Computational Models for the Vibration and Modal Analysis of Silica Nanoparticle-Reinforced Concrete Slabs with Elastic and Viscoelastic Foundation Effects

Abstract

The integration of silica nanoparticles (NS) into cementitious composites has emerged as a promising strategy to refine the microstructure and enhance concrete performance. Beyond their chemical role in accelerating hydration and promoting additional C–S–H gel formation, silica nanoparticles act as physical fillers, reducing porosity and improving interfacial bonding within the matrix. These dual effects result in a denser and more resilient composite, whose mechanical and dynamic responses differ from those of conventional concrete. However, studies addressing the vibrational and modal behavior of nano-reinforced concretes, particularly under elastic and viscoelastic foundation conditions, remain limited. This study investigates the dynamic response of NS-reinforced concrete slabs using a refined quasi-3D plate deformation theory with five (05) unknowns. Different foundation configurations are considered to represent various soil interactions and assess structural integrity under diverse supports. The effective elastic properties of the nanocomposite are obtained through Eshelby’s homogenization model, while Hamilton’s principle is used to derive the governing equations of motion. Navier’s analytical solutions are applied to simply supported slabs. Quantitative results show that adding 30 wt% NS increases the Young’s modulus of concrete by about 26% with only ~1% change in density; for simply supported slender slabs, this results in geometry-dependent increases of up to 18% in the fundamental natural frequency. While the Winkler and Pasternak foundation parameters reduce this frequency, the damping parameter of the viscoelastic foundation enhances the dynamic response, yielding frequency increases of up to 28%, depending on slab geometry.

1. Research Background

In the field of structural engineering, the ongoing quest for materials and methodologies that elevate the performance and durability of concrete structures remains paramount [1]. As urban environments evolve and the imperative for resilient infrastructure intensifies, researchers and engineers continuously seek innovative technologies to bolster traditional concrete [2]. Among these endeavors, the incorporation of silica nanoparticles (NS) into concrete formulations stands out as a pioneering avenue [3]. This integration facilitates the transformation of conventional concrete beams and slabs into advanced, high-performance structures [4], offering enhanced strength [5], durability [6], and resilience to various environmental and loading conditions [7].

In recent research projects, various studies have examined the nuanced effects of incorporating NS into concrete compositions. These studies not only contribute significant findings into the enhancement of mechanical strength and durability but also explore the intricate interplay between different materials within the concrete matrix. Saleh et al. [8] investigated the impact of NS and polystyrene granules on concrete properties. Incorporating NS led to a notable improvement in compressive strength, enhancing the material’s structural robustness. In contrast, the introduction of polystyrene granules resulted in a reduction in compressive strength. However, the combined influence of NS and polystyrene granules synergistically enhanced the concrete’s thermal insulation capabilities, with a thermal conductivity range of (0.43–0.45) W/m. °C, demonstrating a balanced compromise between strength and thermal performance. Abhilash et al. [9] explored the advantages of integrating NS into concrete, emphasizing heightened mechanical and durability characteristics, elevated compressive strength, and strong pozzolanic activity. The study delved into determining the optimal NS dosage for concrete, comparing its pozzolanic activity with other mineral additions, and exploring potential synergies with additional additives to enhance concrete durability. Fallah et al. [10] studied the influence of NS on the mechanical properties and durability of high-strength concrete with silica fume. Results showed that the introduction of NS enhanced concrete mechanical properties and durability, suggesting its potential as a beneficial additive in concrete mix designs. Wu et al. [11] investigated the impact of nano-SiO₂ and carbon fiber on the mechanical and residual properties of concrete after high-temperature exposure. A specific NSCFRC mixture with 0.25% carbon fiber and 1% nano-SiO₂ significantly enhanced compressive, tensile, and flexural strengths compared to standard concrete. Moreover, tests performed using Scanning Electron Microscopy (SEM) revealed improved concrete matrix compactness due to NS filling effect. Behfarnia et al. [12] explored alkali-activated slag concrete as an eco-friendly alternative, replacing part of the slag with NS and micro-silica (MS). While partial replacement with NS increased permeability, the use of MS improved permeability in alkali-activated slag concrete. Wang et al. [13] investigated the impact of nano-SiO₂ on lightweight aggregate concrete (LWAC) properties. Results showed a significant increase in compressive strength with 3% nano-SiO₂, while long-term shrinkage remained relatively unaffected. Higher nano-SiO₂ dosages reduced early-age cracking, attributed to enhanced interfacial transition zones (ITZ) between aggregates and paste. In addition to material modification approaches, meso-scale parameters such as coarse aggregate size also play a critical role in the bond–slip mechanism between steel reinforcement and concrete. Recent experimental studies have demonstrated that increasing aggregate size significantly enhances bond strength due to improved mechanical interlock, whereas pure mortar exhibits only about 39% of the bond capacity of conventional concrete, highlighting the limitations of neglecting aggregate effects in bond models [14]. Khaloo et al. [15] evaluated the impact of low replacement ratios of nano-SiO₂ with varying surface areas on high-performance concrete (HPC). Results indicated that nano-SiO₂ performance depended on specific surface areas and water-to-binder ratio. At lower ratios, nano-SiO₂ with lower specific surface area outperformed finer particles. Microstructural analysis revealed reduced efficiency of higher surface area, affecting mechanical properties.

Numerous other experimental studies have been undertaken to further investigate the influence of NS on the static behavior of concrete structures. Particularly, these researches have focused on various mechanical aspects, including the evaluation of load bearing capacities and strength development through rigorous test methodologies such as uniaxial compression and split tensile tests. In addition, the researchers explored the field of physical and chemical effects induced by the inclusion of nano-SiO₂ particles in concrete matrices. These effects cover crucial parameters such as durability, microstructural alterations and the onset of chemical reactions within the concrete matrix. The culmination of these studies has offered a significant contribution to understand the impact of NS on certain nano-reinforced concrete properties. To provide a comprehensive understanding of the findings from these studies,

Table 1. The effect of NS on the mechanical and chemical properties of concrete.

Concrete Type	NS/MS Content (wt%)	Conclusions	Refs.
-Moderate, high strength concrete	/	-Incorporation of MS and/or NS significantly increases compressive strength. -Elastic modulus and splitting tensile strength are only marginally affected. -Bond strength and slip energy are improved for plain steel rebar, while limited or negative effects are observed for ribbed rebar.	[16]
-Ordinary Concrete	/	-Increasing the quantity of NS can enhance the compressive strength of concrete.	[17]
-Recycled concrete	0, 3 and 6%	-A higher content of NS tends to enhance the relative residual splitting tensile strength of concrete.	[18]
-Geopolymer concrete (GPC)	2%	-Steel fiber did not lead to a significant improvement in compressive strength unless paired with NS.	[19]
-Recycled aggregate geopolymer concrete	1, 2 and 3%	-Mechanical and durability properties of GPC were both improved through the substitution of 1% NS.	[20]
-Fiber-reinforced concrete	1, 2 and 3%	-Concrete containing 10% SiO2 Fumes, and 1% steel fibers exhibits favorable mechanical properties.	[21]
-High-strength concrete	1 and 2%	-Replacing a portion of the cement with 2% NS resulted in enhanced overall strength of the concrete.	[22]
-High-performance concrete	0.5∼3%	-Increasing NS content leads in improved structural behavior. -2% addition of nano-SiO2 was the optimal dosage.	[23,24,25]
-High-volume fly ash concrete	2 and 4%
-High-strength light weight concrete	3%	-In chemical aggressive environments, NS enhances the strength of concrete.	[26]
-Coal and fly ash concrete	1∼5%	-Incorporating NS can significantly enhance the mechanical performance of concrete, as well as its resistance to freeze–thaw cycles and chloride ion penetration.	[27]
-Self-compacting concrete	0∼6%	-Enhancing concrete’s mechanical performance against static loads can be achieved through the incorporation of fibers and NS.	[28]

offers a detailed summary of the effects of NS on the properties of various types of concrete compositions.

When discussing the impact of incorporating NS particles on the dynamic properties of concrete elements, it is notable that fewer investigations have been conducted in this area compared to analyses of their effect on static properties. However, recent research studies have begun to shed light on the potential benefits of adding NS to enhance the dynamic response of concrete structures.

Table 2. The effect of NS on the dynamic properties of concrete elements.

Concrete Type	NS Content (wt%)	Conclusions	Ref.
-Ultra-high performance concrete	3%	-In presence of fiber reinforcement, nano-SiO2 additions appear to have an insignificant influence on the dynamic strengths of the material. However, the strength of the material can be increased with increasing volume dosage of nanomaterials.	[29]
-High volume fly ash concrete	0.12%	-The dynamic compressive strengths of high volume fly ash concrete were higher at 400 and 700 °C.	[30]
-High-volume fly ash concrete with polypropylene fibers	0.12%	-The concrete exhibited improved performance even at early ages regarding dynamic compressive strength, critical strain, damage resistance, and toughness, attributed to the enhanced activity of NS during the heating process.	[31]
-Fibrous ultra-high performance concrete	1%	-Incorporation of 20% silica fume in addition to 2% of steel fibers or 1% of NS resulted in improvement in fiber-matrix bond and dynamic properties of UHPC.	[32]

provides an overview of relevant research findings that support this assertion, suggesting that incorporating NS into concrete has the potential to positively influence its dynamic properties.

Despite the growing body of literature addressing the static performance of nano-modified concretes, their dynamic characteristics, particularly when coupled with complex subgrade conditions, remain insufficiently quantified. This study specifically addresses this research gap by providing a combined analytical investigation of NS-reinforced concrete slabs interacting with complex viscoelastic foundations. While prior work has either focused on material-level enhancements or simplified support conditions, the integrated effects of high NS dosages, slab geometry, and foundation parameters, such as shear layers, spring constants, and damping that simulate various soil–structure interactions, have not been fully explored.

While experimental studies are indispensable, analytical modeling serves as a potent tool to determine the stiffness tensor component for comprehending, predicting, and optimizing plate behavior in engineering applications [33]. Recently, Harrat et al. [34] and Chatbi et al. [35] conducted analytical investigations focused on the static response of a concrete slab infused with varying proportions of NS. Utilizing higher deformation theories, their research notably demonstrates a substantial enhancement in the mechanical behavior of the slab structure under flexural loads. This improvement stems from the heightened elastic properties introduced by these nanoscale elements within the concrete matrix. Complementing these findings, Dine Elhennani et al. [36] explored the dynamic properties of concrete beams reinforced with diverse nano-particles. Employing a refined Quasi-3D beam deformation theory, their research corroborated the observed enhancements in the beam’s dynamic response. Jassas et al. [37] conducted a numerical investigation into the effect of NS on the forced vibration behavior of concrete slabs, and concluded that by increasing the volume percentage of SiO₂ nanoparticles, the linear frequency of the structure is increased and the maximum dynamic deflection is decreased.

In the domain of structural dynamics and mechanical responses, numerous significant studies have deepened our comprehension of intricate systems positioned on viscoelastic foundations and exposed to dynamic loading scenarios. Wang et al. [38] delved into the dynamic behavior of shear deformation beams under high-speed thermal and mechanical loadings. The study derived theoretical equations using the first-order shear de-formation theory and nonlinear strain-displacement relationships. Additionally, the influence of a nonlinear viscoelastic foundation on beam response was considered, yielding novel outcomes characterizing system responses in various scenarios. Mohammadi et al. [39] focused on the dynamic modeling of a double-walled cylindrical functionally graded (FG) micro-shell. Incorporating size effects through modified couple stress theory (MCST) and embedded in a viscoelastic medium, the study revealed that the viscoelastic foundation plays a crucial role in determining the natural frequency of double-walled cylindrical FG micro-shells. Alnujaie et al. [40] explored the forced dynamic response of a thick functionally graded (FG) beam on a viscoelastic foundation with porosity effects. Investigating the impact of the porosity coefficient, distribution, and foundation parameters, the study demonstrated the applicability of functionally graded viscoelastic beams in aerospace, nuclear, power plant structures, and marine applications. Abdelrahman et al. [41] introduced a nonlinear displacement-based finite element model for analyzing the nonlinear dynamic response of a flexible double wishbone vehicle suspension system. Considering damping effects, the model incorporated kinematic nonlinearity through von Kármán strain components, offering insights into the effects of road irregularities, vehicle speed, and material damping coefficients on the nonlinear vibrations response of double wishbone suspension systems. Arefi et al. [42] conducted an investigation into the size-dependent analysis of free vibrations in a sandwich nano-plate resting on a visco–Pasternak foundation. The study delved into the influence of critical parameters, such as applied electric and magnetic potentials, nonlocal parameters, and visco–Pasternak’s parameters. Findings revealed that augmenting the spring and shear parameters of the foundation resulted in higher natural frequencies for the nano-plate, attributed to the in-creased stiffness of the foundation. Arefi et al. [43] undertook a thorough examination of the magneto-electro-thermo-mechanical bending and free vibration characteristics of a sandwich microplate, employing strain gradient theory. The microplate comprised a core and two integrated piezo-magnetic face sheets, subjected to electric and magnetic potentials, thermal loadings, and positioned on a Pasternak’s foundation. The inquiry primarily focused on comprehending how these parameters influence the vibration and bending responses of the sandwich microplate. Additionally, the study concluded that the two parameters of the Winkler–Pasternak foundation significantly affect deflection and electromagnetic outcomes. The findings suggested that an increase in both direct and shear parameters lead to a decrease in all mechanical components, including deflection, maximum electric and magnetic potentials. Zenkour et al. [44] investigated the controlled motion of viscoelastic/fiber-reinforced/magneto-strictive/sandwich plates supported by visco–Pasternak foundations. The study provided numerical validation and highlighted that increasing the thickness ratio of the viscoelastic layer to the magneto-strictive layer improves vibration control, offering valuable information for designing structural control systems in various applications.

Exploring avenues for structural improvement, an intriguing focus centers on the potential synergies between enhancing slabs with NS and their interaction with viscoelastic foundations. The novelty and contributions of the present study are threefold: (1) Scientific Novelty: this work presents the first comprehensive analytical investigation of the dynamic (vibrational and modal) behavior of NS-reinforced concrete slabs, explicitly accounting for their interaction with a viscoelastic foundation; (2) Material Modeling Novelty: the effective elastic properties of the nano-reinforced concrete matrix, considering varying and high weight percentages of NS, their random distribution within the matrix, and the uniform spherical morphology of the nanoparticles, are determined using the advanced Eshelby homogenization model; (3) Methodological Novelty: the study employs an advanced Quasi-3D plate deformation theory with five (05) unknowns, representing a significant refinement over conventional theories, which typically involve six (06) or more unknowns. Importantly, this Quasi-3D theory inherently avoids shear locking, a common numerical issue in lower-order shear deformation theories, by using a higher-order displacement field that satisfies the zero transverse shear stress conditions at the plate surfaces, eliminating the need for empirical shear correction factors. Unlike traditional 2D plate deformation theories such as high order deformation theory (HSDT, [45]), sinusoidal shear deformation theory (SSDT, [46]), trigonometric shear deformation theory (TrSDT, [47]), and simple refined plate theory (RPT, [48]), the Quasi-3D theory achieves higher accuracy by accounting for crucial three-dimensional effects, notably transverse shear deformation, particularly relevant in thin plates. Furthermore, it offers improved prediction of stress and strain distribution across the plate’s thickness, considering variations in bending and shearing effects throughout the thickness direction. Notably, the used Quasi-3D plate theory encompasses the stretching effect occurring in the plate thickness, a factor overlooked in 2D deformation theories. In structural dynamics and vibration analysis, the Quasi-3D deformation plate theory yields more precise modal analysis results, capturing the complete three-dimensional behavior of the plate’s vibration modes. To ascertain the elastic properties of the nano-infused concrete matrix, the Eshelby homogenization model is applied. The equations of motion for the slab are constructed using Hamilton’s principle, and solutions to the equilibrium equations for a simply supported concrete slab are provided through Navier’s analytical methods.

Consequently, the present work aims not only to examine the vibrational response of NS -reinforced slabs but also to establish a coherent computational framework capable of linking nano-scale material design to structural-scale dynamic performance. This multiscale perspective positions the study within emerging research trends in smart and adaptive infrastructure materials.

2. Materials and Methods

This section introduces the analytical formulation developed for a reinforced concrete plate incorporating spherical NS. The formulation is constructed within the framework of the refined quasi-3D deformation plate theory, which employs (05) five independent field variables, following the theoretical foundation originally established by Thai and Kim [49].

2.1. Kinematics of the Reinforced Concrete Plate

The slab is assumed to be fully edge-supported, with dimensions including a length of ‘$L_s$’, a width of ‘$b_s$’, and a total thickness of ‘$h_s$’, the subscript ‘$s$’ designates quantities associated with the slab.

At the representative element volume level (VER), it is assumed that NS infused into the concrete matrix are randomly distributed, as illustrated in Figure 1, wherein the x–axis corresponds to the length of the plate, the y–axis corresponds to the width of the plate, and the z–axis corresponds to the median plane of the plate.

$$x\in \left(0, L_s\right); y\in \left(0,{ b}_s\right); z\in \left(-{\displaystyle \frac{h_s}{2}}, {\displaystyle \frac{h_s}{2}}\right)$$

To analytically model the concrete slabs and precisely analyze their vibrations, the influence of thickness stretching is taken into account through the use of a Quasi-3D plate deformation theory. This involves introducing a third component to the total transverse displacement ‘$U_z$’, initially consisting of bending ($w_b$) and shear ($w_s$) components [48]. This additional third component represents the thickness stretching effect (${\phi }_z$). This approach was initially proposed by Thai and Kim [49].

$$U_z=w_b+w_s+g\left(z\right){\phi }_z$$

(1)

Therefore, the adopted refined Quasi-3D theory can express the transverse displacement ‘$U_z(x,y,z,t)$’ and the axial displacement in the plane ‘$U_{\alpha }(x,y,z,t)$’ according to the following field:

$$\left\{\begin{array}{c}{U}_{\alpha }=u_{\alpha }-z{\displaystyle \frac{\partial w_b}{\partial \alpha }}-f(z){\displaystyle \frac{\partial w_s}{\partial \alpha }}\\ U_z=w_b+w_s+g\left(z\right){\phi }_z\end{array}\right.$$

(2)

Here, the subscript ‘α’ corresponds to the x-direction when the intended displacement is along the length of the plate, and to the y-direction when the intended displacement is along the width of the plate. This displacement field describes the total displacement of a specific material point located at coordinates (x, y, z) inside the plate element.

The function f(z) serves as a shape function expressing the variation of shear stresses in the plate thickness and is expected to be an odd function of ‘z’, while g(z) should be an even function, as follows:

$$f\left(z\right)=z\left({\displaystyle \frac{1}{4}}-{\displaystyle \frac{5z^2}{3h^2}}\right); g\left(z\right)=1-{\displaystyle \frac{\partial f\left(z\right)}{\partial z}}.$$

(3)

Equation (4) furnishes the linear strain components of the plate, linked to the refined Quasi-3D displacement field described in Equation (2):

$$\begin{array}{l}{\epsilon }_{\alpha }={\displaystyle \frac{\partial U_{\alpha }}{\partial \alpha }}={\epsilon }_{\alpha }^0+z{k}_{\alpha }^b+f\left(z\right)k_{\alpha }^s,\left(\alpha =x,y\right)\\ {\gamma }_{xy}={\displaystyle \frac{\partial U_x}{\partial y}}+{\displaystyle \frac{\partial U_y}{\partial x}}={\gamma }_{xy}^0+z{\gamma }_{xy}^b+f\left(z\right){\gamma }_{xy}^s.\\ {\gamma }_{\alpha z}={\displaystyle \frac{\partial U_z}{\partial \alpha }}+{\displaystyle \frac{\partial U_{\alpha }}{\partial z}}=g\left(z\right){\gamma }_{\alpha z}^s.\end{array}$$

(4)

where;

$$\begin{array}{l}{\epsilon }_{\alpha }^0={\displaystyle \frac{\partial u_{\alpha }}{\partial \alpha }};k_{\alpha }^b=-{\displaystyle \frac{{{\partial }^{2}w}_b}{\partial {\alpha }^2}};k_{\alpha }^s=-{\displaystyle \frac{{\partial }^2{w}_s}{\partial {\alpha }^2}};\\ {\gamma }_{xy}^0={\displaystyle \frac{\partial u_x}{\partial y}}+{\displaystyle \frac{\partial u_y}{\partial x}};{\gamma }_{xy}^b=-2{\displaystyle \frac{{{\partial }^{2}w}_b}{\partial x\partial y}};\\ {\gamma }_{xy}^s=-2{\displaystyle \frac{{\partial }^2{w}_s}{\partial x\partial y}};{\gamma }_{\alpha z}^s={\displaystyle \frac{\partial w_s}{\partial \alpha }}.\end{array}$$

(5)

2.2. Stress–Strain Relations

The constitutive stress–strain relations for a concrete slab reinforced with randomly distributed nano-sized silicon dioxide particles (SiO₂) can be defined according to Hooke’s relations for the nano-composite material, presented in matrix form as:

$$\left\{\begin{array}{c}{\sigma }_x\\ {\sigma }_x\\ {\sigma }_z\\ {\tau }_{xy}\\ {\tau }_{xz}\\ {\tau }_{xz}\end{array}\right\}=\left[\begin{array}{ccccc}{\tilde{\aleph }}_{11}& {\tilde{\aleph }}_{12}& {\tilde{\aleph }}_{13}& {\tilde{\aleph }}_{14}& {\tilde{\aleph }}_{15}\\ {\tilde{\aleph }}_{12}& {\tilde{\aleph }}_{22}& {\tilde{\aleph }}_{23}& {\tilde{\aleph }}_{24}& {\tilde{\aleph }}_{25}\\ {\tilde{\aleph }}_{13}& {\tilde{\aleph }}_{23}& {\tilde{\aleph }}_{33}& {\tilde{\aleph }}_{34}& {\tilde{\aleph }}_{35}\\ {\tilde{\aleph }}_{14}& {\tilde{\aleph }}_{24}& {\tilde{\aleph }}_{34}& {\tilde{\aleph }}_{44}& {\tilde{\aleph }}_{45}\\ {\tilde{\aleph }}_{15}& {\tilde{\aleph }}_{25}& {\tilde{\aleph }}_{35}& {\tilde{\aleph }}_{45}& {\tilde{\aleph }}_{55}\end{array}\right]\left\{\begin{array}{c}{\epsilon }_x\\ {\epsilon }_y\\ {\epsilon }_z\\ {\gamma }_{xy}\\ {\gamma }_{xz}\\ {\gamma }_{yz}\end{array}\right\}$$

(6)

At this point, the tensor $\left[\tilde{\aleph }\right]$ represents the reduced elastic stiffness tensor of the RC slab reinforced with NS, obtained through homogenization model.

2.3. Homogenization Model for NS Reinforced Concrete Slab

The present study employs a two-phase homogenization framework to represent the mechanical behavior of reinforced concrete slabs embedded with spherical NS. This framework is developed based on several key hypotheses and physical assumptions that ensure both analytical consistency and physical realism.

• Phase I—Homogeneous concrete matrix: At the initial stage, the concrete is treated as a homogeneous, isotropic material representing conventional unreinforced concrete. This baseline phase serves as a reference for evaluating the effect of NS inclusion on the slab’s overall stiffness and dynamic response.
• Phase II—Bi-phasic composite material: Upon introducing NS particles, the medium is regarded as a two-phase composite consisting of a cementitious matrix and dispersed nanoparticles. The nanoparticle volume fraction varies from 0% (pure concrete) to 30%, allowing the investigation of how different reinforcement levels influence the composite stiffness, flexural rigidity, and vibration characteristics.

The model formulation is further supported by the following physical considerations:

• Spherical morphology of nanoparticles: The NS are assumed to be perfectly spherical for analytical simplicity. Although real nanoparticles may slightly deviate from this ideal shape, such approximations effectively capture the primary mechanisms of particle packing, pore refinement, and matrix densification, with minimal influence on the overall macroscopic response.
• Random distribution within the matrix: The nanoparticles are assumed to be uniformly and randomly dispersed throughout the concrete without forming clusters or agglomerates. This assumption ensures an isotropic reinforcement effect and simplifies the homogenization procedure, while still providing a physically realistic representation of nano-reinforced concrete behavior.

The estimation of the effective stiffness tensor ‘${\tilde{\aleph }}_{ij}$’ is achieved through an Eshelby-type homogenization framework adapted for the present nano-reinforced system. This analytical approach takes into account both the elastic properties and the geometric characteristics of the reinforcing phase, which, in the present case, consists of spherical NS [50]. Although the Eshelby model was originally formulated for ellipsoidal inclusions, its extension to the nanoscale introduces additional factors such as interfacial effects and localized stress concentrations. While these microscale phenomena are not explicitly modeled in the present formulation, their overall influence is implicitly reflected in the effective stiffness parameters. This ensures that the proposed homogenization remains physically representative of the nanocomposite’s mechanical response. The resulting stiffness tensor components ‘${\tilde{\aleph }}_{ij}$’ for the nano-reinforced plate can therefore be expressed in matrix form as expressed in Equation (7). This formulation provides a simplified yet physically consistent route for connecting the nanoscale reinforcement distribution to macroscopic elastic performance, avoiding the computational expense of full-scale micromechanical simulations:

$${\tilde{\aleph }}_{ij}={\left({\left({\aleph }_m^{ij}\right)}^{-1}-V_r{\left\{\left({\aleph }_r^{ij}-{\aleph }_m^{ij}\right)\left[\Gamma -V_r\left(\Gamma -I_d\right)\right]+{\aleph }_m^{ij}\right\}}^{-1}\left({\aleph }_r^{ij}-{\aleph }_m^{ij}\right){\left({\aleph }_m^{ij}\right)}^{-1}\right)}^{-1}$$

(7)

In this analytical homogenization approach, ‘$I_d$’ signifies an identity matrix of dimensions (5 × 5). The designation, ‘V_m’ and ‘V_r’ denote the volume fractions of the concrete matrix and the reinforcement, respectively. Additionally, ‘$\Gamma$’ is the Eshelby tensor directly associated with the elastic Poisson ratios of the nanoparticles.

$$\begin{array}{l}{\aleph }_{\mathsf{{\rm Y}}}^{11}={\aleph }_{\mathsf{{\rm Y}}}^{22}={\aleph }_{\mathsf{{\rm Y}}}^{33}={\displaystyle \frac{\left(1-{\upsilon }_{\mathsf{{\rm Y}}}\right)E_{\mathsf{{\rm Y}}}}{\left(1+{\upsilon }_{\mathsf{{\rm Y}}}\right)\left(1-2{\upsilon }_{\mathsf{{\rm Y}}}\right)}};\\ {\aleph }_{\mathsf{{\rm Y}}}^{12}={\aleph }_{\mathsf{{\rm Y}}}^{13}={\aleph }_{\mathsf{{\rm Y}}}^{23}={\displaystyle \frac{\upsilon E_{\mathsf{{\rm Y}}}}{\left(1+{\upsilon }_{\mathsf{{\rm Y}}}\right)\left(1-2{\upsilon }_{\mathsf{{\rm Y}}}\right)}};\\ {\aleph }_{\mathsf{{\rm Y}}}^{66}={\aleph }_{\mathsf{{\rm Y}}}^{44}={\aleph }_{\mathsf{{\rm Y}}}^{55}={\displaystyle \frac{E_{\mathsf{{\rm Y}}}}{\left(1+{\upsilon }_{\mathsf{{\rm Y}}}\right)}}.\end{array}$$

(8)

In this formulation, ‘$\aleph$_m’ and ‘$\aleph$_r’ denote the stiffness tensors corresponding to the concrete matrix and the NS, respectively. Both phases are modeled as isotropic continua, and their constitutive expressions are presented in Equation (8). The subscript ‘$\mathsf{{\rm Y}}$’ is used to distinguish the material under consideration—taking the value ‘m’ for the matrix and ‘r’ for the reinforcement phase. The parameter ‘E’ refers to Young’s modulus, representing either the concrete or the NS, while ‘υ’ stands for Poisson’s ratio. The indices (1, 2, 3) correspond to the Cartesian directions (x, y, z) associated with the plate’s local coordinate system.

Considering that the NS are randomly dispersed and possess a spherical morphology within the concrete medium, the relevant Eshelby tensor ‘$\Gamma$’ for this configuration can be expressed as [51]:

$$\Gamma =\left[\begin{array}{cccccc}{\kappa }_{1111}& {\kappa }_{1122}& {\kappa }_{1133}& {\kappa }_{1123}& {\kappa }_{1113}& {\kappa }_{1112}\\ {\kappa }_{2211}& {\kappa }_{2222}& {\kappa }_{2233}& {\kappa }_{2223}& {\kappa }_{2213}& {\kappa }_{2212}\\ {\kappa }_{3311}& {\kappa }_{3322}& {\kappa }_{3333}& {\kappa }_{3323}& {\kappa }_{3313}& {\kappa }_{3312}\\ {\kappa }_{2311}& {\kappa }_{2322}& {\kappa }_{2333}& {\kappa }_{2323}& {\kappa }_{2313}& {\kappa }_{2312}\\ {\kappa }_{1311}& {\kappa }_{1322}& {\kappa }_{1333}& {\kappa }_{1323}& {\kappa }_{1313}& {\kappa }_{1312}\\ {\kappa }_{1211}& {\kappa }_{1222}& {\kappa }_{1233}& {\kappa }_{1123}& {\kappa }_{1213}& {\kappa }_{1212}\end{array}\right]$$

(9)

The assumption of random dispersion is justified by experimental observations in nano-modified concretes, where uniform distribution is rarely achieved but still ensures isotropic macroscopic behavior [52,53,54]. This simplification enables the analytical model to remain tractable without compromising physical realism, which is crucial for linking theoretical predictions with laboratory-scale validation [55,56].

Knowing that, the components of this Eshelby tensor are as follows [57]:

$$\begin{array}{l}{\kappa }_{1111}={\kappa }_{2222}={\kappa }_{3333}={\displaystyle \frac{7-5{\upsilon }_r}{15\left(1-{\upsilon }_r\right)}};\\ {\kappa }_{1122}={\kappa }_{1133}={\kappa }_{2233}={\kappa }_{2211}={\kappa }_{3311}={\kappa }_{3322}={\displaystyle \frac{5{\upsilon }_r-1}{15\left(1-{\upsilon }_r\right)}};\\ {\kappa }_{1212}={\kappa }_{1313}={\kappa }_{2323}={\displaystyle \frac{4-5{\upsilon }_r}{15\left(1-{\upsilon }_r\right)}}.\end{array}$$

(10)

The other remaining components of the Eshelby’s tensor are null; $e_{ijij}=0$.

In addition, given the decisive influence of material density on the vibratory characteristics of elements constructed from this material, it is pertinent to determine the density of the nano-composite plate ‘${\rho }_T$’. This assessment is conducted through the application of a straightforward rule of mixture.

$${\rho }_T={\rho }_m{V}_m+{\rho }_r{V}_r$$

(11)

Here, ${\rho }_m$ and ${\rho }_r$ are the masse densities of the matrix and silicon dioxide nano-particles, respectively.

For the purpose of comparison, various homogenization models, including the Voigt model and Hashin–Shtrikaman upper (HS-UB) and lower (HS-LB) bounds, are utilized alongside Eshelby’s model. This comprehensive approach is undertaken to precisely characterize the equivalent elastic properties of a concrete slab infused with diverse proportions of silicon dioxide nanoparticles (SiO₂).

The Voigt homogenization model offers an isotropic approximation by assuming uniform strain within the composite [58,59]. Neglecting the agglomeration effect of NS, the model provides the effective modulus of reinforcements and their surrounding matrix, denoted as ‘E_out’, as defined in [60]:

$$E_{out}={\displaystyle \frac{3}{8}}\left\{{\displaystyle \frac{V_r\left(1-\zeta \right)}{1-\zeta }}{E}_r+\left[1-{\displaystyle \frac{V_r\left(1-\zeta \right)}{1-\zeta }}\right]E_r\right\}+{\displaystyle \frac{5}{8}}\left\{{\displaystyle \frac{E_r{E}_r\left(1-\xi \right)}{\left[\left(1-\xi \right)-V_r\left(1-\zeta \right)\right]E_r+V_r\left(1-\zeta \right)E_m}}\right\}$$

(12)

Here, $\zeta$ and $\xi$ are the parameters that describe the agglomeration of NS in the matrix.

The Hashin–Shtrikaman upper and lower bounds establish the limits within which the effective properties of the nanocomposite are expected to lie [61] by utilizing the variational principle for linear elasticity.

$$\begin{gathered} {\tilde{G}}_{UB}=G_r+\left({\displaystyle \frac{V_m}{{\left(G_m-G_r\right)}^{-1}+\frac{6\left(K_r+2G_r\right)V_r}{\left(5G_r\left(5K_r+4G_r\right)\right)}}}\right) \\ {\tilde{G}}_{LB}=G_m+\left({\displaystyle \frac{V_r}{{\left(G_r-G_m\right)}^{-1}+\frac{6\left(K_m+2G_m\right)V_m}{\left(5G_m\left(5K_m+4G_m\right)\right)}}}\right) \end{gathered}$$

(13)

By considering concrete as homogeneous, the resulting nano-composite is isotropic, and conversion formulas designed for isotropic materials can be used to determine the upper and lower elastic constants.

$$\begin{gathered} E_{UB}^T={2\tilde{G}}_{UB}\left(1+{\upsilon }_T\right) \\ {E}_{LB}^T={2\tilde{G}}_{LB}\left(1+{\upsilon }_T\right) \end{gathered}$$

(14)

where

$${\upsilon }_T={\upsilon }_m{V}_m+{\upsilon }_r{V}_r$$

(15)

2.4. Equations of Motion of the RC Slab

The present analysis employs Hamilton’s principle to derive the five dynamic equations of motion for the RC slab embedded with nano-reinforcements.

$${\int }_{t1}^{t2}\Pi \partial t={\int }_{t1}^{t2}\left({\delta \psi }_p+{\delta \Phi }_{fe}+\delta \chi \right)\partial t=0$$

(16)

In Hamilton’s principle, ${\delta \psi }_p$ stands for the plate’s internal deformation energy, while ${\delta \Phi }_{fe}$ represents the strain energy of the applied foundation. In addition, $\delta \chi$ refers to the kinetic energy generated of the mass system.

The representation of the strain energy ‘${\delta \psi }_p$’ associated with the RC slab can be expressed as follows:

$${\delta \psi }_p={\int }_0^{L_s}{\int }_0^{b_s}{\int }_{-h_s/2}^{h_s/2}\left\{\left({\sigma }_{\alpha }\delta {\epsilon }_{\alpha }\right)+\left({\sigma }_z\delta {\epsilon }_z\right)+\left({\tau }_{xy}\delta {\gamma }_{xy}\right)+\left({\tau }_{\alpha z}\delta {\gamma }_{\alpha z}\right)\right\}dAdz$$

(17)

Upon substituting Equations (4) and (6) into Equation (17), the expression for the internal strain energy is derived as follows:

$$\begin{gathered} {\delta \psi }_p={\int }_0^{L_s}{\int }_0^{b_s}\left[N_{\alpha }{\displaystyle \frac{\partial \delta u_{\alpha }}{\partial \alpha }}-M_{\alpha }^b{\displaystyle \frac{{\partial }^2\delta w_b}{{\partial \alpha }^2}}-M_{\alpha }^s{\displaystyle \frac{{\partial }^2\delta w_s}{{\partial \alpha }^2}}+N_{xy}\left({\displaystyle \frac{\partial \delta u_x}{\partial y}}+{\displaystyle \frac{\partial \delta u_y}{\partial x}}\right)-2M_{xy}^b{\displaystyle \frac{{\partial }^2\delta w_b}{\partial x\partial y}}-2M_{xy}^s{\displaystyle \frac{{\partial }^2\delta w_s}{\partial x\partial y}} \\ +Q_{\alpha z}\left({\displaystyle \frac{\partial \delta w_s}{\partial \alpha }}+{\displaystyle \frac{\partial \delta {\phi }_z}{\partial \alpha }}\right)-R_z\delta {\phi }_z\right]dxdy\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \end{gathered}$$

(18)

The terms $(N,M_b,M_s,Q,R_z)$ denote the plate’s stress resultants, where $N_x,N_y,N_{xy}$ are the in-plane normal forces; $M_x^b,M_y^b$ are the bending moments about the local $y$– and $x$–axes; $M_x^s,{ M}_y^s$ are the shear moments along the local $z$–axis; $Q_{xz},Q_{yz}$ are the torsional moments; and $R_z$ represents the rotational moment about the local $z$–axis.

These stress resultants can be defined analytically as:

$$\begin{array}{l}\left\{N_{\alpha },N_{xy}\right\}={\int }_{-h_s/2}^{h_s/2}\left\{{\sigma }_{\alpha },{\tau }_{xy}\right\}dz\\ \left\{M_{\alpha }^b,M_{xy}^b\right\}={\int }_{-h_s/2}^{h_s/2}z\left\{{\sigma }_{\alpha },{\tau }_{xy}\right\}dz\\ \left\{M_{\alpha }^s,M_{xy}^s\right\}={\int }_{-h_s/2}^{h_s/2}f\left(z\right)\left\{{\sigma }_{\alpha },{\tau }_{xy}\right\}dz\\ Q_{\alpha z}={\int }_{-h_s/2}^{h_s/2}g\left(z\right){\tau }_{\alpha z}dz\end{array}$$

(19)

Through subsequently integrating Equations (4) and (6) into Equation (19), the stress resultants generated in the plate can be formulated in relation to material stiffness and displacement components. The corresponding expressions are provided in appendix.

The analysis of the concrete slab’s dynamic behavior necessitated a realistic model for the soil-structure interaction (SSI). To this end, a supportive three-parameter visco–Winkler–Pasternak model was employed [62,63], which is a key distinguishing feature of this study. This advanced model goes beyond the limitations of the basic elastic Winkler foundation by accurately representing the physical characteristics of a real subgrade, which is essential for dynamic and vibration analysis. Specifically, the model is termed “three-parameter” because it accounts for three crucial physical phenomena: (1) the elasticity of the subgrade (Winkler springs), (2) the inter-layer shear interaction (Pasternak shear layer), and (3) the viscous damping properties of the soil medium. The load-displacement relationship between the plate and this foundation model (${\delta \Phi }_{fe}$) is established on the basis of these parameters, which allows for a more comprehensive and realistic simulation of energy dissipation and dynamic response. This model characterizes the interaction pressure, $fe\left(x,y\right)$, exerted by the foundation on the bottom surface of the slab based on the plate’s transverse deflection, $\delta U_z$. The pressure is expressed by the following constitutive Equation (20):

$$\begin{gathered} {\delta \Phi }_{fe}=-{\int }_0^A\left[\delta U_z\right]dA \\ {\delta \Phi }_{fe}=-{\int }_0^A\left[k_w{U}_z{\delta U}_z-k_s\left({\displaystyle \frac{\partial }{\partial x^2}}-{\displaystyle \frac{\partial }{\partial y^2}}\right)U_z{\delta U}_z+c_d{\displaystyle \frac{\partial }{\partial t}}\left(U_z{\delta U}_z\right)\right]dA \end{gathered}$$

(20)

In this context, ${\delta \Phi }_{fe}$ represents the underlying foundation response per unit area, while ‘$k_w$’ and ‘$k_s$’ signify the Winkler’s and Pasternak’s stiffness values for the elastic foundation, where ‘$k_s$’ represents the shear layer stiffness, and ‘$k_w$’ represent the springs constant. Moreover, with the introduction of the damping coefficient, annotated ‘$c_d$’, the foundation starts to function as a viscoelastic foundation [64]. This study explores various foundation types using Equation (20). Such a formulation captures the coupled shear and compressive actions within the foundation medium, which are crucial in describing the load transfer mechanism between the slab and the supporting soil layer. The inclusion of viscous damping extends the model’s realism by reflecting time-dependent energy dissipation observed in real subgrade materials.

Depending on the selected parameters, different foundation models can be identified as follows:

• Winkler elastic foundation: $k_w\ne 0$, $k_s=0$, $c_d=0$.
• Winkler–Pasternak elastic foundation: $k_w\ne 0$, $k_s\ne 0$, $c_d=0$.
• Visco–Pasternak foundation: $k_w=0$, $k_s\ne 0$, $c_d\ne 0$.
• Visco–Winkler–Pasternak foundation (three-parameter viscoelastic model): $k_w\ne 0$, $k_s\ne 0$, $c_d\ne 0$.

It is worth mentioning that the choice of a three-parameter viscoelastic foundation reflects the need to capture realistic soil–structure interactions [65,66]. This model accommodates both instantaneous elastic response and delayed viscous effects, which become significant in subgrades with moisture-dependent or time-dependent characteristics [67]. Such complexity is essential to accurately simulate real foundation–slab coupling phenomena [68].

The variation of the kinetic energy ‘$\delta \chi$’ of the RC slab, is calculated using the following expression:

$$\delta \chi ={\int }_0^{L_s}{\int }_0^{b_s}{\int }_{-h_s/2}^{h_s/2}\left[{\dot{U}}_{\alpha }\delta {\dot{U}}_{\alpha }+{\dot{U}}_z\delta {\dot{U}}_z\right]{\rho }_{T}dAdz$$

(21)

where the dot-superscript signifies the differentiation with the respect to time ‘t’. Substituting the displacement expression of the nano-reinforced concrete slab, accounting for the stretching effect evolving through the slab thickness in Equation (2), into Equation (21) yields:

$$\begin{gathered} \delta \chi ={\int }_0^L\left\{j_0\left[{\dot{u}}_x\delta {\dot{u}}_x+{\dot{u}}_y\delta {\dot{u}}_y+\left(({\dot{w}}_b+{\dot{w}}_s\right)(\delta {\dot{w}}_b+\delta {\dot{w}}_s)\right]-j_1\left(\dot{u_x}\partial \delta {\dot{w}}_{b,x}+\delta \dot{u_x}\partial {\dot{w}}_{b,x}+\dot{u_y}\partial \delta {\dot{w}}_{b,y}+\delta \dot{u_y}\partial {\dot{w}}_{b,y}\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.33em}+j_2\left(\partial {\dot{w}}_{b,x}\partial \delta {\dot{w}}_{b,x}+{\dot{w}}_{b,y}\partial \delta {\dot{w}}_{b,y}\right)-{\mathcal{l}}_1\left({\dot{u}}_x\partial \delta {\dot{w}}_{s,x}+\delta {\dot{u}}_x\partial {\dot{w}}_{s,x}+{\dot{u}}_y\partial \delta {\dot{w}}_{s,y}+\delta {\dot{u}}_y\partial {\dot{w}}_{s,y}\right) \\ +{\mho }_2\left(\partial {\dot{w}}_{s,x}\partial \delta {\dot{w}}_{s,x}+\partial {\dot{w}}_{s,y}\partial \delta {\dot{w}}_{s,y}\right)\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \\ +{\mathcal{l}}_2\left(\partial {\dot{w}}_{b,x}\partial \delta {\dot{w}}_{s,x}+\partial {\dot{w}}_{s,x}\partial \delta {\dot{w}}_{b,x}+\partial {\dot{w}}_{b,y}\partial \delta {\dot{w}}_{s,y}+\partial {\dot{w}}_{s,y}\partial \delta {\dot{w}}_{b,y}\right)\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \\ +{\mathcal{l}}_0\left(\left(\dot{w_b}+\dot{w_s}\right)\delta \dot{{\phi }_z}+\dot{{\phi }_z}\left(\delta \dot{w_b}+\delta \dot{w_s}\right)\right)-{\mho }_0\dot{{\phi }_z}\delta \dot{{\phi }_z}\right\}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.33em} \end{gathered}$$

(22)

where;

$$\begin{gathered} \left\{I_0,I_1,I_2,J_0,J_1,J_2,K_0,K_2\right\}={\int }_{-h_s/2}^{h_s/2}\left\{1,z,z^2,g,f,zf,g^2,f^2\right\}{\rho }_{T}dz \\ \left\{j_0,j_1,j_2,{\mathcal{l}}_0,{\mathcal{l}}_1,{\mathcal{l}}_2,{\mho }_0,{\mho }_2\right\}={\int }_{-h_s/2}^{h_s/2}\left\{1,z,z^2,g,f,zf,g^2,f^2\right\}{\rho }_{T}dz \end{gathered}$$

(23)

Upon collecting and substituting Equations (18), (20) and (22) into Equation (16), integrating the resulted equation by parts, and collecting the coefficients of $\delta u_{\alpha }$, $\delta w_b$, $\delta w_s$, and $\delta {\phi }_z$, the five equations of motion are obtained and expressed as follows:

$$\begin{array}{l}\delta u_x : {\displaystyle \frac{\partial N_x}{\partial x}}+{\displaystyle \frac{\partial N_{xy}}{\partial y}}=j_0{\ddot{u}}_x-j_1\left({\displaystyle \frac{{\partial \ddot{w}}_b}{\partial x}}\right)-{\mathcal{l}}_1\left({\displaystyle \frac{\partial w_s}{\partial x}}\right)\\ \delta u_y : {\displaystyle \frac{\partial N_y}{\partial y}}+{\displaystyle \frac{\partial N_{xy}}{\partial x}}=j_0{\ddot{u}}_y-j_1\left({\displaystyle \frac{{\partial \ddot{w}}_b}{\partial y}}\right)-{\mathcal{l}}_1\left({\displaystyle \frac{\partial w_s}{\partial y}}\right)\\ \delta w_b : {\displaystyle \frac{{\partial }^2{M}_x^b}{\partial x^2}}+2{\displaystyle \frac{{\partial }^2{M}_{xy}^b}{\partial x\partial y}}+{\displaystyle \frac{{\partial }^2{M}_y^b}{\partial y^2}}-fe=j_0\left({\partial \ddot{w}}_b+{\partial \ddot{w}}_s\right)+{\mathcal{l}}_0\left(\partial {\ddot{\phi }}_z\right)+j_1\left({\displaystyle \frac{\partial {\ddot{u}}_x}{\partial x}}\right)-j_2\left({\displaystyle \frac{{\partial }^2{\ddot{w}}_b}{\partial x^2}}\right)-j_2\left({\displaystyle \frac{{\partial }^2{\ddot{w}}_s}{\partial x^2}}\right)\\ \delta w_s : {\displaystyle \frac{{\partial }^2{M}_x^s}{\partial x^2}}+2{\displaystyle \frac{{\partial }^2{M}_{xy}^s}{\partial x\partial y}}+{\displaystyle \frac{{\partial }^2{M}_y^s}{\partial y^2}}-{\displaystyle \frac{\partial Q_{xz}}{\partial x}}-{\displaystyle \frac{\partial Q_{yz}}{\partial y}}+fe\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} =j_0\left({\partial \ddot{w}}_b+{\partial \ddot{w}}_s\right)+{\mathcal{l}}_0\left(\partial {\ddot{\phi }}_z\right)+j_1\left({\displaystyle \frac{\partial {\ddot{u}}_y}{\partial y}}\right)-j_2\left({\displaystyle \frac{{\partial }^2{\ddot{w}}_b}{\partial y^2}}\right)-j_2\left({\displaystyle \frac{{\partial }^2{\ddot{w}}_s}{\partial y^2}}\right)\\ \delta {\phi }_z : {\displaystyle \frac{\partial Q_{xz}}{\partial x}}+{\displaystyle \frac{\partial Q_{yz}}{\partial y}}-R_z=j_0\left(\ddot{w_s}+\ddot{w_b}+{\ddot{\phi }}_z\right)+{\mho }_0\left({\ddot{\phi }}_z\right)\end{array}$$

(24)

2.5. Closed Form Solutions for a Simply Supported RC Slab

Navier’s approach is employed to derive the closed-form solutions of the five equations of motion in Equation (24) for a simply supported RC slab.

$$\begin{gathered} u_x=w_b=w_s={\displaystyle \frac{d{w}_b}{dx}}={\displaystyle \frac{d{w}_s}{dx}}=0, at (x,0) \\ {u}_x=w_b=w_s={\displaystyle \frac{d{w}_b}{dx}}={\displaystyle \frac{d{w}_s}{dx}}=0, at \left(x,b\right) \\ {u}_y=w_b=w_s={\displaystyle \frac{d{w}_b}{dy}}={\displaystyle \frac{d{w}_s}{dy}}=0, at \left(0,y\right) \\ {u}_y=w_b=w_s={\displaystyle \frac{d{w}_b}{dy}}={\displaystyle \frac{d{w}_s}{dy}}=0, at \left(a,y\right) \end{gathered}$$

(25)

At the line-edge boundaries; (x = 0, a) and (y = 0, b), the following Navier’s admissible displacement functions, expressed as trigonometric series can satisfying the boundary conditions in Equation (25):

$$\begin{gathered} u_x\left(x,y,t\right)=\sum _{s=1}^{\infty }\sum _{p=1}^{\infty }{U}_{xsp}cos\left(\lambda x\right)sin\left(\mu y\right)e^{i\omega t} \\ {u}_y\left(x,y,t\right)=\sum _{s=1}^{\infty }\sum _{p=1}^{\infty }{U}_{ysp}sin\left(\lambda x\right)cos\left(\mu y\right)e^{i\omega t} \\ {w}_b\left(x,y,t\right)=\sum _{s=1}^{\infty }\sum _{p=1}^{\infty }{W}_{bsp}sin\left(\lambda x\right)sin\left(\mu y\right)e^{i\omega t} \\ {w}_s\left(x,y,t\right)=\sum _{s=1}^{\infty }\sum _{p=1}^{\infty }{W}_{ssp}sin\left(\lambda x\right)sin\left(\mu y\right)e^{i\omega t} \\ {\phi }_z\left(x,y,t\right)=\sum _{s=1}^{\infty }\sum _{p=1}^{\infty }{W}_{zsp}sin\left(\lambda x\right)sin\left(\mu y\right)e^{i\omega t} \end{gathered}$$

(26)

where, $\lambda =s\pi /a$, $\mu =p\pi /b$, and $i=\sqrt{-1}$, ‘$\omega$’ is the eigenvalue of natural frequency, and (U_xsp, U_ysp, W_bsp, W_ssp, W_zsp) are the arbitrary parameters to be determined. Finally, to obtain the analytical solutions and calculate the eigenvalues ‘$\omega$’ of the plate element, the results of the substitution can be arranged in the following matrix form:

$$\left(\left[\begin{array}{ccccc}{\mathcal{S}}_{11}& {\mathcal{S}}_{12}& {\mathcal{S}}_{13}& {\mathcal{S}}_{14}& {\mathcal{S}}_{15}\\ {\mathcal{S}}_{12}& {\mathcal{S}}_{21}& {\mathcal{S}}_{23}& {\mathcal{S}}_{24}& {\mathcal{S}}_{25}\\ {\mathcal{S}}_{13}& {\mathcal{S}}_{23}& {\mathcal{S}}_{33}& {\mathcal{S}}_{34}& {\mathcal{S}}_{35}\\ {\mathcal{S}}_{14}& {\mathcal{S}}_{24}& {\mathcal{S}}_{34}& {\mathcal{S}}_{44}& {\mathcal{S}}_{45}\\ {\mathcal{S}}_{15}& {\mathcal{S}}_{25}& {\mathcal{S}}_{35}& {\mathcal{S}}_{45}& {\mathcal{S}}_{55}\end{array}\right]-{\omega }^2\left[\begin{array}{ccccc}{\varsigma }_{11}& {\varsigma }_{12}& {\varsigma }_{13}& {\varsigma }_{14}& {\varsigma }_{15}\\ {\varsigma }_{12}& {\varsigma }_{21}& {\varsigma }_{23}& {\varsigma }_{24}& {\varsigma }_{25}\\ {\varsigma }_{13}& {\varsigma }_{23}& {\varsigma }_{33}& {\varsigma }_{34}& {\varsigma }_{35}\\ {\varsigma }_{14}& {\varsigma }_{24}& {\varsigma }_{34}& {\varsigma }_{44}& {\varsigma }_{45}\\ {\varsigma }_{15}& {\varsigma }_{25}& {\varsigma }_{35}& {\varsigma }_{45}& {\varsigma }_{55}\end{array}\right]\right)\left\{\begin{array}{c}{U}_{xsp}\\ U_{ysp}\\ W_{bsp}\\ W_{ssp}\\ W_{zsp}\end{array}\right\}=\left\{\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\end{array}\right\}$$

(27)

The explicit forms of the stiffness and mass matrix components used in the analytical formulation of the nano-composite RC slab are provided in Appendix B for completeness.

3. Results and Discussions

3.1. Material Characterization

To establish the credibility of our material characterization, it is imperative to carry out a comprehensive comparative study. This study aims to validate the analytical homogenization model (Eshelby’s) and confirm the calculated equivalent elastic properties of the nanocomposite. These equivalent properties are determined from both the concrete and NS components within the nano-composite and the elastic properties of each component are depicted in

Table 3. The elastic properties of the matrix and reinforcement used in the study.

Elastic Properties	Refs.	Concrete Matrix	NS
Young’s modulus (GPa)	[69,70]	$E_m=20$	$E_r=70$
Poisson’s ration	[69,71]	${\upsilon }_m=0.2$	${\upsilon }_r=0.3$
Density (Kg/m3)	[72,73]	${\rho }_m=2400$	${\rho }_r=2334$
Shear modulus (GPa)	$G_{\propto }={\displaystyle \frac{E_{\propto }}{\left[2\left(1+{\upsilon }_{\propto }\right)\right]}}$ * [74]	$G_m=8.33$	$G_r=26.92$
Bulk modulus (GPa)	$K_{\propto }={\displaystyle \frac{E_{\propto }}{\left[3\left(1-2{\upsilon }_{\propto }\right)\right]}}$ * [74]	$K_m=11.10$	$K_r=58.33$

* “$\propto$” refers to either ‘m’ for the matrix properties, or ‘r’ for the nano-reinforcement properties.

.

A conversion formula specifically designed for isotropic materials is used for this purpose to calculate the elastic constants of the nanocomposite material, as expressed in Equation (28):

$$\begin{array}{l}{\tilde{\aleph }}_{11}={\tilde{\aleph }}_{22}={\tilde{\aleph }}_{33}={\displaystyle \frac{\left(1-{\upsilon }_T\right)E_T^{11}}{\left(1+{\upsilon }_T\right)\left(1-2{\upsilon }_T\right)}}\\ {\tilde{\aleph }}_{12}={\tilde{\aleph }}_{13}={\tilde{\aleph }}_{23}={\displaystyle \frac{{\upsilon }_T{E}_T^{11}}{\left(1+{\upsilon }_T\right)\left(1-2{\upsilon }_T\right)}}\\ {\tilde{\aleph }}_{66}={\tilde{\aleph }}_{44}={\tilde{\aleph }}_{55}={\displaystyle \frac{E_T^{11}}{\left(1+{\upsilon }_T\right)}}\end{array}$$

(28)

The comprehensive study included a comparison of the estimated elastic moduli (E_T) for a concrete matrix reinforced with NS using the Eshelby homogenization model and a similar matrix using the Voigt homogenization model. It is important to note that our analysis intentionally excluded the impact of agglomeration of SiO₂ nanoparticles in the concrete matrix [34]. In addition, our study also incorporated the Hashin–Shtrikman (HS) model, which establishes upper (HS-UB) and lower (HS-LB) bounds. Using the HS model enabled us to obtain a complete range and define limit values [61]. To proceed this comparison, we have included Figure 2, which presents the collective elastic moduli of a concrete slab infused with NS using three different homogenization models.

Curves in Figure 2, consistently show an increase in elastic constants (E_T) upon incorporation of NS, a trend that aligns with experimental evidence [35,36]. The increase in elastic properties suggests a positive impact of nanoparticles on the concrete matrix, attributed to the higher elastic properties of NS compared to those of the concrete matrix. Physically, the observed increase in elastic constants can be attributed to the intrinsic stiffness of the silica nanoparticles and their ability to efficiently transfer stress within the cementitious matrix, thereby reinforcing the concrete at the microstructural level. However, subtle variations in the estimated elastic constants are apparent, probably due to the different assumptions of each model, particularly Eshelby’s, which incorporates the shape of the nanoparticles—a factor not present in the other models. Importantly, based on the Eshelby model used in our analysis, the estimated modulus of elasticity of the nanocomposite falls within the range established by Hashin–Shtrikman, confirming the reliability of our characterization. Overall, our study of reinforced concrete properties using various homogenization models reinforces our understanding of the nano-material, confirming the beneficial impact of NS on the elastic performance of the concrete matrix.

The comparison across different homogenization schemes not only validates the theoretical assumptions but also illustrates the sensitivity of elastic constants to the homogenization strategy. This insight is critical for engineers aiming to tailor nanocomposite concrete properties based on available reinforcement volume and processing methods.

To consolidate our use of the Eshelby model, a detailed comparison of the elastic stiffness ‘${\tilde{\aleph }}_{ij}$’ of a concrete matrix infused with NS using the Voigt and Hashin–Shtrikman homogenization models is established in Table 2. This further analysis strengthens the validity of our results, and gives us even greater confidence in the accuracy and practicality of the Eshelby model for characterizing the elastic properties of nanocomposite materials.

The data in

Table 4. Elastic stiffnesses (${\tilde{\aleph }}_{ij}$) of a concrete matrix reinforced with different proportions of NS.

Volume Percentage of NS Reinforcement in a Concrete Matrix	Analytical Homogenization Approaches	$\mathbf{Nano}-\mathbf{Composite} \mathbf{Elastic} \mathbf{Stiffnesses} {\tilde{\mathit{\aleph }}}_{\mathit{i}\mathit{j}}$
		${\tilde{\mathit{\aleph }}}_{11}={\tilde{\mathit{\aleph }}}_{22}={\tilde{\mathit{\aleph }}}_{33}$	${\tilde{\mathit{\aleph }}}_{12}={\tilde{\mathit{\aleph }}}_{13}={\tilde{\mathit{\aleph }}}_{23}$	${\tilde{\mathit{\aleph }}}_{66}={\tilde{\mathit{\aleph }}}_{44}={\tilde{\mathit{\aleph }}}_{55}$
0%	Eshelby’s approach	22.22	5.56	16.67
	Voigt approach	22.22	5.56	16.67
	$\left({\mathrm{H}\mathrm{S}}^{\mathrm{*}} \mathrm{U}\mathrm{B}+{\mathrm{H}\mathrm{S}}^{\mathrm{*}} \mathrm{L}\mathrm{B}\right)/2$	22.22	5.56	16.67
10%	Eshelby’s approach	24.89	6.32	19.18
	Voigt approach	25.37	6.34	19.03
	$\left({\mathrm{H}\mathrm{S}}^{\mathrm{*}} \mathrm{U}\mathrm{B}+{\mathrm{H}\mathrm{S}}^{\mathrm{*}} \mathrm{L}\mathrm{B}\right)/2$	25.41	6.75	18.65
20%	Eshelby’s approach	27.94	7.26	21.89
	Voigt approach	28.70	7.18	19.03
	$\left({\mathrm{H}\mathrm{S}}^{\mathrm{*}} \mathrm{U}\mathrm{B}+{\mathrm{H}\mathrm{S}}^{\mathrm{*}} \mathrm{L}\mathrm{B}\right)/2$	29.02	8.18	20.83
30%	Eshelby’s approach	31.47	8.41	24.80
	Voigt approach	32.26	8.07	24.20
	$\left({\mathrm{H}\mathrm{S}}^{\mathrm{*}} \mathrm{U}\mathrm{B}+{\mathrm{H}\mathrm{S}}^{\mathrm{*}} \mathrm{L}\mathrm{B}\right)/2$	33.14	9.90	23.24

* Hashin–Shtrikman approach.

clearly illustrate the beneficial effects of NS on the structure of concrete. In particular, the stiffness of nano-reinforced concrete consistently increases as the volume percentage of NS increases. While there may be slight variations due to varying hypotheses, this trend is observed in all homogenization models. The observed increase in elastic stiffness with higher NS content can be physically attributed to the densification of the cementitious matrix and the improved load transfer through the Interfacial Transition Zone (ITZ), which collectively enhance the effective mechanical properties of the nanocomposite. This upward trajectory in stiffness is a key indicator of the potential improvements in mechanical properties that can be expected from these reinforcements.

On another point, as our analysis focuses on assessing the dynamic behavior of concrete slabs reinforced with NS, it is crucial to evaluate the influence of this reinforcement on the overall density of the nanocomposite. Density, directly related to the mass of the nanocomposite slab, plays a central role in dynamic considerations [75].

In Figure 3, the impact on the density of the nanocomposite concrete matrix is illustrated as the volume percentage of nano-SiO₂ varies, using the simple mixing rule. Figure 3 illustrates trend where the incorporation of these reinforcements tends to decrease the density of the nanocomposite, a favorable result when analyzing the element dynamically. This reduction in density can be physically explained by the low intrinsic density of nano-silica compared to the cementitious matrix, which slightly lightens the overall composite without compromising stiffness, thereby contributing to improved dynamic performance.

After carefully confirming the estimated equivalent elastic properties through the trusted Eshelby model, we can now proceed into the mechanical aspect—dynamic analysis. Our focus shifts to examining the vibration performance of NS-aided concrete slabs in this next stage of investigation. The reduction in density associated with NS addition may also imply enhanced vibration performance, as lower mass can favor higher frequency responses for the same stiffness configuration. This interrelation between stiffness and density highlights the multi-scale influence of nanoscale additives on global dynamic characteristics.

3.2. Validation of the Plate Theory

Due to the significant contribution of deformation plate theories in predicting the natural frequencies of nano-composite plates, it is crucial that we thoroughly validate our analytical modelling. This is especially important through the use of the refined Quasi-3D plate theory, which is known to account for the stretching effect that evolves through the plate thickness.

In this section, the following dimensionless parameters are utilized to present and standardize the equations, facilitating a more meaningful comparison of results, regardless of their units.

• For the dynamic analysis:

$${\omega }_n=10h\sqrt{{\displaystyle \frac{E_m}{{\rho }_m}}}w\left({\displaystyle \frac{L_s}{2}},{\displaystyle \frac{b_s}{2}},0\right)$$

(29)

• For the different foundations:

$$k_w={\displaystyle \frac{{L_s}^4}{D_m}}{\tilde{k}}_w ; k_s={\displaystyle \frac{{L_s}^2}{D_m}}{\tilde{k}}_s ;c_d={\tilde{c}}_{d}h\sqrt{{\displaystyle \frac{h_s}{{\rho }_m{D}_m}}} ; D_m={\displaystyle \frac{{E_m{h}_s}^3}{12\left(1-{{\upsilon }_m}^2\right)}}$$

(30)

As there is a lack of existing data on the dynamic behavior of concrete plates reinforced with NS, we have adopted similar material and geometric parameters as those utilized by Thai and Choi [76] in their study of functionally graded material plates. This serves as a valuable method to validate our mathematical approach.

As illustrated in the

Table 5. Comparison of the current theory with other theories in the literature (L_s/h_s = 10, L_s = b_s).

Theory	Shape Function f(z)	Power Law Index ‘P’
		0	0.5	1	2	5
Refined Quasi-3D theory (Present)	$f\left(z\right)=-{\displaystyle \frac{z}{4}}+{\displaystyle \frac{5z^3}{3h^2}}$	0.1085	0.0924	0.0836	0.0761	0.0715
Refined plate theory (RPT) [76]	$f\left(z\right)=-{\displaystyle \frac{z}{4}}+{\displaystyle \frac{5z^3}{3h^2}}$	0.1135	0.0964	0.0869	0.0788	0.0740
Trigonometric shear deformation theory (TrSDT) [77]	$f\left(z\right)=-{\displaystyle \frac{z}{4}}+{\displaystyle \frac{5z^3}{3h^2}}$	0.1134	0.0975	0.0891	0.0819	0.0767
First order shear deformation theory (FSDT) [78]	$f\left(z\right)=z$	0.1133	0.0963	0.0868	0.0789	0.0744
Classical plate theory (CPT) [79]	$f\left(z\right)=0$	0.1164	0.0986	0.0888	0.0807	0.0765

, the results demonstrate notable consistency in dynamic analysis. The refined Quasi-3D deformation theory employed yields outcomes closely aligned with other deformation theories found in the literature, showcasing only minor discrepancies. This slight difference can be attributed to the Quasi-3D theory’s inclusion of the stretching effect, a factor not considered in the refined plate theory (RPT) and trigonometric shear deformation theory (TrSDT), despite utilizing the same shape function f(z). The agreement observed across these theories significantly affirms the validity of our modeling approach using the refined Quasi-3D in accurately predicting the dynamic behavior of plates.

3.3. Dynamic Analysis of RC Slabs Reinforced with SiO₂ Nanoparticles

Up to this point, we have substantiated our estimation of the elastic properties of the nanocomposite through the Eshelby homogenization approach and validated our analytical modeling for predicting the dynamic response of plates using refined Quasi-3D slab theory. In the subsequent sections, our focus turns to examining the dynamic behavior by analyzing the vibrational response of simply supported square concrete slabs infused with different amounts of NS.

A comparison of the fundamental nondimensional frequencies (${\omega }_n$) obtained through the application of the refined Quasi-3D slab theory and other theories from existing literature is presented in

Table 6. Nondimensional fundamental frequency (${\omega }_n$) of simply supported RC plate infused with NS, (L_s/h_s = 10, L_s = b_s).

Theory	Volume Percentage of NS Reinforcement ‘Vr’
	0	5	10	15	20	25	30
Refined Quasi-3D theory (Present)	0.6719	0.6947	0.7178	0.7413	0.7653	0.7897	0.8147
Refined plate theory (RPT)	0.6867	0.7101	0.7339	0.7581	0.7830	0.8084	0.8346
Trigonometric shear deformation theory (TrSDT)	0.6867	0.7101	0.7339	0.7581	0.7830	0.8084	0.8346
First order shear deformation theory (FSDT) *	0.6867	0.7100	0.7338	0.7581	0.7830	0.8084	0.8346
Classical plate theory (CPT)	0.6986	0.7223	0.7465	0.7711	0.7963	0.8222	0.8488

* Shear correction factor for the first order shear deformation theory is (5/6).

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The findings depicted in Table 6 clearly demonstrate that, with an increasing volume percentage ‘V_r’ of NS-reinforcement, the frequencies of concrete slabs tend to increase, highlighting the rigidification effect induced by these nano-sized entities on the concrete slab. The natural frequency of the plate has been increased by up to 22% by incorporating 30 wt% of nano-SiO₂ compared with an unreinforced plate. Further analysis of results across different shear deformation theories reveals a slight discrepancy between the refined Quasi-3D theory and the refined plate theory (RPT) as well as the trigonometric theory (TrSDT). This discrepancy is directly attributed to the fact that the Quasi-3D theory takes into account the stretching effect (${\epsilon }_z\ne 0$), which evolves through the plate thickness ‘h_s’. The discrepancy is particularly noticeable despite the application of the same shape function f(z). The increase in natural frequencies with higher NS content is primarily due to the stiffening effect imparted by the nanoparticles, while the slight discrepancy between the Quasi-3D theory and other deformation theories arises from its consideration of the stretching effect through the plate thickness, which is neglected in the other models.

Determining the proper mode shapes of a plate for free vibration is crucial for understanding its dynamic behavior and ensuring structural integrity. Figure 4 presents a comparison of mode shapes derived from the free vibration analysis of a simply supported concrete slab reinforced with varying volumes of NS. The concrete plate is simulated using the refined Quasi-3D plate theory. These mode shapes are contrasted with those of an unreinforced concrete slab (V_r = 0%), emphasizing the discernible vibratory alterations introduced by the inclusion of NS. The depicted mode shapes, referred to as coupled modes according to Equation (2), demonstrating the flexure deformations of the plate.

The incorporation of NS enhances the effective elastic and shear moduli of concrete through two key mechanisms: the filler effect of nanoparticles densifies the cementitious matrix, and the refinement of the interfacial transition zone (ITZ) strengthens the weak link between aggregates and paste. For a given NS dosage, all analyzed mode shapes exhibit a consistent percentage increase in natural frequencies, reflecting the uniform stiffening effect imparted by the nanoparticles. Consequently, the mode shapes visually confirm the expected deformation patterns, while the frequency analysis quantitatively captures this enhancement. Together, these observations demonstrate the improved dynamic performance of the NS-reinforced concrete slab compared to its unreinforced counterpart.

3.4. The Effect of Winkler–Pasternak’s Elastic Foundation

To take account of the complex interactions between the reinforced concrete slab and the ground, elastic foundations are incorporated. This consideration encompasses scenarios in which the slab is subjected to vertical and shear forces from the environment. In the following analysis, the Winkler–Pasternak elastic foundation model is used, which comprises a series of Winkler springs with the stiffness ‘k_w’ and a Pasternak shear layer with the stiffness ‘k_s’, as illustrated in Figure 5a.

Figure 6 depicts the influence of the Winkler springs constant ‘${\tilde{k}}_w$’, also known as the spring constant, on the nondimensional frequency (${\omega }_n$) of a concrete slab reinforced with various volumes of NS.

The findings presented in Figure 6 demonstrates that the ‘${\tilde{k}}_w$’ spring constant has a weakening impact on the nondimensional frequencies (${\omega }_n$), causing them to decrease as the value of the spring constant increases. This pattern remains consistent regardless of the quantity of NS reinforcements used. This behavior can be attributed to the increased flexibility of the slab-foundation system induced by higher Winkler stiffness, which reduces the overall dynamic stiffness of the plate, thus lowering the natural frequencies.

In a similar context, Figure 7 illustrates the influence of Pasternak’s shear layer constant ‘${\tilde{k}}_s$’ on the nondimensional fundamental frequency of a concrete slab infused with NS. Analogous to the spring constant, the shear layer constant exhibits a decreasing effect on the plate’s frequencies. However, the impact of the shear layer constant is more pronounced, leading to a substantial increase in frequencies. Overall, this decrease in calculated frequencies is primarily attributed to the role of these constants in distributing shear across the plate geometry. The increase in Pasternak shear stiffness affects the internal shear distribution within the slab, thereby modifying its dynamic response and influencing the natural frequencies according to the slab–foundation interaction.

Table 7. The effect of the Winkler–Pasternak foundation on the natural frequency of a concrete plate reinforced with NS (L_s/h_s = 10, L_s = b_s, ${\tilde{c}}_d$ = 0).

${\check{\mathit{k}}}_{\mathit{w}}$	${\check{\mathit{k}}}_{\mathit{s}}$	Volume Percentage of NS Reinforcement ‘Vr’
		0	5	10	15	20	25	30
10	0	0.6719	0.6947	0.7178	0.7413	0.7653	0.7897	0.8147
	5	0.6659	0.6889	0.7122	0.7359	0.7600	0.7845	0.8097
	10	0.6033	0.6285	0.6538	0.6794	0.7054	0.7317	0.7585
20	10	0.5335	0.5616	0.5897	0.6179	0.6462	0.6748	0.7036
	20	0.5259	0.5544	0.5828	0.6113	0.6399	0.6687	0.6978
30	30	0.3433	0.3851	0.4247	0.4627	0.4996	0.5357	0.5715
50	30	0.2211	0.1368	0.1141	0.2155	0.2857	0.3447	0.3976

illustrates the influence of the Winkler–Pasternak elastic foundation swapping various configurations, on the vibrational response of RC slabs infused with different proportions of silicon dioxide nanoparticles (SiO₂).

From a structural design standpoint, the observed frequency enhancement with higher NS ratios implies that such modifications could be strategically employed to mitigate resonance phenomena in slabs exposed to repetitive dynamic loading, such as vehicular or machinery-induced vibrations.

As observed earlier, Table 7 corroborates the previous findings that these elastic foundation patterns have a decreasing effect on the natural frequencies of the RC slabs. This trend is consistently observed across different elastic foundation patterns.

3.5. The Effect of the Visco–Pasternak Foundation

To study the influence of the visco–Pasternak foundation, a damping coefficient is introduced to Pasternak shear layer as demonstrated in Figure 5b.

The next stage of analysis focuses on the damping coefficient ‘${\tilde{c}}_d$’, introduced within the viscoelastic layer, to quantify its contribution to energy dissipation and frequency modulation in nano-reinforced slabs. Figure 8 is presented. Different volume percentages of NS are studied and compared with an unreinforced slab. In contrast to the elastic foundation, the damping parameter in this type of foundation tends to increase the slab’s nondimensional frequencies, whatever the percentage of reinforcement, and this is demonstrated consistently in all cases of reinforcement. The damping coefficient enhances the dynamic response by dissipating vibrational energy, which effectively increases the natural frequencies of the slab, highlighting the interplay between foundation damping and NS-induced stiffness.

3.6. Effect of Visco–Winkler–Pasternak Foundation

To comprehensively incorporate the three parameters of the viscoelastic foundation, namely Winkler springs ‘${\tilde{k}}_w$’, Pasternak’s shear layer ‘${\tilde{k}}_s$’, and the damping coefficient ‘${\tilde{c}}_d$’, all three are simultaneously introduced to form a viscoelastic foundation, as depicted in Figure 5c.

Table 8. Comparison of nondimensional fundamental frequency of a concrete plate impregnated with NS and connected to the Winkler–Pasternak-visco foundation. (L_s/h_s = 5, L_s = b_s, V_r = 30%).

${\check{\mathit{k}}}_{\mathit{s}}$	${\check{\mathit{k}}}_{\mathit{w}}$	$\mathbf{Damping} \mathbf{Parameter} ‘{\tilde{\mathit{c}}}_{\mathit{d}}’$
		1	3	6	10	15	20	30
0	10	3.9539	4.9843	5.4385	5.8839	6.0557	6.1463	6.2401
	20	3.9409	4.9776	5.4343	5.8818	6.0543	6.1452	6.2394
	50	3.9013	4.9573	5.4216	5.8754	6.0500	6.1421	6.2373
10	10	3.6821	4.8478	5.3535	5.8414	6.0276	6.1253	6.2262
	20	3.6674	4.8407	5.3491	5.8392	6.0262	6.1243	6.2255
	50	3.6228	4.8191	5.3358	5.8327	6.0219	6.1210	6.2233
20	10	3.3721	4.7017	5.2647	5.7980	5.9991	6.1041	6.2122
	20	3.3551	4.6941	5.2601	5.7958	5.9976	6.1031	6.2115
	50	3.3031	4.6708	5.2463	5.7891	5.9933	6.0998	6.2093
30	10	3.0037	4.5439	5.1719	5.7537	5.9702	6.0827	6.1981
	20	2.9828	4.5356	5.1671	5.7515	5.9687	6.0817	6.1974
	50	2.9186	4.5103	5.1526	5.7446	5.9643	6.0784	6.1952

elucidates the influence of the three-parameter viscoelastic foundation on the natural frequency of thick concrete slabs (a/h = 5), infused with various volumes ‘V_r’, of nano-sized NS. In this table, various viscoelastic configurations are examined to thoroughly analyze the impact of this foundation type on the nondimensional frequency of RC slabs. Notably, the influence of the damping parameter ‘${\tilde{c}}_d$’, is considerably more pronounced than the other two parameters (${\tilde{k}}_w$, ${\tilde{k}}_s$). With an increase in the damping coefficient ‘${\tilde{c}}_d$’, the nondimensional frequency (${\omega }_n$), tends to intensively rise, imparting greater stiffness and rigidification to the plate. Consistent with earlier investigations, the Winkler and Pasternak constants (${\tilde{k}}_w$, ${\tilde{k}}_s$), continue to exhibit a weakening effect, albeit less apparent in the presence of the damping parameter.

It is also relevant to examine the impact of the visco–Winkler–Pasternak foundation on the non-dimensional frequency of different types of concrete plates, both thin and thick. The thickness-to-length geometry ration of the plate (a/h), is then variated to mainly showcase the influence of the damping parameter ‘${\tilde{c}}_d$’.

Figure 9 demonstrates the impact of a viscoelastic foundation on plates reinforced with ‘V_r = 30%’ NS. The influence of this viscoelastic foundation becomes more pronounced with increasing plate thickness, as depicted in the figure. However, it is noteworthy that, regardless of whether the plate is thin or thick, this type of foundation exerts a reinforcing effect. This effect stems from the viscoelastic foundation’s ability to dissipate energy as the plate vibrates, resulting in damping effects. Consequently, damping reduces the vibration amplitude and alters the dynamic response of the plate.

4. Conclusions

Integrating NS into concrete brings substantial benefits, improving its strength and longevity while potentially reducing maintenance costs and advancing sustainability. This study carefully examined their influence on concrete dynamics, employing refined Quasi-3D deformation theory and Eshelby’s model to determine elastic properties. Exploring various elastic foundation configurations, including viscoelastic models, alongside comprehensive parametric studies, assessed nanoparticle volume effects, plate geometry, and mode shapes.

Our investigation revealed that:

• Consistent with existing understanding; our analytical homogenization revealed that the addition of NS to concrete matrices significantly enhances their elastic properties, a correlation observed in relation to the volume of these reinforcements.
• Infusing high volumes of NS into a concrete matrix (V_r = 30 wt%) can increase the elastic properties of the nano-infused matrix by up to 26% and also enhance the elastic stiffness of the nano-infused concrete matrix by up to 30%.
• Dynamic analysis has shown a notable enhancement in plate dynamics with the incorporation of nano-SiO₂, with the most significant improvement observed when incorporating 30 wt% of NS, resulting in an increase in plate natural frequency by up to 18%.
• Incorporating NS as reinforcement in concrete slabs significantly impacts and alters the inherent mode shapes of the plates, rendering them more valuable, Optimal amount of NS (V_r = 30 wt%) results in a significant increase of up to 32% in all studied mode shapes of the nano-composite plate.
• Contrasting the shear layer and spring constants of the elastic foundation, the damping parameter within the viscous foundation significantly contributes to increased frequencies, thereby enhancing plate stability.

Despite the successful validation and application of the framework, several limitations remain. The use of Navier’s analytical method assumes idealized simply supported boundary conditions and a geometrically regular slab, which limits direct applicability to real structures with complex geometries and support conditions. The Eshelby homogenization model assumes linear elasticity and dilute nanoparticle concentrations and does not account for nanoparticle agglomeration, which may affect real-world material performance. Additionally, both the structural model and the three-parameter foundation formulation are restricted to linear elastic and viscoelastic behavior, neglecting possible nonlinear, inelastic, or transient responses under extreme dynamic loading conditions. Future research should address these limitations by employing numerical techniques such as FEM or GDQM to handle complex boundary conditions, incorporating nonlinear material models for concrete, and adopting advanced constitutive models for soils. Experimental validation of predicted natural frequencies would further strengthen the reliability of the multiscale approach.

The results of this study demonstrate that NS incorporation enhances dynamic performance, increases vibration resistance, and extends the service life of concrete slabs. The combination of nano-reinforcements and viscoelastic foundations provides a clearer understanding of soil–structure interaction, enabling the design of more resilient and sustainable infrastructure. The developed model also serves as a practical tool for preliminary design and optimization of slab performance under dynamic and soil-supported conditions, with potential extensions to temperature-dependent viscoelasticity and interfacial damage models.