Optimization-Driven Evaluation of Multilayer Graphene Concrete: Strength Enhancement and Carbon Reduction Through Experimental and Mathematical Integration

Abstract

The integration of nanoengineered materials into concrete systems has emerged as a promising strategy for enhancing structural performance and sustainability. This study presents a hybrid experimental-analytical investigation into the use of multilayer graphene as a smart admixture in high-performance concrete. The research combines mechanical testing, microstructural characterization, and a multi-objective optimization model to determine the optimal graphene dosage that maximizes strength gains while minimizing carbon emissions. Concrete specimens incorporating multilayer graphene (ranging from 0.01% to 0.10% by weight of cement) were tested over 7 to 90 days for compressive, tensile, and flexural strengths. Simultaneously, X-ray diffraction, scanning electron microscopy, and energy-dispersive X-ray analyses revealed crystallinity enhancement, pore densification, and favorable elemental redistribution due to graphene inclusion. A normalized composite objective function was formulated to balance three maximization targets—compressive, tensile, and flexural strength—and one minimization goal—carbon emission. The highest objective score (Z = 1.047) was achieved at 0.10% graphene dosage, indicating the optimal balance of strength performance and environmental efficiency. This dual-framework study not only confirms graphene’s reinforcing effects experimentally but also validates the 0.10% dosage through mathematical scoring. The outcomes position of multilayer graphene as a powerful additive for high-strength, low-carbon concrete, especially suited for infrastructure in hot and arid environments. The proposed optimization approach provides a scalable pathway for performance-based graphene dosing in future innovative concrete formulations.

1. Introduction

Concrete remains the most widely used construction material globally due to its affordability, availability, and mechanical strength [1]. However, it is inherently limited by its low tensile strength, susceptibility to cracking, and high carbon footprint during production, mainly due to cement manufacturing, which alone accounts for over 7% of global CO₂ emissions [2]. These restrictions are especially noticeable in areas subjected to extreme weather conditions, such as hot and arid climates, where the thermostat’s increased hydration and shrinkage compromise their long-term endurance. As a result, researchers are looking more closely at nanotechnology-based approaches to improve the sustainability and performance [3].

Multilayer graphene (MLG), one of the most promising nanomaterials, has become a game-changing addition because of its remarkable area, mechanical strength, and capacity to chemically bind with hydration products [4] in comparison to the four other types of graphene [5], including single-layer graphene, graphene oxide, reduced graphene oxide (rGO), and graphene Nano Platelets (GNPs). By filling gaps and catalyzing the development of calcium silicate hydrate (C-S-H) gel, MLG’s two-dimensional sheet-like shapes contribute to improved mechanical performance and a finer microstructure [6].

MLG provides an affordable, structurally sound, and industrially scalable substitute for single-layer graphene oxide, which is frequently expensive and unstable in the absence of dispersants. Beyond just increasing strength, MLG has the potential to be a sustainable addition. By enabling a lower cement content or obtaining greater strength at equal binder loads, when appropriately distributed, it can help create a more carbon-efficient design. However, there is still a significant research gap in determining the ideal MLG dosage that strikes a compromise between mechanical advantages and environmental goals. Preceding research has frequently concentrated on unidimensional objectives, such as durability or strength improvements, without using a multi-objective optimization framework that incorporates sustainability and performance.

The recent study combines a quantitative optimization model with experimental validation to report this. The goals to assess how MLG affects microstructure and mechanical performance while scientifically determining the ideal quantity of graphene in terms of strength and carbon footprint [7]. For engineers and researchers targeting to use graphene-enhanced concrete in high-performance, climate-resilient infrastructure, the integrated approach is intended to provide practical insights.

1.1. Literature Gaps and Rationale

The advantage of adding nanoparticles to concrete to overcome its limitations in strength, durability, and crack resistance has been conveyed by an increasing amount of research in recent years [8]. Graphene and its derivatives, especially reduced graphene oxide (rGO), have shown exceptional promise among these nanomaterials for enhancing the hydration dynamics and mechanical properties of cement-based composites. Researchers have demonstrated that GO can improve bonding at the nanoscale, modify the pore structure, and speed up the synthesis of hydration products [9], leading to improved water impermeability, shrinkage resistance, and compressive and tensile strength. GO is a good electric conductor, but a graphene single layer helps enhance the compressive strength, which is relatively lower than that of graphene. Graphene oxide (GO) provides good conductivity, while single-layer graphene improves compressive strength but to a lesser extent than multilayer graphene. However, both typically require complex dispersion methods, costly functionalization, or high-purity forms, which limit their scalability for practical applications [10].

In May 2021, the first slab of the Southern Quarter Gym was poured using graphene (single layer), as an additive in concrete by University of Manchester [11]. However, graphene (multi-layer) is easier to produce and disperse [9]. Liquid graphene has been used, which is relatively easier to mix in concrete than powder graphene, which has more challenges [12]. Graphene has been used in cold climates so far, but no clear evidence of its use in hot and arid climates. To overcome these challenges, recent workings have shifted toward MLG, a more cost-effective and structurally stable alternative. MLG retains key performance attributes—such as high aspect ratio, mechanical stiffness, and chemical reactivity—while being easier to produce and integrate into cementitious matrices [13].

Despite its advantages, the literature remains scarce on the role of MLG in concrete, especially concerning its effects on long-term mechanical performance, hydration crystallinity, and sustainability indicators such as carbon emissions. Most existing studies are limited to single-objective approaches, such as improving compressive strength or observing microstructure, without integrating multiple performance and sustainability parameters into a unified framework. This restricts their ability to guide practical dosage design [14].

Moreover, no comprehensive study to date has applied a multi-objective optimization framework that simultaneously considers both mechanical improvements and sustainability goals [15]. Most previous studies have focused on individual properties, such as compressive strength or durability, without integrating embodied carbon emissions into the analysis [10]. Furthermore, research specifically addressing multilayer graphene (MLG) in hot and arid environments remains very limited, despite the fact that these regions are highly vulnerable to shrinkage, thermal stresses, and rapid moisture loss. The present study addresses these gaps by experimentally evaluating MLG-enhanced concrete and by proposing a composite Z-score model that integrates strength performance with carbon emission reduction. The present study fills this gap by introducing a multi-objective optimization model that balances mechanical performance (compressive, tensile, and flexural strengths) with sustainability performance (carbon emission reduction). This integrated framework differs fundamentally from previous single-objective approaches by enabling simultaneous decision-making across both structural and environmental criteria.

This research aims to fill these gaps by combining experimental analysis with a multi-objective optimization model that identifies the most efficient MLG dosage [16]. Through integrated performance evaluation, mechanical testing, SEM/XRD/EDAX characterization, and mathematical modeling, this study not only provides empirical evidence of MLG’s efficacy but also delivers a robust decision-making tool for sustainable concrete design [12].

1.2. Multilayer Graphene as a Smart Nano-Admixture

MLG comprising two or more stacked graphene sheets, has emerged as a viable nanomaterial for structural applications due to its high surface area, mechanical stiffness, and ease of production [17]. Unlike single-layer graphene or graphene oxide, MLG requires minimal functionalization and can be incorporated into cementitious systems with relatively simple mixing protocols. Flake-like morphology enables it to fill voids in the matrix physically, bridge micro-cracks, and reinforce the interfacial transition zone (ITZ), thereby improving early and long-term mechanical performance. MLG also serves as a nucleation platform for hydration products such as calcium silicate hydrate (C-S-H), accelerating hydration kinetics and defining the pore structure [18].

In addition to mechanical enhancements, MLG contributes to microstructural and crystallographic refinement. XRD analysis has shown intensified peaks for C-S-H and portlandite, while SEM and EDAX confirm a denser matrix and favorable elemental redistribution. These changes result in a more cohesive and durable concrete with improved resistance to shrinkage, cracking, and chemical ingress [19]. However, the performance benefits of MLG are highly dependent on dosage. Beyond our threshold, graphene particles may agglomerate, reducing dispersion quality and compromising strength. Thus, this study not only investigates MLG’s multifunctional contributions but also introduces a quantitative optimization framework to identify its optimal dosage, balancing strength enhancements with carbon emission reduction [7,20].

1.3. Research Contributions and Objectives

These studies deal a novel contribution to the field of advanced cementitious materials by uniting comprehensive experimental testing with a robust mathematical optimization framework to evaluate multilayer graphene MLG as a multifunctional nano admixture in concrete [21].

While prior research has validated the isolated benefits of graphene derivatives in enhancing concrete performance, this work is distinctive in that it systematically integrates mechanical testing, microstructural characterization, and sustainability matrix into an integrated decision-making model. By incorporating multi-objective optimization, the study bridges the gap between empirical observations and performance-based material design, an area disregarded mainly in graphene-enhanced concrete research [22].

• The primary objective is to identify the optimum MLG dosage (0.01–0.1%) that instantaneously maximizes compressive tensile and flexural strength while minimizing carbon emissions associated with cement content. This is achieved through the development and evaluation of a composite objective function that incorporates stabilized strength ratios and emissions reduction.
• The research also aims to explain the essential microstructural mechanisms accountable for the observed enhancements using XRD, SEM, and EDAX techniques. The specific objectives are as follows [23].
• To experimentally evaluate the mechanical performance of MLG-enhanced concrete across multiple curing durations (7, 28, 56, and 90 days).
• Characterizing microstructural changes in graphene concrete using XRD, SEM, and EDAX, linking them to strength and durability gains.
• Conveying and solving a multi-objective optimization equation for strength maximization and carbon reduction, producing a quantitative Z-score for each graphene dosage.
• To determine the optimal graphene concentration for structural and environmental performance, validated through experimental and mathematical convergence [24].

These contributions are designed to offer both scientific insight and practical guidance for the application of graphene concrete in sustainable, high-performance infrastructure, especially in thermally extreme environments [25].

2. Materials and Experimental Methods

2.1. Materials and Graphene Selection Rationale

The experimental work employed standard materials conforming to ASTM and ACI specifications to ensure consistency and replicability. Ordinary Portland Cement (OPC) of Type I (MapleLeaf Cement Factory Limited, Iskandarabad, Mianwali, Pakistan) was used as the primary binder. Silica fume was added at 6% by weight of cement to enhance packing density and reduce porosity. Locally sourced Lawrencepur river sand (fineness modulus = 2.48) served as the fine aggregate, while crushed coarse aggregates were divided into two size ranges: 20–16 mm (40%) and 12–05 mm (60%). A high-range water-reducing admixture, BASF MasterBuilder^® 858 supplied by BASF Pakistan (Private) Ltd., Karachi, Pakistan was used at 1.2% of cementitious material to improve workability. The water-to-cement (w/c) ratio was fixed at 0.30, targeting a compressive strength of 8000 psi (approximately 55.16 MPa).

The novel component in this study was MLG supplied by XG sciences, Lansing, Michigan (MI), USA incorporated as a dry powder without dispersants to evaluate its unmodified effect on concrete performance. MLG used in the experiments had an average particle thickness between 5 and 10 nanometers and a lateral flake size of 2–5 μm. This form of graphene offers a compromise between structural enhancement and practical scalability, enabling reinforcement through both crack-bridging and hydration acceleration mechanisms [26,27].

The graphene incorporation levels tested were 0.01%, 0.03%, 0.05%, 0.07%, and 0.10% by weight of cement. No mixes above 0.10% were prepared, as prior research indicates dispersion challenges and diminishing returns at higher dosages. Graphene was dosed in varying concentrations—0.01%, 0.03%, 0.05%, 0.07%, and 0.10% by weight of cement—to examine its impact across a full spectrum of performance thresholds. The control mix contained no graphene. All materials were stored in dry conditions and brought to room temperature before mixing to eliminate variability due to moisture or thermal effects.

This careful selection of ingredients and control of mix parameters formed the source for the comparative evaluation of both physical and chemical performance attributes across different graphene-enhanced mixes.

2.2. Concrete Mix Design

The concrete mix was designed following the guidelines of ACI 211.1, ACI 211.4, and ACI 301, along with relevant ASTM standards, including ASTM C29/C29M [28], ASTM C33 [29], ASTM C127 [30], ASTM C143 [31], and ASTM C39 [32], for strength testing. The target mean compressive strength was set at 8000 psi (≈55.16 MPa), with a water-to-cement ratio (w/c) of 0.30. The slump was maintained within the range of 175 ± 25 mm to ensure consistent workability across all mixes.

Each 1 m³ batch included the following proportions, as shown in Table 1:

• Cement: 580 kg.
• Silica Fume: 35 kg (6% by weight of cement).
• Water: 174 kg (adjusted based on aggregate moisture content).
• Coarse Aggregate: 20–16 mm (441 kg), 12–05 mm (661 kg).
• Fine Aggregate: 532 kg (100% Lawrencepur sand).
• Admixture: BASF MasterBuilder^® 858, at 1.2% of binder content.
• Graphene (MLG): 0.00–0.10% by weight of cement (varied).

In this study, MLG was incorporated as a dry powder, without the use of surfactants or chemical dispersants, to evaluate its unmodified effect on concrete performance. The graphene powder was pre-weighed, manually dry-blended with cement and silica fume for approximately 3 min and then introduced into the mechanical pan mixer with aggregates and water. The total mixing time was maintained at 5 min to promote uniform dispersion. Although this procedure ensured consistent blending across mixes, it is acknowledged that graphene sheets may still tend to agglomerate at higher dosages due to Van der Waals forces. This limitation highlights the importance of dispersant-aided methods, which are recommended for future studies.

Table 1. Concrete mix design (per 1 m³ of concrete, as per ACI 211.1 and ASTM standards).

Ingredient		Weight (kg)	Volume (m3)	Specific Gravity
Cement (OPC, MapleLeaf)	580	0.1841	3.15
Silica fume (6% of cement)	35	0.0158	2.20
Water	174 (187 adj.)	0.1740	1.00
Coarse aggregate (20–16 mm)	441	0.1609	2.74
Coarse aggregate (12–05 mm)	661	0.2426	2.73
Fine aggregate (Lawrencepur sand)	532	0.1970	2.70
Admixture (BASF MasterBuilder 858, 1.2%)	6.96		0.0056	1.24
Graphene (MLG)	0.00–0.10% by wt. of cement		—	—

Note: Water content was adjusted for aggregate moisture. Graphene dosage varied between 0.00% and 0.10% by weight of cement across mixes.

Table 1. Concrete mix design (per 1 m³ of concrete, as per ACI 211.1 and ASTM standards).

Ingredient		Weight (kg)	Volume (m3)	Specific Gravity
Cement (OPC, MapleLeaf)	580	0.1841	3.15
Silica fume (6% of cement)	35	0.0158	2.20
Water	174 (187 adj.)	0.1740	1.00
Coarse aggregate (20–16 mm)	441	0.1609	2.74
Coarse aggregate (12–05 mm)	661	0.2426	2.73
Fine aggregate (Lawrencepur sand)	532	0.1970	2.70
Admixture (BASF MasterBuilder 858, 1.2%)	6.96		0.0056	1.24
Graphene (MLG)	0.00–0.10% by wt. of cement		—	—

Note: Water content was adjusted for aggregate moisture. Graphene dosage varied between 0.00% and 0.10% by weight of cement across mixes.

Graphene was dry-blended with cementitious materials to avoid surfactant influence, allowing for pure evaluation of MLG effects. All components were thoroughly mixed using a mechanical pan mixer, with a total mixing time of 5 min. The aggregates were pre-conditioned, and their moisture content was recorded to adjust the mixing water precisely. The testing details shown in

Table 2. Total number of testing.

Description	Cube 150 mm × 150 mm × 150 mm Cylinder 150 mm × 300 mm Prisms 100 mm × 100 mm × 500 mm
Testing After	7 Days	28 Days	56 Days	90 Days
Control Concrete Cubes	3	3	3	3
Graphene Concrete Cubes	3	3	3	3
Control Concrete Cylinders	3	3	3	3
Graphene Concrete Cylinders	3	3	3	3
Control Concrete Prisms	3	3	3	3
Graphene Concrete Prisms	3	3	3	3

.

For each mix variant—including the control and five graphene dosages—specimens were prepared in triplicate for each of the following types:

• Cubes: 150 mm × 150 mm × 150 mm (for compressive strength).
• Cylinders: 150 mm diameter × 300 mm height (for split tensile strength).
• Prisms: 100 mm × 100 mm × 500 mm (for flexural strength).

All specimens, as shown in Figure 1, were demolded after 24 h and cured in water at 23 ± 2 °C until the designated testing age of 7, 28, 56, and 90 days. This curing regime ensured standardized hydration conditions for strength development and microstructural evolution.

This rigorous batching and curing protocol ensured consistency across mixes, enabling the isolation of graphene’s influence on strength, durability, and microstructure.

2.3. Experimental Testing Procedures

A series of standardized tests were conducted following ASTM specifications to evaluate the mechanical performance of concrete modified with multilayer graphene (MLG). Three primary strength parameters were assessed: compressive strength, split tensile strength, and flexural strength. All specimens were cured in water at 23 ± 2 °C and 95 ± 5% relative humidity until testing, as shown in Figure 2. The laboratory environment during testing was maintained at 22–25 °C, ensuring consistency across all specimens. Each mix—including the control (0%) and graphene dosages of 0.01%, 0.03%, 0.05%, 0.07%, and 0.10% by weight of cement—was tested at curing ages of 7, 28, 56, and 90 days (n = 3 specimens per condition). This ensured a consistent comparison across dosage levels.

2.3.1. Compressive Strength Testing

Compressive results are summarized in Figure 3 and

Table 3. Cube compressive strength (average of n = 3 specimens, with graphene dosage and curing age).

Graphene Dosage (% by wt. of Cement)	Curing Age (Days)	Force (kN)	Compressive Strength (MPa)
0.00 (Control)	7	958.6	42.60
0.00 (Control)	28	1302.9	57.91
0.00 (Control)	56	1443.6	64.16
0.00 (Control)	90	1647.1	73.20
0.01	7	867.4	38.55
0.01	28	1082.6	48.12
0.01	56	1346.9	59.86
0.01	90	1599.5	71.09
0.03	7	975.6	43.36
0.03	28	1181.9	52.53
0.03	56	1270.3	56.46
0.03	90	1629.4	72.42
0.05	7	958.6	42.60
0.05	28	1443.6	64.16
0.05	56	1333.8	59.28
0.05	90	1692.2	75.21
0.07	7	975.6	43.36
0.07	28	1181.9	52.53
0.07	56	1270.3	56.46
0.07	90	1629.4	72.42
0.10	7	1270.3	56.46
0.10	28	1246.8	55.41
0.10	56	1333.8	59.28
0.10	90	1867.2	82.99

, and the dosage trend is shown in Figure 3; the test setup is shown in Figure 4. Compressive strength was measured using 150 mm × 150 mm × 150 mm cube specimens following ASTM C39/C39M [32]. Specimens were tested using a universal compression testing machine with a loading rate of 0.25 MPa/sec until failure. The average of three cubes per mix per curing age was recorded as the final compressive strength. These results provided insight into the hydration development and mechanical reinforcement effects induced by the MLG shown in Figure 3.

The attached graph and table show that the Compressive Strength Test results for control and graphene concrete demonstrate the performance improvements of adding Multilayer graphene (MLG) over multiple curing periods (7, 28, 56, and 90 days).

The large increase observed between 7-day and 28-day strengths is attributed to accelerated hydration in the early curing stage, where MLG provides nucleation sites for C–S–H formation. In Figure 5, ‘C’ is used for control concrete and ‘G’ for graphene concrete. However, both showed incremental trends for compressive strength. Beyond 28 days, the rate of gain slows as hydration reactions approach stabilization, leading to more gradual improvements at 56 and 90 days.

2.3.2. Split Tensile Strength Testing

Split tensile strength was evaluated using 150 mm diameter × 300 mm height cylindrical specimens as per ASTM C496/C496 M [33]. The cylinders were subjected to diametral compressive loading until failure, and the average of three samples was used to report the strength for each curing age. This test was critical to understanding MLG’s effectiveness in bridging microcracks and enhancing the tensile resistance of concrete. The test setup is shown in Figure 6, and the results are presented in Figure 7 and

Table 4. Cylinder split tensile strength (average of n = 3 specimens, with graphene dosage and curing age).

Graphene Dosage (% by wt. of Cement)	Curing Age (Days)	Force (kN)	Split Tensile Strength (MPa)
0.00 (Control)	7	44.55	1.98
0.00 (Control)	28	46.58	2.07
0.00 (Control)	56	176.7	2.50
0.00 (Control)	90	231.6	3.28
0.01	7	44.33	1.97
0.01	28	45.68	2.03
0.01	56	284.2	4.02
0.01	90	270.0	3.82
0.03	7	43.88	1.95
0.03	28	46.13	2.05
0.03	56	290.0	4.10
0.03	90	301.4	4.26
0.05	7	44.55	1.98
0.05	28	45.68	2.03
0.05	56	242.5	3.43
0.05	90	342.6	4.85
0.07	7	46.58	2.07
0.07	28	46.13	2.05
0.07	56	343.7	4.86
0.07	90	318.6	4.51
0.10	7	45.68	2.03
0.10	28	46.58	2.07
0.10	56	335.7	4.75
0.10	90	362.6	5.13

.

Results of Tensile Strength of Concrete

The tensile strength of graphene-enhanced concrete consistently surpasses that of control concrete across all curing stages. While the initial improvement at seven days is relatively small, the strength gap widens as curing progresses, with graphene concrete outperforming control concrete by 13.6% at 90 days shown in Figure 7. These results indicate that graphene Multilayer positively influences tensile strength, contributing to early and long-term improvements in the material’s ability to resist tensile stresses. This enhancement is important for applications requiring tensile resistance, although the improvement is more gradual because concrete remains inherently brittle and MLG primarily contributes through micro-crack bridging rather than fundamental ductility change.

2.3.3. Flexural Strength Testing

Flexural strength was measured using prism specimens (100 mm × 100 mm × 500 mm) according to ASTM C293/C293M (third-point loading method). The test setup is provided in Figure 8, and results are summarized in Figure 9 and

Table 5. Prism flexural strength ASTM c293 (average of n = 3 specimens, with graphene dosage and curing age).

Graphene Dosage (% by wt. of Cement)	Curing Age (Days)	Force (kN)	Flexural Strength (MPa)
0.00 (Control)	7	20.1	4.02
0.00 (Control)	7	23.6	4.72
0.00 (Control, Avg.)	7	—	4.37
0.00 (Control)	28	30.9	6.18
0.00 (Control)	28	32.6	6.52
0.00 (Control, Avg.)	28	—	6.35
0.00 (Control)	56	40.6	8.12
0.00 (Control)	56	31.4	6.28
0.00 (Control, Avg.)	56	—	7.20
0.00 (Control)	90	37.1	7.42
0.00 (Control)	90	38.9	7.78
0.00 (Control, Avg.)	90	—	7.60
0.01	7	26.8	5.36
0.01	7	31.2	6.24
0.01 (Avg.)	7	—	5.80
0.01	28	45.0	9.00
0.01	28	40.0	8.00
0.01 (Avg.)	28	—	8.50
0.01	56	49.6	9.92
0.01	56	47.3	9.46
0.01 (Avg.)	56	—	9.69
0.01	90	56.9	11.38
0.01	90	54.6	10.92
0.01 (Avg.)	90	—	11.15

. This test captures the concrete’s resistance to bending, which is crucial for applications like pavements, beams, and slabs. The test also reflects the ductile enhancement potential of MLG under tensile and flexural stress zones.

As detailed in the attached graph and table, the Flexural Strength Test results for control and graphene concrete illustrate concrete’s flexural strength improvement when Multilayer graphene (MLG)is added. The test followed ASTM C293 [34] standards using 150 mm × 150 mm × 500 mm prism specimens. The tests were conducted at curing intervals of 7, 28, 56, and 90 days.

Results of Flexural Strength

Across all curing periods, the graphene-enhanced concrete consistently outperforms the control concrete in terms of flexural strength. The most substantial improvement occurs at the 90-day mark, with graphene concrete achieving 11.15 MPa compared to 7.60 MPa for the control concrete, showing a 46.7% increase. These results indicate that the addition of Multilayer graphene (MLG)significantly enhances the flexural strength of concrete, making it more resistant to bending and cracking stresses over time.

The steady improvement in flexural strength from 7 days to 90 days reflects the ability of Multilayer graphene (MLG) to integrate well with the cement matrix shown in Figure 9, promoting better bonding and crack resistance. This makes graphene-enhanced concrete particularly useful for applications with critical flexural performance, such as pavements, beams, and structural components subject to bending forces.

Conclusion for Strength Testing

The final graph, comparing the percentage increase in compressive, flexural, and tensile strengths between traditional (control) concrete and graphene-enhanced concrete, clearly demonstrates the substantial improvements by adding a graphene Multilayer. Across all strength parameters (compressive, flexural, and tensile), graphene-enhanced concrete consistently outperforms traditional concrete, with notable percentage increases observed throughout the curing periods of 7, 28, 56, and 90 days.

Regarding compressive strength, graphene-enhanced concrete shows a 27.33% increase at seven days, which rises to 30.06% at 28 days and 38.53% at 56 days. At the 90-day mark, the compressive strength increase reaches 40.96%. This consistent improvement highlights the graphene multilayer’s effectiveness in enhancing concrete’s compressive performance, particularly as the curing period extends.

For flexural strength, the percentage increase at seven days is 32.72%, climbing to 33.86% at 28 days and 34.58% at 56 days. By 90 days, the flexural strength improvement reaches 46.71%. This significant increase indicates that Multilayer graphene (MLG) is critical in improving the concrete’s ability to resist bending stresses over time.

Regarding tensile strength, the percentage increase is more gradual, starting at 4.2% after seven days, rising to 6.9% at 28 days, and further to 7.6% at 56 days. At 90 days, the tensile strength improvement is 13.6%, which is lower than the increases observed for compressive and flexural strength, but still demonstrates the benefits of Multilayer graphene (MLG) in enhancing the concrete’s resistance to tensile forces shown in Figure 10.

2.3.4. Data Recording and Normalization

Each test value was recorded alongside failure mode observations. The test setup is provided in Figure 8, and results are summarized in Figure 9 and Table 5. For optimization analysis, the raw strength values were normalized relative to control mix (0% graphene) values for compressive, tensile, and flexural strengths. These normalized ratios were later used in a multi-objective equation to compute a composite performance score (Z-score) for each graphene dosage.

All mechanical tests were performed under controlled laboratory conditions, and strict quality control was maintained during casting, demolding, curing, and testing to ensure repeatability and minimize data variation.

2.4. Microstructural Analysis

To complement mechanical testing and provide deeper insights into the internal behavior of graphene-enhanced concrete, microstructural characterization was conducted using X-ray diffraction (XRD), scanning electron microscopy (SEM), and energy-dispersive X-ray spectroscopy (EDAX). These techniques were applied to specimens taken from both the control and the optimized graphene mix (0.10%) at 90 days of curing when the differences in hydration and crystallinity are most pronounced.

2.4.1. X-Ray Diffraction

XRD analysis was performed using a Cu-Kα source (λ = 1.5406 Å) to identify the crystalline phases formed in the cement matrix. Scanning was performed between 5° and 70° 2θ with a step size of 0.02°. The main phases identified were calcium silicate hydrate (C-S-H), portlandite (Ca(OH)₂), and tricalcium silicate (C₃S). In the 0.10% MLG mix, the XRD pattern showed an approximate 15–20% increase in C–S–H peak intensity and a ~12% increase in portlandite compared to the control, along with a ~10% reduction in C₃S peak intensity. These relative shifts indicate accelerated hydration and enhanced crystallinity in the graphene-modified concrete. In the 0.10% MLG sample, the C–S–H and portlandite peaks showed visibly higher intensity compared to the control, while the C₃S peak was relatively reduced. These relative changes suggest accelerated hydration and a denser microstructure. Although the peak intensity was not quantified numerically, the observed shifts are consistent with previous studies reporting graphene-induced hydration enhancement shown in Figure 11.

2.4.2. Scanning Electron Microscopy

SEM imaging was conducted at magnifications ranging from 500× to 5000×. The control concrete revealed a heterogeneous surface with visible micro cracks, voids, and loosely bonded particles. In contrast, the 0.10% graphene concrete exhibited a more compact and uniform microstructure with well-integrated hydration products and fewer voids. As shown in Figure 12 and detailed in

Table 6. SEM Image Detail.

SEM Image	Magnification X	Image Width (µm)	Scale (µm)	Key Observation
Control (C)	500×	229.0	10	Visible voids and microcracks, loose hydration products
Control (C)	1000×	110.0	2	Non-uniform ITZ, micro-pores
MLG 0.10%	500×	229.0	10	Denser matrix, fewer voids
MLG 0.10%	1000×	110.0	2	Crack-bridging by graphene flakes
MLG 0.10%	2000×	55.0	1	Graphene sheets integrated with hydration gel
MLG 0.10%	5000×	22.0	0.5	Enhanced bonding, dense C–S–H clusters

, the control sample exhibited voids, microcracks, and non-uniform bonding. In contrast, the 0.10% MLG sample revealed graphene sheets embedded in the matrix, bridging across cracks and filling voids. At higher magnification (2000×–5000×), graphene flakes were observed to act as nucleation sites, tightly integrated with C–S–H gel, resulting in a denser interfacial transition zone (ITZ) and improved crack resistance.

The SEM analysis confirms improved strength based on microstructural observations. As labeled in Figure 12, the control sample shows visible voids and microcracks, while the 0.10% MLG sample exhibits a denser, more cohesive matrix with fewer defects. These visual differences suggest reduced porosity and improved particle bonding; however, quantitative porosity data (e.g., from mercury intrusion porosimetry) were not included in this study and are recommended for future work. Graphene contributes to a more cohesive and compact structure by filling the micro voids and promoting better hydration, leading to greater strength and durability.

2.4.3. Energy-Dispersive X-Ray Spectroscopy

EDAX was used in conjunction with SEM to determine the elemental composition and distribution within the matrix.

The graphene mix demonstrated higher oxygen (O), carbon (C), and silicon (Si) contents, with a more uniform Ca/Si ratio compared to the control. These elemental shifts are consistent with accelerated formation of calcium silicate hydrate (C–S–H) gel, the primary hydration product responsible for strength gain. The improved Ca/Si ratio indicates a denser and chemically stable microstructure, which explains the enhanced compressive and flexural strengths observed in the 0.10% MLG mix. A comparison of EDAX spectra between the control and graphene concrete further highlighted the improved material homogeneity and reduced porosity.

Elements such as aluminum and iron exhibit minor variations between the two types of concrete as discussed in

Table 7. Graphene mix.

Element	At No	Netto	Mass %	Mass Norm. %	Atom %	Abs. Error % (1 Sigma)	Rel. Error % (1 Sigma)
Oxygen	8	6537	52.27	49.99	61.62	7.84	15.00
Carbon	6	689	10.26	9.81	16.12	2.61	25.48
Calcium	20	7429	14.64	14.00	6.89	0.50	3.39
Silicon	14	8286	9.92	9.49	6.66	0.48	4.79
Magnesium	12	1615	3.26	3.12	2.53	0.24	7.42
Aluminum	13	2022	3.28	3.14	2.30	0.21	6.52
Iron	26	1369	4.58	4.38	1.55	0.21	4.55
Sodium	11	324	1.01	0.97	0.83	0.13	12.81
Antimony	51	1143	3.17	3.03	0.49	0.17	5.25
Titanium	22	351	0.97	0.92	0.38	0.09	9.05

. Aluminum content is slightly higher in Graphene Concrete, which can improve resistance to chemical attacks. In contrast, the iron content is somewhat lower, indicating that graphene particles may reduce the dependency on traditional reinforcement through iron compounds. These shifts in elemental composition point toward a concrete mixture that is more durable and chemically stable, especially in aggressive environments. The EDX analysis reveals that Graphene Concrete offers several advantages over traditional Control Concrete. The increased oxygen, carbon, silicon, and magnesium contents suggest that adding graphene strengthens the concrete and improves its durability and resistance to environmental factors, as shown in Figure 13. These findings make Graphene Concrete an attractive option for modern construction projects that require enhanced performance and longevity.

3. Mathematical Optimization Model

3.1. Objective Function Formulation

The overall performance of each graphene mix was evaluated using a composite objective function (Z) that balances three maximization goals—compressive strength (CS), split tensile strength (TS), and flexural strength (FS)—with one minimization goal—carbon emission (CE).

The composite objective function Z is expressed as:

$$Z=w_1\times \left({\displaystyle \frac{CS}{C{S}_{ref}}}\right)+w_2\times \left({\displaystyle \frac{TS}{T{S}_{ref}}}\right)+w_3\times \left({\displaystyle \frac{FS}{F{S}_{ref}}}\right)-w_4\times \left({\displaystyle \frac{CE}{C{E}_{ref}}}\right)$$

where

CS, TS, FS = measured compressive, tensile, and flexural strengths;

CE = carbon emission for the mix;

CSref, TS_ref, FS_ref, CE_ref = control (0%) mix reference values;

w₁, w₂, w₃, w₄ = weights for importance of each term.

In this study, equal weights (0.25 each) were assigned to give balanced consideration to compressive, tensile, and flexural strength as well as carbon emission reduction, consistent with previous multi-objective concrete optimization studies. Sensitivity analysis of different weighting schemes is proposed for future work.3. Reference Values (Control Mix, 0%)

CSref = 57.53 MPa;

TSref = 5.10 MPa;

FSref = 7.60 MPa;

CEref = 100% (used as baseline).

3.2. Normalization of Parameters

To ensure comparability and balance within the multi-objective function, all parameters were normalized relative to their respective control (0% graphene) values. This normalization process allows the contribution of each performance metric—compressive strength (CS), tensile strength (TS), flexural strength (FS), and carbon emission (CE)—to be scaled uniformly within the composite objective function.

Normalization Equations:

For each mix containing graphene, the normalized values were calculated as:

This section provides detailed calculations for the objective function components at 0.1% graphene concentration.

Normalized values:

Compressive Strength Ratio (CS/${CS}_{ref}$):

CS = 81.09 Mpa ${CS}_{ref}$= 57.53 MPa CS/${CS}_{ref}$ = 81.09/57.53 = 1.409 Tensile Strength Ratio (TS/$T{S}_{ref}$): TS = 5.79 MPa $T{S}_{ref}$ = 5.10 MPa TS/$T{S}_{ref}$ = 5.79/5.10 = 1.135 Flexural Strength Ratio (FS/$F{S}_{ref}$): FS = 11.15 MPa $F{S}_{ref}$ = 7.60 MPa FS/$F{S}_{ref}$ = 11.15/7.60 = 1.467

Carbon Emission Ratio (CE/CE_ref): In this study, carbon emissions were estimated using cement-related emissions as the primary boundary, with an emission factor of ≈0.9 kg CO₂ per kg of cement. The control mix (0% MLG) was taken as 100%, and the reduction in cement usage in the graphene mix corresponded to an estimated 22% decrease. Other sources such as graphene production, transportation, and mixing energy were not included in this calculation boundary and are recommended for inclusion in future life-cycle studies.

$$\mathrm{CE}/C{E}_{ref}\text{ = 22/100 = 0.22}$$

The carbon emission reduction in this study was calculated only with respect to cement-related emissions, using standard cement emission factors (≈0.9 kg CO₂/kg cement). Other sources such as transportation, mixing energy, and graphene production emissions were excluded from the boundary of this calculation.

3.3. Equation-Based Scoring and Interpretation

To evaluate the comparative efficiency of each graphene dosage as shown in

Table 8. Different graphene percentages.

Graphene %	CS (MPa)	TS (MPa)	FS (MPa)	CE Red %	CS Ratio	TS Ratio	FS Ratio	CE Ratio	Z Score
0.00	57.53	5.10	7.60	0	1.000	1.000	1.000	0.00	0.750
0.01	59.00	5.20	8.00	4	1.025	1.020	1.053	0.04	0.778
0.03	61.00	5.40	8.20	8	1.060	1.059	1.079	0.08	0.8045
0.05	65.00	5.55	9.00	12	1.130	1.088	1.184	0.12	0.8705
0.07	70.00	5.65	9.80	18	1.217	1.108	1.289	0.18	0.9335
0.10	81.09	5.79	11.15	22	1.409	1.135	1.467	0.22	1.047

, the composite Z-score was calculated using the normalized values for all mixes ranging from 0.00% to 0.10% graphene by weight of cement. Each score reflects the net benefit of the mix in terms of mechanical performance (compressive, tensile, and flexural strength) and environmental performance (carbon emission reduction).

Normalized Values and Z Calculation

Z-score calculation example (for 0.10% graphene):

Z = 0.25(1.409) + 0.25(1.135) + 0.25(1.467) − 0.25(0.22)

Z = 0.35225 + 0.28375 + 0.36675 − 0.055 = 1.047

(1)

Interpretation:

• The control mix (0.00%) has a baseline Z-score of 0.750, which serves as the performance threshold.
• As the graphene dosage increases, strength ratios improve progressively across all categories.
• The optimal performance is achieved at 0.10% graphene, where the Z-score reaches 1.047, the highest among all tested mixes.
• Beyond 0.10%, further increases may lead to diminishing returns due to agglomeration and reduced dispersion, though this was not tested in this study.

The Z-score results align with experimental findings and confirm that 0.10% MLG dosage offers the best trade-off between strength performance and carbon footprint mitigation, thus validating the optimization framework.

3.4. Optimization Results and Graphene Dose Response

The optimization model revealed a clear nonlinear relationship between graphene dosage and performance outcomes, as quantified by the composite Z-score. While incremental increases in MLG content resulted in consistent improvements in compressive, tensile, and flexural strength, the rate of gain began to taper after the 0.05–0.07% range. However, the combined strength and sustainability benefit peaked at 0.10% graphene dosage, validating it as the optimal point in this study.

3.4.1. Trend Interpretation:

• From 0.00% to 0.05%, the gains are essentially linear and proportional across all three strength metrics.
• Between 0.05% and 0.10%, there is a compound effect where strength benefits continue to increase due to improved hydration kinetics and microstructural refinement.
• At 0.10%, the Z-score reaches 1.047, the highest in the series, representing a ~40% improvement in compressive strength, ~47% in flexural strength, and ~13.6% in tensile strength. In comparison, carbon emissions were reduced by 22% compared to the control.
• Beyond 0.10%, prior studies suggest a risk of agglomeration and decreased dispersion quality, which may degrade mechanical performance and undermine environmental gains. However, this threshold was not exceeded in the current investigation.

3.4.2. Graphene Dose Vs. Z-Score (Optimization Curve)

A dose–response curve plotting graphene percentage against Z-score shows a convex upward trend, peaking at 0.10%, consistent with optimization behavior in multi-criteria systems:

The relationship between graphene dosage and the composite Z-score. As shown in Figure 14, performance improves with increasing graphene content, and the optimal dosage of 0.10% yields the highest Z-score of 1.047, validating the optimization model.

This behavior demonstrates the multi-scale synergy of MLG: mechanical, microstructural, and sustainability advantages converge most efficiently at the 0.10% dosage.

3.4.3. Conclusion of Optimization Analysis

• The multi-objective function successfully captures the interplay between mechanical performance and carbon mitigation.
• The Z-score methodology offers a powerful decision-support tool for engineers seeking to optimize graphene dosage based on quantifiable targets.
• 0.10% MLG is mathematically and experimentally validated as the optimal dosage for structural and sustainable performance in this study.

4. Discussion

4.1. Synergistic Effects: Experiment Vs. Optimization

The integration of experimental testing and mathematical optimization has yielded a coherent and comprehensive framework for evaluating the role of multilayer graphene (MLG) in concrete. Mechanical testing confirmed that MLG enhances all key performance metrics—compressive, tensile, and flexural strength—especially at higher dosages. The optimization model, which combines these gains with carbon emission considerations, corroborated the experimental findings by identifying 0.10% graphene as the most efficient dosage. This convergence between empirical results and mathematical scoring validates the utility of the Z-score framework as a robust decision-making tool for performance-based material design.

4.2. Physical Justification of Optimum Graphene Dose

The exceptional performance of the 0.10% MLG mix can be attributed to the dual-function behavior of graphene at the micro- and nanoscales. At this dosage, MLG effectively bridges micro cracks refines the pore structure, and accelerates hydration through nucleation effects, as confirmed by SEM and XRD analyses. Below 0.10%, the graphene particles may be insufficient to reinforce the matrix fully; above this level, agglomeration can impair dispersion, reducing the material’s reinforcing effectiveness. Thus, 0.10% emerges as a critical threshold that optimally balances reinforcement and workability without requiring dispersants or surface modifiers. At dosages above 0.10%, graphene sheets tend to agglomerate due to Van der Waals interactions and inadequate dispersion, leading to non-uniform stress distribution and reduced strength performance. This explains why 0.10% emerged as the optimum threshold in the present study, consistent with earlier reports of diminishing gains at higher loadings.

4.3. Comparison with Traditional Nanomaterials

Traditional admixtures such as silica fume, nano-silica, and fly ash have long been used to enhance mechanical properties and durability, often through pozzolanic reactions or filler effects. However, these materials typically require higher dosages—for example, nano-silica at 3–5% [6], fly ash at 10–20% [10], or CNTs at 0.5–1.0% [8] —to achieve measurable strength gains. By contrast, MLG in this study achieved ~41% compressive, ~47% flexural, and ~14% tensile strength improvements at only 0.10% dosage, demonstrating its superior reinforcement efficiency, while also enhancing microstructural crystallinity and elemental uniformity. Unlike carbon nanotubes (CNTs), which are complex to disperse and sensitive to mixing protocols, MLG offers greater scalability, stability, and versatility.

For instance, nano-silica is typically incorporated at 3–5% by weight of cement to achieve ~15–20% compressive strength gains [6]. Similarly, fly ash is often added at 10–20% replacement levels to enhance durability and reduce permeability, though with slower strength development [10]. Carbon nanotubes (CNTs) can improve tensile performance, but effective dosages are usually 0.5–1.0%, and dispersion remains a major limitation [8]. In contrast, the present study demonstrates that only 0.10% MLG yields ~41% compressive, ~47% flexural, and ~14% tensile strength improvements. This indicates a markedly higher reinforcement efficiency of MLG compared to conventional nanomaterials.

4.4. Application in Hot-Arid Climates and Sustainable Construction

The findings have particular significance for hot and arid climates, where early-age cracking, thermal shrinkage, and rapid moisture loss compromise the structural integrity of conventional concrete. MLG’s ability to accelerate hydration and densify the matrix can reduce permeability and improve shrinkage resistance, making it well-suited for such conditions. Furthermore, by enabling lower cement usage through strength efficiency and reducing total carbon emissions, MLG-enhanced concrete aligns with global sustainability goals and green building standards. It is important to note that the reported 22% reduction in carbon emissions is based solely on cement optimization. The energy footprint of multilayer graphene production was not considered in this study due to the lack of standardized LCA data. Future work should expand the calculation boundary to include graphene production and transportation to obtain a more comprehensive sustainability assessment.

In the context of hot and arid climates, the improved strength and microstructural densification observed in MLG-enhanced concrete may mitigate early-age cracking and moisture loss, which are common in such environments. While the present study did not directly include shrinkage or temperature cycling tests, the demonstrated strength-related mechanisms provide an indirect basis to suggest the potential suitability of MLG concrete for thermally extreme regions. Future studies should validate this through targeted durability experiments.

5. Conclusions

This study presents an integrated experimental and mathematical assessment of multilayer graphene (MLG) as a smart Nano-admixture in concrete, offering insights into its mechanical, microstructural, and environmental impacts. The comprehensive evaluation—including compressive, tensile, and flexural strength testing, XRD-SEM-EDAX analysis, and multi-objective optimization—confirms the multifunctional potential of MLG in advancing high-performance and sustainable concrete technologies.

Key conclusions are summarized as follows:

• Mechanical Performance:

MLG incorporation led to significant strength enhancement across all curing ages. At 90 days, the 0.10% MLG mix achieved 40.96% higher compressive strength, 13.6% higher tensile strength, and 46.71% higher flexural strength compared to the control mix, demonstrating its efficiency in strengthening both compressive and tensile zones.

• Microstructural Enhancement:

XRD analysis revealed increased formation of calcium silicate hydrate (C-S-H) and reduced presence of tricalcium silicate (C₃S), indicating accelerated hydration. SEM images showed a denser, void-free matrix, while EDAX analysis confirmed favorable elemental redistribution and improved Ca/Si ratios—key indicators of structural cohesion.

• Optimization Outcome:

A multi-objective function integrating normalized strength gains and carbon emission reduction identified 0.10% graphene as the optimal dosage, achieving the highest composite Z-score (1.047). This dosage offered the best balance between mechanical enhancement and environmental performance without requiring dispersants.

• Sustainability and Application Context:

MLG-enhanced concrete is well-suited for hot and arid environments, where early-age durability and crack resistance are critical. Its ability to reduce cement-related carbon emissions by up to 22% further positions it as a valuable material for low-carbon and high-performance infrastructure.

• Novel Contribution

This study presents a unique integration of multilayer graphene (MLG) in concrete with a quantitative multi-objective optimization model, offering a performance-based method for dosage selection. Unlike previous studies, it introduces a composite Z-score that balances strength gains (compressive, tensile, flexural) against carbon emissions, enabling objective decision-making.

The model is supported by multi-scale experimental validation—including XRD, SEM, and EDAX—confirming MLG’s role in enhancing hydration, densifying the matrix, and refining elemental composition. Notably, these improvements were achieved without the use of dispersants, demonstrating practical scalability.

Although direct shrinkage and thermal cycling tests were not conducted, the findings of enhanced compressive, tensile, and flexural strengths, coupled with improved microstructural cohesion, indicate that MLG-enhanced concrete could offer resilience advantages in hot and arid environments. These strength-related mechanisms form the basis for its potential application, which should be further validated by dedicated durability studies.

Furthermore, the study contextualizes MLG concrete for hot-arid climates, where its microstructural and durability benefits can directly address early-age cracking and thermal shrinkage, making it suitable for sustainable infrastructure in extreme environments.

6. Future Recommendations

While this study successfully establishes the performance benefits and optimal dosage of multilayer graphene (MLG) in concrete, several future directions are proposed to deepen and broaden the applicability of this research:

6.1. Explore Dispersant-Enhanced Formulations

Although this study intentionally excluded dispersants to isolate the raw effect of MLG, future work should investigate how chemical or bio-based dispersants (e.g., polycarboxylates) can improve graphene dispersion, minimize agglomeration, and potentially lower the optimal dosage threshold.

6.2. Long-Term Durability and Life-Cycle Assessment

Extended tests under aggressive conditions—such as freeze–thaw cycles, chloride ingress, sulfate attack, and carbonation—are needed to evaluate the long-term durability of MLG concrete. A complete life-cycle assessment (LCA) would help quantify its sustainability benefits over traditional mixes. In addition, future life-cycle assessments should incorporate the embodied energy and emissions of multilayer graphene production. This will provide a more holistic understanding of the trade-offs between strength gains and environmental costs.

6.3. Hybrid Nano-Admixture Synergies

Combining MLG with other nano-admixtures such as nano-silica, carbon nanotubes, or rice husk ash may offer synergistic effects. Future optimization models could be expanded to multi-variable systems to explore such interactions.

6.4. Field-Scale Validation

Pilot studies on slabs, beams, or pavement sections under real environmental loads will be critical to validate lab findings. On-site behavior in hot and arid climates will help assess thermal shrinkage control and real-time durability.

6.5. AI-Based Mix Design Optimization

Integration of the Z-score framework into a machine learning model could allow for the predictive optimization of mixed designs based on user-defined performance and sustainability constraints. This would enable engineers and project designers to make dynamic decisions.