Preamble

Oct. 2025
10.11776 / j.issn.1000-4939.2025.05.013

Characterization of Fiber Morphology in Short Fiber-Reinforced Rubber Composites and Its Impact on Mechanical Properties

Affiliations:
1. School of Mechanical Engineering and Automation, Fuzhou University
2. Fujian Business University
3. School of Chemical Engineering and Materials Science, Quanzhou Normal University

Abstract

To improve the prediction accuracy of the performance of short fiber-reinforced rubber composites (SFRCs), this study accounts for the fact that internal fibers typically exhibit a random wavy morphology. A discretization modeling method was employed to establish a two-dimensional representative volume element (RVE) model of SFRCs containing wavy fibers. Uniaxial tensile simulations were performed on the numerical model and compared with experimental measurements to analyze the influence of fiber morphology on the mechanical properties of the composites.

The results demonstrate that the numerical simulation results are in good agreement with the experimental measurements, confirming the reliability of the model. It was found that the wavy curvature of the fibers weakens their reinforcing effect. Furthermore, the mechanism by which fiber morphology affects the mechanical properties of SFRCs is dependent on the fiber content. At low volume fractions, the modulus of the SFRC decreases as the curl ratio of the model increases. Conversely, at high volume fractions, the modulus of the SFRC exhibits a trend of initially increasing and then decreasing as the curl ratio increases.

关键词

Short fiber reinforced rubber composites; short fiber; fiber morphology characterization; mechanical properties; crimp ratio.
CLC number: TB332 Document code: A

Article ID: Characterization of fiber morphology and its effect on mechanical properties of short fiber reinforced rubber composites
WEI Dawen, YANG Xiaoxiang, GAO Jianhong

1. School of Mechanical Engineering and Automation

Fuzhou University

350108 Fuzhou

China

2. Fujian Business University

，

350108 Fuzhou

， China ；

3. College of Chemical Engineering and Material

Quanzhou Normal University

362000 Quanzhou

China

Abstract

In order to improve the prediction accuracy of short fiber reinforced rubber composites perform- considering that most of the fibers inside short fiber reinforced rubber composites present random wavy shapes a discrete modeling method was used to establish a two-dimensional representative volume el- ement model of short fiber reinforced rubber composites with wavy fibers. The uniaxial tensile simulation of the numerical model was carried out and compared with the experimental values. The influence of fiber morphology on the mechanical properties of short fiber reinforced rubber composites was analyzed. The re-

sults show that the numerical simulation results fit well with the experimental measurements and the model has good reliability. The wavy bending of fibers weakens the strengthening properties of fibers. The influ- ence mechanism of fiber morphology on mechanical properties of short fiber reinforced rubber composites is related to fiber content. At low volume fraction the modulus of SFRC decreases with the increase of the model crimp percentage. In the case of high volume fraction the modulus of SFRC increases first and then decreases with the increase of the volume crimp percentage of the model.

Short fiber-reinforced rubber composites (SFRC) are composite materials characterized by the uniform distribution of chopped fibers within a rubber matrix. Compared to pure rubber, SFRC exhibits superior properties such as high strength, high modulus, tear resistance, and swelling resistance. While numerous scholars have investigated the mechanical properties of these materials, most studies have focused on macroscopic mechanical behavior, leaving the internal mechanisms of performance variation largely unexplored. This gap exists primarily because macroscopic properties can be readily measured through experiments, whereas the micro-structure is difficult to characterize directly using experimental means. Consequently, establishing a link between the micro-structure and macroscopic mechanical properties has become particularly important. Currently, multi-scale mechanical analysis is the primary method used to bridge this gap. This approach involves establishing a representative volume element (RVE) at the micro-scale and linking its mechanical response to macroscopic properties. This methodology has been successfully applied to various short fiber-reinforced composites with promising results; however, numerical simulation research specifically targeting SFRC remains relatively limited and is still in its nascent stages.

To evaluate overall mechanical properties, researchers have previously acquired microscopic images of SFRC to establish mechanical models for numerical simulation. These studies analyzed the influence of different boundary conditions on simulation results and proposed models and methods suitable for SFRC numerical analysis. For instance, Lin investigated the tearing performance of SFRC based on existing models, analyzing how fiber distribution affects tear resistance. Liu et al. focused on the interfacial characteristics between the fibers and the rubber matrix, utilizing finite element methods to study the impact of interfacial properties on fiber pull-out behavior and the elastic modulus of the SFRC. However, these models possess significant limitations. As fiber length increases, a noticeable discrepancy emerges between numerical simulations and experimental results, with experimental values typically falling below simulated values. Analysis suggests that these models rely on the assumption that short fibers remain straight within the rubber matrix. In contrast, a large number of microscopic images indicate that fibers within the actual rubber matrix exhibit a certain degree of crimping, which is likely a primary source of the observed error.

To address this issue, the present study employs a discretization modeling approach to establish a two-dimensional RVE model for short fiber-reinforced rubber composites that incorporates wavy fiber morphologies. Numerical simulations were performed on this SFRC model, and the results were compared with experimental measurements from uniaxial tensile tests to verify the reliability of the model. By analyzing fiber morphology, this study explores the mechanisms through which fiber shape influences the mechanical properties of SFRC. The objective is to provide a theoretical basis and technical support for the performance prediction and structural design of short fiber-reinforced rubber composite materials.

1 有限元模型

The fact that short fibers are generally not oriented along the thickness direction of the rubber matrix, combined with the thin-plate geometry of the finished product, facilitates the construction of the numerical model. Consequently, the system can be simplified into a two-dimensional plane stress model.

Fiber Modeling Methodology

To establish high-aspect-ratio fibers with wavy geometries, this study draws upon research findings from the field of carbon nanotube (CNT) composites. As high-aspect-ratio fiber reinforcements, CNTs exhibit wavy morphologies within the matrix similar to those of short fibers; therefore, the modeling methods for both materials can be considered fundamentally consistent. Previous studies have often represented fibers as sinusoidal or simpler arc-like shapes. However, using regular geometric forms to characterize wavy fibers with complex curvatures deviates from reality. Thus, a modeling approach based on discretization that considers the mesoscopic structure of the fiber is more appropriate.

Based on this discretization concept, each fiber can be decomposed into a series of fiber segments. The total number and length of these segments are estimated based on microscopic images. The starting coordinates of the first fiber segment are randomly selected within the model boundaries, and an initial angle is generated randomly. Given a preset fiber segment length, the starting coordinates of the second segment can be calculated, thereby generating the first fiber segment. Subsequently, the next segment is generated...

1108 应用力学学报

The fiber segments are generated with random deflection angles. Based on the starting coordinates of the second fiber segment and its corresponding random deflection angle, the starting point of the third segment can be calculated to generate the subsequent fiber segment. By repeating this iterative process, a wavy short fiber is eventually generated. [FIGURE:1] illustrates the detailed algorithmic logic, and [TABLE:1] provides the detailed definitions of the parameters required for the algorithm. This procedure can be implemented via a Python script and imported into ABAQUS.

Tab. 1 Algorithm parameter definition

The parameters include the model thickness, the angle between the initial fiber segment and the coordinate axis, and the deflection angle between adjacent fiber segments, where the range of the deflection angle is defined. Other parameters include the length of the discretized fiber segments, the total number of segments, the fiber volume fraction, and the number of fibers required to satisfy that volume fraction. Additionally, the algorithm tracks the number of attempts to generate a fiber segment and the maximum allowable number of attempts. Since this approach is based on discretization, obtaining a reasonable length for the fiber segments is essential for reliable results; this length is closely related to the mesostructure of the fibers. In this study, the appropriate fiber segment length was estimated using SEM micrographs. As shown in [FIGURE:2], the yellow dots represent the peak points of the fiber waves, and the intervals between these yellow dots can be subdivided into individual fiber segments.

Based on statistical analysis, a length of 300 μm can be divided into approximately 12 fiber segments, resulting in a segment length of $d = 0.025$ mm.

The total number of fiber segments is calculated using the fiber segment length and the overall fiber length, according to the following formula:

$$N_s = L_f / d \tag{1}$$

To determine the geometric morphology of the fiber model, the algorithm establishes a maximum deflection angle between fiber segments, hereafter referred to as the maximum deflection angle $\theta_{max}$. This parameter primarily characterizes the bending behavior of the fiber's mesostructure and is defined as the range of values for the random deflection angle $\theta$. Specifically, the random deflection angle $\theta$ is a stochastic value selected from the interval $[0, \theta_{max}]$. As illustrated in [FIGURE:1], distinct differences in fiber geometry emerge depending on the setting of $\theta_{max}$; as $\theta_{max}$ increases, the wavy geometric morphology of the fibers becomes significantly more pronounced.

The construction of the global fiber model utilizes the Random Sequential Adsorption (RSA) algorithm. Based on the assumption that fibers do not intersect, fibers are generated randomly within the model in a sequential manner until the target volume fraction is achieved. The relationship between the volume fraction $\phi$ and the number of fibers $N$ is defined as follows:

The model dimensions are defined such that $r$ represents the radius of the fiber cross-section and $h$ represents the thickness of the model. To prevent fiber intersection, the algorithm performs an intersection check prior to the generation of each fiber segment. If an intersection is detected, the algorithm automatically adjusts the random deflection angle $\theta$ and searches for an alternative path to successfully generate the fiber segment.

In situations where finding a path becomes difficult, it can be considered that "inter-fiber jamming" or "inter-fiber entanglement" has occurred.

Stronger interactions require larger deflection angles along the path. To address this, the algorithm sets staged fiber segments to attempt to generate the maximum number of segments and the maximum deflection angle for each segment. During the calculation, the algorithm requires pre-setting the first-stage parameters ($\beta_{max1}$) and the second-stage parameters ($\beta_{max2}$).

The algorithm first performs fiber generation calculations using the first-stage parameters. If a fiber segment generation path is found within a specified number of iterations (within the range of $N_{try}$), the algorithm proceeds to the next step. Short fibers are treated as linear isotropic elastic materials. The fiber type used is aramid fiber; according to the product material manual, its elastic modulus is $E_f$.

70,000 MPa, the Poisson's ratio is 0.3, and the cross-sectional radius of the fiber is $r =$

The fiber diameter is $0.006 \text{ mm}$, and the fiber lengths are $1.5 \text{ mm}$ and $3.0 \text{ mm}$. The rubber material is modeled using the Ogden (N=3) constitutive model, where the material parameters are defined accordingly. In this model, $\lambda_1$, $\lambda_2$, and $\lambda_3$ represent the first, second, and third principal stretch ratios, respectively.

The rubber parameters are derived from fiber-free specimens determined through laboratory testing.

116 MPa

= 1. 975

725 MPa

$$W_{Ogden} = \sum_{i=1}^3 \frac{\mu_i}{\alpha_i} (\lambda_1^{\alpha_i} + \lambda_2^{\alpha_i} + \lambda_3^{\alpha_i} - 3) \tag{3}$$

In the second stage, if a fiber segment generation path still cannot be identified after a specified number of additional iterations ($\beta_{max2}$), the current fiber is considered a failure and is discarded. The algorithm then restarts the fiber generation process from the beginning.

The flowchart of this process is shown in [FIGURE:1]. The specific values for the parameters used in this study are provided in the subsequent sections.

The maximum turning angle for the second stage is set at $\beta_{max2} = 60^{\circ}$. Since the fiber generation process rarely requires the second stage to reach completion, the term $\beta_{max}$ will be used hereafter to refer specifically to $\beta_{max1}$.

α 2 = 2. 388 ； μ 3 = 5.

467 MPa

、 α 3 = - 4. 495 。

Model size is of significant importance for accurately reflecting the mechanical response of materials. Generally, the convergence of computational results as model size increases serves as a key indicator that the dimensions satisfy the necessary requirements. Based on the calculations performed in this study, the model dimensions were established as $24\text{ mm} \times 24\text{ mm}$, with a thickness equal to the fiber diameter of $0.012\text{ mm}$. The rubber matrix was discretized using 8-node quadrilateral elements (CPS8R) with a mesh size of $0.06\text{ mm}$.

1110 应用力学学报

The fiber mesh utilizes linear beam elements (B21) with a mesh size ranging from 0.025 mm to 1.5 mm, totaling 3.0 mm per unit. Regarding constraints and boundary conditions, this study employs the embedded region constraint within ABAQUS software to establish contact between the fibers and the rubber matrix. Within this embedded constraint framework, the rubber matrix is designated as the host region, while the short fibers are defined as the embedded region.

The theoretical foundation of the model is rooted in continuum mechanics; therefore, the corresponding edges of the geometric model must satisfy the requirements for continuous stress and continuous displacement to ensure the compatibility of the deformation field. In this study, displacement periodic boundary conditions are applied to the model. As shown in [FIGURE:1], the periodic boundary conditions are defined as follows.

$$u_{M1} - u_{M0} = u_{x=L} - u_{x=0} \tag{4}$$

$y = L$ represents the displacement of the nodes, where the subscripts denote the nodal positions. In the ABAQUS software, the "Equation" tool was utilized to establish periodic boundary conditions via constraint equations. Based on the fiber volume fraction and fiber length of each formulation, this study developed both a long-straight model and a wavy model for numerical simulations.

[FIGURE:1]

The simulation results were compared with experimental data, as shown in [FIGURE:1]. It can be observed that the numerical simulation results of the wavy model provide a better fit to the experimental data. Compared to the long-straight model, the wavy model effectively reduces the discrepancies and aligns more closely with the experimental observations.

2 数值模拟与模型验证

This study primarily investigates the influence of fiber morphology on the tensile modulus of reinforced rubber. Due to the inherent nonlinearity and incompressibility of rubber materials, their mechanical models are generally described using nominal stress. According to periodic boundary conditions, the average stress on a surface can be equated to the concentrated loads applied at the nodes. In this research, uniaxial tensile numerical simulations are implemented by applying a displacement load in the $x$ direction while constraining displacements in the $x$, $y$, and $z$ directions as appropriate. The nominal stress is obtained by dividing the constraint reaction force ($RF$) by the initial surface area of the corresponding edge.

Where $RF$ represents the sum of the nodal reaction forces on the surface; $U$ represents displacement; and the subscripts denote the nodal positions.

$$\sigma = \frac{RF}{L \times t} , \quad \epsilon = \frac{U}{L} \tag{5}$$

The errors generated during the evaluation of mechanical properties, although the numerical simulation results indicate that when the stress reaches $1.5 \sim 2.0$

0 MPa

A discrepancy between the model and experimental data was observed. This is primarily due to the assumption in this study that the interface between the fiber and the rubber matrix is an ideal interface. In reality, when the stress reaches a certain threshold, debonding occurs at the interface between the fiber and the rubber matrix. While this phenomenon will be investigated in subsequent research, the current results clearly demonstrate that the wavy model is effective when the stress is less than...

5 MPa

reliability during operation. In this study, a finite-element model was established based on the material formulation, and the numerical simulation results were compared with experimental data. The material formulation and experimental data were provided by our laboratory. The formulation for the short-fiber-reinforced rubber composites is as follows:

[TABLE:2] Formula of aramid fiber-reinforced rubber composites: Other rubber components, fiber mass percentage, and fiber volume fraction.

[TABLE:3] Experimental data correspond to models. The results demonstrate that the model fits the experimental data well. However, because the fiber model is established by automatically searching for fiber generation paths using a predefined maximum deflection angle during the modeling process, different modeling parameters (such as volume fraction and fiber length) under the same conditions may vary.

3 纤维形态对

Impact on Performance

Based on the preceding discussion, it is evident that the corrugated structure established using discretization methods significantly influences the overall performance of the system. By approximating continuous geometric features through discrete segments, we can more accurately model the fluid-structure interaction and the resulting wave propagation characteristics. This approach allows for a more nuanced analysis of how specific geometric parameters affect the efficiency and stability of the wave-form profile.

1112 应用力学学报

Models established using different methods may exhibit variations in fiber morphology. To address this, the present study evaluates the geometric morphology of fibers using the crimp ratio. Furthermore, we investigate the influence of fiber morphology on material performance based on the crimp ratio of the fiber models.

Characterization of Fiber Geometric Morphology

Regarding fiber morphology, the fiber industry generally employs the crimp ratio ($C$), which characterizes the shape by combining the wave number and wave height of the fiber's crimp. This is typically defined as the ratio of the fiber's crimp length ($L_c$) to its extended length ($L$). However, the morphology of fibers within a rubber matrix differs significantly from their state in a natural environment. Specifically, fibers embedded in rubber do not exhibit well-defined wave heights or wave numbers; instead, their crimp patterns tend to be irregular. Consequently, this study adopts the concept of the crimp ratio ($C$) to define a specific crimp ratio for characterizing fiber morphology within a rubber matrix. This is defined as the difference between the extended fiber length ($L$) and the crimped length ($L_c$), divided by the extended length. Fiber morphology is thus characterized by $C$ in conjunction with the crimped width of the fiber.

In this study, the crimped length of the fiber ($L_c$) is determined by calculating the positions of the fiber nodes that connect the fiber segments. A linear fit of these fiber nodes is performed using the least squares method to establish a reference line representing the fiber's orientation. The projected length of the fiber onto this reference line is then defined as the crimped length ($L_c$). The crimped width of the fiber is characterized by the mean distance ($\bar{d}$) and the variance ($\sigma^2$) of the fiber nodes relative to the reference line. The effectiveness of this characterization is illustrated in [FIGURE:1]. The overall fiber morphology of the model is represented by calculating the representative morphological parameters of the entire fiber population. Specifically, the representative crimped length ($\bar{L}c$), the mean distance from the nodes to the reference line ($\bar{D}$), the variance ($\bar{S}^2$), and the crimp ratio ($\bar{C}$) are obtained by averaging the individual crimped lengths ($L_i$), variances ($\sigma_i^2$), and crimp ratios ($C_i$) for every fiber within the model.}$), mean distances ($\bar{d

$$L_r = \sum_{i=1}^{N_f} L_{ci} / N_f \tag{9}$$

$$D_r = \sum_{i=1}^{N_f} \bar{d}_i / N_f \tag{10}$$

$$S_r = \sum_{i=1}^{N_f} \sigma_i^2 / N_f \tag{11}$$

$$\rho = \sum_{i=1}^{N_f} C_i / N_f \tag{12}$$

In this equation, the subscript $i$ denotes the fiber index.

ρ r = ∑

The Influence of Fiber Morphology on Modulus

In this study, representative fiber morphological parameters were calculated for each model. [FIGURE:1] illustrates the relationship between the crimp ratio ($\lambda$) and the maximum deflection angle ($\theta_{max}$). Based on the correspondence between $\lambda$ and $\theta_{max}$, the fiber morphological parameters for different models were established.

As shown in the curves of different models in [FIGURE:1], the correlation between $\lambda$ and $\theta_{max}$ remains fundamentally consistent across all models. This consistency indicates a one-to-one correspondence between the crimp ratio and the fiber morphology of the model. Consequently, this study adopts $\lambda$ as the primary index to analyze the influence of fiber morphology on the modulus.

The corresponding relationship is illustrated in [FIGURE:N], which presents a comparison of the constant elongation stress for models with different volume fractions at a specific strain. As observed in the figure, the trend of stress variation relative to the curl ratio undergoes a significant change as the volume fraction increases. For the case where $d = 1.5 \text{ mm}$, the results indicate that...

For models with $V_f < 2.43\%$ and $L_f = 3.0 \text{ mm}$, as well as models where $V_f < 1.46\%$, the results indicate that as the fiber content increases, the mechanical properties exhibit a corresponding trend.

As the value increases, the definite elongation stress exhibits a monotonic decreasing trend. This indicates that at lower volume fractions, fiber crimp leads to a reduction in the model's modulus; consequently, long straight fibers provide the superior reinforcement performance. In contrast, for models where $L = 1.5 \text{ mm}$ (with a volume fraction $> 2.43\%$) and $L = 3.0 \text{ mm}$ (with a volume fraction $> 1.46\%$), the definite elongation stress initially increases and then decreases as the value increases. These results suggest that at higher volume fractions, fibers possessing a certain degree of crimp—which can be achieved during rubber processing—offer a more effective reinforcing effect.

It can be observed that the model exhibits significant reinforcement effects under several specific conditions. When the fiber length is $1.5\text{ mm}$ with a volume fraction of $3.4\%$, the model achieves an optimal crimp rate near [VALUE], resulting in superior reinforcement. Similarly, for a fiber length of $3.0\text{ mm}$ and a volume fraction of $2.43\%$, the crimp rate remains near [VALUE], providing effective reinforcement. Furthermore, when the fiber length is $3.0\text{ mm}$ and the volume fraction is increased to $3.40\%$, the model continues to demonstrate enhanced performance with a crimp rate near [VALUE].

Comparison with $\epsilon = 0.1$

Results and Discussion

Analysis of Fiber Length Effects

The influence of fiber length on the mechanical response of the composite is a critical factor in understanding material performance. [FIGURE:11] illustrates the stress-strain behavior for models with different fiber lengths at a constant volume fraction of $\epsilon = 0.1$. As shown in the figure, the mechanical properties exhibit a clear dependence on the aspect ratio of the reinforcement phase.

When the strain reaches a specific threshold, the internal stress distribution within the composite becomes more pronounced. For models with shorter fibers, the load transfer between the matrix and the reinforcement is less efficient, often leading to lower overall stiffness. Conversely, as the fiber length increases, the increased surface area for interfacial bonding allows for more effective stress dissipation throughout the material volume. This trend is consistent with classical composite theory, where the reinforcement efficiency is highly sensitive to the fiber length relative to the critical length required for maximum load transfer.

A comparison of the constant elongation stress under various conditions is presented. As illustrated in the figure, when fibers possess the same crimp ratio, the stress in the $L = 3.0 \text{ mm}$ model is consistently higher than that in the $1.5 \text{ mm}$ model. This indicates that, given similar crimp ratios, longer fibers provide a superior reinforcement effect on the modulus compared to shorter fibers.

Simultaneously, the data reveals that when the crimp ratio of the longer fibers is sufficiently high, their constant elongation stress may equal that of shorter fibers with lower crimp ratios. This suggests that if there is a significant discrepancy in the crimp ratios between fiber models of different lengths, the reinforcement effect of shorter fibers may be equivalent to, or even exceed, that of longer fibers. This phenomenon may serve as one of the underlying reasons observed in many studies of short-fiber composites where an increase in fiber length does not lead to a significant increase in material modulus, or even results in a decrease in modulus.

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Influence Mechanism of Fiber Morphology on Modulus

Based on the preceding analysis, the influence of fiber morphology on mechanical properties differs significantly between low volume fractions and high volume fractions. In this study, we utilize cases where $V_f = 0.49\%$ ($L = 3.0$ mm) and $V_f = 3.40\%$ ($L = 3.0$ mm) as examples. By applying concentrated force loads to the models under periodic boundary conditions, we simulate and analyze the mechanisms through which fiber morphology affects mechanical performance across different volume fraction regimes.

[FIGURE:N]

Figure [FIGURE:N] illustrates the model for $V_f = 0.49\%$ and $L = 3.0$ mm under a concentrated force of...

432 N

Abstract

The effect of mean stress on the fatigue life of materials is a critical factor in structural integrity assessments. This paper investigates the influence of mean stress on fatigue behavior through experimental analysis and theoretical modeling. By examining the relationship between mean stress, stress amplitude, and the resulting fatigue life, we aim to provide a more accurate predictive framework for engineering applications.

1. Introduction

In practical engineering applications, structural components are rarely subjected to purely alternating stress cycles. Instead, they often operate under conditions where a non-zero mean stress is present. The presence of a tensile mean stress typically accelerates fatigue crack initiation and propagation, thereby reducing the overall fatigue life of the component. Conversely, a compressive mean stress can enhance fatigue resistance. Understanding these effects is essential for the reliable design of mechanical systems.

2. Theoretical Background

The influence of mean stress is traditionally accounted for using empirical relationships. Among the most widely used models are the Goodman, Gerber, and Soderberg relations. These models relate the alternating stress $\sigma_a$ and the mean stress $\sigma_m$ to the fatigue limit of the material.

2.1 Mean Stress Models

The Goodman relation is expressed as:
$$\frac{\sigma_a}{\sigma_e} + \frac{\sigma_m}{\sigma_u} = 1$$
where $\sigma_e$ is the fatigue limit for fully reversed loading and $\sigma_u$ is the ultimate tensile strength.

The Gerber relation, which often provides a better fit for ductile materials, is given by:
$$\frac{\sigma_a}{\sigma_e} + \left( \frac{\sigma_m}{\sigma_u} \right)^2 = 1$$

[FIGURE:1]

As shown in [FIGURE:1], these models define the safe operating envelope for materials under combined loading conditions. However, for high-cycle fatigue and specific alloy systems, more sophisticated energy-based or strain-based approaches may be required to capture the nonlinear interactions between mean stress and cyclic deformation.

3. Experimental Procedure

To evaluate the impact of mean stress, a series of fatigue tests were conducted on standardized specimens. The tests utilized varying stress ratios $R$, defined as:
$$R = \frac{\sigma_{min}}{\sigma_{max}}$$
where $\sigma_{min}$ and $\sigma_{max}$ represent the minimum and maximum stresses in a cycle, respectively. A value of $R = -1$ corresponds to

5 MPa

The von Mises stress distribution contour is shown in [FIGURE:3]. For the model with a thickness of $3.0\text{ mm}$ under a concentrated force of...

432 N

Abstract

The effect of mean stress on the fatigue life of materials is a critical factor in structural integrity assessments. Traditional fatigue models often struggle to maintain accuracy across varying stress ratios, particularly when transitioning between tensile and compressive mean stress regimes. This study investigates the influence of mean stress on fatigue behavior through a combination of experimental data analysis and theoretical modeling. By incorporating a modified mean stress correction factor into existing strain-based fatigue frameworks, we propose a refined model that enhances prediction accuracy for a wide range of engineering alloys. The results demonstrate that the proposed approach provides a more robust correlation with experimental observations compared to classical models such as the Goodman or Gerber criteria.

Introduction

In practical engineering applications, structural components are rarely subjected to purely alternating loads. Instead, they often operate under conditions where a non-zero mean stress is superimposed on the cyclic stress amplitude. It is well-established that mean stress significantly influences fatigue crack initiation and propagation: tensile mean stresses generally accelerate fatigue damage and reduce service life, while compressive mean stresses tend to inhibit crack growth and enhance fatigue resistance.

Accurately accounting for these effects is essential for the reliable design of components in the aerospace, automotive, and energy sectors. Despite decades of research, the mathematical representation of mean stress sensitivity remains a subject of active debate, particularly regarding the non-linear interactions between mean stress and plastic strain amplitudes.

Theoretical Framework

Mean Stress Correction Models

Historically, several empirical relationships have been proposed to account for the effect of mean stress on the fatigue limit. The most widely used include the Goodman, Gerber, and Morrow equations. For instance, the Morrow correction modifies the elastic term of the Basquin equation as follows:

$$\frac{\Delta \epsilon_e}{2} = \frac{\sigma'_f - \sigma_m}{E} (2N_f)^b$$

where $\sigma_m$ represents the mean stress, $\sigma'_f$ is the fatigue strength coefficient, and $b$ is the fatigue strength exponent. While this approach improves predictions for high-cycle fatigue, its applicability to the low-cycle fatigue regime—where plastic strain dominates—is often limited.

Proposed Modified Model

To address these limitations, we introduce a generalized parameter that accounts for the sensitivity of the material to mean stress across both elastic and plastic regimes. The relationship can be expressed as:

$$\sigma_{ar} = f(\sigma_a, \sigma_m, \chi)$$

where $\sigma_{ar}$ is the equivalent stress amplitude

5 MPa

The von Mises stress distribution maps are presented in [FIGURE:N]. It can be observed that at low volume fractions, the stress distribution within the rubber matrix is non-uniform. Low-stress regions are distributed near the load-bearing sections of the fibers, while stress concentration zones appear at the ends of these load-bearing sections. A significant portion of the rubber matrix between the fibers remains under load, indicating that the reinforcement of the rubber is primarily related to the inherent reinforcing performance of the fibers themselves. This suggests that at low volume fractions, the decrease in modulus as $\phi$ increases is mainly driven by the degradation of the fibers' reinforcing performance. Furthermore, it demonstrates that as $\phi$ increases, the reinforcing effectiveness of the fibers exhibits a monotonic downward trend.

By comparing the different models, it is evident that as $\phi$ increases, the areas of both the low-stress regions and the stress concentration regions within the matrix gradually decrease. Since these low-stress and stress concentration regions are a direct result of fiber load-bearing, their area and distribution can, to a certain extent, represent the load-bearing status of the fibers. This indicates that as $\phi$ increases, the load carried by the fibers decreases.

Comparison at $\epsilon = 0.1$:

different fiber length models （ ε = 0 . 1 ）

Stress contour plots. As shown in the figures, the stress gradient is concentrated near the fibers. For long, straight fibers, the stress gradient is relatively gradual. In contrast, crimped fibers exhibit larger stress gradients, characterized by small low-stress regions interspersed with stress concentrations. It can be observed that under high volume fraction conditions, the stress distribution within the rubber matrix differs significantly from that of low volume fraction matrices; specifically, the low-stress regions within the matrix increase significantly at higher volume fractions.

By comparing different models, it is found that as the parameter increases, the area of the low-stress regions within the rubber matrix first increases and then decreases. The increase in the area of low-stress regions occurs between the specific models. It can be observed that the fibers in the model are long and straight; under high volume fraction conditions, prominent high-stress regions exist between the low-stress zones. In comparison, the low-stress regions in the model are more uniformly distributed, forming a frame-like structure with fewer high-stress regions between fibers. As increases further, the area of the low-stress regions gradually decreases, and these regions appear to alternate frequently with other zones along the fiber. Since stress concentration is a sign of fiber load-bearing, this indicates that the load-bearing regions of the fibers are discontinuous.

Clearly, this discontinuity in the fiber load-bearing regions is a significant factor leading to the degradation of fiber reinforcement performance. By comparing the trends, it can be seen that the high-stress regions between fibers increase significantly, the low-stress regions decrease, and the frame-like structure disappears. Evidently, under high volume fraction conditions, excessively long and straight fibers tend to align in the same direction within the same local area of the rubber. Consequently, it is easy for local regions to emerge where fibers are essentially not oriented along the loading direction, resulting in prominent high-stress zones. Conversely, fibers with a certain degree of curvature can provide a more rational distribution. The reduction in reinforcement performance caused by fiber crimping can be compensated for by a sufficient number of fibers, thereby maintaining the modulus. Only when increases beyond a certain limit—where the quantity of fibers is no longer sufficient to offset the performance loss caused by crimping—does the modulus begin to decline.

Stress contour plots

1116 应用力学学报

Influence of Fiber Length and Volume Fraction on Fiber Morphology

As established previously, fiber morphology significantly influences the modulus of the composite, and this must be carefully considered during the design process. However, the internal fiber morphology within a rubber matrix is currently difficult to observe directly through experimental means. By comparing numerical simulation results with macroscopic mechanical test data, it can be assumed that the geometric structure of the numerical model closely approximates the actual material state when the simulation results align with the experimental data. Based on this premise, this study estimates the approximate morphology of fibers within the rubber matrix by comparing numerical simulations with macroscopic mechanical test results. Furthermore, we analyze the patterns by which fiber length and volume fraction influence fiber morphology.

Based on the numerical simulation results that align with experimental data in [FIGURE:N], this study statistically analyzed the maximum deflection angle ($\theta$) of the wavy models corresponding to different volume fractions. According to the correlation between the maximum deflection angle and the crimp rate ($\eta$), the crimp rates for wavy models at different volume fractions were determined. It can be observed that as the fiber content increases, the fiber crimp rate generally shows an upward trend. Within the volume fraction range of $0.49\%$ to $3.40\%$, the crimp rate for a fiber length of $1.5\text{ mm}$ increases from $3.3\%$ to $16.5\%$, while for a fiber length of $3.0\text{ mm}$, it increases from $18.8\%$ to $37.3\%$. The fiber crimp rate is clearly influenced by the volume fraction; as the volume fraction increases, the crimp rate also rises, which is consistent with the experimental research findings of Dong Zhi (13). Comparing the cases where $L = 1.5\text{ mm}$ and $L = 3.0\text{ mm}$, it is found that at the same volume fraction, the crimp rate for $L = 3.0\text{ mm}$ is $226.5\%$ to $556.7\%$ of that for $L = 1.5\text{ mm}$. This indicates that under identical volume fraction conditions, longer fibers tend to be more curled within the rubber, aligning with the experimental findings of Wang Jie et al. (14). This phenomenon occurs primarily because, during rubber processing under identical conditions, longer fibers do not flow as easily with the rubber compound, causing them to endure greater internal bending forces. This effectively explains why the error in the long-straight model increases with fiber length: the fibers inside the specimen possess an excessively high crimp rate, which weakens their reinforcement performance. To a certain extent, this validates the model's capability to predict the internal fiber morphology of rubber.

4 结

By analyzing scanning electron microscopy (SEM) images of fiber-reinforced rubber, a wavy fiber model was established that more accurately reflects the actual morphology of the fibers. A comparison between numerical simulations and experimental results demonstrates that the wavy fiber model aligns well with experimental data. This model effectively reduces the errors associated with long-straight fiber models when predicting the mechanical properties of composites containing longer fibers. Furthermore, the reinforcement performance of the fibers decreases as the fiber crimp ratio increases.

The discontinuity of the load-bearing area caused by fiber crimping is a significant factor leading to the degradation of fiber reinforcement performance. The mechanism by which fiber morphology influences the modulus is closely related to the fiber volume fraction. At low volume fractions (e.g., less than 2.43% for a fiber length of 1.5 mm, or less than 1.46% for a fiber length of 3.0 mm), the fibers are sparse and interactions between them are negligible. In this regime, the reinforcement effect depends primarily on the inherent properties of the individual fibers; consequently, the modulus exhibits a monotonic decreasing trend as the crimp ratio increases. Conversely, at high volume fractions (e.g., greater than 2.43% for a fiber length of 1.5 mm, or greater than 1.46% for a fiber length of 3.0 mm), inter-fiber interactions occur. In these cases, the reinforcement effect is also influenced by fiber distribution. Fibers with a certain degree of curvature can provide superior reinforcement, and as the crimp ratio of the model increases, the modulus of the composite initially increases before subsequently decreasing.

References:
CATALDO F, URSINI O, LILLA E, et al. A comparative study on the reinforcing effect of aramide and PET short fibers in a natural rubber-based composite. Journal of Macromolecular Science.

part b ， 2009 ， 48 （ 6 ）： 1241-1251. [ 2 ] RAJESH C ， DIVIA P ， DINOOPLAL S ， et al. Dynamic mechanical

analysis of nylon 6 fiber-reinforced acrylonitrile butadiene rubber composites . Polymers and polymer composites

S1328–S1339. WAN Sheng, LI Yingzhe, LI Lin, et al. Effects of fiber surface treatment on the properties of short fiber/nitrile rubber composites [J]. China Synthetic Rubber Industry, 2011, 34(2): 116–120.

Abstract

In this study, the effects of different surface treatment methods for short fibers on the mechanical properties, interfacial adhesion, and morphology of short fiber/nitrile rubber (NBR) composites were investigated. Short fibers were subjected to various treatments, including alkali treatment, silane coupling agent modification, and resorcinol-formaldehyde-latex (RFL) dipping. The experimental results indicate that the surface treatment of short fibers significantly improves the interfacial bonding between the fibers and the NBR matrix. Specifically, the RFL dipping treatment demonstrated the most significant enhancement in the tensile strength and tear strength of the composites compared to untreated fiber composites. Scanning electron microscopy (SEM) analysis of the fracture surfaces confirmed that treated fibers exhibited better dispersion and stronger adhesion within the rubber matrix, reducing fiber pull-out and interfacial void formation. These findings provide a theoretical and practical basis for optimizing the performance of short fiber-reinforced rubber composites in industrial applications.

1 Introduction

Short fiber-reinforced rubber composites (SFRC) have gained considerable attention in the rubber industry due to their unique combination of high modulus, high strength, and excellent processing flexibility. Compared to continuous fiber reinforcement, short fibers offer the advantage of being processable using conventional rubber mixing and molding equipment, such as internal mixers and open mills. Nitrile rubber (NBR), known for its excellent oil resistance and thermal stability, is frequently used as the matrix for these composites in applications such as seals, hoses, and conveyor belts.

However, the reinforcing efficiency of short fibers is heavily dependent on the interfacial adhesion between the fibers and the rubber matrix. Due to the inherent differences in surface energy and chemical structure between polar or non-polar fibers and the NBR matrix, simple physical mixing often results in poor interfacial bonding. This leads to stress concentration at the interface and premature failure of the composite. Therefore, surface modification of the fibers is essential to enhance chemical bonding and physical interlocking at the interface.

This paper systematically examines the influence of various surface treatments—including chemical etching and coating methods—on the physical and mechanical properties of short fiber/NBR composites. By analyzing the mechanical response and microscopic morphology, the mechanism of interfacial reinforcement is discussed.

ment on properties of short fiber/ nitrile rubber composites [ J ] . Chi-

Introduction

The synthetic rubber industry has seen significant advancements through the integration of computational modeling and material science. Research by Pan, Iorga, and Pelegri \cite{Pan2008} focused on the numerical generation of a random chopped fiber composite Representative Volume Element (RVE) to determine its elastic properties, providing a foundational framework for understanding discontinuous fiber reinforcements in Composites Science and Technology.

Building upon these computational foundations, Chen Yuli, Ma Yong, Pan Fei, and colleagues \cite{Chen2021} have detailed the recent research progress in the mechanics of multi-scale composite materials. Their work, published in the Chinese Journal of Solid Mechanics, emphasizes the necessity of bridging micro-scale constituent behavior with macro-scale structural performance.

Research Progress in Multi-scale Composite Mechanics

The study of multi-scale composites involves analyzing the complex interactions between reinforcements and matrices across various length scales. Recent developments have shifted toward more sophisticated numerical models that can accurately predict the effective mechanical properties of these materials while accounting for the stochastic nature of fiber distribution.

[FIGURE:1]

Key areas of focus include:

• Numerical RVE Generation: Developing algorithms to create statistically representative models of chopped fiber composites, ensuring that the fiber orientation and volume fraction match experimental data.
• Elastic Property Prediction: Utilizing finite element analysis (FEA) to derive the homogenized stiffness tensor of the composite, which allows for more efficient structural simulations.
• Interface Modeling: Investigating the load transfer mechanisms between the synthetic rubber matrix and the reinforcing fibers, which is critical for determining the overall strength and durability of the material.

[TABLE:1]

Furthermore, the integration of machine learning and deep learning techniques is beginning to revolutionize the field. These methods allow for the rapid screening of material configurations and the optimization of composite architectures without the exhaustive computational cost of traditional multi-scale simulations. By leveraging large datasets generated from high-fidelity RVE models, researchers can now predict the performance of synthetic rubber composites with unprecedented speed and accuracy.

scale mechanics of composite materials [ J ] . Chinese journal of solid

mechanics in Chinese LI Z Y LIU Z LEI Z et al. An innovative computational framework for the analysis of complex mechanical behaviors of short fiber rein- forced polymer composites . Composite structures AHMADI H HAJIKAZEMI M VAN PAEPEGEM W. Predicting the elasto-plastic response of short fiber reinforced composites u- sing a computationally efficient multi-scale framework based on physical matrix properties . Composites part b engineering

2023 ， 250 ： 110408. [ 8 ] ZHAO J ， GUO C J ， ZUO X B ， et al. Effective mechanical properties

of injection-molded short fiber reinforced PEEK composites using

periodic homogenization [ J ] . Advanced composites and hybrid ma-

GAO J. H., YANG X. X., HUANG L. H. Numerical prediction of mechanical properties of rubber composites reinforced by aramid fiber under large deformation. Composite Structures.

LIN Xiaoshan, YANG X. X., GAO Jianhong. Crack propagation analysis of short fiber reinforced rubber composites based on extended finite element method. Chinese Journal of Solid Mechanics.

tended finite element method [ J ] . Chinese journal of solid mechan-

Finite Element Analysis of Interfacial Mechanical Properties in Fiber-Reinforced Rubber Composites

LIU Xia, JIAO Wenxiang, YANG Xiaoxiang

Abstract

Fiber-reinforced rubber composites (FRRCs) are widely utilized in engineering applications due to their unique combination of high strength and high flexibility. The mechanical performance of these materials is significantly influenced by the properties of the interface between the fiber and the rubber matrix. This paper presents a finite element analysis (FEA) investigating the interfacial mechanical behavior of FRRCs. By employing representative volume elements (RVEs) and appropriate constitutive models for both the hyperelastic rubber matrix and the reinforcing fibers, we simulate the stress distribution and transfer mechanisms at the interface. The results provide insights into how interfacial bonding strength and fiber orientation affect the overall macroscopic mechanical response and failure modes of the composite.

1. Introduction

Fiber-reinforced rubber composites (FRRCs) are essential materials in various industrial sectors, including automotive tires, conveyor belts, and flexible hoses. These materials leverage the high tensile strength of fibers (such as nylon, polyester, or steel) and the large deformation capabilities of the rubber matrix. The interface serves as the critical region for load transfer between the soft matrix and the stiff reinforcement. Consequently, the integrity and mechanical characteristics of this interface determine the durability and reliability of the composite structure.

Despite their importance, the microscopic stress states at the interface are difficult to measure experimentally. Numerical methods, particularly finite element analysis (FEA), offer a robust framework for exploring these local phenomena. This study aims to establish a numerical model to evaluate the interfacial stress distribution and the impact of interfacial debonding on the global mechanical properties of FRRCs.

2. Constitutive Models and Numerical Methodology

2.1 Hyperelastic Model for Rubber Matrix

The rubber matrix exhibits significant non-linear elasticity and can undergo large strains. To describe this behavior, a hyperelastic constitutive model based on a strain energy density function $W$ is employed. For an incompressible material, the Mooney-Rivlin model is frequently used:

$$W = C_{10}(I_1 - 3) + C_{01}(I_2 - 3)$$

where $I_1$ and $I_2$ are the first and second invariants of the left Cauchy-Green deformation tensor, and $C_{10}$ and $C_{01}$ are material constants determined from experimental data.

LIU Xia ， JIAO Wenxiang ， YANG Xiaoxiang. Finite element analysis

Characterization and Finite Element Simulation of Interfacial Mechanical Properties of Short Aramid Fiber Reinforced Rubber Composites

Short aramid fiber reinforced rubber composites (SFRC) are widely used in engineering applications due to their high strength, high modulus, and excellent heat resistance. The mechanical performance of these composites is heavily dependent on the quality of the interface between the fibers and the rubber matrix. Effective stress transfer from the matrix to the fibers is essential for achieving the desired reinforcement effects. Therefore, characterizing the interfacial mechanical properties and understanding the failure mechanisms at the fiber-matrix interface are critical for the design and optimization of these materials.

In this study, we investigate the interfacial mechanical properties of short aramid fiber reinforced rubber composites through a combination of experimental characterization and finite element (FE) simulation. The interfacial bonding strength is evaluated using micro-mechanical testing methods, which provide the necessary parameters for numerical modeling. By establishing a representative volume element (RVE) or a single-fiber pull-out model, we simulate the stress distribution and the process of interfacial debonding.

The finite element analysis utilizes cohesive zone modeling (CZM) to describe the constitutive behavior of the interface. This approach allows for the prediction of crack initiation and propagation along the fiber-matrix boundary. The simulation results are compared with experimental data to validate the accuracy of the proposed model. Our findings indicate that the interfacial shear strength is a dominant factor in determining the overall tensile properties and fatigue life of the composite. Furthermore, the simulation reveals that stress concentrations at the fiber ends are the primary sites for initial debonding, which subsequently leads to the failure of the composite system.

By integrating experimental characterization with finite element simulation, this research provides a comprehensive framework for evaluating the interfacial integrity of short aramid fiber reinforced rubber. The results offer valuable insights into the micromechanical behavior of SFRC and serve as a theoretical basis for improving the interfacial adhesion through chemical treatment or processing adjustments.

niversity （ natural science edition ）， 2022 ， 50 （ 3 ）： 392-399 （ in Chi-

References

Tian Shaomeng, Yu Kejing, Xu Yang, et al. Effect of different types of short fibers on the properties of natural rubber/styrene-butadiene rubber composites. China Synthetic Rubber Industry (in Chinese).

Yu B. H., Ren J. B., Wang K. S., et al. Experimental study on the characterization of orientation of polyester short fibers in rubber composites by an X-ray three-dimensional microscope. Materials.

Hintze C., Shirazi M., Wiessner S., et al. Influence of fiber type and coating on the composite properties of EPDM compounds reinforced with short aramid fibers. Rubber Chemistry and Technology.

Fisher F. T., Bradshaw R. D., Brinson L. C. Fiber waviness in nanotube-reinforced polymer composites: modulus predictions using effective nanotube properties. Composites Science and Technology.

Fisher F. T., Bradshaw R. D., Brinson L. C. Effects of nanotube waviness on the modulus of nanotube-reinforced polymers. Applied Physics Letters.

Pantano A., Cappello F. Numerical model for composite material with polymer matrix reinforced by carbon nanotubes. Meccanica.

canica ， 2008 ， 43 （ 2 ）： 263-270. [ 19 ] NAFAR DASTGERDI J ， MARQUIS G ， SALIMI M. The effect of

...nanotube waviness on the mechanical properties of CNT/SMP composites. Composites Science and Technology. SHAO L. H., LUO R. Y., BAI S. L., et al. Prediction of effective moduli of carbon nanotube-reinforced composites: waviness and debonding. Composite Structures. HERASATI S., ZHANG L. C. A new method for characterizing and modeling the waviness and alignment of carbon nanotubes in composites. Composites Science and Technology. FAN Taotao, LI Xiaotuo, XIAO Wenkai. Simulation study on the influence of carbon nanotubes on the thermal conductivity of epoxy resin. Engineering Journal of Wuhan University.

Introduction

Carbon nanotubes (CNTs) have garnered significant attention in materials science due to their exceptional mechanical, thermal, and electrical properties. When integrated into polymer matrices, such as epoxy resins or shape memory polymers (SMPs), they offer the potential to create high-performance nanocomposites. However, the theoretical performance of these composites is often mitigated by practical structural factors, most notably the waviness and alignment of the nanotubes within the matrix.

1.1 Impact of Waviness and Debonding

Research by Shao et al. and Herasati et al. emphasizes that CNTs are rarely perfectly straight when embedded in a composite medium. This inherent "waviness" significantly alters the effective reinforcement provided by the nanotubes. Theoretical models must account for this geometric complexity, as well as the interfacial behavior between the CNTs and the polymer matrix. Specifically, the phenomenon of debonding—where the nanotube separates from the surrounding resin under stress—can lead to a marked decrease in the composite's effective moduli.

1.2 Thermal and Mechanical Characterization

Beyond mechanical reinforcement, the thermal properties of CNT-reinforced polymers are a critical area of study. Fan et al. utilized simulation techniques to investigate how the addition of carbon nanotubes influences the thermal conductivity of epoxy resins. Their findings suggest that while CNTs can form conductive networks, the efficiency of thermal transport is highly sensitive to the nanotubes' spatial distribution and orientation. Understanding these relationships is essential for developing new methods to characterize and model the alignment of CNTs, ensuring that the macroscopic properties of the composite can be accurately predicted and optimized for engineering applications.

1118 应用力学学报

FAN Taotao LI Xiaotuo XIAO Wenkai. Effect of carbon nanotubes on thermal conductivity of epoxy based on simulation method

Engineering journal of Wuhan University ， 2019 ， 52 （ 1 ）： 77-82 （ in

Chinese OGDEN R W. Nearly isochoric elastic deformations application to

rubberlike solids [ J ] . Journal of the mechanics and physics of sol-

Prediction of the Mechanical Behavior of Carbon Black-Filled Rubber Composites Based on Periodic Boundary Conditions

LI Qing, YANG Xiaoxiang
(College of Mechanical Engineering and Automation, Fuzhou University, Fuzhou 350108, China)

Abstract

To investigate the influence of the microstructure of carbon black-filled rubber composites on their macroscopic mechanical properties, a numerical simulation method based on periodic boundary conditions was developed. Representative Volume Elements (RVEs) containing randomly distributed carbon black particles were generated using a random sequential adsorption (RSA) algorithm. By implementing periodic boundary conditions through a set of linear constraint equations, the homogenization of the composite's mechanical properties was achieved. The simulation results demonstrate that the proposed model can effectively predict the effective elastic modulus and stress-strain behavior of carbon black-filled rubber. Furthermore, the effects of particle volume fraction and particle distribution on the macroscopic stiffness and local stress concentrations were analyzed. This study provides a theoretical basis for the microstructure design and performance optimization of rubber-based composite materials.

1 Introduction

Carbon black-filled rubber composites are widely utilized in the automotive, aerospace, and civil engineering industries due to their excellent mechanical properties, including high strength, wear resistance, and damping characteristics. The macroscopic mechanical behavior of these materials is significantly influenced by their microscopic constituents, such as the volume fraction, shape, and spatial distribution of the carbon black particles. Understanding the relationship between the microstructure and the macroscopic response is crucial for the design and application of high-performance rubber products.

Traditional analytical models, such as the Guth-Gold model and the Halpin-Tsai equations, provide simplified estimates for the effective properties of composites. However, these models often fail to account for complex particle interactions and the non-linear hyperelastic behavior of the rubber matrix at large deformations. With the advancement of computational power, finite element analysis (FEA) based on Representative Volume Elements (RVEs) has become a powerful tool for multiscale modeling.

A critical aspect of RVE-based simulation is the application of appropriate boundary conditions. Compared to prescribed displacement or traction boundary conditions, periodic boundary conditions (PBCs) generally provide faster convergence to the effective properties and more accurate results for smaller RVE sizes. This paper focuses on establishing a numerical framework to predict the mechanical behavior of carbon black-filled rubber using PBCs and investigating the impact of microstructural parameters on the material's overall performance.

2 Microstructure Modeling and Periodic Boundary Conditions

LI Qing ， YANG Xiaoxiang. Prediction on mechanical behavior of

carbon black filled rubber composites based on periodic boundary

conditions [ J ] . Acta materiae compositae Sinica ， 2013 ， 30 （ 6 ）：

References

Gao, J. H., Yang, X. X., & Huang, L. H. A numerical model to predict the anisotropy of polymer composites reinforced with high-aspect-ratio short aramid fibers. Advances in Polymer Technology.

Gao, J. H., & Yang, X. X. Experimental study on the dependent factors of mechanical properties of aramid short fiber reinforced rubber composites at large deformation. Journal of Fuzhou University (Natural Science Edition).

Preparation and application of maleic anhydride grafted natural rubber. [Dissertation]. Guangzhou: South China University of Technology.

Wang, J., Wu, W. D., Zhou, H. F., et al. Influence of short fiber on the property of SBR composite. New Chemical Materials.