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Abstract

Unidirectional fiber-reinforced polymer matrix composites (PMCs) have great potential for engineering applications. The effective properties can be further tailored by incorporating a secondary reinforcement into these materials. This study investigates the damping characteristics of PMCs reinforced by both unidirectional glass fibers and hollow glass microspheres (HGMs) using a micromechanics-based finite element method (FEM). To account for the complex nature of the filler/matrix dissipation mechanisms, a thin, lossy interphase layer is considered for both the fibers and HGMs inside the representative volume element (RVE) of the PMCs. The numerical homogenization approach is first validated against analytical, numerical, and experimental data from the literature. Subsequently, a comprehensive parametric study is conducted to examine the influence of microstructural features on the composite’s damping behavior. Specifically, the effects of thickness, stiffness, and damping properties of the interphase, as well as fiber volume fraction (FVF), HGM volume fraction (HVF), and HGM thickness on the directional loss factors are evaluated using the strain energy approach. The findings indicate that while an increase in FVF generally reduces the damping performance of the composite, a weak interfacial bond between the fillers and the matrix can enhance the damping properties of hybrid filler-reinforced PMCs.

Keywords

Polymer matrix composite hybridized fiber/hollow microsphere damping interphase micromechanical finite element modeling

Introduction

The development of advanced composites to supersede conventional materials with superior multifunctionality has been an area of wide research interest.^1–4 Many applications, including aerospace structures, sporting equipment, automotive and marine industries, require materials with excellent stress wave attenuation and vibration damping capabilities.^2,3 Material damping is a critical design consideration, as it mitigates vibration amplitudes, extends fatigue life, and improves impact resistance.⁴ The effective damping capacity of a composite material is not only dependent on the inherent damping properties of its constituent materials but is also influenced by microstructural features. These include the volume fraction, thickness, shape, and orientation of the reinforcement, as well as the interfacial bonding properties between the reinforcement and matrix.^5–7 The primary mechanisms contributing to damping and energy dissipation in fiber-reinforced PMCs include the viscoelastic nature of the polymer matrix and fibers, damping from the interfacial region and matrix micro-cracks, and viscoplastic damping.^8–10 The viscoelasticity of the polymer matrix is the most significant contributor among these parameters. Furthermore, significant relative displacement at the interface can lead to debonding and friction, which represents a fundamental mode of energy dissipation during vibration.¹¹

Micromechanics methods have been widely utilized to predict the effective properties of composite materials.^12–14 However, their application to the prediction of composite damping has been less explored.¹⁵ Several micromechanical approaches have been used to predict the damping behavior of composites, including viscoelastic constitutive models and the strain energy approach. In the first approach, at least one component of the composite, typically the matrix, is modeled as a viscoelastic material. Its behavior is characterized in the time or frequency domain using either a Prony series^16–20 or a complex modulus at a specific frequency, often derived from dynamic mechanical analyzer (DMA) measurements.^4,21,22 In the second approach, the effective damping of the composite is calculated as the ratio of the dissipated strain energy to the stored strain energy.^15,23 Micromechanics is employed to approximate the dissipated strain energy by the homogenizing the local strain energy multiplied by the local damping coefficient.^24,25 In both methods, it is crucial to account for the contributions of the reinforcement/matrix interfacial slip, friction, and shear loss. This is achieved by adding these effects as external forces in mathematical models^26–28 or by modeling a weakened interphase region.^16,17,23,24

Engineering continually strives to develop high-performance composites with enhanced multifunctional capabilities. In this front, hybridization of reinforcing phase offers another approach for introducing new functionalities and enhancing the properties of single-fiber-reinforced composites in terms of strength, damping, and ductility.^29–32 Composites containing reinforcing phase that provide desirable thermal, mechanical, and electrical properties at a lower weight and cost, are particularly favored for industrial applications. HGMs have emerged as key materials for creating lightweight, multifunctional composites due to their low weight, desirable thermal properties, and low permittivity.³³ The properties of HGM-filled hybrid composites have been the focus of several studies.³⁴ For instance, Ferreira et al. experimentally investigated the mechanical behavior of glass and carbon fiber-reinforced epoxy composites, examining the effects of HGMs and fiber volume fraction on flexural stiffness, fracture toughness, compressive and impact behavior,³⁵ and storage and loss modulus.³⁶ Biarjemandi et al.³⁷ and Moradi et al.³⁸ used micromechanics-based FEM to investigate the mechanical properties, thermal conductivity, and thermal expansion coefficient of hollow sphere/glass fiber-reinforced polymer hybrid composites, focusing on the effects of FVF, HVF, and HGM thickness. Although the three-phase RVEs adopted in the previous studies provided reasonable predictions for mechanical and thermal properties, the interphase cannot be neglected when modeling damping owing to its more complex and interfacial nature.

Despite extensive research on the damping properties of conventional fibrous and particulate composites, some notable gaps need to be addressed. The study of the damping behavior of advanced PMCs, particularly those based on unidirectional fiber/HGM hybrids, is an area that warrants further investigation. This study aims to fill this gap by using finite element (FE) and micromechanical approaches to investigate the effects of interphase properties and reinforcement volume fractions on the damping properties of hybrid filler-reinforced epoxy (HFRE) composites. To this end, RVEs consisted of unidirectional glass fiber, HGM, and epoxy matrix with the appropriate boundary conditions are constructed. This paper is organized as follows: Section Methodology outlines the fundamental principles of damping evaluation based on the strain energy approach. Section Numerical homogenization details the FE considerations for achieving efficient and accurate damping models for composites. In Section Results and discussion, the computational framework is validated. Finally, Section Conclusion presents the results of a parametric study of micromechanical features on the overall damping behavior of the hybrid PMC. The findings of numerical simulations of damping coefficients can provide a valuable foundation for designing the composite structures subjected to vibrational loadings.

Methodology

The micromechanical damping analysis of unidirectional fiber-reinforced composites containing HGMs requires determining the contributions of five constituents: the matrix, fibers, microspheres, and interphase regions of both the fiber/matrix and microsphere/matrix. Based on the concept of energy dissipation, the specific damping capacity ( $ψ$ ) of a material in vibration is defined as the ratio of the dissipated energy ( $D$ ) to the stored energy ( $U$ ) per cycle of vibration, as expressed in equation (1)²⁵:

ψ = \frac{D}{U}

(1)

As established by Rikards et al.,³⁹ the loss factor ( $η$ ) is related to the specific damping capacity by equation (2):

η = \frac{ψ}{2 π}

(2)

For the five-phase hybrid composite in this study, the overall loss factor can be expressed in terms of the reinforcement, matrix, and interphase components as equation (3):

η = \frac{η_{f} U_{f} + η_{f, i} U_{f, i} + η_{p} U_{p} + η_{p, i} U_{p, i} + η_{m} U_{m}}{U_{f} + U_{f, i} + U_{p} + U_{p, i} + U_{m}}

(3)

where

η_{f}

η_{f, i}

are the loss factors of fiber and its interphase, respectively;

η_{p}

η_{p, i}

are the loss factors of the HGM and its interphase, respectively; and

η_{m}

is the loss factor of the matrix.

U_{f}

U_{f, i}

are the strain energies stored in the fiber and its interphase, respectively;

U_{p}

U_{p, i}

are the strain energies stored in the HGMs and their interphases, respectively; and

U_{m}

is the strain energy stored in the matrix.

To evaluate the overall HFRE composite loss factors, the RVE is subjected to normal and shear loadings in the principal material directions (XYZ or 123). The resulting stress and strain distributions within the RVE are then calculated through micromechanical analysis. The strain energy of each constituent is obtained by summing the induced energy over the volume of each component, as shown in equation (4)²⁵:

U = \frac{1}{2} \int_{V} [σ_{11} ε_{11} + σ_{22} ε_{22} + σ_{33} ε_{33} + σ_{23} γ_{23} + σ_{13} γ_{13} + σ_{12} γ_{12}] d V

(4)

Isotropic damping properties are assumed for the reinforcement fillers, matrix, and interphases, with the corresponding values listed in Table 1. Due to the difficulty in precisely estimating interphase properties,^17,24 two isotropic materials are considered for the interphase: “Hard interphase” has an elastic modulus 10 times that of the matrix and a damping coefficient half that of the matrix; and “Soft interphase” has a low stiffness (0.1 of the matrix) and high damping (5 times that of the matrix). A hard interphase typically arises from strong chemical bonding, surface treatments, or nanoparticle coatings that stiffen the region around the reinforcement. This enhances load transfer and improves stiffness and strength, but reduces energy dissipation and damping. Conversely, a soft interphase, resulting from weak bonding, poor adhesion, or interfacial damage, lowers stiffness but increases damping capacity. This trade-off between stiffness and damping highlights the challenge of simultaneously optimizing both properties in composites. It should be noted that for hybrid composites with a uniform distribution of HGMs and assuming the fiber is oriented in the x-direction, only four independent loss factors (

η_{11}

η_{22}

η_{12}

η_{23}

) need to be calculated.²⁵

Table 1.

Material properties of HFRE constituents.²⁴

Properties	E-glass fiber	HGM	Epoxy	Soft interphase	Hard interphase
$E (G P a)$	72.4	72.4	2.76	0.276	27.6
$ν$	0.2	0.2	0.35	0.35	0.35
$η$	0.0018	0.0018	0.015	0.075	0.0075

Numerical homogenization

The RVEs, which consist of the unidirectional glass fibers and HGMs, are generated using COMSOL Multiphysics software. The fibers are modeled with a radius of 0.37 mm and default volume fraction of 43%. E-glass hollow spheres, with a radius of 0.1 mm, thickness of 0.05 mm, and a default volume fraction of 5%, are randomly distributed throughout the epoxy matrix. It is assumed that there is no intersection or collision between the fibers and HGMs. Given the thin nature of the interphase region (e.g., 1/70th the radius of unidirectional carbon fibers as noted in Ref. 40), the effect of the interphase is modeled using the thin layer feature in COMSOL. This approach allows the interphase effect to be considered as a thin analytical surface with specified elastic and damping properties, avoiding the need to modify the RVE geometry by adding a physical thin layer around the reinforcements.^41,42 The main advantage of this method is that it increases the accuracy of the 3D model by accounting for the interphase region without the computational burden associated with a large number of small mesh elements that would be required for a thin physical layer.

The use of a single-fiber RVE is a common simplification that reduces computational cost and isolates fundamental micromechanical mechanisms. However, it neglects microstructural randomness, fiber clustering, and misalignment. Furthermore, the model assumes perfect bonding between reinforcements and matrix, and therefore nonlinear damping mechanisms such as debonding, interfacial friction, and stick-slip phenomena^8,15 are not captured. The damping formulation in this work is based on the strain energy method, assuming frequency-independent loss factors calculated as weighted averages of constituent loss factors and strain energies. In reality, polymer matrix composites exhibit strong viscoelastic behavior, with properties varying significantly with frequency and strain rate.

Periodic boundary conditions (PBCs), widely accepted as the most appropriate for analyzing geometrically periodic RVEs,⁴³ are applied to the nodal points on the surface of the RVE. Six load cases are applied, of which two are expected to yield similar results. The models are discretized with tetrahedral elements, comprising an average of 57,837 elements and 276,888 degrees of freedom, as shown in Figure 1. These mesh sizes are selected based on convergence studies, which ensured less than 1% deviation in strain energy predictions. Stationary FE simulations are then performed to predict the strain energy and effective damping properties of the HFREs. As detailed in Table 2, four distinct effective loss factors ( $η_{i j}^{e f f}$ ) of the HFREs are evaluated by applying the corresponding boundary conditions to each face of the RVE. The strain, $ε_{0}$ , is set to 0.01 of the RVE length ( $L_{R V E} = 1 m m$ ). The boundary conditions are similar to those reported in Ref. 44 for evaluating axial ( $C_{11}^{e f f}, C_{22}^{e f f}$ ) and shear ( $C_{44}^{e f f}, C_{66}^{e f f}$ ) components of stiffness matrix. However, in this study, the induced elastic and dissipated energies are used to evaluate the loss factor of the HFRE composites instead of directly calculating the stiffness components. For simplicity, the effective loss factor of the HFRE composites in different directions ( $η_{i j}^{e f f}$ ) is presented without “ $e f f$ ” superscript. Following the application of the boundary conditions specified in Table 2, the distributions of stored elastic energy density and displacement within the RVE are presented in Figure 2. The red arrows represent the displacement field, while the color bar illustrates the spatial distribution of energy density. As observed, except for the 11 direction-where the majority of the energy is stored in the fiber phase-the energy in the other three directions is predominantly distributed within the matrix. This suggests that damping behavior is fiber-dominated in the fiber direction, whereas it is matrix-dominated in the transverse and shear directions. This observation is further explored by the subsequent results.

Figure 1.

3D FE model of the RVE with arrangement of HGMs throughout the RVE in adopted Cartesian coordinate system.

Table 2.

Boundary conditions for calculating the effective loss factors of HFREs.

Loss factor	Boundary conditions	Dissipated energy
$η_{11}^{e f f}$	$u_{1} (x = 0) = 0$	$D = \frac{V}{2} η_{11}^{e f f} C_{11}^{e f f} ε_{0}^{2}$
	$u_{1} (x = 1) = ε_{0} \times L_{R V E}$
	$u_{2} (y = 0 . y = 1) = 0$
	$u_{3} (z = 0 . z = 1) = 0$
$η_{22}^{e f f}$	$u_{1} (x = 0 . x = 1) = 0$	$D = \frac{V}{2} η_{22}^{e f f} C_{22}^{e f f} ε_{0}^{2}$
	$u_{2} (y = 0) = 0$
	$u_{2} (y = 1) = ε_{0} L_{R V E}$
	$u_{3} (z = 0 . z = 1) = 0$
$η_{23}^{e f f}$	$u_{1} (x = 0 . x = 1) = 0$	$D = \frac{V}{2} {η_{12}^{e f f} C_{44}^{e f f} γ}_{0}^{2}$
	$u_{1} (y = 0 . y = 1) = 0$
	$u_{3} (y = 0 . y = 1) = 0$
	$u_{1} (z = 0 . z = 1) = 0$
	$u_{2} (z = 0) = 0$
	$u_{2} (z = 1) = {2 ε}_{0} L_{R V E}$
$η_{12}^{e f f}$	$u_{2} (x = 0 . x = 1) = 0$	$D = \frac{V}{2} η_{12}^{e f f} C_{66}^{e f f} γ_{0}^{2}$
	$u_{3} (x = 0 . x = 1) = 0$
	$u_{1} (y = 0) = 0$
	$u_{1} (y = 1) = {2 ε}_{0} L_{R V E}$
	$u_{3} (y = 0 . y = 1) = 0$
	$u_{3} (z = 0 . z = 1) = 0$

Figure 2.

Stored elastic energy density (J/mm³) and displacement fields (mm) in the RVE under four periodic loads for effective loss factor calculations in: (a) 11, (b) 22, (c) 12, and (d) 23 directions.

In order to investigate the effect of the number of HGMs on the directional damping properties, RVEs with a constant HVF of 5% and a hollow spheres thickness of 0.05 mm are considered with different lengths ( $L_{R V E}$ ). It is found that increasing the number of HGMs from 9 to 12, and from 12 to 15, resulted in changes in the directional loss factors within a range of $\pm 0.6 %$ . Therefore, to maintain a balance between accuracy and computational efficiency, an $L_{R V E}$ of 1 mm and 12 randomly dispersed HGMs are used in the FE simulations.

Results and discussion

This section validates the importance of considering the interphase in modeling the damping properties of composites by comparing the results of this study with numerical and experimental data from the literature. After verifying the accuracy of the computational framework, the effects of the interphase properties, FVF, and HVF on the damping of the HFRE composites are discussed.

Validation

First, due to the insufficient direct experimental data, the homogenization results from the proposed micromechanics-based FE model are compared with the numerical results of an E-glass fiber-reinforced epoxy composite (without hollow spheres) from the study by Chandra et al.⁴⁵ The damping and elastic properties of the constituents are similar to those presented in Table 1, and perfect bonding is assumed between the fiber and the matrix. For comparison, the FE results are benchmarked against two different analytical damping models: one based on Eshelby’s method and another on unified micromechanics, as proposed by and Zhao and Weng⁴⁶ and Saravanos and Chamis,⁴⁷ respectively. The detailed formulations are presented in Ref. 45; therefore, only the final results are reported here. The comparison studies are depicted in Figure 3(a)–(d). The present FE model shows strong agreement with Eshelby’s approach,⁴⁶ with a predictive error margin of less than 5%, which demonstrates its reliability in estimating the damping properties of composites. Additionally, when compared to the FE method results from Ref. 45, the present model offers an acceptable agreement. As illustrated in Figure 3(a), the Saravanos-Chamis approach⁴⁷ accurately predicts the loss factor along the fiber direction ( $η_{11}$ ), but its predictions for $η_{22}$ , $η_{23}$ , and $η_{12}$ are not consistent with other models.

Figure 3.

Variation of loss factor (a) $η_{11}$ , ((b)) $η_{22}$ , ((c)) $η_{23}$ , and (d) $η_{12}$ of unidirectional composites with FVF for different models.

While a two-phase composite model has previously been sufficient for predicting the damping of E-glass/epoxy composites, this study further validates the current model by comparing its predictions against the experimental findings of Saravanos and Chamis⁴⁷ for a different system: E-glass/IMLS (polyester) unidirectional composites. The specific damping capacities of IMLS and E-glass in the axial directions (11, 22) are reported as 8.5% and 1.1%,⁴⁷ respectively. In this case, a hard interphase layer with a thickness of $r_{i} = 0.1 r_{f}$ , elastic modulus of $E_{i} = 10 E_{m}$ , Poisson’s ratio of $υ_{i} = υ_{m}$ , and specific damping capacity of $ψ_{i} = 2 ψ_{f}$ is considered. Where, the subscript $i$ , $m$ , and $f$ denote the interphase, matrix and fiber properties, respectively. The elastic properties of polymer matrix are considered as $E_{m} = 2.8$ GPa and $v_{m} = 0.4$ , and E-glass properties are taken from Table 1. The evolution of normalized damping capacity of the composite is exhibited in Figure 4. As depicted in Figure 4(a), the present FE model accurately predicts the damping capacity of the composite at higher FVFs, with an error of less than 5%. It is found from Figure 4(b) that when the FVF is higher than 30%, the predictions of transverse specific damping capacity of the composite by the model are in good agreement with the experimental data.⁴⁷ Overall, these results validate the capability of the strain energy method, coupled with a thin-layer interphase, for predicting the damping properties of composites. The results indicate that the values of specific damping capacity of the composite in longitudinal and transverse directions are decreased as the FVF increases.

Figure 4.

Comparison of predicted and experimental normal (a) longitudinal and (b) transverse specific damping capacity of the composite.

Numerical results of HFRE composites

Figure 5 illustrates the evolution of the directional loss factors of the HFRE composite at default volume fractions with respect to relative interphase thickness ( $r_{i} / r_{f}$ ) on a logarithmic scale. For the soft interphase, the loss factors of the considered hybrid PMCs increase as the interphase thickness increases. The enhancement in the values of $η_{22}$ , $η_{23}$ , and $η_{12}$ is more significant than that in the $η_{11}$ . The axial loss factor ( $η_{11}$ ) in the case of hard interphase follows the same trend, yet with a more gradual incline as compared to the soft interphase case. However, an increase in the fiber interphase thickness has a negligible effect on the transverse normal and shear loss factors.

Figure 5.

Predicted directional loss factors of the HFRE composite as a function of relative interphase thickness around fibers for both hard and soft interphases: (a) $η_{11}$ , (b) $η_{22}$ , (c) $η_{23}$ , (d) $η_{12}$ .

The effects of interphase stiffness and its loss factor at default volume fractions and relative interphase thickness of 0.1 are also investigated, and the numerical results are illustrated in Figure 6. In Figure 6(a), the relative interphase damping is held at 0.5, whereas in Figure 6(b), the relative stiffness is set to 10. As shown in Figure 6(a), the directional loss factor coefficients $η_{11}$ , $η_{23}$ of the hybrid PMCs are enhanced by the increase of relative interphase stiffness. However, the loss factor coefficients $η_{22}$ , $η_{12}$ initially decrease by increasing relative interphase stiffness up to 0.1, but begin to increase thereafter. Additionally, the directional loss factors of the HFRE composite increase with an increase in the loss factor of the interphase as can be seen in Figure 6(b). However, the loss factor in the longitudinal direction is significantly affected by changes in the interphase damping.

Figure 6.

Predicted directional loss factors of the HFRE composite as a function of the relative (a) stiffness, and (b) loss factor of the fiber interphase.

Figure 7 shows the effects of interphase around the HGMs on the directional loss factors with changes in their relative interphase thickness. Due to the low HVF, the changes in the directional loss factor with respect to the relative interphase thickness are insignificant compared to those for the fiber’s interphase (see Figure 5). However, for the soft interphase, $η_{22}$ and $η_{12}$ show a marginal increase with an increase in the interphase thickness. The interphase characteristics are assumed to be similar for both fibers and HGMs, since they are E-glass-based fillers. It is noteworthy to state that the trade-off between damping and stiffness arises from changes in reinforcement interphase properties from soft to hard. This behavior can be tailored through practical methods such as functionalization of reinforcements, controlled polymerization, and adjustment of matrix viscosity or coating characteristics to achieve a desired combination of stiffness and damping.

Figure 7.

Predicted directional loss factors of the HFRE composite as a function of the relative interphase thickness around the HGMs for both hard and soft interphases. (a) $η_{11}$ , (b) $η_{22}$ , (c) $η_{23}$ , (d) $η_{12}$ .

Figure 8 a and b depict the direction loss factors of the HFRE composite as a function of the relative stiffness and loss factor of the interphase around HGMs. As before, in Figure 8(a) the relative interphase damping is held at 0.5, and in Figure 8(b), the relative stiffness is set to 10. As expected, increasing the damping of the HGM interphase, slightly increases the overall loss factors of the studied composite.

Figure 8.

Predicted directional loss factors of the HFRE composite as a function of the relative (a) stiffness, and (b) loss factor of the HGMs’ interphase.

In general, the damping behavior along the fiber direction is predominantly determined by the fiber’s damping properties, while in other directions, it is dominated by the matrix. Moreover, the addition of high-modulus fibers to a polymer matrix tends to reduce the damping characteristics of the resulting composites.⁸ Figure 9 illustrates the decrease in directional loss factor of HFRE composites with an increase in both FVF and HVF. In Figure 9(a), increasing the FVF from 10% to 50%, lead to 49.8% reduction in $η_{11}$ , an 11.8% reduction in $η_{22}$ , and decreases of 6.1% and 10.4% in $η_{23}$ and $η_{12}$ , respectively. In Figure 9(b), the FVF is kept constant at its default value of 43%, and the HVF is varied by changing the number of dispersed HGMs in the RVE. Basically, increasing the number of HGMs increases the volume fraction of lossy interphase around the HGMs. In addition, due to higher surface-to-volume ratio of HGMs compared to the fiber, an increase in the volume fraction of HGMs from 1.25% to 10% results in a 6.3% increase in the normal longitudinal loss factor ( $η_{11}$ ). However, the other three loss factors of the considered hybrid PMC decrease by approximately 4.1% with this increase in HVF.

Figure 9.

Predicted directional loss factors of the HFRE composite as a function of (a) FVF, and (b) HVF.

Incorporating HGMs into the RVE introduces a distinct mechanism for tuning directional damping. While HGMs increase the interface area between reinforcement and matrix, strong bonding between silica particles and the polymer matrix can limit energy dissipation.⁴⁸ By carefully adjusting HGM content, size, and interphase characteristics, it is possible to selectively enhance damping along specific directions, offering directional tailoring not achievable with uniform fibers alone. In comparison, recent studies using carbon nanomaterials^49,50 also improve damping, but do so primarily by introducing additional energy-dissipating mechanisms at the nanoscale without significantly affecting stiffness or weight. These reinforcements promote damping enhancement via synergistic multi-scale reinforcement interactions.

Herein, the FEM is used to evaluate the role of fiber elastic modulus in the damping characteristics of selected PMCs reinforced by the hybrid fillers. The variation of directional loss factors of the composite system with respect to the relative fiber elastic modulus is investigated, and the results are presented in Figure 10. As previously discussed, $η_{11}$ is fiber-dominated and decreases with increasing fiber elastic modulus. In contrast, the other three loss factors of the HFRE composite increase due to the higher contribution of stored elastic energy in the matrix as the fiber stiffness increases. This trend is particularly evident in the case of hard interphase with ( $\frac{E_{f, i}}{E_{m}} = \frac{E_{p, i}}{E_{m}} = 10, \frac{η_{f, i}}{η_{m}} = \frac{η_{p, i}}{η_{m}} = 0.5$ ).

Figure 10.

Predicted directional loss factors of the HFRE composite as a function of relative fiber elastic modulus.

The effect of HGM thickness is also investigated. For an HGM radius of 0.1 mm, the thickness is varied from 0.02 mm to 0.1 mm. A thickness of 0.1 mm corresponds to the case in which there is no void, and the matrix is filled with solid spherical particles. As shown in Figure 11, an increase in the thickness of the HGMs leads to a marginal improvement in the $η_{11}$ and $η_{23}$ . However, the loss factors $η_{22}$ and $η_{12}$ vary non-monotonically with HGM thickness. This behavior can be attributed to the interplay between the effective volume, and the absorbed strain energy in the HFRE composite constituents.

Figure 11.

Predicted directional loss factors of the HFRE composite as a function of HGM thickness.

The mechanical and physical properties of composite materials are affected by their microstructures.^38,51–54 The damping behavior of the composite is strongly influenced by the low loss factor of fibers and their dominant contribution along the fiber direction. Incorporating HGMs into the RVE enables tailoring of damping performance in different directions and under varying loading conditions. The interphase plays a key role, with soft and thick interphases enhancing directional damping coefficients. Additionally, HGM thickness, FVF, and HVF are critical parameters, whose effects on mechanical properties have been confirmed experimentally through the literature.^8,36 By optimizing interphase properties, reinforcement fractions, and HGM thickness, a desired balance of strength and damping can be achieved in the proposed HFRE composite.

Conclusion

This study presented a comprehensive investigation of the damping behavior of HFRE composites using a micromechanics-based finite element approach. This work proposed a detailed examination of the role of the interphase in composites containing both fibers and HGMs, a topic that has received limited attention in the literature to date. The findings demonstrated that the damping properties of these hybrid composites are intricately linked to a variety of microstructural parameters. A significant conclusion was that while increasing the FVF generally led to a reduction in the overall damping of the composite, particularly in the fiber direction, the presence of a “soft” or weakened interphase can counteract this effect and even enhance damping performance. This is a critical insight for the design of composite materials where a balance between stiffness and damping is required. Furthermore, this study highlighted the distinct contributions of the fiber and HGM interphases to the overall damping behavior of the composite. The fiber interphase was found to have a more pronounced effect on the directional damping, whereas the HGM interphase properties, although less influential due to the lower volume fractions considered, still play a role in tuning the directional damping characteristics. By elucidating the complex relationships between microstructure and damping, this paper paves the way for the development of next-generation composites with tailored performance characteristics. Future research could extend this work to explore the effects of non-uniform particle distributions, employing different nano to micro-scaled reinforcements with varying geometries on the damping behavior of these advanced materials.

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of generative AI and AI-assisted technologies in the writing process

During the preparation of this work the authors used Gemini 2.5 Pro in order to improve the readability and language of the manuscript. After using this tool/service, the authors reviewed and edited the content as needed and take full responsibility for the content of the published article.

ORCID iDs

Hasan Seraj

Reza Ansari

Mohammad Kazem Hassanzadeh-Aghdam

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Micromechanics-based finite element modeling of damping in hollow spheres/fiber-reinforced polymer composites

Abstract

Keywords

Introduction

Methodology

Numerical homogenization

Results and discussion

Validation

Numerical results of HFRE composites

Conclusion

Footnotes

Declaration of conflicting interests

Funding

Declaration of generative AI and AI-assisted technologies in the writing process

ORCID iDs

References