Numerical prediction of elastic properties of filler-modified asphalt binders using finite element analysis

Abstract

This study presents a numerical approach to estimate the elastic properties of filler-modified asphalt binders using 2D and 3D micro-mechanical modeling and homogenization principles. A three-phase Representative Volume Element (RVE) model was developed to evaluate the elastic modulus and Poisson’s ratio of glass fiber-reinforced polymer (GFRP) powder filler-modified asphalt binders. Periodic boundary conditions were applied to the RVEs, and linear elastic simulations were conducted for 5wt.%, 10wt.%, and 15wt.% filler contents. The results show that elastic modulus is more sensitive to boundary conditions than Poisson’s ratio. Additionally, elastic modulus increases with mesh density, while Poisson’s ratio remains relatively unaffected by mesh size. The choice of mesh type also significantly impacts the elastic properties. High stress concentrations were identified around the glass fiber particles, suggesting potential failure zones. The estimated elastic modulus values for 5 wt.%, 10 wt.%, and 15 wt.% filler content are 3305.42, 3342.72, and 3380.95 MPa, respectively, with corresponding Poisson’s ratio values of 0.3474, 0.3448, and 0.3421. The Halpin-Tsai model, considered more accurate in the literature, shows good agreement with the FEM results, indicating reasonable accuracy.

Keywords:
Asphalt binder; Fillers; Finite element analysis; Elastic properties

1. INTRODUCTION

Filler particles are commonly incorporated into asphalt to enhance specific mechanical properties and act as void-filling agents; Some of the commonly used fillers are limestone, Portland cement, fly ash, slag ash and many others [¹, ²]. Recent research shows that, Asphalt binders modified with filler powders, particularly Glass Fiber Reinforced Polymer (GFRP), exhibit improved elasticity performance at high temperatures, making them more resistant to rutting damage [³, ⁴]. Moreover, these fillers contribute to the binder’s resistance to both short-term and long-term aging, effectively extending the lifespan of pavement structures [⁵]. JOENCK et al. [⁶] conducted a study investigating the self-healing capabilities of various asphalt binder modifiers such steel fibers, steel slag and graphite filler; there study found that graphite fillers presented a significant improvement on the asphalt self-healing capabilities after microwave heating, these characteristics were found to improve with higher filler weight ratio. Elastic parameters such as elastic modulus and Poisson’s ratio are crucial in the design of pavement layers, and these values are typically obtained through experimental testing, such as the uniaxial compressive test [⁷, ⁸]. However, experimental methods can be time-consuming and costly, prompting the adoption of finite element analysis (FEA) methods as a more flexible and cost-effective alternative [⁹].

Asphalt binder is often treated as a homogeneous material at the macroscale level in finite element analysis (FEA) due to its relatively uniform mechanical properties under various loading conditions and tests [¹⁰, ¹¹]. However, the incorporation of aggregates, modifiers, and fillers into the binder leads to the formation of inhomogeneities at various geometrical levels of specimens; through application of finite element methods, particularly the use of Representative Volume Elements (RVEs), enables the homogenization of asphalt mixtures [¹²]. Homogenization refers to the process of averaging the properties of the constituent phases such as asphalt binder, aggregates, and fillers into a single, effective material model suitable for larger-scale simulations [¹³]. A Representative Volume Element is defined as the smallest sub-region of a material that is large enough to statistically represent the material’s mechanical, thermal, and other physical properties for a given scale [¹⁴, ¹⁵]. The use of RVEs in FEA facilitates the homogenization of asphalt mixtures by capturing the material’s heterogeneous nature and enabling the calculation of macroscale properties based on microscale behavior hence reducing computation costs [¹⁶]. By applying homogenization principles within FEA, the microscale properties of asphalt mixtures can be estimated and subsequently used for simulations at the macroscale and mesoscale levels.

The accurate selection of the RVE size is crucial for ensuring the reliability of the results; the RVE must encompass a sufficient number of microstructural features to provide a statistically representative sample, ensuring that the FEM analysis reflects real-world scenarios effectively [¹⁷]. OZER et al. [¹⁸] utilized digital image analysis of actual asphalt concrete specimen images to estimate the size of the Representative Volume Element; there study concluded that to achieve statistical homogeneity, the selected window sizes should be equal to or greater than two to three times the nominal maximum aggregate size in the asphalt concrete; Additionally, the study emphasizes that, for more accurate results, three-dimensional RVEs should be considered. The accuracy of finite element method (FEM) analysis can also be influenced by mesh density; previous study by TURON et al. [¹⁹], concluded that for progressive delamination analysis, more precise results can be obtained when the mesh is ten times finer than the baseline. Similarly, ZERFU and EKAPUTRI [²⁰], conducted a non-linear finite element analysis to investigate the element size effect in reinforced geopolymer concrete beams; it was found that mesh size in three-dimensional models significantly impacts the results; For instance, reducing the mesh size from 25 mm to 10 mm resulted in a difference of less than 0.5 mm between the maximum deflection values of the FEM results and experimental data. Boundary conditions play a crucial role towards the accuracy of FEA results; these enable the accurate representation RVE microstructure and also significantly influence the stress distribution and deformation within the model; leading to potential inaccuracies in the predicted properties [²¹, ²²]. Research on the use of GFRP fillers in asphalt binders is still in its early stages; however, existing literature suggests promising improvements in asphalt performance [²³,²⁴,²⁵].

Addressing this research gap, the current study aims to present methods for estimating the elastic properties of a three-phase composite GFRP filler modified asphalt binder. A 2D and 3D finite element approach is proposed to predict the elastic properties of GFRP-modified asphalt binders while also examining the effects of key attributes, such as RVE geometry, boundary conditions, and mesh size, on the material’s elastic behavior. Additionally, the FEM results will be compared with established analytical methods, including Ruess, Voigt, Halpin-Tsai, and Paul’s model, to assess the accuracy of these methods in predicting the elastic properties of the modified binder.

2. MATERIALS AND METHODS

2.1. Estimating glass fiber and epoxy resin proportions of GFRP fillers

Glass Fiber Reinforced Polymer (GFRP) powders are typically produced by processing waste GFRP segments from wind turbine blades into powder; mechanical methods are typically used such as hammer mills, ball mills, dust separators, and mesh sieves [²⁶]. The resulting GFRP powders primarily consist of glass fibers and epoxy resin particles. The proportions of these components can be directly adopted from the volume fractions of epoxy resin and glass fiber used in the production of the wind turbine blades for instance XU et al. [²⁷] outlines that a typical wind turbine blade contains approximately 76.49% glass fiber and 23.51% epoxy resin. However, the specific proportions of these constituents may vary depending on the recycling methods used to process the waste GFRP into filler powder.

In a study by LAN et al. [²⁵], mechanical processing, specifically hammer milling, was employed to extract both fibers and filler powder from GFRP waste. This method resulted in a reduced proportion of glass fibers in the extracted filler powder. Additionally, the microstructure of the filler particles was analyzed using Scanning Electron Microscopy (SEM), revealing that glass fibers were present as short, cylindrical-shaped particles, while epoxy resin particles appeared as irregular, broken pieces. Based on these observations, the current study adopts their SEM image to estimate the glass fiber-to-epoxy ratio.

To perform FEM homogenization of GFRP filler-modified asphalt mixtures, it is essential to estimate the proportions of glass fiber and epoxy resin particles in the GFRP filler, in order to determine the specific phase volume fractions. These proportions are determined through the following stages: image preparation using Microsoft Photo Viewer, vectorization with the Trace Bitmap feature in Inkscape, and the subsequent area estimation of the vectorized regions. During the image preparation phase, the SEM image was opened in Microsoft Photo Viewer, where the Generative Erase tool was used to remove unwanted white labels, thereby minimizing errors during the vectorization process. Furthermore, the image’s brightness was set to 100, contrast to −31, and highlights to 80. These adjustments enhanced the visibility of the GFRP particles, improving the accuracy of the vectorization process.

The SEM image was imported into Inkscape software, where the Trace Bitmap feature and the brightness cut-off function (as shown in Figure 1) were used to convert the regions containing GFRP particles into vector geometry. Using the Measure Path extension in Inkscape, the area of the vectorized GFRP region was calculated to be 16,397.90 mm². The regions representing glass fiber particles in the SEM image were fewer in number and therefore traced manually using the Draw Bezier Curve and Straight Lines tools. The area of the resulting vectorized glass fiber regions was found to be 1,426.494 mm². The proportion of glass fiber particles in the GFRP was then calculated as 0.087. By subtracting this value from 1, the proportion of epoxy resin particles was estimated to be 0.913. Consequently, the estimated ratio of glass fiber particles to epoxy resin particles in the GFRP powder is approximately 0.087:0.913.

Figure 1
Stages for estimating glass fiber to epoxy resin particles in GFRP.

2.2. Finite element method (FEM)

In the current study, Finite Element Analysis methods were implemented using Abaqus software, utilizing the Representative Volume Element approach; these methods have been extensively used in various studies. Filler particles in asphalt binders, at both microscale and macroscale, can be observed as multiple, randomly distributed particles with complex and varying geometries (as shown in Figure 2). Analyzing the exact geometry of these filler particles is often a time-consuming task and, in some cases, a nearly impossible process. Consequently, most researchers simplify the geometry of these particles, representing them as either circular inclusions for 2D analysis or spherical inclusions for 3D analysis [¹³, ²⁸].

Figure 2
Homogenization process for 2D micro-scale structure.

While simplifying the geometry of filler particles generally has a minimal impact on elastic properties as long as the filler volume fractions are accurately represented, it can significantly influence the results when modeling damage using RVEs [²⁹]. This is because the particle geometry plays a crucial role in stress concentration and, therefore, in the overall structural damage response under loading conditions. FEM homogenization involves positioning the various constituent phases of the composite material within a matrix or larger domain and applying an appropriate mesh and boundary conditions to simulate the behavior of the combined system. The response of the homogenized model is then analyzed to determine the effective mechanical properties of the material.

2.2.1. Abaqus 2D RVE model development

A square diamond-shaped 2D representative volume element measuring 100 µm on each side was utilized for the two-dimensional finite element method analysis. The elastic properties were examined for filler to binder weight ratio of 5wt.%, 10wt.%, and 15wt.%. GFRP powder filler consists of glass fiber and epoxy resin particles, with an estimated proportion of 0.087:0.913. The particle radius in the specified phase was modified to represent the change in filler content. Particle radius was calculated using Equation (1), whereby Re and Rf represent the radii of the epoxy and fiber regions for a particular filler content. In Equation (1) and (2), S denotes the side of the RVE, Gv represents the current filler quantity (5, 10, 15), Pe indicates the ratio of epoxy resin particles (0.913), and Pf signifies the ratio of fiber particles (0.087).

(1) $$R_e={(S^2\cdot \frac{G_v}{100.\pi }\cdot p_e)}^{\stackrel{⃜}{2}}$$

(2) $$R_f={\left(S^2\cdot \frac{G_v}{100.\pi }\cdot p_f\right)}^{\stackrel{⃜}{2}}$$

The 2D Representative Volume Element (RVE) was created in Abaqus software as a two-dimensional deformable shell part. The calculated radii for the glass fiber and epoxy resin particles are provided in Table 1. The geometry was constructed in the form of a square with corner coordinates (−50, −50) and (50, 50). Particle inclusions were incorporated into the geometry using Abaqus’ partitioning feature, whereby the particle geometries were projected onto the 2D square shell component. The material phases glass fiber, epoxy resin, and matrix were organized into separate sets: S1, S2, and S3, respectively. Figure 3 shows the developed Abaqus RVE models for varying GFRP content.

Table 1
Epoxy resin and glass fiber region radius for the various GFRP filler contents.

GFRP CONTENT (%)	5%	10%	15%
Rf	3.721	5.262	6.445
Re	12.054	17.047	20.879

Figure 3
2D RVE models at different filler contents.

2.2.2. Abaqus 3D RVE model development

A cubic 3D Representative Volume Element (RVE) with side lengths of 100 µm was selected for the three-dimensional finite element method (FEM) analysis. The RVE contains a single spherical epoxy resin particle at the center, surrounded by eight spherical glass fiber particles placed at specific coordinates near the corners. This particle configuration was adopted from a previous study by LING et al. [³⁰], who used an identical RVE setup. Their study primarily focused on developing a novel micromechanical model to investigate the uniaxial compressive and tensile strength characteristics of semi-flexible pavements. The results from their microstructural model were found to closely align with the corresponding experimental findings.

(3) $$R_f={\left(\frac{3\cdot S^3\cdot G_v\cdot P_f}{3200\cdot \pi }\right)}^3$$

(4) $$R_e={\left(\frac{3\cdot S^3\cdot G_v\cdot P_e}{400\cdot \pi }\right)}^3$$

Three-dimensional RVEs were constructed to evaluate the elastic properties of glass fiber reinforced polymer (GFRP) at filler to binder weight ratios of 5%, 10%, and 15%; these specific filler volumes fractions were achieved by altering the radius of the particle inclusions. The radius of each glass fiber particle was determined using equation (3), while the radius of the epoxy resin particles was calculated using equation (4). In these equations, S represents the side of the cubical RVE, Gv denotes the current GFRP filler content, Pf indicates the ratio of fiber in GFRP filler powder (0.087), Pe signifies the ratio of epoxy resin particles in GFRP filler powder (0.913), Rf is the radius of an individual glass fiber particle, and Re is the radius of a single epoxy resin particle. The parameters are imported into Abaqus during the modeling of the 3D Representative Volume Element part.

Table 2 illustrates the coordinates used to translate fiber glass particles to designated positions in the assembly module. Calculated radius of epoxy particles (Re) and the radius of fiberglass particles (Rf) used for model construction are shown in Table 3.The epoxy resin, glass fiber particles, and cube matrix phase were integrated within the Abaqus assembly module, forming a single composite part instance. Sets were established for the various phases and labeled as S1, S2, and S3 for the asphalt, epoxy resin and glass fiber phases. Figure 4 illustrates the various phases constituting the 3D RVE.

Table 2
Translation coordinates for the 8-fiber glass spherical particles in the Abaqus assembly module.

X	Y	Z	GLASS FIBER PARTICLE
35	35	35	1
−35	−35	−35	2
−35	35	35	3
35	−35	35	4
35	35	−35	5
−35	−35	35	6
35	−35	−35	7
−35	35	−35	8

Table 3
Epoxy resin and glass fiber particle radius at different GFRP filler contents for 3D RVE.

GFRP CONTENT (%)	5%	10%	15%
Rf	5.0633	6.3794	7.3026
Re	22.1709	27.9336	31.9760

Figure 4
Constituent phases in the GFRP filler-modified asphalt binder 3D RVE.

2.2.3. Meshing, materials and boundary conditions assignment

Mesh properties of the RVEs are assigned in the Abaqus Mesh module, and for this analysis, plane stress element types are considered. After the generation of the 2D and 3D base models, an appropriate mesh type was selected; for the 3D RVEs, quadratic tetrahedral elements of type C3D10 were selected. Due to the complex nature of the geometry, the tetrahedral elements allow for proper meshing of the RVE while preventing the formation of bad elements. Bad elements are those that have poor geometrical features, which may lead to inaccuracies in the analysis. For 2D analysis, two element types were studied, namely linear quadrilateral, type CPS4R, and linear triangle, type CPS3. To analyze the effects of mesh size on the elastic properties, 3 kinds of mesh sizes were selected, namely 7, 8, and 9 μm. In Table 4, the number of elements generated for 3D and 2D RVE models at 7 μm mesh size is presented. SBS-modified asphalt binder type PG76-22 Superpave specification is selected for this study. Asphalt binder, epoxy resin and glass fiber material response behavior are considered linear elastic. Materials properties used in the Abaqus simulations are illustrated in Table 5.

Table 4
Number of elements used in each of the RVEs.

	ELEMENT TYPE	NO. OF ELEMENTS
RVE TYPE		5%	10%	15%
2D	quadrilateral	472	303	326
	Triangular	826	640	584
3D	Tetrahedral	31356	30370	26444

Table 5
Elastic material properties are considered.

MATERIAL	ELASTIC MODULUS (MPA)	POISSON’S RATIO	TEMPERATURE (°C)	SOURCE
SBS-Asphalt binder	3269.1	0.35	−10	[31]
Epoxy resin	3420	0.3	~	[32] [33]
Glass fiber	86000	0.21	~	[34]

Dirichlet boundary conditions were applied to the 3D and 2D RVEs are shown in Figure 5 (a) and (b), respectively. The displacement of nodes on the front face of the 3D RVE was linked to the reference point using the equation constraint feature in Abaqus; a similar technique was used to link nodes on the front edge of the 2D RVE. A test load of 10 μm was applied to the RVE through the reference point and the following parameters were obtained for post-processing, namely, the reaction force, lateral displacement, applied displacement and cross-section area of the RVE.

Figure 5
(a) boundary conditions applied on the 3D RVE (b) boundary conditions applied on the 2D RVE.

Lateral displacement is determined by averaging the displacements of the upper nodes of the RVE; thereafter, the lateral strain is computed by dividing the lateral displacement by the undeformed side length of the RVE. The applied stress is determined by dividing the reaction force by the cross-sectional area of the representative volume element; subsequently, the applied strain is computed by dividing the applied displacement load by the undeformed side length of the RVE. The elastic modulus is the ratio of applied stress to applied strain, while Poisson’s ratio is the ratio of lateral strain to applied strain. The elastic properties of the 2D and 3D RVEs were also evaluated utilizing the Easy-PBC plugin program; the main difference between both techniques lies in the boundary conditions implemented and the type of step employed. The Easy PCB plugin utilizes a static general step with non-linear geometry disabled, while imposing equation constraints on all surface or edge nodes of the RVE to ensure the periodicity of the RVE system. The Easy PBC plugin’s development and implementation have been further examined by OMAIREY et al. [³⁵] and utilized in the current study for comparative analysis. In the present study, Young’s modulus (E11) and Poisson’s ratios (V12, V13) are gathered for further analysis.

2.3. Mathematical and analytical methods

2.3.1. Rule of mixture models (ROM)

Rule of mixtures (ROM) models comprise Reuss and the Voight model; these two formulations can be used to estimate theoretical lower and upper bounds for the effective elastic modulus of the composite materials, respectively [³⁶]. These models utilize the elastic properties and volume fractions of the different phases to compute the effective elastic parameters. ROM models have been applied by several researchers who quantified their accuracy [³⁷,³⁸,³⁹]; regardless, it’s crucial not to solely rely on their values, for instance. SUHU and SREEKANTH [⁴⁰] compared Young’s modulus results from Voight’s model, Reuss’s model, and FEA results, and they found that experimental results were nearly double the predictions from the ROM analytical model, whereas the experimental results were nearly closer to the FEA results. In the previous section, it was outlined that GFRP fillers consist of fiberglass and epoxy particles; when the GFRP filler powder is added to the asphalt binder, the final composite is a three-phase material. The commonly used equations for Reuss and the Voight analytical models only encompass two-phase composite materials; hence, under the current study, these equations are extended to cover the three-phase GFRP filler-modified asphalt binders as shown in equations (5) and (6).

(5) $$E=E_m{V}_m+E_f{V}_f+E_e{V}_e$$

(6) $$\frac{1}{E}=\frac{V_m}{E_m}+\frac{V_f}{E_f}+\frac{V_e}{E_e}$$

Where V is the volume fraction, E is the elastic modulus, and subscripts f, e, and m are used to state the particular phase the property belongs to; “f” stands for fiber glass, “e” stands for epoxy resin, and m stands for matrix (asphalt binder). In equations (5) and (6), by replacing the elastic modulus factor with Poisson’s ratio for the different phases, the effective composite Poisson’s ratio can be calculated; this similar approach is applied to other models.

2.3.2. Halpin-Tsai model

The Halpin-Tsai model developed by Halpin and Tsai consists of a set of semi-empirical mathematical equations used to predict the effective elastic properties of composite materials by taking into account the reinforcement ratio, elastic modulus, and geometrical aspects of the constituent phases [⁴¹]. This analytical model has been found to predict elastic properties more accurately at lower volume fractions, with lesser accuracy as the inclusion volume fraction increases [⁴²]. A basic two-phase Halpin-Tsai model equation is presented in the study by YUN et al. [³⁸], which was extended to simulate three-phase composites by MAREŞ et al. [⁴³], and the modified equation (7) is presented. The value of Ec can be obtained by substituting values from equations (8) and (9) into Equation (7); Where Ec stands for the composite’s effective elastic modulus, S represents the reinforcing factor, and the ratio of h to w is assumed to be (1) for spherical inclusions.

(7) $$E_c=E_m\cdot \frac{1+2\cdot S_f\cdot q_f\cdot V_f}{1-q_f\cdot V_f}\cdot \frac{1+2\cdot S_e\cdot q_e\cdot V_e}{1-q_e\cdot V_e}$$

(8) $$q_{f,e}=(\frac{E_{f,e}}{E_m})\cdot \frac{1}{\frac{E_{f,e}}{E_m}+2\cdot S_{f,e}}$$

(9) $$S_{f,e}=\frac{h}{w}$$

2.3.3. Paul model

Paul’s model presents a more simplified approach for estimating the effective elastic properties of composites, particularly when compared to the Halpin-Tsai model. Unlike the Halpin-Tsai model, which accounts for the effects of inclusion geometry, Paul’s model does not consider these effects. Instead, it relies on the volume fractions and elastic moduli of the constituent phases. The standard form of Paul’s model is designed to describe the elastic behavior of two-phase composites. To extend this model for use with three-phase composites, the equation is modified, as shown in Equation (10), which was adopted from the work of MAREŞ et al. [⁴³].

(10) $$E_c=E_m\cdot \frac{E_m\cdot \left(E_f+E_m\right)\cdot \sqrt{V_f^2}+\left(E_e-E_m\right)+\sqrt{V_e^2}}{E_m+\left(E_f-E_m\right)\sqrt{V_f^2\cdot \left(1-\sqrt{V_f^2}\right)}+\left(E_e-E_m\right)\cdot \sqrt{V_e^2\cdot \left(1-\sqrt{V_e}\right)}}$$

3. RESULTS

3.1. Comparison between Easy PBC results and Dirichlet boundary condition

The elastic modulus and Poisson’s ratio for both 2D and 3D Representative Volume Elements (RVEs) were calculated under various boundary conditions. The commonly used Easy-PBC Abaqus plugin applies periodic boundary conditions (PBC) to the RVE, and its results were compared with those obtained using Dirichlet boundary conditions. The results illustrating the effects of boundary conditions on elastic modulus and Poisson’s ratio values are presented in Figure 6. In Figure 6(a) and Figure 6(b), a comparison of the 2D RVE results is shown. It can be observed that, as the GFRP filler content increases, the elastic modulus of the RVE also increases. The effects of the applied boundary conditions have minimal or negligible influence on the elastic modulus, with the difference between the values from the two methods ranging from 0.04 to 0.07 MPa for 2D analysis. This difference increases as the GFRP filler content rises.

Figure 6
Comparison of Elastic property results RVE Dirichlet boundary conditions Vs Easy-PBC results; (a) Elastic modulus E-PBC vs 2D-RVE, (b) Poisson’s ratio E-PBC vs 2D-RVE, (c) Elastic modulus E-PBC vs 3D-RVE (b) Poisson’s ratio E-PBC vs 3D-RVE.

When using Dirichlet boundary conditions, the elastic modulus increases by an average of 27.37 MPa for every 5% increase in GFRP filler content, whereas the Easy-PBC plugin yields an average increase of 27.35 MPa. The Poisson’s ratio results for the 2D RVE analysis are shown in Figure 6(b). It is evident that Poisson’s ratio decreases as the GFRP filler content increases under both boundary conditions. The difference in Poisson’s ratio values between the two methods is minimal, indicating that the results are nearly identical. Under the Easy-PBC plugin, for every 5% increase in GFRP filler content, a 0.002475 decrease in Poisson’s ratio is observed, whereas under Dirichlet boundary conditions, the decrease is 0.002403.

Results from 3D RVE analysis are presented in Figure 6 (c) and (d); it can be observed that similar to the 2D RVE analysis, an increase in GFRP filler content equally leads to a gradual increase in elastic modulus. It can also be observed that boundary condition types had a less or negligible effect on the elastic modulus value, regardless the difference in values from both methods ranges within 0.27 to 2.09 MPa. For Easy-PBC elastic modulus results, every 5% increase in GFRP filler content, an average increase in elastic modulus of 37.764 MPa is attained, whereas for Dirichlet boundary conditions, an average increase of 36.86 MPa is attained. In Figure 6 (d), as the GFRP filler content increases, the value of Poisson’s ratio gradually decreases. For Easy-PBC results, an average decrease in Poisson’s ratio of 0.0027 is attained, whereas for Dirichlet boundary conditions, an average decrease of 0.0025 is attained. Therefore, in general, utilizing Easy-PBC or Dirichlet boundary conditions to estimate elastic properties has negligible effects on measured results. WU et al. [⁴⁴] conducted a study to investigate the effects of boundary conditions on the elastic properties of glass fiber reinforced composites. Their study found that the elastic modulus (E_x) varies with changes in the boundary conditions, a conclusion that is consistent with the findings of the present section; they also noted that the primary cause of this variation is the distribution of inhomogeneities within the material.

3.2. Effects of mesh size of elastic properties of RVE

The mesh density of the RVE was varied in order to analyze its effects on the values of elastic modulus and Poisson’s ratio; the elastic properties of the RVEs were measured for RVEs with mesh sizes of 9, 8, and 7 μm; results from this analysis are summarized in Figure 7. Effects of mesh size under 2D analysis are illustrated in Figure 7 (a) and (b); it can be observed for 2D analysis that the effect of mesh size has minimal effects on the measure value of elastic modulus relative to each GFRP filler content; regardless, it’s noted that as mesh size decreases, the values of elastic modulus decrease by a value less than 0.6 MPa. The effects of mesh size on Poisson’s ratio are shown in Figure 7 (b); it’s noted that the effects of mesh size are as well negligible at each GFRP filler content. Results from the 3D analysis are presented in Figure 7 (c) and (d) for elastic modulus and Poisson’s ratio, respectively. It’s noted that for 3D RVEs, as the mesh size decreases, the value of elastic modulus decreases; the average decrease in elastic modulus at 5%, 10%, and 15% is 0.12, 0.165, and 0.675 MPa, respectively. Change in mesh size has negligible effects on the measured value of Poisson’s ratio, i.e., the average decrease in Poisson’s ratio is less than 0.0001. It can therefore be stated that for both 3D and 2D RVE analyses, the elastic modulus value decreases with decreasing mesh size, whereas the Poisson’s ratio is less affected by mesh size.

Figure 7
Effects of mesh size elastic properties; (a) 2D elastic modulus (b) 2D Poisson’s ratio (c) 3D elastic modulus (d) 3D Poisson’s ratio.

Rasoolpoor et al. (2021) employed Representative Volume Elements to study the effects of low-velocity impact on nano-modified composites; they investigated the influence of mesh size on the peak loads and deflection. Their results demonstrated that an increase in mesh size led to a decrease in deflection and an increase in the peak load of the RVEs [⁴⁵]. In the current study, deflection and peak load are key parameters for computing the elastic properties of the FEM models, with a decrease in deflection indicating higher stiffness or Youngs modulus of the RVE. Thus, the findings of this section reflect a typical RVE response to variations in mesh size.

3.3. Effects of mesh type on elastic properties of RVE

The effects of the mesh type of RVE elastic property results were investigated under 2D analysis. The effect of mesh type on elastic modulus and Poisson’s ratio is illustrated in Figure 8 (a) and (b), respectively. Figure 8 (c) summarizes the percentage error in elastic property values obtained from either mesh type. The percentage error is obtained by firstly computing the difference in two values and then dividing it by the value estimated by the quadrilateral mesh and multiplying it by 100. According to Figure 8 (a), the elastic modulus estimated by the triangular mesh is higher in comparison to that estimated by the quadrilateral mesh, whereas from Figure 8 (b) the Poisson’s ratio estimated by the tri mesh type is lower in comparison to that estimated by the quad mesh type. It’s also observed in Figure 8 (c) that the percentage error for the elastic modulus value estimated under the two mesh types rapidly increases with increase GFRP filler content; for 5%, 10% and 15% the error percentage in elastic modulus values is estimated as 0.07%, 0.16%, 0.22%. The percentage error of the Poisson’s ratio value has a gradual increase from 5% to 10% and a slight decrease at 15% GFRP filler content. Overall, it can be concluded that the selected mesh type has a significant effect on the estimated values of elastic modulus and Poisson’s ratio whereby RVEs with triangular mesh possess the highest elastic modulus whereas those with Quadrilateral mesh type possess the highest Poisson’s ratio; this may be due to the difference in stress transfer within the individual elements [⁴⁶].

Figure 8
Effects of mesh type on elastic properties of GFRP filler asphalt, (a) Elastic modulus for Quad and tri mesh (b) Poisson’s ratio for Quad and tri mesh type, (c) Mesh error percentage on Elastic modulus (E) and Poisson’s ratio (V).

3.4. Comparison of FEM effective elastic properties and analytical method results

Results obtained from the current study are limited to Abaqus finite element analysis due to the absence of experimental results; therefore, in order to check the reliability of the results, the well-known analytical methods for estimating elastic properties are utilized. These include Ruess, Voight, Paul and the Halpin-Tsai model. Figure 9 illustrates a comparison of the elastic modulus estimated by 2D and 3D FEM in comparison with analytical methods. In Figure 9, the FEM elastic modulus is closely equal to those from the Ruess model at all GFRP filler contents. The elastic modulus estimated by the Voigt’s model is the highest for all GFRP filler contents. It can also be noted that the elastic modulus estimated by all methods increases with an increase in GFRP filler content. According to Figure 9, Voight and Ruess’s model values of elastic modulus presented the highest and lowest estimations, respectively, and such findings align with other researchers in which they found these models typically predict upper and lower bounds of elastic modulus [³⁶, ³⁸]. Previous research findings by MAREŞ et al. [⁴³] showed that Halpin-Tsai model elastic modulus predictions were closer to experimental results compared to Paul’s model.

Figure 9
Comparison between elastic modulus results estimated by FEM vs analytical method.

Figure 10 illustrates a comparison of Poisson’s ratio estimated by 2D and 3D FEM in comparison with results from analytical methods. It’s was also observed that in all methods, as the GFRP filler content increases, the Poisson’s ratio decreases; and that the Poisson’s ratio estimated by 2D finite element methods is the highest at all GFRP filler contents, whereas values from Ruess’s model are the lowest. From both Figure 9 and Figure 10, it can be observed that the FEM results are estimates that are in close range with the Halpin-Tsai Model analytical estimates; hence, it can be assumed that the FEM results possess a certain level of accuracy. On another note, both FEM and analytical results possess similar trend at different GFRP filler content; hence, the FEM methods can effectively be implemented to study the effects of GFRP filler content on elastic properties.

Figure 10
Comparison between Poisson’s ratio estimated by FEM vs analytical method.

3.5. Stress and deformation contour maps

Contours for deformation magnitude (U) are illustrated in Figure 11. The applied boundary conditions for all cases have been shown in order to visualize their effects on the deformation of the RVE. In Figure 11 (a), it is observed that high deformation occurs at the face on which load is applied; the deformation decreases towards the edge that lies between the two faces with applied boundary restrictions. In Figure 11 (b), the boundary constraints applied by the Easy-PBC plugin differ from those utilized in Figure 11 (a); it can be noticed by the difference in their deformation contour; the easy-PBC RVE deformation contour exhibits high deformation at the back that gradually reduces to the front face. In Figure 11 (c) and (d), the deformation contours are similar regardless of the mesh type; therefore, it can be stated that the deformation of the RVE is primarily dependent on the applied boundary conditions. In Figure 11 (e), the deformation of a 2D RVE analyzed with the easy-PBC plugin is shown. It can be observed that deformation is high at the outer edges and decreases towards the center of the RVE.

Figure 11
Deformation contour (magnitude U) (a) 3D Dirichlet boundary conditions (b) 3D Easy-PBC plugin (c) 2D Dirichlet boundary conditions (tri) (d) 2D Dirichlet boundary conditions (quad) (e) 2D Easy-PBC plugin.

Figure 12 illustrates the Mises stress contours for 3D and 2D representative volume elements (RVEs). In Figure 12 (a), the mises stress corresponding to the GFRP filler particles is illustrated; the central large particle represents the epoxy resin phase, while the eight smaller surrounding particles represent the glass fiber phase particles. It can be observed that the stresses are lesser within the epoxy resin phase and much higher glass fiber phase; this is because the glass fiber phase possess higher stiffness, compared to its neighboring phase. This leads to the generation of high resistance to deformation, hence the large stress regions. The high stress concentration around the glass fiber particles can also be observed in the 2D RVEs shown in Figure 12 (c) and (d). The stress distribution for 2D RVEs with quadrilateral mesh is more uneven compared to that of triangular mesh type; this could be attributed to the high number of elements in the triangular element RVE. A high number of elements in a model is largely attributed to better stress transfer between elements, hence a more even stress contour.

Figure 12
Mises stress contours (a) Stress on 3D GFRP particles (b) 3D RVE (c) Stresses on Quad mesh 2D RVE (d) Stress on tri mesh 2D RVE.

4. CONCLUSION

The elastic modulus and Poisson’s ratio of GFRP filler-modified asphalt binder were estimated using 2D and 3D finite element analysis (FEA). Dirichlet boundary conditions and the Abaqus plugin (Easy-PBC) were applied, and the results from both methods were compared. The effects of mesh type and mesh size on elastic properties were also investigated. Additionally, deformation and von Mises stress contour plots were generated to analyze the response of different RVE phases. Analytical models, including Ruess, Voigt, Paul, and the Halpin-Tsai model, were used to estimate the elastic properties, and their results were compared with those from FEA. A summary of the key findings and conclusions is provided below:

• Boundary conditions: The variation in boundary conditions had a more significant impact on the elastic modulus estimates obtained from the 3D analysis compared to those from the 2D analysis. The difference in elastic modulus values between the two methods under 2D analysis ranged from 0.040 MPa to 0.065 MPa, whereas in the 3D analysis, the difference ranged from 0.272 MPa to 2.079 MPa. In contrast, the boundary conditions had a negligible effect on the estimation of Poisson’s ratio.

• Mesh size: An increase in mesh size leads to higher estimates of elastic modulus, whereas Poisson’s ratio is less sensitive to changes in mesh size. The effect of mesh size was more pronounced in the 3D analysis compared to the 2D analysis.

• Mesh type: The elastic modulus estimated using the triangular mesh was relatively higher compared to that obtained with the quadrilateral mesh. The error between the mesh types at 5%, 10%, and 15% filler content was found to be 0.07%, 0.16%, and 0.22%, respectively. In contrast, the Poisson’s ratio estimated with the triangular mesh was lower than that obtained with the quadrilateral mesh. These findings indicate that the choice of mesh type has a significant impact on the estimated values of both elastic modulus and Poisson’s ratio.

• FEM VS Analytical: The trends observed in both FEM results and analytical values are similarly influenced by the GFRP filler content, suggesting that FEM methods can effectively be used to study the impact of GFRP filler on elastic properties. Additionally, the Halpin-Tsai model has been shown in the literature to closely match experimental results. Given that the FEM results align closely with the analytical values under this model, it can be inferred that the FEM results exhibit a reasonable degree of accuracy. However, experimental testing remains crucial for further validation.

• Stress and deformation plots: The deformation of the Representative Volume Element (RVE) is primarily influenced by the applied boundary conditions. It is also observed that the glass fiber phase exhibits higher stress concentrations compared to other phases. Furthermore, the stress distribution within the RVE is significantly affected by the number of elements in the model. A higher element count generally facilitates better stress transfer between elements, leading to a more uniform stress contour.

• Elastic modulus and Poisson’s ratio: Finally, the estimated values for elastic modulus for 5wt.%, 10wt.% and 15wt.% are 3305.42,3342.72 and 3380.95 MPa whereas the values for poisons ratio at 5wt.%, 10wt.% and 15wt.% are 0.3474, 0.3448 and 0.3421.

Overall, the incorporation of filler material increases the stiffness of the asphalt binder. The findings are based on both FEM analysis and analytical model calculations. However, to fully understand the real-world behavior, experimental methods are recommended for future studies to validate the results and analyze actual scenarios. Nonetheless, the current findings provide valuable preliminary insights and contribute to understanding the trends in elastic properties at various filler contents.

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Publication Dates

• Publication in this collection
  23 May 2025
• Date of issue
  2025

History

• Received
  18 Feb 2025
• Accepted
  16 Apr 2025