectively, for static elastic modulus calculation of concrete (maximum aggregate size: 37.5 mm). For mortar (maximum aggregate size: 4.8 mm), $\beta$ ranges from 4131 to 4763 [62]. The coefficient $\beta$ decreases with reduced aggregate size. Simultaneously, dynamic modulus tests on mortar specimens (40 mm × 40 mm × 80 mm, 100 mm × 100 mm × 100 mm, and $\varphi$5 mm × 10 mm) yielded $\beta$ values of 5408, 5238, and 4982 [62]. This indicates negligible size effect, thus validating the applicability of these $\beta$ values at the mesoscopic scale. Herein, $\beta$ ≈ 4000 was adopted to calculate static elastic moduli for M1–M6, as tabulated in

Table 7. Compressive strength and static elastic modulus of NP in groups M1–M6.

Items	M1	M2	M3	M4	M5	M6
Compressive Strength/MPa	62.5	56.6	46.2	72.4	60.4	50.8
Static Elastic Modulus/GPa	31.6	30.1	27.2	34.0	31.1	28.5

.

$$E=\beta \sqrt{f_c}$$

(3)

where $E$ is the elastic modulus (MPa); $f_c$ is the compressive strength (MPa); $\beta$ is a coefficient.

Given pronounced size effects on compressive strength and elastic modulus of cement paste, the specimen dimension for $E$ in Equation (3) was strictly matched to that for $f_c$ (10 mm × 10 mm × 15 mm). Consequently, prismatic specimens measuring 10 mm × 10 mm × 20 mm were modeled for elastic modulus simulation tests per GB/T 50081-2019 [63] and ABNT NBR 8522 [64], maintaining identical aspect ratios to standard specimens in GB/T 50081-2019 [63]. The testing sequence initiated with one eccentricity-compensating loading cycle to establish axial load equilibrium, followed by two precompression cycles and formal testing, as shown in Figure 7, with loading/unloading rates maintained at 1.5 MPa/s, consistent with compressive tests. Stress–strain data at points A and B were recorded, and elastic moduli were calculated via Equation (4).

$$E={\displaystyle \frac{F_{1/3}-F_0}{A}}\times {\displaystyle \frac{L}{{\epsilon }_B-{\epsilon }_A}}$$

(4)

where $E$ is the static elastic modulus (MPa); $F_{1/3}$ is the load at one-third of the cube compressive strength (N); $F_0$ is the load at 0.5 MPa (N); $A$ is the specimen bearing area (mm²); $L$ is the longitudinal gauge length (mm); ${\epsilon }_B$ and ${\epsilon }_A$ are the longitudinal deformations at points B and A, respectively (mm).

$E^{\ast }$ and ${\overline{E}}^{\ast }$ were scaled synchronously to discrete multiples (0.00, 0.25, 0.50, 0.75, 1.00, 1.50, 2.00, 3.00) of their initial values in the M1 group. Elastic modulus simulation tests were performed, yielding a correlation curve between effective elastic modulus and specimen elastic modulus, as shown in Figure 8. Specimen elastic moduli for M1–M6 from Table 6 were substituted into this curve, thereby determining calibrated $E^{\ast }$ and ${\overline{E}}^{\ast }$ values for all M1–M6 groups.

(2)

Calibration of $\overline{c}$

Shear simulation tests were conducted on 10 mm × 10 mm × 15 mm specimens at a loading rate of 0.375 MPa/s. With zero normal stress ${\sigma }_n$ applied to the shear plane, Mohr–Coulomb yield criterion dictates that cohesion $\overline{c}$ equals shear stress ${\tau }_n$, as shown in Equation (5). Consequently, specimen shear strength was governed solely by $\overline{c}$. The bond strength ratio ${\overline{\sigma }}_c/\overline{c}$ was held constant to preserve failure mode. Subsequently, $\overline{c}$ and ${\overline{\sigma }}_c$ were scaled incrementally to discrete multiples (0.00, 0.25, 0.50, 0.75, 1.00, 1.50, 2.00, 2.50, 3.00) of their initial values, yielding a correlation curve between shear strength and bond parameter scaling factor, as illustrated in Figure 9. Cohesion (i.e., experimental shear strength) from Table 6 was substituted into this curve, determining scaling factors and corresponding $\overline{c}$ values for M1–M6. All scaling factors for M1–M6 clustered around 2.005.

$${\tau }_n=\overline{c}+{\sigma }_n\tan \overline{\varphi }$$

(5)

(3)

Calibration of ${\overline{\sigma }}_c$

Splitting tensile simulations were performed on 10 mm × 10 mm × 30 mm specimens at 0.15 MPa/s. Given that parallel bonds fail exclusively through $\overline{c}$-governed shear failure or ${\overline{\sigma }}_c$-governed tensile failure, and $\overline{c}$ had been determined previously, only ${\overline{\sigma }}_c$ required adjustment. ${\overline{\sigma }}_c$ was scaled to discrete multiples (0.00, 0.25, 0.50, 0.75, 1.00, 1.50, 2.00, 2.50, 3.00) of initial values across groups, yielding a splitting tensile strength versus scaling factor correlation curve, as illustrated in Figure 10. Tensile strengths (experimental splitting strengths) from Table 6 were substituted into this curve, determining ${\overline{\sigma }}_c$ scaling factors and corresponding values for M1–M6, with all scaling factors converging near 1.523.

The bond parameter calibration results for NP are presented in

Table 8. Calibration results for bonding parameters of NP.

Parameters		M1	M2	M3	M4	M5	M6
Linear Group	Effective Modulus $E^{\ast }$/GPa	35.8	32.2	26.3	42.3	34.5	28.8
Parallel-Bond Group	Bond Effective Modulus ${\overline{E}}^{\ast }$/GPa	35.8	32.2	26.3	42.3	34.5	28.8
	Tensile Strength ${\overline{\sigma }}_c$/MPa	9.58	8.37	6.34	11.38	8.92	7.51
	Cohesion $\overline{c}$/MPa	33.6	28.2	22.0	38.8	29.4	23.8

.

3.2. Parameter Assignment and Calibration for ITZ and RA

Calibration parameters for the ITZ mirrored those of NP as specified in Table 6. Given that both NP–NA and NP–OP interfacial behaviors were predominantly governed by NP, their parameters were assigned identical values. Calibration was thus restricted to NP–NA and OP–NA interfaces. The ITZ mechanical model employed zero-thickness bonds, while nanoindentation tests yielded indentation depths and widths ranging from 3.5 to 5.1 μm, as shown in Figure 11—orders of magnitude smaller than the diameters of ITZ particle element. Consequently, measured indentation elastic moduli were directly adopted as effective elastic moduli for ITZ.

Initial attempts to calibrate ${\overline{\sigma }}_c$ and $\overline{c}$ for NP–NA interfaces using NP methodologies revealed that the roughness of aggregate cutting surface in fabricated ITZ specimens was difficult to replicate. Physical interlocking between paste and aggregate was compromised by smooth cutting surfaces, resulting in measured ITZ splitting tensile strengths of 0.180–0.253 times and shear strengths of 0.176–0.226 times those of corresponding pastes—ratios below conventional values. Consequently, reduced ${\overline{\sigma }}_c$ and $\overline{c}$ values derived from NP parameters were adopted for NP–NA interfaces. Ratios of ITZ indentation elastic modulus to corresponding paste indentation elastic modulus were directly utilized as reduction factors, as summarized in

Table 9. Indentation elastic modulus and reduction factors for ITZ and corresponding NP.

Items	M1	M2	M3	M4	M5	M6	OP
NP Elastic Modulus/GPa	40.5	31.1	26.5	44.6	37.8	28.6	36.4
ITZ Elastic Modulus/GPa	32.2	22.7	17.5	38.6	31.8	19.4	30.8
Reduction Factor	0.795	0.730	0.660	0.865	0.841	0.678	0.846

.

The bond parameter calibration results for NP–NA are presented in

Table 10. Calibration results for bonding parameters of NP–NA.

Parameters		M1-NA	M2-NA	M3-NA	M4-NA	M5-NA	M6-NA
Linear Group	Effective Modulus $E^{\ast }$/GPa	32.2	22.7	17.5	38.6	31.8	19.4
	Reference Gap $g_r$/mm	−0.05	−0.05	−0.05	−0.05	−0.05	−0.05
Parallel-Bond Group	Bond Effective Modulus ${\overline{E}}^{\ast }$/GPa	32.2	22.7	17.5	38.6	31.8	19.4
	Tensile Strength ${\overline{\sigma }}_c$/MPa	7.62	6.11	4.19	9.85	7.50	5.09
	Cohesion $\overline{c}$/MPa	26.7	20.6	14.5	33.6	24.7	16.1
Other Parameters		Take the corresponding value of the relevant NP

.

Proceeding to the calibration of parameters for OP and OP–NA interfaces, comprehensive parameter lists, initial values, and calibration status are documented in

Table 11. Parameter list, initial values, and calibration status for OP and OP–NA.

Parameters		OP	OP–NA
Linear Group	Effective Modulus $E^{\ast }$/GPa	34.5	31.8
	Friction Coefficient $\mu$	0.813	0.813
	Reference Gap $g_r$/mm	−0.05	−0.05
Parallel-Bond Group	Bond Effective Modulus ${\overline{E}}^{\ast }$/GPa	34.5	31.8
	Tensile Strength ${\overline{\sigma }}_c$/MPa	8.92	7.50
	Cohesion $\overline{c}$/MPa	29.4	24.7
	Friction Angle $\overline{\varphi }$/°	39.1	39.1
Other Parameters		Take the corresponding value of NP

Note: Bolded values are parameters to calibrate; others are fixed.

. Given close alignment of indentation modulus between OP and M5, as shown in Table 9, initial effective elastic moduli, tensile strengths, and cohesion values for OP and OP–NA were respectively assigned calibrated values of M5 in Table 8 and M5-NA in Table 10. Internal friction angles for OP and OP–NA were approximated by the measured repose angle of OP powder, while friction coefficients were set as tangents of these angles to ensure continuous frictional transition upon bond failure.

Abrasion testing and simulation of RA were conducted, as shown in Figure 12. RAs were placed in a Los Angeles abrasion machine without steel balls. After 100 revolutions, detached OP mass was quantified to calculate OP detachment rate. The bond strength ratio ${\overline{\sigma }}_c/\overline{c}$ was held constant to preserve failure mode, while cohesion ($\overline{c}$) and tensile strength (${\overline{\sigma }}_c$) of OP and OP–NA were synchronously scaled to discrete multiples (0.00, 0.25, 0.50, 0.75, 1.00, 1.50, 2.00, 2.50, 3.00) of initial values. This yielded an OP detachment rate versus bond parameter scaling factor correlation curve, as illustrated in Figure 12. Substituting the experimental OP detachment rate (7.47%) into this curve determined RAs’ bond parameter scaling factor and corresponding $\overline{c}$ and ${\overline{\sigma }}_c$ values. The scaling factor was 0.895—notably low due to two factors: (1) higher microcrack density in OP reduced overall performance versus M5; (2) the model’s inherent lack of damage accumulation due to time-invariant parameters. To replicate 100-revolution abrasion effects within initial simulated cycles, bond strength necessitated reduction below experimentally measured initial values.

Following the calibration of RA bond parameters, the tensile strength and cohesion of OP were calibrated to 7.98 MPa and 26.3 MPa, respectively, while those of the OP–NA interface were determined as 6.71 MPa and 22.1 MPa, respectively.

4. Mesoscopic DEM Modeling of MRC

Six groups of MRC specimen models with varying porosities were established using M1–M6. Specimens measured 100 mm × 100 mm × 100 mm, designated as: M1–20%, M2–25%, M5–30%, M3–20%, M4–20%, and M6–20%, following the “paste group name–porosity” nomenclature. All porosities fell within the conventional 15–35% range [12]. Identical modeling procedures were implemented across groups; M1–20% serves as the exemplar for DEM modeling methodology in this section. All modeling steps were executed via custom scripting in the FISH language.

4.1. RA Placement

RA placement was performed in two stages: NA was initially placed, then replaced in situ by RA. Linear contact model parameters for NA–wall and NA–NA interactions were approximated per

Table 12. Linear contact model parameters for NA–wall and NA–NA.

Parameters	NA–wall	NA–NA
Effective Modulus $E^{\ast }$/GPa	1.0	56.2
Stiffness Ratio $k^{\ast }$	1.00	0.25
Friction Coefficient $\mu$	0.479	0.875
Other Parameters	Default Values

. With the exception of stiffness ratio and NA–wall effective elastic modulus, all parameters were experimentally measured. The effective modulus for NA–NA contacts was assigned the nanoindentation modulus of NA. The NA–wall friction coefficient was derived from spring dynamometer measurements between loading platens and MRC specimens. The NA–NA friction coefficient was approximated as the tangent of the measured NA powder angle of repose, i.e., tan 41.2°.

NA was placed within the 100 mm × 100 mm × 100 mm target domain according to experimental gradation. Particles were allowed to settle under gravity, after which the time step was amplified to 1 s to enhance displacement under unbalanced forces—simulating vibration effects that densified the NA assembly. Surface compaction was applied at a force of 50 N, after which excess NA was removed. This yielded a densely packed NA specimen model, as shown in Figure 13. Cross-sectional analysis confirmed high compactness, with a simulated porosity of 42.3%, closely aligned with the measured value of 41.5%.

Gradation of simulated NA was verified, as shown in Figure 14, with particle sizes ranging from 9.56 mm to 36.15 mm and mean diameter of 16.22 mm. The particle size distribution closely aligned with test data.

Following the RA modeling methodology in Section 2, all NA were replaced in situ by RA of identical size and shape, as shown in Figure 15. OP exhibited random spatial distribution. OP content was 23.1%, represented by 84,708 discrete ball elements.

Parameters for linear contact models governing OP–wall, OP–OP, and OP–NA interactions were assigned as follows: (1) OP–wall contacts adopted parameters identical to those of NA–wall in Table 12. (2) OP–OP contacts utilized relevant parameters from OP in Table 11. (3) OP–NA contacts employed averaged values of OP–OP parameters and NA–NA parameters from Table 12. Bond parameters within OP and at OP–NA interfaces were assigned per calibration results in Section 3.2. Critical emphasis was placed on bond gap determination: bond gaps were determined individually per RA to ensure bonds existed exclusively between adjacent elements. Bond gaps between ball elements on RA surfaces were set to ball diameters. Per Table 2, five discrete diameters existed: 0.6 mm, 0.8 mm, 1.0 mm, 1.4 mm, and 2.0 mm, thus defining five distinct bond gaps. Their spatial distribution is illustrated in Figure 16, with total bonds numbering 214,037.

4.2. NP Placement

Voids within the RA simulation specimen were filled with 2.00 mm-diameter balls representing NP. NP particles were then randomly deleted via coordinate sampling until specimen porosity reached the design value. NP elements contacting OP and NA were designated as ITZ entity elements for OP–NP and NA–NP interfaces, respectively. All bonds adjacent to these ITZ elements were assigned corresponding ITZ parameters, constituting the mechanical ITZ model. Parallel bond model was applied to all NP-related bonds, with parameters assigned per calibration results in Section 3. Figure 17 displays the MRC specimen model of group M1–20%, comprising 15,741 NP balls, 84,708 OP balls, and 349,239 bonds.

The MRC model is composed of NA, OP, NP, and three types of ITZs derived from pairwise combinations of these phases, comprehensively encompassing all primary constituents of MRC specimens. Nearly all parameters were calibrated against experimental measurements, ensuring high accuracy. The cross-section of the specimen in Figure 18 exhibits sparse NP elements resulting from MRC’s macroporous structure necessitating thin internal NP layers, which causes the majority of NP elements to be converted into ITZ elements.

5. Investigation of the Compressive Behavior of MRC

5.1. Failure Mode Analysis

Uniaxial compression test simulations were performed on all established MRC specimen models. Similar failure modes were observed across groups; M1–20% serves as the representative case for analysis, as shown in Figure 19. Post-failure examination revealed extensive spalling of surface paste and aggregates, which was consistent with the phenomenon observed in actual compression tests.

Analysis of internal stress chains demonstrated progressive stress accumulation with increasing strain, as shown in Figure 20. Peak internal stress occurred at approximately 0.008 strain—coinciding with maximum specimen strength—after which localized stress residuals persisted despite overall stress reduction. The coefficient of variation (COV) for internal stress increased overall with rising strain. Due to the presence of residual stress, the COV remained substantial after peak strain. The mean COV value was 1.777, reflecting considerable heterogeneity in stress distribution within MRC.

Stress chain distribution at 0.008 strain exhibited concentration in central and peripheral arcuate zones, corresponding well with experimental failure patterns, as shown in Figure 21. Minimal stress was observed near top and bottom regions due to end constraints.

The analysis of bond failure mode revealed the following progression, as shown in Figure 22: The proportion of tensile failure increased progressively with strain, reaching 87.0% at 0.006 strain before decelerating. By 0.012 strain, tensile failures constituted 91.3% versus 8.7% shear failures. Consequently, tensile failure dominated specimen failure. Total bond failure rate accelerated nonlinearly with strain, exhibiting near-quadratic growth indicative of second-order effects. At 0.008 strain—corresponding to peak strength—the proportion of bond failure measured 1.4%. Thus, in the present model, critical bonds governing specimen strength represented merely 1.4% of total bonds, primarily concentrated in high-stress chain zones identified in Figure 21.

Failure progression analysis of distinct bond types is presented in Figure 23. Failure increments among distinct bond types exhibit differences with increasing strain, primarily attributed to significant initial variations in bond numbers. Subsequent analysis of bond failure proportion demonstrated near-identical failure levels across bond types below 0.004 strain. Beyond this threshold, failure severity followed the hierarchy: NP–OP > NP–NA > NP–NP > OP–OP > OP–NA. This sequence directly corresponds to the influence ranking of MRC components on compressive strength. The highest failure proportion in NP–OP interfacial bonds identifies the new–old paste interface as the weakest link, while minimal OP–NA bond failure indicates negligible influence of old paste–natural aggregate interfaces on compressive strength.

Collectively, all NP-associated bonds occupy higher positions in the failure severity hierarchy compared to other bond types. Consequently, NP exerts greater influence on compressive strength than RA. Mechanistically, significant failure of OP–OP bonds was initiated only at 0.008 strain—marking the onset of OP failure. Thus, OP contributed negligibly to strength prior to peak load. This limitation fundamentally arises because RA possesses no intrinsic strength under uniaxial compression without NP bonding; consequently, NP and its associated ITZ invariably sustain initial loading.

5.2. Deformation and Strength Analysis

Stress–strain curves of all MRC specimen groups were comparatively analyzed, as shown in Figure 24, where Point A denotes the elastic limit and Point B the peak stress. Figure 24a compares the simulated curves with the experimental curves obtained from the compression tests conducted in accordance with GB/T 50081-2019 [63]. Experimental curves revealed that both points A and B occurred later in MRC than in conventional concrete. For MRC, point A corresponded to 51.8–55.5% of peak stress, whereas it was approximately 40.0% in conventional concrete [65]—steel fiber-reinforced concrete, despite enhanced ductility, attained only 50.0% [66]. Peak strains ranged from 0.006 to 0.009 for MRC, contrasting with 0.002 [65] and 0.003 [66] for conventional and steel fiber-reinforced concretes, respectively. This behavior is attributable to progressive closure of abundant pores in MRC under compression, prolonging the elastic phase and elevating peak strain. Beyond point B, MRC exhibited a steeper post-peak descending portion than conventional concrete due to rapid crack propagation through high-porosity networks. Aside from these distinctions, the stress–strain development of MRC generally aligned with conventional concrete patterns.

Simulated curves exhibited delayed elastic limit point ${\mathrm{A}}^{\prime }$ at 79.6–88.2% of peak stress. This discrepancy arises because all interparticle contacts were purely elastic, with plasticity represented solely through spatially heterogeneous distribution of particle elements. Consequently, more particles can better replicate specimen plasticity. Despite employing 100,449 ball elements, the model inadequately replicated plasticity, resulting in abbreviated plastic phase A-B. The simulated and experimental curves exhibited close agreement before point B, while certain deviations existed after point B—consistent with mesoscale DEM simulation results for conventional concrete [36]. Overall, the two curves demonstrated consistent trends, indicating that the simulation curves can reflect reality to some extent.

Per ASTM C1018 [67] and the literature [66], toughness index and brittleness index were defined to evaluate MRC compressive toughness, as presented in Equations (6) and (7). The stress–strain curve area was partitioned into four zones (S₁–S₄) as diagrammed in Figure 24b. Higher toughness index denotes superior post-cracking load-bearing and energy absorption capacity. Higher brittleness index corresponds to slower decay of energy absorption capacity in MRC after cracking, indicating better preservation of toughness. Elevation of both indices enhances MRC service life.

$$I={\displaystyle \frac{S_1+S_2+S_3}{S_1}}={\displaystyle \frac{{\displaystyle {\int }_0^{3{\epsilon }_c}\sigma d\epsilon }}{{\displaystyle {\int }_0^{{\epsilon }_c}\sigma d\epsilon }}}$$

(6)

$$B={\displaystyle \frac{S_1+S_2+S_3+S_4}{S_1+S_2}}={\displaystyle \frac{{\displaystyle {\int }_0^{4{\epsilon }_c}\sigma d\epsilon }}{{\displaystyle {\int }_0^{2{\epsilon }_c}\sigma d\epsilon }}}$$

(7)

where $I$ is the toughness index; $B$ is the brittleness index; $\sigma$ is stress (MPa); $\epsilon$ is strain; ${\epsilon }_c$ is peak strain.

Analysis employed simulated values, with computational results presented in Figure 25. Conventional concrete and steel fiber-reinforced concrete typically exhibit toughness indices of 1–2 and 3–4, respectively [66]. The MRC demonstrated toughness indices of 2.234–2.948 and brittleness indices of 1.205–1.392, indicating favorable toughness and toughness preservation capacity. Toughness index ranking: M3–20% > M5–30% > M2–25% > M6–20% > M1–20% > M4–20%. Brittleness index ranking: M5–30% > M2–25% > M3–20% > M6–20% > M1–20% > M4–20%. Overall, both indices increased with rising porosity and declining compressive strength. This occurs because post-peak MRC does not disintegrate abruptly—crushed RA gradually fills pores, providing buffering. Buffering capacity increases with porosity. At identical porosity, lower MRC strength reflects inferior NP strength since skeletal structure strength remains constant. Consequently, a lower peak strength of MRC corresponds to a relatively higher contribution of strength provided by the skeletal structure in the post-peak regime. However, these trends are non-absolute; porosity and strength exhibit combinatorial effects. The M3–20% group, characterized by a porosity of 20% and a compressive strength of 10.2 MPa, achieved the highest toughness index among all tested groups.

Research has established a strong correlation between peak stress and peak strain in concrete [65,68]. This study performed regression calibration based on the classical power function equation [69], as illustrated in Figure 26. The predictive formula for peak strain in MRC is given by Equation (8), with calculation errors ranging from 0.5% to 5.3%.

$${\epsilon }_c=205f_c^{0.5}\times {10}^{-5}, R^2=0.916$$

(8)

where ${\epsilon }_c$ is peak strain; $f_c$ is compressive strength (MPa).

Compressive strength, elastic modulus, and Poisson’s ratio for all specimen groups are presented in Figure 27. Per ASTM C469/C469M-22 [70], elastic modulus adopted the secant modulus at 40% compressive strength. Test and simulated values demonstrated close agreement for both compressive strength and elastic modulus. Compressive strength errors measured 4.1% (M1–20%), 3.7% (M2–25%), and 3.6% (M5–30%), while elastic modulus errors were 23.4%, 14.2%, and 16.9%, respectively. The elastic modulus exhibits a higher error compared to the compressive strength. This discrepancy can also be attributed to the limited number of particles, which constrains the relative proportions of the elastic and plastic stages prior to peak stress.

Analysis utilized simulated values. Rankings for compressive strength and elastic modulus were: M4–20% > M1–20% > M6–20% > M2–25% > M3–20% > M5–30%. Comparing compressive strengths: M1–20% exceeded M3–20% by 5.0 MPa; M4–20% exceeded M6–20% by 3.8 MPa. A high water–binder ratio weakened strength, though high silica fume content alleviated this adverse effect. Conversely, M1–20% measured 0.9 MPa lower than M4–20%; M3–20% measured 2.1 MPa lower than M6–20%. High silica fume content enhanced strength, while low water–binder ratio mitigated the strengthening effect of silica fume. Elastic modulus followed identical trends. Poisson’s ratio ranking was: M1–20% ≈ M3–20% ≈ M4–20% ≈ M6–20% > M2–25% > M5–30%. MRC exhibited lower Poisson’s ratios (0.12–0.17) versus conventional concrete (about 0.20) [71]. Poisson’s ratio was strongly correlated with porosity—higher porosity led to a lower Poisson’s ratio, as pore structures constrained capacity for transverse deformation, promoting compaction-dominated densification with reduced lateral strain.

Equation (3) proved inapplicable for elastic modulus calculation in MRC, exhibiting no stable $\beta$ value. This limitation arises because Equation (3) neglects the influence of pores on the elastic modulus of MRC, necessitating new formulations. Linear regression of test and simulated compressive strengths/elastic moduli for M1–20%, M2–25%, and M5–30% yielded Equations (9) and (10). Regression-derived approximations for M3–20%, M4–20%, and M6–20% served as test proxies in

Table 13. Regression and calculation of compressive strength and elastic modulus across MRC groups.

Groups	Compressive Strength/MPa		Elastic Modulus/MPa			Calculation Error of Modulus/%	Mean Error/%
	Test	Simulation	Test	Simulation	Calculation
M1–20%	14.6	15.2	3728.0	2856.4	3782.3	1.5	3.9
M2–25%	10.9	11.3	2343.0	2010.5	2541.5	8.5
M5–30%	8.3	8.6	1866.5	1550.9	1756.3	5.9
M3–20%	(9.8)	10.2	(2230.6)	1784.5	2087.2	6.4
M4–20%	(15.5)	16.1	(4187.1)	3349.7	4139.1	1.1
M6–20%	(11.8)	12.3	(2756.8)	2205.4	2763.9	0.3

Note: The values in parentheses represent the regression-derived estimates.

.

$$E_{reg}=1.250E_{sim}, R^2=0.996$$

(9)

$$f_{reg}=0.963f_{sim}, R^2=0.999$$

(10)

where $E_{reg}$ and $E_{sim}$ are the regressed and simulated elastic modulus of the MRC specimen, respectively (MPa); $f_{reg}$ and $f_{sim}$ are the regressed and simulated cube compressive strength of the MRC specimen, respectively (MPa).

Based on Equation (3), regression analysis was performed on the tested values of compressive strength and elastic modulus for all specimen groups; thus, a formula for calculating the elastic modulus of MRC, namely Equation (11), was proposed. The calculated elastic modulus values for each group were entered into Table 13, with errors ranging from 0.3% to 8.5% and a mean value of 3.9%. Equation (11) demonstrated high accuracy.

$$E=\alpha f_c^{1.5}\left(1+V\right)$$

(11)

where $E$ is the calculated elastic modulus of the MRC specimen (MPa); $f_c$ is the cube compressive strength of the MRC specimen (MPa); $\alpha$ is a coefficient (taken as 56.5 herein); $V$ is the porosity of the MRC specimen.

Based on Equation (11), a positive correlation was observed between porosity and elastic modulus. This phenomenon occurred because both elastic modulus and compressive strength decreased with increasing porosity; however, compressive strength primarily depends on the integrity of the specimen, while elastic modulus primarily depends on the stiffness and continuity of the skeleton. Consequently, the magnitude and sensitivity of the reduction in elastic modulus were typically lower than those of compressive strength. In Equation (11), porosity was employed as a dimensionless number to compensate for this difference in reduction magnitude; the higher the porosity, the greater this difference became, and thus the larger the required compensation value.

6. Conclusions

In this study, PFC 3D was utilized to establish a randomly polyhedral RA composite model and an MRC model with stabilized bond quantities. A systematic methodology for parameter testing and calibration was proposed. Compressive behavior of MRC was investigated based on the developed models, with key conclusions summarized below:

(1)

The DEM modeling approach demonstrates high compatibility with MRC. It can simulate all components of MRC, including three types of ITZs, readily replicate pore structures, and achieve dense packing with an aggregate volume fraction of 57.7%. Errors in simulating compressive strength and elastic modulus were 3.8% and 18.2%, respectively.

(2)

The stress–strain curves of MRC and conventional concrete share similar trends; nevertheless, MRC exhibits larger strain and a steeper post-peak descending portion. At peak stress, stress is primarily concentrated in the central region and the arc-shaped zones surrounding it (COV = 1.729). After peak stress, significant localized residual stress persisted within the specimen, leading to a slight increase in the COV. Overall, the post-peak toughness and toughness retention capacity of MRC increase with rising porosity and decreasing compressive strength.

(3)

Failure of MRC specimens is dominated by tension rather than shear. Critical bonds determining specimen strength account for only 1.4% of total bonds, predominantly distributed in regions with dense force chains and high load concentrations. The influence ranking of different MRC components on compressive strength is: NP–OP > NP–NA > NP–NP > OP–OP > OP–NA. The influence of NP on compressive strength is more significant than that of RA.

(4)

The Poisson’s ratio of MRC (0.12–0.17) is lower than that of conventional concrete and negatively correlates with porosity. The prediction formula for peak strain and the calculation formula for elastic modulus (with porosity integrated as a parameter) were established for MRC, with respective errors of 2.6% and 3.9%.

Three limitations were noted in the present model: (1) damage accumulation was not considered; (2) the parallel bond model exhibited inherent limitations in replicating the plasticity of MRC; (3) the particle sizes of NP (2.0 mm) and ITZs (0.6–2.0 mm) constrained simulation accuracy and resolution. Nevertheless, this model established a novel pathway for investigating mechanical properties and mesoscopic damage mechanisms in MRC. Future studies could extend this framework to conduct mesoscopic simulations of MRC durability—including fatigue, freeze–thaw cycles, carbonation, and corrosion—as well as simulations under complex coupled loading conditions.