# A complete approach for evaluating the vibrational behavior of RTM-manufactured functional composites under random vibrations

Sirine Ben Ameur, Mohamed Mtibaa, Ahmed Yaich, Moez Beyaoui, Abdelkhalak El Hami, Abdelghani Saouab, Mohamed Haddar, Mohamed El Haddar

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Sirine Ben Ameur, Mohamed Mtibaa, Ahmed Yaich, Moez Beyaoui, Abdelkhalak El Hami, et al. A complete approach for evaluating the vibrational behavior of RTM-manufactured functional composites under random vibrations. Composites Communications, 2025, 58, pp.102530. 10.1016/j.coco.2025.102530. hal-05497347

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# A complete approach for evaluating the vibrational behavior of RTM-manufactured functional composites under random vibrations

Sirine Ben Ameur<sup>a,b,c,*</sup>, Mohamed Mtibaa<sup>a,b,c</sup>, Ahmed Yaich<sup>d</sup>, Moez Beyaoui<sup>b</sup>, Abdelkhalak EL. Hami<sup>a</sup>, Abdelghani Saouab<sup>c</sup>, Mohamed Haddar<sup>b</sup>

## ARTICLE INFO

### Keywords:

Functional composites

Vibrational response

RTM

Power spectral density (PSD)

## ABSTRACT

This paper introduces an innovative methodology for evaluating the vibrational behavior of functional composites reinforced with nanoparticles under random vibration loads. The novelty of this work lies in providing a comprehensive PSD-based frequency-domain approach, rather than conventional temporal methods, to characterize the dynamic response of such composites. The proposed model integrates: 1) a numerical simulation of nanoparticle-filled resin injection in a fibrous preform, validated against experimental data; 2) a homogenization model to determine the composite's mechanical properties based on constituent volume fractions; and 3) a finite element model coupled with PSD analysis to assess the vibrational response. The addition of nanoparticles enhances out-of-plane mechanical properties (E<sub>22</sub>, G<sub>12</sub>, G<sub>23</sub>) while increasing weight and having minimal impact on E<sub>11</sub>. Modal analysis reveals lower natural frequencies in bending modes but higher in torsional modes. Spectral analysis shows that acceleration and stress PSD peaks occur at modes 1 and 3 with functional composites exhibiting higher PSD values. A parametric study varying particles (0%, 20%, 40%) and fiber volume fractions (45%, 50%, 55%) indicates that increasing nanoparticles degrades vibrational performance, while a higher fiber fraction improves response with negligible impact on failure risk. This study provides a modern and effective framework for optimizing functional composites subjected to stochastic vibrations, addressing the lack of PSD-based frequency-domain analyses for nanoparticle-reinforced composites, which have previously only been studied in temporal methods.

## 1. Introduction

Functional composites are attracting growing interest in the composites industry, especially in aeronautics and space sectors. The purpose of functionalizing composite materials is either to improve their properties or to attribute a specific function, which can be achieved by adding fillers or nanofillers as micrometric or nanometric particles to the composite during manufacture [1]. The specific properties that can be achieved [2–4] include fire resistance, abrasion resistance, electromagnetic shielding, and self-healing [5,6], while the properties that can be enhanced are mechanical [7,8], thermal [9], electrical, or electromagnetic. Among the most widely used particles are alumina (Al<sub>2</sub>O<sub>3</sub>) particles, due to their superior mechanical strength, thermal stability, and cost-efficiency. Several studies have demonstrated significant

improvements in mechanical strength, thermal properties, and impact resistance in composites containing epoxy reinforced with Al<sub>2</sub>O<sub>3</sub> particles [10–15].

Functional composites can be manufactured either by depositing the particles directly on the surface of the preform (preform functionalization), or more commonly, by injecting a particle-filled resin (suspension) into the preform (resin functionalization) [9]. The latter method benefits from the advantages of Liquid Composite Molding (LCM) processes, which are widely used in the manufacture of advanced composites in several fields. For example, using Resin Transfer Molding (RTM) or Vacuum Resin Transfer Molding (VARTM) processes with a particle filled resin suspension provides specific properties to composites [6,7, 16]. These include spherical and cylindrical particles, as well as conventional glass or carbon fibers.

<footnote><sup>a</sup> Laboratory of Mechanics of Normandy, INSA Rouen Normandy, University of Rouen, 76800, France</footnote>

<footnote><sup>b</sup> Laboratory of Mechanics, Modeling and Production, National Engineering School of Sfax, Sfax, 3000, Tunisia</footnote>

<footnote><sup>c</sup> CNRS, LOMC, Normandy University, Université du Havre, Le Havre, 76600, France</footnote>

<footnote><sup>d</sup> Institute of Clement Ader (ICA), Toulouse university, CNRS/INSA/ISAE/Mines Albi/UPS, Toulouse, 31400, France</footnote>

<footnote>* Corresponding author. Department of Mechanical Engineering, INSA Rouen Normandie, 76800, France.</footnote>

<footnote>E-mail address: sirine.ben_ameur@insa-rouen.fr (S.B. Ameur).</footnote>
In addition, the presence of particles in the resin affects the impregnation of the suspension during LCM processes. This leads to increased suspension viscosity, particle filtration by the fibrous medium, and non-uniform particle distribution in the final part [5,17–19]. Consequently, the suspension flow slows down, leading to dry spots, poor fiber saturation, or longer impregnation cycles. Particle filtration is a complex phenomenon influenced by several parameters related to the process and the materials involved. Several authors have worked on this subject, such as Erdal et al. [20], Lefevre et al. [21], Abliz et al. [22], Reia Da Costa et al. [23], Haji et al. [24], Sas et al. [25], Mtibaa et al. [26]. They studied the impregnation of a fibrous medium by a suspension, proposing a coupling between flow modeling and filtration modeling. These authors proposed macroscopic models to describe the flow of suspension and particle filtration by the fibrous medium. These models have been applied to RTM, CRTM, and VARTM processes. Furthermore, many studies concerning the mechanical properties of functional composites have been carried out [27,28], while their dynamic response to random vibrations, which represent a common loading condition in real applications, has remained insufficiently investigated. Existing PSD-based random vibration analyses are limited to unfilled laminates or traditional composites, and do not address nanoparticle-reinforced composites [29,30]. Random vibration loads, which are frequently encountered in aircraft components, wind turbines, and industrial machinery, can induce fatigue and have a significant impact on structural integrity [31–34]. Several studies have contributed to advancing the field of vibrational analysis of composites. For example, Zhuo et al. [35] developed a model to analyze the nonlinear vibration characteristics of fiber-reinforced composite truncated conical shells through both theoretical and experimental investigation, and Jian et al. [36] examined the fatigue behavior of carbon fiber-reinforced polymer plates under high-cycle random vibration. Additionally, Singh et al. [37] studied the vibration behavior of CNT composite plates using non-polynomial shear deformation, though this approach requires more computational resources compared to the finite element method. Some studies have analyzed the impact of ply orientation on vibrational characteristics of fiber-reinforced composites [38], while others have improved damage detection in laminated composites under free and forced vibrations [39]. Recent modeling advances have further enhanced the predictive capabilities in this area. Cohesive-zone models capture static and fatigue-driven delamination with high accuracy [40], and damage-surface formulations likewise predict intralaminar cracking under impact loads [41]. However, to the best of our knowledge, this study is the first to develop a fully integrated methodology that combines RTM process modeling, mechanical property homogenization, and PSD-based vibrational analysis for nanoparticle-reinforced composites.

This research addresses that gap by employing numerical modelling and PSD analysis to evaluate the dynamic performance of functional composites under white noise excitation, a prevalent source of random vibrations. PSD is widely recognized as a powerful tool for characterizing random processes [42], and for determining the stress distribution over the entire structure. Notably, the first advancements in fatigue analysis based on PSD were made by NASA to evaluate the fatigue damage induced by vibration and aerodynamic buffeting of spacecraft.

The originality of this study lies in the development of a comprehensive computational framework that links the manufacturing process to the vibrational response of functional composites. It integrates a numerical simulation of the RTM process to model the impregnation of glass fiber preforms with alumina-filled epoxy resin, a homogenization-based analytical model to predict the mechanical properties of the resulting composite, and a finite element model to study the vibratory response of the composites under random vibration using modal and spectral analyses. This approach enables a direct evaluation of how particle content and fiber volume fraction influence vibrational performance. A comparative study is conducted between functional and classical composites, quantifying the effect of nanoparticle addition on

mechanical behavior, natural frequencies, and PSD-based stress distribution during vibration.

## 2. Material and methodology

### 2.1. Modeling of suspension flow through a fibrous medium in the RTM process

The combination of Darcy's law (1) with the conservation equation (2) provides the governing pressure field equation (3). This combination describes the flow of the suspension through the fibrous medium.

$$ \vec{V} = -\frac{1}{\mu}(K \cdot \nabla p) \quad (1) $$

$$ \nabla \cdot \vec{V} = 0 \quad (2) $$

$$ \nabla \cdot \left( \frac{1}{\mu} (K \cdot \nabla p) \right) = 0 \quad (3) $$

where $\vec{V}$ is the flow velocity, $K$ represents the permeability tensor of the medium, $\mu$ is the suspension viscosity and $P$ denotes the pressure.

The filter medium (fibrous medium + particles) porosity $\epsilon$, as well as its permeability, decrease due to the particles' filtration phenomenon during the suspension impregnation through this medium. Thus, these particles are detected in two states: retained particles ($\sigma$) and mobile particles ($\epsilon C$). Where $C$ represents the particles concentration. This porosity $\epsilon$ depends on the retention $\sigma$ and the initial porosity of the fibrous medium $\epsilon_0$, as defined in equation (4) [21].

$$ \epsilon = \epsilon_0 - \sigma \quad (4) $$

According to the particle mass conservation equation, the distribution of particles in the porous medium during the suspension flow is expressed by equation (5) [20], whereas the kinetics of retention, and eventually of resuspension of particles by the fibrous medium, can be described by equation (6):

$$ \frac{\partial[\epsilon C]}{\partial t} + \nabla \cdot \{VC\} + \frac{\partial \sigma}{\partial t} = 0 \quad (5) $$

$$ \frac{\partial \sigma}{\partial t} = \alpha VC \left( 1 - \frac{\sigma}{\sigma_u} \right) \quad (6) $$

where $\sigma_u$ is the maximum filtration capacity of the particle deposit by the porous medium and $\alpha$ represents the filtration coefficient. This coefficient varies as the porous medium was impregnated [24] and depends on the permeability $K$ and porosity $\epsilon$, as follows:

$$ \alpha = \alpha_0 \left[ \left( \frac{K}{K_0} \right)^{-\frac{1}{2}} \left( \frac{\epsilon}{\epsilon_0} \right)^{\frac{3}{2}} \left( \frac{1 - \epsilon}{1 - \epsilon_0} \right)^{-1} \right]^{a_1} \quad (7) $$

where $\alpha_0$ is the initial filtration coefficient and $a_1$ is the model positive constant.

The evolution of the permeability of the filter medium can be estimated based on the model presented in Ref. [25]. This model depends on porosity $\epsilon$ (Eq. (4)) and the initial fibrous medium permeability $K_0$:

$$ K = K_0 \left[ \left( \frac{\epsilon}{\epsilon_0} \right) \left( \frac{1 - \epsilon}{1 - \epsilon_0} \right)^{-2} \right] \quad (8) $$

(Here, $K$ denotes permeability). The viscosity of the suspension depends on the particle concentration evolution ($C$), viscosity of neat resin ($\mu_0$) and empirical constant ($A$). It is described by relation (9) [25].

$$ \mu = \mu_0 \left[ 1 - \frac{C}{A} \right]^{-2} \quad (9) $$
## 2.2. Predicting mechanical properties with homogenization models

The mechanical properties of composites are critically important in the manufacture of materials. These properties can be predicted using analytical models based on the homogenization approach. This approach considers the composite material as a continuous homogeneous medium whose elastic properties are periodic functions of space. Based on the results of numerical modeling of the RTM process (section 3.1), which are the volume fraction of each phase in the composite: particles (C.ε + σ), resin (1 - (C.ε + σ) - V<sub>ffinale</sub>) and fibers (V<sub>ffinale</sub> = 1 - ε<sub>ffinale</sub>), coupled with the mechanical properties of the composite components (Young's modulus E and Poisson's ratio v), it is possible to determine the distribution of mechanical properties in the final part. In this paper, the approach proposed by Ahmed et al. [43] to calculate the mechanical properties of composite material containing three distinct phases: resin, particles, and unidirectional fibers is applied. This approach is based on two-stage homogenization process in order to obtain the properties of the final composite material. In the first stage, the Mori-Tanaka's model (Appendix A) is applied to determine the mechanical behavior of the suspension containing the resin and particles, known as "composite 1". This model is used to homogenize the properties of these components and to calculate the effective properties of this composite. In the second stage, "composite 2" is considered, which is obtained by adding unidirectional fibers to composite 1. The properties of this reached composite are calculated using the Hashin and Rosen model (Appendix A), coupled with the generalized self-coherent (Christensen-Lo, GSC) model (Appendix A). This coupling makes it possible to homogenize the properties of composite 1 with the fibers to determine the effective properties of composite 2. This approach, which involves two homogenization stages, provides a consistent way of calculating the mechanical properties of the final composite material, considering the interaction between the different phases. The necessary steps of this approach to predicting the mechanical properties of a functional composite are illustrated in Fig. 1. The mathematical formulations of these models are detailed in Appendix A.

For this purpose, it is essential to obtain specific information about the volume fractions of each component of the function composite, as well as their mechanical properties, particularly Young's modulus and Poisson's ratio. These properties and volume fractions are presented in Table 1.

## 2.3. Random vibration methodology

After determining the mechanical properties of the composite, the

<table><caption>**Table 1**

Mechanical properties and volume fractions of the constituents.</caption><thead><tr><th></th><th></th><th>Volume fraction</th><th>Young's Modulus</th><th>Poisson's ratio</th></tr></thead><tbody><tr><td rowspan="2">Glass fiber</td><td></td><td>V<sub>fr</sub> = 0.45</td><td>75 GPa</td><td>0.2</td></tr><tr><td>Suspension V<sub>fr</sub> = 1 - V<sub>fr</sub></td><td>V<sub>fr</sub> = Cε + σ</td><td>500 GPa</td><td>0.2</td></tr><tr><td></td><td>Alumina particles</td><td>V<sub>fr</sub> = 1 - V<sub>fr</sub></td><td>5 GPa</td><td>0.3</td></tr></tbody></table>

next step is to analyze its response under random vibratory excitations. This section describes the methodology used to evaluate its dynamic behavior under random vibration loads. This is based on the application of random vibration theory and finite element analysis to study the impact of particle and fiber volume fractions on the natural frequencies, modal shapes, and stress-displacement responses under white noise excitation.

### 2.3.1. Finite element analysis for classical and functional composites

To evaluate the dynamic response of composites, a distinct modeling approach was adopted for classical and functional composites, using finite element analysis (FEA) in Ansys APDL. Both models represent a rectangular composite plate with dimensions 200 mm × 100 mm × 10 mm (length × width × thickness), discretized using SHELL181 elements, which are well-suited for layered composite structures. For the classical composite model, homogeneous material properties were assigned, including Young's moduli (E<sub>11</sub> and E<sub>22</sub>), shear moduli (G<sub>12</sub> and G<sub>23</sub>), Poisson's ratios (ν<sub>12</sub> and ν<sub>23</sub>), as well as density (ρ). For the functional composite, the same geometry and element type were used, however the material properties were spatially varied based on the particle distribution predicted by the RTM simulation. Specifically, local nanoparticles volume fractions were mapped to each element, and a loop in Ansys APDL was used to assign individual mechanical properties (Young's moduli E<sub>11</sub>(x) and E<sub>22</sub>(x), shear moduli G<sub>12</sub>(x) and G<sub>23</sub>(x), Poisson's ratios ν<sub>12</sub>(x) and ν<sub>23</sub>(x), density ρ(x)) to each SHELL181 element, based on the corresponding local particle content. Then, to assemble these rectangles into a single geometric piece, a function built into Ansys APDL was used. A uniform thickness and constant mesh size were used for modal and spectral analysis. Both models share the same geometry and mesh configurations, however, the functional composite model explicitly accounts for the heterogeneous material architecture resulting from the manufacturing process.

### 2.3.2. Modal analysis implementation

In the modal analysis process, the equation of motion, derived from

<figure><img src="image_3.png" /><figcaption>**Fig. 1.** Mechanical properties prediction approach.</figcaption></figure>
Newton's second law, is expressed as [44]:

$$[M]\{u\} + [C]\{u\} + [K]\{u\} = \{F\} \quad (10)$$

where [M] is the structural mass matrix, {$u$} is the nodal acceleration vector, [C] is a structural damping matrix, {$u$} is the nodal velocity vector, [K] is a stiffness matrix, {$u$} is the nodal displacement vector, and {$F$} is an applied load vector.

This analysis employed the Block Lanczos method to extract the first five natural frequencies and corresponding mode shapes for both classical and functional composites. The eigenvalue problem was solved as follows:

$$(K - \omega_t^2) \{\emptyset_t\} = 0 \quad (11)$$

### 2.3.3. Spectral analysis: power spectral density analysis

A structure's response to random vibration is described within the framework of random process theory. Suppose physical parameter is recorded n times; the resulting ensemble of time histories {$f(t)$} forms a random process. For this process, the mean value (Eq. (12)), variance (Eq. (13)), autocorrelation function (Eq. (14)), and covariance matrix (Eq. (15)) can be calculated.

$$E[f(t_1)] = \int_{-\infty}^{+\infty} f(t_1)p(f(t_1))df \quad (12)$$

$$\sigma_f^2 = E[f^2] - (E[f])^2 \quad (13)$$

$$R_f(t_1, t_1 + \tau) = E[(f(t_1)).(f(t_1 + \tau))^T] \quad (14)$$

$$\sum_f (t_1, t_1 + \tau) = E[(f(t_1) - \overline{f(t_1)}).(f(t_1 + \tau) - \overline{f(t_1 + \tau)})^T] \quad (15)$$

Where $p(f(t_1))$ is the probability density of $f(t)$.

When all these statistical properties are invariant with respect to time and its overall statistical properties are equal to the temporal properties, then the process is considered stationary, Gaussian, and ergodic.

According to the Wiener-Khinchin theorem, the Fourier transform of the autocorrelation function yields the power spectral density (PSD):

$$\Phi_f(f) = \int_{-\infty}^{+\infty} R_f(\tau)e^{-j2\pi\tau f} d\tau \quad (16)$$

The PSD describes how the process energy is distributed over frequency. If $f$ is expressed in Hz then the PSD has units of m²/Hz. Alternatively, if the angular frequency $w = 2\pi f$ is the independent variable, then the PSD has units of m²/(rad/s). Thus, to convert frequencies $f$ and $w$, the PSD value must also be scaled [12]:

$$\Phi_l(w)dw = 2\pi\Phi_l(f)df \quad (17)$$

Following the modal analysis, a spectral analysis is conducted, in which a stationary, zero-mean Gaussian band-limited white noise with a flat PSD excitation level is applied to assess the composite's behavior under random vibration conditions. The PSD thus plays a dual role; as the input, representing the random load, and as the output, characterizing vibrational response in displacement, stress and acceleration.

Using the statistics defined above, the 1-sigma displacement, stress and acceleration PSDs are obtained. Each PSD has a specific unit depending on the physical quantity considered: acceleration PSD in (m/s²)²/Hz or G²/Hz, describing the mean-square acceleration per unit frequency bandwidth, and stress PSD is expressed in Pa²/Hz, representing the mean-square value of stress fluctuations per unit frequency. A white noise excitation, which is a type of random signal that has a flat PSD over all frequencies, is fixed at 1G²/Hz (1G = 9.81 m/s²) in the [10, 2000] Hz frequency band, where G represents gravitational acceleration. This PSD excitation is applied at the clamped edge of the composite

in the transverse (Z) direction. Results are extracted at the selected point (A) (Fig. 2). This location at the free edge is chosen for its maximum response amplitude and high sensitivity to resonances.

The PSD stress response (Pa²/Hz) is computed from the PSD stress matrix. Under plane stress conditions, this matrix is expressed as [32]:

$$\Phi_S = \begin{bmatrix} \Phi_{Sxx-Sxx} & \Phi_{Sxx-Syy} & \Phi_{Sxx-Sxy} \\ \Phi_{Syy-Sxx} & \Phi_{Syy-Syy} & \Phi_{Syy-Sxy} \\ \Phi_{Sxy-Sxx} & \Phi_{Sxy-Syy} & \Phi_{Sxy-Sxy} \end{bmatrix} \quad (18)$$

The stress PSD matrix (Eq. (18)), calculated via Python-ANSYS coupling, captures both auto- and cross-spectral densities of stress components, revealing resonance peaks and the distribution of energy across the frequency spectrum.

This frequency-domain analysis enables precise identification of critical frequencies and modes, even in highly heterogeneous materials. By determining the signal power distribution over the frequency range, this approach highlights resonance peaks and key vibratory characteristics under random vibration loads.

### 2.4. Proposed complete approach

This research presents a comprehensive approach to understanding the influence of alumina particles on the mechanical properties of parts manufactured using the RTM process, as well as on their vibratory behavior under random vibrations. To achieve these objectives, three different models were developed and coupled.

- RTM numerical model: This model simulates the suspension flow through the fibrous medium during the RTM process, determines the particle distribution in the manufactured part and calculates the density evolution of the obtained composite part. This density is determined using the following equation:

$$Den = (C.\varepsilon + \sigma)\rho_p + (1 - \varepsilon)\rho_f + (1 - (C.\varepsilon + \sigma) - (1 - \varepsilon))\rho_r \quad (19)$$

Table 2 lists the densities of the materials used in this study.

- Homogenization model: This model calculates the mechanical properties of the manufactured part, including Young's moduli ($E_{11}$, $E_{22}$), shear moduli ($G_{12}$, $G_{23}$), and Poisson's ratio ($\nu_{12}$, $\nu_{23}$), based on analytical models.

- Modal and Spectral analysis model: This numerical model employs the finite element method, using ANSYS APDL software, to perform random vibration analysis of the obtained composites.

Fig. 3 illustrates the workflow between these three models.

<figure><img src="image_4.png" /><figcaption>Fig. 2. Finite element model.</figcaption></figure>
<table><caption>**Table 2**

Densities of the different materials used.</caption><thead><tr><td>Material</td><td>Density (kg/m3)</td></tr></thead><tbody><tr><td>Epoxy resin (ρ<sub>r</sub>)</td><td>1175[43]</td></tr><tr><td>Glass fiber (ρ<sub>f</sub>)</td><td>2540[45]</td></tr><tr><td>Alumina particles (ρ<sub>p</sub>)</td><td>4000 [46]</td></tr></tbody></table>

# 3. Results and discussion

## 3.1. Numerical results of the injection

### 3.1.1. Validation of the RTM process modeling

To validate the numerical model of the RTM process, a comparison was conducted between our numerical results (flow front kinetics and final particle concentration distribution) and experimental data from the literature ("Test 1" [22] and "Test 2" [21]). The two experimental tests are based on a linear injection of a particle-filled resin into the preform under an imposed pressure. In test 1 [21], the suspension is constituted of an unsaturated polyester (UP) resin and spherical micro-beads with a diameter of 12 µm. A 1.5 % additive (BYK-W 980, BYK-Chemie) is added to limit sedimentation and aggregation of the filler. A synthetic PET fiber felt is used as reinforcement. The experimental setup includes rigid tooling (steel half mold and thick PMMA top plate) with a cavity measuring 90 mm in width, 4 mm in thickness and 400 mm in length. The mold is fed at a constant pressure of 0.2 MPa using a pressure bucket equipped with a motorized mixer to maintain a homogeneous blend. The flow is monitored through the transparent PMMA top plate, allowing observation and recording of the flow front kinetics. After polymerization, samples are cut from the composite part at known locations, weighed, measured, and burned at 500 °C to determine the filler content as a function of position. In the test 2 [22], the suspension is composed of epoxy resin and spherical boehmite nanoparticles with an average diameter of 104 µm. A quasi-unidirectional carbon fabric serves as reinforcement. The injection experiments are performed using a custom-developed RTM-tool that enables extraction of fluid samples during impregnation and observation of flow fronts. A textile preform (10 cm × 20 cm) is placed on a fine-polished steel tool and compacted

with a glass-PMMA sandwich plate and a metal stiffening frame. Injection is carried out at a constant pressure of up to 3 bar, depending on the flow velocity of the suspension. The injected suspension is extracted at five locations spaced 5 cm apart between the inlet and the outlet. Particle concentration in the extracted samples is determined using thermogravimetric analysis (TGA Q5000). The material properties and test conditions for both experiments are summarized in Table 3.

Fig. 4 and Fig. 5 present a comparison between the numerical and experimental results, showing (a) the flow front advancement and (b) the final particle concentration evolution. In both tests, a strong agreement is observed.

### 3.1.2. Suspension flow simulation

In this simulation, a rectilinear injection of the suspension (epoxy resin + alumina particles) through a fibrous medium (glass fiber) was carried out under an imposed pressure $P_0$. This suspension was injected along the entire left edge of the preform, as shown in Fig. 6. The suspension flow was assumed to be unidirectional, and the injection process terminated once the preform was fully saturated. This preform has a rectangular form with a length $L_0 = 0.2$ m, a width $L_1 = 0.1$ m and a thickness $H_0 = 0.01$ m. The equations detailed in Section 3.1, which describe the injection of the suspension through the fibrous medium during RTM process, were implemented and discretized using a one-dimensional finite difference method.

The initial and boundary conditions of our model are described in Table 4.

<table><caption>**Table 3**

Parameters used in the simulation.</caption><thead><tr><th></th><th>Test 1 [22]</th><th>Test 2 [21]</th></tr></thead><tbody><tr><td>C<sub>0</sub>: Initial concentration (%vol)</td><td>5</td><td>40.5</td></tr><tr><td>μ<sub>0</sub>: Initial viscosity of neat resin (Pa. s<sup>-1</sup>)</td><td>0.04</td><td>0.2</td></tr><tr><td>K<sub>0</sub>: Initial fiber permeability (10<sup>-10</sup>) (m<sup>2</sup>)</td><td>0.85</td><td>4.92</td></tr><tr><td>V<sub>0</sub>: Fiber's volume fraction (%)</td><td>50</td><td>18.7</td></tr><tr><td>α<sub>0</sub>: Filtration coefficient (m<sup>-1</sup>)</td><td>0.2703</td><td>0.27</td></tr><tr><td>α<sub>1</sub>: Model constant</td><td>1</td><td>1</td></tr><tr><td>P<sub>0</sub>: Injection pressure (Pa<sup>-1</sup>)</td><td>1 10<sup>5</sup></td><td>2 10<sup>5</sup></td></tr></tbody></table>

<figure><img src="image_5.png" /><figcaption>**Fig. 3.** Complete model algorithm.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Fig. 4. Comparison and validation of the numerical model with experimental data [22]. (a) front advancement and (b) concentration prediction.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 5. Comparison and validation of the numerical model with experimental data [21]. (a) front advancement and (b) concentration prediction.</figcaption></figure>

The material properties and simulation parameters are presented in Table 5.

### 3.1.3. Evolution of particle concentration and density during the injection process

Fig. 7a presents the initial injected particle concentration profiles $C_0$, for the classical composite (with 0% of $C_0$) and the functional composite (with 20% of $C_0$). Fig. 7c illustrates the density evolution along the length of the obtained composites. This evolution is proportional to the final particles' distribution in the manufactured parts (Fig. 7b).

### 3.1.4. Impact of alumina particles on the composite mechanical properties

Fig. 8 demonstrates that adding alumina particles leads to significant improvements in the mechanical properties of the composite. Specifically, the longitudinal Young's modulus ($E_{11}$) was the least affected

<table><caption>Table 4
Initial and boundary conditions for the RTM simulation.</caption><tr><td rowspan="2">Initial conditions</td><td>Pressure (P)</td><td>P(x, 0) = 0</td></tr><tr><td>Retention (σ)</td><td>σ(x, 0) = 0</td></tr><tr><td rowspan="5">Boundary conditions</td><td>Porosity (ε)</td><td>ε(x, 0) = ε<sub>0</sub></td></tr><tr><td>Particle concentration (C)</td><td>C(x, 0) = 0</td></tr><tr><td>Pressure at inlet (x = 0)</td><td>P(x = 0, t) = P<sub>0</sub></td></tr><tr><td>Concentration at inlet (x = 0)</td><td>C(x = 0, t) = C<sub>0</sub></td></tr><tr><td>Pressure at front (x = L(t))</td><td>P(x = L(t), t) = 0</td></tr><tr><td></td><td>Retention at front (x = L(t))</td><td>σ(x = L(t), t) = 0</td></tr></table>

<figure><img src="image_8.png" /><figcaption>Fig. 6. Illustration of the RTM process [26].</figcaption></figure>
<table><caption>**Table 5**

Physical properties of materials and parameters used for the simulation model.</caption><tr><td>μ<sub>0</sub>: Initial viscosity (Pa.s)</td><td>0.1</td></tr><tr><td>K<sub>0</sub>: Initial permeability (m<sup>2</sup>)</td><td>2 10<sup>-10</sup></td></tr><tr><td>C<sub>0</sub>: Initial concentration (%)</td><td>20</td></tr><tr><td>ε<sub>0</sub>: Initial porosity (%)</td><td>55</td></tr><tr><td>α<sub>0</sub>: Filtration coefficient (m<sup>-1</sup>)</td><td>1</td></tr><tr><td>Δx: Space step (m)</td><td>6.66 10<sup>-4</sup></td></tr><tr><td>σ<sub>u</sub>: Ultimate specific deposit</td><td>0.5</td></tr><tr><td>A: Empirical constant</td><td>0.68</td></tr><tr><td>P<sub>0</sub>: Imposed injection pressure (Pa)</td><td>3 10<sup>5</sup></td></tr><tr><td>a<sub>1</sub>: Parameter model</td><td>1</td></tr></table>

property, with an increase equivalent to 4 %. In contrast, the transverse Young's modulus ($E_{22}$) showed a 32 % increase, while the $G_{12}$ shear modulus improved by 34.5 % and the $G_{23}$ shear modulus by 34 %. These mechanical properties were directly proportional to the particle distribution profile shown in Fig. 7b.

On the other hand, the Poisson ratios decrease significantly, thus demonstrating an inversely proportional relationship with the final particle distribution. In fact, $Nu_{12}$ decreases by 2.5 % (Fig. 8e) and $Nu_{23}$ decreases by 6.3 % (Fig. 8f). Since $Nu_{12}$ reflects the ratio of transverse to longitudinal strain, while $Nu_{23}$ represents the ratio of through-thickness to transverse strain, these reductions indicate that the composite becomes more resistant to deformation. The presence of alumina particles mainly enhances transversal and out-of-plane mechanical properties,

rather than longitudinal ones.

## 3.2. Vibrational response of classical and functional composites under random vibration loads

### 3.2.1. Mesh sensitivity analysis

A mesh convergence study was conducted for both composite models. Refined meshes were generated using the ESIZE command. For each mesh, the first five natural frequencies were extracted, and the relative error:

$$ \Delta f_i = \frac{|f_i(N_k) - f_i(N_{ref})|}{f_i(N_{ref})} \times 100 \quad (20) $$

was evaluated with respect to the finest mesh as reference. Convergence was considered acceptable when $|\Delta f_i| \le 0.1$. This threshold was selected to ensure a high level of accuracy in the computed natural frequencies. Fig. 9 (a) and (b) illustrate the evolution of the relative error with respect to mesh density for the classical and functional composites, respectively. Using the finest mesh (180 000 elements) as reference, the error falls below 0.1 % when the model contains 45 000 elements, corresponding to element size of 0.6 mm. A similar trend is observed for both composites, beyond this point, the first five modes become mesh independent. Consequently, the 0.6 mm mesh is adopted for the subsequent modal and random vibration analyses.

<figure><img src="image_9.png" /><figcaption>**Fig. 7.** Evolution of key parameters during the injection process: (a) Initial particle concentration (%), (b) Particle volume per mold cavity volume (%), (c) Density (kg.m⁻³).</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Fig. 8. Evolution of mechanical properties as a function of nondimensional length: (a) E₁₁ (GPa), (b) E₂₂ (GPa), (c) G₁₂ (GPa), (d) G₂₃ (GPa), (e) Nu₁₂ and (f) Nu₂₃, as a function nondimensional length.</figcaption></figure>

### 3.2.2. Modal analysis results

Fig. 10 presents the first five mode shapes and their corresponding natural frequencies for both classical and functional composites. The results indicate that the natural frequencies of modes 1 and 3, which are dominated by bending, are higher in the classical composite, whereas those of modes 2, 4 and 5, associated with torsional deformation, are higher in the functional one. This difference can be explained by the following semi-analytical expression based on Euler-Bernoulli beam theory:

$$f = \frac{\alpha}{2\pi} \sqrt{\frac{Eh^2}{\rho L^4 (1 - v^2)}} \qquad (21)$$

where the constants h, L, and α remain identical for both composites. The ratio $\frac{E}{\rho(1-v^2)}$ becomes the crucial factor. The bending modes are influenced by the ratio $\frac{E_{11}}{\rho(1-v_{12}^2)}$. This ratio is higher for the classical composite (as shown in Table 6), which explains the higher natural frequencies observed for bending modes. On the other hand, the torsional modes, dominated by the ratio $\frac{E_{22}}{\rho(1-v_{12}^2)}$, show higher values for
<figure><img src="image_11.png" /><figcaption>Fig. 9. Relative error of the first five natural frequencies as a function of element number for (a) the classical composite and (b) the functional composite (20 wt % Al₂O₃).</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 10. Vibration modes of classical composite (a), and composite with initial injected particles concentration (C₀) equals to 20 % Al₂O₃ (b).</figcaption></figure>

<table><caption>**Table 6**

Normalized elastic moduli ratios of pure and filled composite.</caption><thead><tr><th rowspan="2"></th><th colspan="2">Ratio</th></tr><tr><th><i>E</i><sub>11</sub><br>ρ(1 − v<sub>12</sub><sup>2</sup>)</th><th><i>E</i><sub>22</sub><br>ρ(1 − v<sub>12</sub><sup>2</sup>)</th></tr></thead><tbody><tr><td>Pure composite</td><td>0.0218</td><td>0.0073</td></tr><tr><td>Filled composite</td><td>0.0195</td><td>0.0083</td></tr></tbody></table>

the functional composite due to the addition of alumina particles, which significantly increase transverse stiffness (E<sub>22</sub>). These results indicate that the addition of alumina particles enhances torsional performance by improving transverse stiffness, while the associated increase in density leads to a reduction in the natural frequencies of bending modes.

### 3.2.3. Spectral analysis results

Using the natural frequencies and mode shapes identified in the
<figure><img src="image_13.png" /><figcaption>Fig. 11. PSD response of functional and classical composites.</figcaption></figure>

<table><caption>**Table 7**
PSD acceleration peaks value.</caption><thead><tr><th></th><th>Functional composite</th><th>Classical composite</th></tr></thead><tbody><tr><td>Mode 1</td><td>6150 G<sup>2</sup>/Hz</td><td>6080 G<sup>2</sup>/Hz</td></tr><tr><td>Mode 3</td><td>1785 G<sup>2</sup>/Hz</td><td>1705 G<sup>2</sup>/Hz</td></tr></tbody></table>

modal analysis, the PSD acceleration response at point A (Fig. 2.) is analyzed over a frequency range of 10 Hz–2000 Hz for both the classical and functional composites, as shown in Fig. 11. This PSD plot reveals two resonance peaks in both composites, corresponding to the natural frequencies of modes 1 and 3. These peaks indicate frequencies with higher power levels, with mode 1 (first bending mode) contributing approximately three times more to the vibration response than mode 3 (second bending mode), making it the dominant mode. The absence of a peak at the frequency of mode 2 is attributed to its shear-torsion nature, which is less efficiently excited by the transverse excitation applied at the clamped edge. Consequently, only the bending modes 1 and 3 are

visible in the PSD spectrum. A comparison of PSD amplitude values, presented in Table 7, shows that the functional composite has slightly higher peak values for mode 1 and mode 3 compared to the classical one. This suggests that while the addition of alumina particles improves the mechanical properties of the composite, it also increases its mass, thereby reducing the natural frequencies of the critical modes (1 and 3) and negatively impacting the vibrational behavior.

Fig. 12a, 12.b and 12.c show the PSD stress distribution: $\Phi_{Sxx\_Sxx}$, $\Phi_{Syy\_Syy}$ and $\Phi_{Sxy\_Sxy}$, respectively, at point A across various frequencies. These frequencies were applied in the vibration analysis for both composites (classical and functional). These figures illustrate how vibration energy is distributed across the frequency spectrum in terms of stress. Higher PSD values correspond to frequencies where stress fluctuations are more intense, potentially indicating regions susceptible to failure. These peaks are associated with the bending modes (1 and 3). The PSD stress values for both composites are summarized in Table 8. The results demonstrate that the functional composite has higher PSD stress values for all components ($\Phi_{Sxx\_Sxx}$, $\Phi_{Syy\_Syy}$ and $\Phi_{Sxy\_Sxy}$) in both modes when compared to the classical one. This outcome is attributed to the lower natural frequencies of the functional composite in these modes.

### 3.3. Parametric study

#### 3.3.1. Impact of particle percentage variation on 1-sigma stress

In the spectral analysis, 1σ displacement, 1σ velocity, and 1σ unit static solutions are employed for fatigue evaluations. These results indicate that stress levels remain below 1σ for 68.2 % of the time, between 1σ and 2σ for 27.2 % of the time, and exceed 3σ for less than 0.3 % of the time [47]. Fig. 13a, 13.b and 13.c illustrate the distribution of 1σ stress values along the composite plate under random vibratory loading for different percentages (0 %, 20 % and 40 %) of injected particle concentration ($C_0$). These figures show that the addition of particles affects the distribution of 1σ stress, particularly the maximum stress value (SMX). Specifically, for the classical composite ($C_0$ = 0%, Fig. 13a), the SMX value is about 23.2 MPa. This value increases to 26.2.

<figure><img src="image_14.png" /><figcaption>Fig. 12. PSDs stresses: (a) $PSDS_{xx\_xx}$, (b) $PSDS_{yy\_yy}$, (c) $PSDS_{xy\_xy}$.</figcaption></figure>
<table><caption>**Table 8**
Comparison of PSD stress components value for reinforced and neat plates in modes 1 and 3.</caption><thead><tr><th rowspan="2"></th><th colspan="2">PSDS<sub>xx,xx</sub> (Pa<sup>2</sup>/Hz)</th><th colspan="2">PSDS<sub>yy,yy</sub> (Pa<sup>2</sup>/Hz)</th><th colspan="2">PSDS<sub>xy,xy</sub> (Pa<sup>2</sup>/Hz)</th></tr><tr><th>Mode 1</th><th>Mode 3</th><th>Mode 1</th><th>Mode 3</th><th>Mode 1</th><th>Mode 3</th></tr></thead><tbody><tr><td>Functional composite</td><td>13 10<sup>13</sup></td><td>12 10<sup>11</sup></td><td>4.3 10<sup>11</sup></td><td>4 10<sup>9</sup></td><td>10<sup>12</sup></td><td>8 10<sup>9</sup></td></tr><tr><td>Classical composite</td><td>9 10<sup>13</sup></td><td>7.6 10<sup>11</sup></td><td>1.4 10<sup>11</sup></td><td>1.5 10<sup>9</sup></td><td>4 10<sup>11</sup></td><td>3 10<sup>9</sup></td></tr><tr><td>Difference (%)</td><td>30.7</td><td>36.67</td><td>67.44</td><td>62.5</td><td>60</td><td>62.5</td></tr></tbody></table>

<figure><img src="image_15.png" /><figcaption>**Fig. 13.** 1-sigma stress results with different percentage of Al2O3.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>**Fig. 14.** PSD acceleration response of the classical composite and the functional composites with varying fiber volume fraction.</figcaption></figure>

MPa for the functional composite with 20 % of $C_0$ (Fig. 13), and further to 29.4 MPa for the composite with 40 % of $C_0$ (Fig. 13c). Based on these observations, increasing the particle volume fraction leads to higher stress fluctuations, which in turn increases the risk of fatigue failure by accelerating damage accumulation.

### 3.3.2. Impact of fibers percentage variation on 1-sigma stress

To investigate the effect of fiber volume fraction ($V_f$) on the vibrational behavior of the functional composites (with $C_0 = 20$ %), a parametric study was conducted by varying $V_f$ to 45 %, 50 % and 55 %. Fig. 14 shows the PSD acceleration response at point A for different values of $V_f$. The results demonstrate that bending modes (modes 1 and 3) are the critical modes, and that the natural frequency values increase proportionally with the fiber volume fraction. This indicates that the functional composite with 55 % $V_f$ has the optimal vibrational response within the studied range.

Fig. 15a, 15.b, and 15.c show the 1σ stress distribution of functional composites with 45 %, 50 %, and 55 % $V_f$, respectively. The results indicate that increasing $V_f$ leads to a slight increase in 1σ stress, suggesting that its effect on failure risk is negligible.

## 4. Conclusion

A comprehensive approach is developed to analyze the vibrational behavior of composites manufactured using the RTM process. This approach integrates: 1) a numerical model that simulates the injection of particle-filled resin (suspension) through a preform (fibrous medium), 2) a homogenization model to calculate the mechanical properties of the resulting composite and 3) a numerical finite element model to analyze the modal and spectral response of the composites.

- The RTM process model is validated by comparing the numerical results with experimental data available in the literature.

- The addition of 20 wt% alumina particles improves the out-of-plane properties of the functional composite. Specifically, $E_{22}$ increases by 32 %, $G_{12}$ by 34.5 %, and $G_{23}$ by 34 %, while the longitudinal Young's modulus shows only a slight improvement of 4 %. However, the weight in this case becomes heavier. Based on these results and the modal analysis, it can be concluded that torsional modes (2, 4 and 5) of the functional composite have higher natural frequencies, while bending modes (1 and 3) are lower than those of the classical composite. The spectral analysis shows that bending modes (1 and 3) are critical for both composites, with the functional composite exhibiting significantly higher PSD stress responses. For example, in mode 1, the PSDS<sub>xx,xx</sub> value increases by 30.7 %, while PSDS<sub>yy,yy</sub> and PSDS<sub>xy,xy</sub> components increase by 67.4 % and 60 %, respectively, compared to the classical composite. These results confirm that increasing the particle content negatively affects the vibratory behavior of the composite.

- In contrast, increasing the fiber volume fraction in functional composites leads to a significant improvement in the overall mechanical properties, but at the same time the weight of the composite becomes higher. The results of modal and spectral analyses show an improvement in vibration performance, for example, the first natural frequency increases from 173 Hz to 184 Hz, and the third mode from 1054 Hz to 1123 Hz as the fiber volume fraction increases from 45 % to 55 %. Additionally, the 1σ stress increases slightly from 26.2 MPa to 28.1 MPa, confirming that this parameter has a negligible effect on fatigue failure risk.
<figure><img src="image_17.png" /><figcaption>Fig. 15. 1-sigma stress results for different fiber volume fractions.</figcaption></figure>

### CRediT authorship contribution statement

**Sirine Ben Ameur:** Writing – review & editing, Writing – original draft, Software, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. **Mohamed Mtibaa:** Writing – review & editing, Software, Methodology, Formal analysis, Data curation, Conceptualization. **Ahmed Yaich:** Writing – review & editing, Visualization, Validation, Methodology. **Moez Beyaoui:** Visualization, Validation, Supervision. **Abdelkhalak EL. Hami:** Visualization, Validation, Supervision. **Abdelghani Saouab:** Supervision, Methodology, Conceptualization. **Mohamed Haddar:** Project administration.

### Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Appendix A

### Suspension homogenization (Mori-Tanaka model) [48]

The Mori-Tanaka model estimates the shear modulus ($k^{MT}$) and compressibility ($G^{MT}$) of composites materials as follows:

$$ 
\begin{aligned}
k^{MT} &= k_m \left( 1 + \frac{f_{ph}(k_i - k_m)}{k_m + \left( \frac{3k_m}{3k_m + 4G_m} \right) \left( 1 - f_{ph} \right) (k_i - k_m) } \right) \\
G^{MT} &= G_m \left( 1 + \frac{f_{ph}(G_i - G_m)}{G_m + \left( \frac{6(k_m + 2G_m)}{5(3k_m + 4G_m)} \right) \left( 1 - f_{ph} \right) (G_i - G_m) } \right)
\end{aligned}
\quad (A.1) $$

Where $k_m$ and $G_m$ are the bulk and shear moduli of the matrix, respectively, $k_i$ and $G_i$ are the bulk and shear moduli of the inclusions, and $f_{ph}$ is the volume fraction of the inclusions.

From these moduli, the Young's modulus ($E^{MT}$) and Poisson's ratio ($\nu^{MT}$) of the composite can be derived as follows:

$$ 
\begin{aligned}
E^{MT} &= \frac{9k^{MT}G^{MT}}{3k^{MT} + G^{MT}} \\
\nu^{MT} &= \frac{3k^{MT} - 2G^{MT}}{2(3k^{MT} + G^{MT})}
\end{aligned}
\quad (A.2) $$

### Homogenization of the final composite material (Hashin and Rosen model + GSC) [49]

Once the properties of the suspension have been determined, the Hashin-Rosen model is applied to calculate the mechanical properties of the unidirectional composite within the homogenized matrix.

This model calculates the longitudinal Young's modulus ($E_{11}$), shear modulus ($G_{12}$), compressibility modulus ($k_{23}$), and Poisson's ratio ($\nu_{12}$) as follows:
$$
\begin{align}
E_{11} &= f_{ph} E_i + (1 - f_{ph}) E_m + \frac{4 f_{ph} (1 - f_{ph}) (v_i - v_m)^2 G_m}{\left( \left( (1 - f_{ph}) G_m \right) / (k_i + G_i / 3) + \left( (f_{ph} G_m) / (k_m + G_m / 3) \right) \right) + 1} \nonumber \\
v_{12} &= f_{ph} v_i + (1 - f_{ph}) v_m + \frac{f_{ph} (1 - f_{ph}) (v_i - v_m) (G_m / (k_m + G_m / 3) - G_m / (k_i + G_i / 3))}{\left( \left( (1 - f_{ph}) G_m \right) / (k_i + G_i / 3) + \left( (f_{ph} G_m) / (k_m + G_m / 3) \right) \right) + 1} \tag{A.3} \\
k_{23} &= k_m + \frac{G_m}{3} + \frac{f_{ph}}{1 - \left( k_i - k_m + \frac{1}{3} (G_i - G_m) \right) + \left( (1 - f_{ph}) / \left( k_m + \frac{4}{3} G_m \right) \right)} \nonumber \\
G_{12} &= G_m \left( \frac{G_i (1 + f_{ph}) + G_m (1 - f_{ph})}{G_i (1 - f_{ph}) + G_m (1 + f_{ph})} \right) \nonumber
\end{align}
$$

Where 1 indicates the fiber axis.

Following this, the Generalized Self-Consistent (GSC) Christensen-Lo model is used to calculate the shear modulus ($G^{GSC}$) through the quadratic equation [50]:

$$A_{GSC} \left(\frac{G^{GSC}}{G_m}\right)^2 + B_{GSC} \left(\frac{G^{GSC}}{G_m}\right) + C_{GSC} = 0 \qquad (A.4)$$

With the coefficients defined as:

$$A_{GSC} = 3f_{ph}(1-f_{ph})^2 \left(\frac{G_i}{G_m} - 1\right) \left(\frac{G_i}{G_m} + 3-4v_i\right) + \left(\frac{G_i}{G_m}(3-4v_m) + (3-4v_i)(3-4v_m) - \left(\frac{G_i}{G_m}(3-4v_m) - (3-4v_i)\right)f_{ph}\right)^3 \left(f_{ph}(3-4v_m)\left(\frac{G_i}{G_m} - 1\right) - \left(\frac{G_i}{G_m}(3-4v_m) + 1\right)\right) \quad (A.5)$$

$$B_{GSC} = -3f_{ph}(1-f_{ph})^2 \left(\frac{G_i}{G_m} - 1\right) \left(\frac{G_i}{G_m} + 3-4v_i\right) + \frac{1}{2} \left(\frac{G_i}{G_m}(3-4v_m) + \left(\frac{G_i}{G_m} - 1\right)f_{ph} + 1\right) \left(\left(3-4v_m\right) - 1\right) \left(\frac{G_i}{G_m} + 3-4v_i\right) - 2 \left(\frac{G_i}{G_m}(3-4v_m) - (3-4v_i)\right) f_{ph}^3 \\ + \frac{f_{ph}}{2} \left(\left(3-4v_m\right) + 1\right) \left(\frac{G_i}{G_m} - 1\right) \left(\frac{G_i}{G_m} + 3-4v_i + \left(\frac{G_i}{G_m}(3-4v_m) - (3-4v_i)\right)f_{ph}\right)^3 \quad (A.6)$$

$$C_{GSC} = 3f_{ph}(1-f_{ph})^2 \left(\frac{G_i}{G_m} - 1\right) \left(\frac{G_i}{G_m} + 3-4v_i\right) + \left(\frac{G_i}{G_m}(3-4v_m) + \left(\frac{G_i}{G_m} - 1\right)f_{ph} + 1\right) \left(\frac{G_i}{G_m} + 3-4v_i + \left(\frac{G_i}{G_m}(3-4v_m) - (3-4v_i)\right)f_{ph}\right)^3 \quad (A.7)$$

## Data availability

Data will be made available on request.

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