Combined Effects of Particle Size and Walnut Shell Powder Content on the Tensile and Compressive Mechanical Properties of Epoxy Composites: Experimental Study and Micromechanical Modeling

Table 1. Chemical composition of walnut shell (Juglans regia L.) based on literature data

Figure 1. Walnut shell preparation process: (a) cleaning and drying, (b) crushing, (c) powder drying to reduce moisture, and (d) sieving

2.2 Composite processing

Composites were prepared with 10, 25, and 40 wt.% filler for each particle size, plus neat epoxy (0%). Powder was manually mixed into resin with a spatula for 10 min; hardener was added (2:1) and mixed for 5 min. The mixture was cast into silicone molds (ASTM D638 and D695) [20, 21]. After casting, bubbles were allowed to rise, and a hot air torch (15–20 cm) was passed over the surface to remove them. No vacuum degassing was used. Dispersion was visually inspected; no macroscopic agglomerates were observed. Curing: 23 ± 2 ℃ for 72 h, then 7-day conditioning. Figure 2 shows the fabrication steps.

2.3 Conversion of mass fractions to volume fractions

Volume fractions were calculated using Eq. (1).

$V_f=\frac{\omega_f / \rho_f}{\left[\left(\frac{\omega_f}{\rho_f}\right)+\left(\frac{\omega_m}{\rho_m}\right)\right]}$             (1)

where, V_f is the volume fraction of the reinforcement, ω_f is the mass fraction of the reinforcement, ω_m is the mass fraction of the matrix, ρ_f is the density of the reinforcement, and ρ_m is the density of the matrix. The volume fraction of the matrix is given by Eq. (2).

$V_m=1-V_f$              (2)

Figure 2. Composite fabrication process: (a) manual mixing of resin and walnut shell powder, (b) casting into silicone molds, and (c) demolded final specimens after curing

Table 2. Coding of the formulations and corresponding volume fractions

The coding of the formulations was defined according to the filler weight percentage and the powder particle size, as presented in Table 2.

2.4 Mechanical testing

2.4.1 Tensile tests

Tensile tests are performed according to ASTM D638 [20] on type I specimens, at a crosshead speed of 5 mm·min⁻¹ on a 30 kN capacity machine, under controlled conditions. Five specimens per formulation (n = 5) are tested until failure. The tensile strength at break, Young's modulus (determined by linear regression between 0.05% and 0.25% strain), and elongation at break are calculated from the stress strain curves. Results are reported as mean ± standard deviation.

2.4.2 Compression tests

Compression tests are performed according to ASTM D695 [21] on an RP 25 ATF machine (25 kN capacity), at a crosshead speed of 1 mm·min⁻¹, on cylindrical specimens (25.4 mm length × 12.7 mm diameter) cast in silicone molds at 23 ± 2 ℃. Figure 3 shows a typical specimen mounted between the grips of the universal testing machine during compression testing. Five specimens per formulation (n = 5) are tested until failure.

Figure 3. Compression test on a cylindrical specimen according to the ASTM D695 standard [21] using a universal testing machine

2.5 Statistical analysis

Two-way analysis of variance (ANOVA) was applied to maximum compressive stress due to non-monotonic trends for 400 µm particles and ductile–brittle transition. Factors: filler content (10, 25, 40%) and particle size (100, 200, 400 µm), with n = 5 (45 observations). Normality (Shapiro–Wilk) and variance homogeneity (Bartlett) were checked at α = 0.05. ANOVA with interaction was followed by Tukey's honestly significant difference (HSD) post-hoc test. Letter groupings (a, b, c) indicate no significant difference at 5%. For tensile tests, a descriptive analysis was preferred because the study focuses on micromechanical modeling rather than statistical inference. Tensile trends are monotonic with low standard deviations relative to differences between formulations. Moreover, the compression ANOVA confirmed the significance of both factors and their interaction, and the same factors govern tensile behaviour. A full statistical evaluation of tensile data will be included in future work.

2.6 Theoretical modeling

2.6.1 Prediction of the elastic modulus: Halpin–Tsai model

The elastic modulus of the composite is modeled using the Halpin–Tsai model [22]. This model was selected for its simplicity and its low data requirements, which are advantageous in the absence of detailed microstructural characterization. The reinforcement geometry is incorporated through a shape parameter ξ, with spherical particles assumed (ξ = 2), a value validated by recent studies [23]. Young's modulus is given by Eq. (3).

$E_c=E_m\left(\frac{1+\xi \eta V_f}{1-\eta V_f}\right)$              (3)

with:

$\eta=\frac{\frac{E_f}{E_m}-1}{\frac{E_f}{E_m}-\xi}$              (4)

The epoxy matrix has a Young’s modulus E_m = 524.6 MPa and a density ρ_m = 1.10 g·cm⁻³. The walnut shell particles are spherical, with a modulus E_f = 5.2 GPa [19] and a true density ρ_f  = 0.692 g·cm⁻³ [18]. The mass fractions studied are 10, 25, and 40%. The assumptions of the study include spherical particles, homogeneous dispersion, perfect interfacial adhesion, and an isotropic composite.

It should be noted, however, that the actual particles are angular and irregular, and the interface is untreated, meaning these assumptions are not fully met. These simplifications tend to overestimate the modulus, especially at high filler contents, as discussed in Section 4.3.

2.6.2 Prediction of the tensile strength: Pukánszky model

Tensile strength is modeled using Pukánszky’s semi-empirical model [24]. This model was chosen because it is designed to analyze the ultimate tensile properties of filled polymer composites and is formulated for quasi-spherical particle fillers, which aligns with the assumptions of this study. Eq. (5) represents the stress, where σ_m is the tensile strength of the pure epoxy matrix, and σ_c is that of the composite.

$\sigma_c=\sigma_m\left(\frac{1-V_f}{1+2.5 V_f} \exp \left(B V_f\right)\right)$               (5)

The parameter B is not a conventional constant; it is influenced by several factors such as the interface, charge transfer, particle aggregation, and the microstructure of the composite [25]. Recent studies have correlated it with interfacial shear strength (IFSS) [25, 26]. For methodological reasons, a parameter B specific to each particle size (100, 200, and 400 µm) was determined by linear regression (ordinary least squares method) based on the experimental results of tensile tests. The values obtained are presented in Table 3.

Table 3. The interaction parameter B in the Pukánszky model, expressed in terms of particle size

The values of B were substituted into Eq. (5) to calculate the theoretical resistances of the composites for each particle size.

4.1 Stiffness strength trade-off and the role of the interface

Walnut shell powder creates a trade-off between stiffness and tensile strength. Fine 100 µm particles increase modulus by up to 77% at 40 wt.%, but tensile strength declines from 10 wt.% for all sizes, indicating insufficient interfacial adhesion without surface treatment. This contrasts with improvements after alkaline treatment [9] and with the optimum at 10 wt.% for date palm nut powder [15], suggesting surface chemistry and morphology influence adhesion. Our results align with untreated walnut shells [12], where strength decreases with filler content, especially for larger particles. Pukánszky parameter B decreases from 1.689 (100 µm) to 1.222 (200 µm) and 0.827 (400 µm). B integrates interface quality, dispersion, and porosity [24] and correlates with IFSS [25, 26]. Its decline reflects deteriorating load transfer due to weaker adhesion and stress concentrations. For 400 µm particles, lower specific surface area reduces contact with the matrix, while angular shape promotes local stress concentrations at corners. These effects initiate debonding at lower strains, explaining the early strength drop. In contrast, 100 µm particles, though untreated, offer larger specific surface area and more uniform distribution, delaying debonding and allowing better stress transfer up to higher filler contents.

4.2 Ductile-to-brittle transition in compression

Under compression, the response is governed by coupled particle size and filler content, confirmed by significant ANOVA interaction (p < 0.001). 100 µm formulations retain ductile behaviour with lateral buckling and superficial cracking but no sudden failure. This ductility allows the high-performance group (57.9–62.9 MPa) to exceed neat epoxy (55.34 MPa) due to relatively homogeneous dispersion and effective load transfer, delaying damage localization. In contrast, 400 µm particles at 10 and 40 wt.% trigger brittle shear failure with oblique cracking at ~45° and a sharp strength drop (33.3–35.0 MPa). Coarse particles act as stress concentrators, promoting early interfacial crack initiation and coalescence into a macroscopic shear plane. Macroscopic failure modes in Section 3.4 support this: ductile 100 µm formulations distribute damage without a continuous crack path, while brittle E40-400 forms a clean oblique fracture plane, suggesting coarse particles localise damage and allow rapid crack propagation. Although no SEM images confirm microscale mechanisms, macroscopic evidence is consistent with proposed roles of particle size and interfacial quality. E25-400 (51.30 MPa) is a notable exception (p < 0.05), falling in the intermediate group. At V_f = 0.346, favourable particle arrangement and possibly reduced local porosity may offset interfacial defects. However, absence of direct microstructural evidence limits confirmation—a limitation for future work.

4.3 Validation of micromechanical models

Halpin–Tsai model [22, 23] increasingly overestimates Young's modulus with filler content, with deviations up to −55% at 40 wt.%. This cannot be attributed solely to agglomeration or porosity. The model assumes perfect interfacial adhesion and spherical particles (ξ = 2), whereas walnut shell powder is angular and irregular, and the untreated interface provides limited stress transfer. These factors explain much of the deviation. The model was chosen for simplicity and low data requirements, advantageous without detailed microstructural characterization. However, the magnitude of deviation at high loadings suggests it is inappropriate beyond a certain volume fraction. Alternative models, such as Lewis–Nielsen with maximum packing fraction and interface-based homogenization models [27], could be explored in future work.

Pukánszky model [24] correlates much better with tensile strength. Deviations range from +23.3% (underestimation) to −11.8% (overestimation), with near-perfect prediction for E10-400 (+0.3%). It performs well for coarse particles at low content, close to its assumptions (quasi-spherical particles, weak inter-particle interactions). Underestimation for fine particles at 10% suggests more effective load transfer than predicted by B, possibly due to high specific surface area, while overestimation at 40% reflects imperfections (agglomerates, porosities) reducing actual strength.

4.4 Limitations and outlook

Several limitations should be acknowledged. First, absence of filler surface treatment limits tensile performance; alkalization or silanization could enhance interfacial adhesion and raise B. Second, microstructural characterization was restricted to macroscopic observations. No SEM imaging, porosity mapping, or quantitative particle size distribution after processing was possible due to equipment limitations. These analyses are essential to confirm proposed mechanisms and will be addressed in future work. Third, particle dispersion was assessed only visually; no microscopic or experimental quantification was possible. Fourth, rheological control of dispersion and optimization of mixing protocol require further investigation to push back the agglomeration threshold at high volume fractions. Future work should incorporate porosity and a dispersion parameter into Halpin–Tsai and Pukánszky models to improve accuracy for these bio-based composites. A systematic study of surface treatments across particle sizes will be a logical next step toward high-performance materials balancing stiffness, strength, and durability.

The authors would like to express their gratitude to their thesis director and co-supervisors for their advice, encouragement, and insightful criticism during this study. Their knowledge and counsel were very helpful in advancing and finishing this project.