# A hybrid approach for predicting fatigue life of fiber-reinforced polypropylene composite (PPGF40): integrating micromechanical modelling

Mohammadali Shirinbayan, Samia Nouira, Jihed Zghal, Joseph Fitoussi

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Mohammadali Shirinbayan, Samia Nouira, Jihed Zghal, Joseph Fitoussi. A hybrid approach for predicting fatigue life of fiber-reinforced polypropylene composite (PPGF40): integrating micromechanical modelling. Composites Part C: Open Access, 2025, 18, pp.100660. 10.1016/j.jcomc.2025.100660. hal-05307179

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Submitted on 10 Oct 2025

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# A hybrid approach for predicting fatigue life of fiber-reinforced polypropylene composite (PPGF40): Integrating micromechanical modelling

Mohammadali Shirinbayan<sup>a, *</sup>, Samia Nouira<sup>b</sup>, Jihed Zghal<sup>c, ID</sup>, Joseph Fitoussi<sup>a</sup>

## ARTICLE INFO

### Keywords:

Short fiber-reinforced polypropylene

Fatigue

Damage

Micromechanical modeling

## ABSTRACT

This paper presents a hybrid approach for predicting the fatigue life of PPGF40. The approach combines micromechanical modeling with empirical techniques, based on an intrinsic relationship. Micromechanical modeling is used to analyze the material's monotonic behavior. The study presents a micromechanical model, based on Mori and Tanaka's approach, for simulating damage at the fiber-matrix interface. The model incorporates a local criterion and linearizes the plastic behavior of the matrix using the secant modulus method. The model parameters are identified by comparing them with experimental stiffness reduction results, and S-N curves for different modeled orientations (0°, 45°, and 90°) are presented. The study concludes by establishing the Tsai-Wu fatigue failure criterion based on hybrid modeling results, demonstrating its usefulness in designing structures such as tailgates. The versatility of the micromechanical model extends to other microstructures upon validation. This methodology provides a framework for linking process, microstructure, and properties, and can be coupled in the future with microstructure prediction tools, such as Moldflow, to support fatigue optimization in PPGF40 and similar materials.

## 1. Introduction

In the design of short-fiber composite materials for both monotonic and cyclic loads, industry professionals must provide relevant sizing for each component in accordance with customer specifications. In the automotive sector, such as for tailgates, these specifications typically include a cycle count (N) for opening and closing operations, a temperature usage range from -30 °C to 80 °C, and standard loading profiles [1-3].

Automotive equipment manufacturer study offices widely adopt structured methodologies for optimal sizing. The task involves designing and pre-sizing the component, selecting the material/process formulation, simulating the manufacturing process (injection, compression) using software like MoldFlow to estimate the spatial distribution of the microstructure, conducting a characterization test campaign in monotonic and fatigue, simulating a fatigue life cycle using calculation software such as Abaqus to deduce the maximum stress, verifying the

material strength at the given stress on the Wöhler curve, and finally, optimizing the design [4].

However, study offices are confronted with the challenge of repeating these steps for each new project or modification, particularly in the case of experimental characterization campaigns. The campaigns above are frequently arduous and expensive, yet indispensable for establishing material laws and rupture criteria. A minor alteration in the design of the composite structure can result in a change in the distribution of the microstructure induced by the process, which in turn affects its mechanical response. In this context, it is essential to establish sizing methodologies in monotonic and fatigue that are both rapid, pragmatic, and capable of identifying the 'material map', which comprises the behavior law and rupture criterion. These methodologies must be compatible with a decoupled approach, ensuring engineers' ease of use, regardless of the induced microstructures. This paper proposes pragmatic and user-friendly methods capable of predicting the composite's lifespan based on its microstructure.

<footnote><sup>a</sup> Arts et Metiers Institute of Technology, CNAM, CNRS, PIMM, HESAM University, F-75013 Paris, France</footnote>

<footnote><sup>b</sup> Université de Lorraine, Laboratoire LEM3 UMR 7239, Metz F-57045, France</footnote>

<footnote><sup>c</sup> Laboratoire Energétique Mécanique Electromagnétisme (LEME), France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail addresses: mohammadali.shirinbayan@ensam.eu (M. Shirinbayan), samia.nouira@univ-lorraine.fr (S. Nouira), jzghal@parisnanterre.fr (J. Zghal), joseph.fitoussi@ensam.eu (J. Fitoussi).</footnote>

<footnote>https://doi.org/10.1016/j.jcomc.2025.100660</footnote>
In the case of fiber-reinforced composites, damage is observed from the initial fatigue cycles, with the size of the damaged zones increasing throughout the test. The nature of the damage may evolve, for example, from minor matrix cracks to delamination and fiber breakage. To predict the state of damage and the number of cycles to failure, it is necessary to model and understand the complete damage path. This is because the gradual deterioration during fatigue causes a continuous redistribution of stress and a reduction in stress concentrations within the material. Therefore, it is important to have a clear understanding of the complete damage path. Fong [5] explains that damage modeling is challenging due to two technical reasons. Firstly, there are different scales of damage, ranging from the atomic to the specimen scale. Secondly, it is impossible to reproduce the microstructure of the specimens identically.

Several studies have been conducted to extend the case of constant uniaxial loading to more complex loading, such as block type and spectrum loading.

Classifying the various existing models in a rigorous manner is proving to be a challenging task. A representative classification can be made based on Sendeckyj's [6] fatigue criteria. Sendeckyj's criteria can be categorized into four groups: macroscopic strength criteria, criteria based on residual strength, criteria that can be classified on residual stiffness, and criteria based on the damage mechanism.

Other authors have classified existing models into three categories.

* The first category includes models that do not consider damage mechanisms but use S-N curves or Goodman diagrams with a fatigue failure criterion.

* The second category includes phenomenological residual stiffness/ residual strength models.

* The third category includes progressive damage models that use one or more variables linked to damage measurements on the material, such as transverse cracking of the matrix or delamination size.

While the fatigue behavior of composite materials differs from that of metals, several models are still based on S-N curves [7]. These models fall under the category of fatigue life models, which require a significant amount of experimental work and do not consider damage mechanisms such as matrix cracking or fiber breakage. The second category of models is known as phenomenological residual stiffness or residual strength models. These models propose an evolutionary law that describes the gradual deterioration of the composite's stiffness or strength in terms of the observed macroscopic properties. Unlike the third category of models, where the damage law is directly linked to the damage phenomenon, residual stiffness models are based on the degradation of residual stiffness during fatigue. Residual stiffness can be measured frequently throughout the fatigue test without reporting additional degradation. The model can be deterministic or statistical. The deterministic model predicts a single stiffness value, assuming it is the same throughout the material. The statistical model predicts a stiffness distribution. The other approach, residual strength, is based on the remaining strength of the material. In a multitude of composite material applications, it is of paramount importance to ascertain the residual strength of the structure. This information is essential for estimating the lifespan of the structure under external loading. The models were developed with this objective in mind, namely to describe the deterioration of initial strength during fatigue. Since their inception, these models have been based on statistical principles. Typically, two Weibull functions are employed to delineate the residual strength and probability of failure for laminates following a specified number of cycles.

For example, as Joeun Choi et al. [8], PP fiber reinforcement exhibited crack propagation due to fiber failure or pullout under fatigue loading, with ductile fracture behavior decreasing as the fiber volume fraction was reduced. These mechanisms are similarly relevant to PPGF40, where the glass fiber reinforcement significantly influences fatigue behavior, particularly regarding fiber failure and pullout driving crack propagation. Furthermore, Selmy et al. [9] investigate the number

of cycles to failure for the composite laminates under varying flexural loads and they contribute to understanding how the structure and material properties affect the fatigue performance of glass fiber-reinforced composites [10].

In terms of fatigue life prediction, models such as Manson-Coffin and Smith Watson Topper (SWT) have been widely used to describe the fatigue behavior of fiber-reinforced materials under complex loading conditions [11,12]. For example, Branco et al. [13] proposed a modified Manson-Coffin model using the equivalent strain energy density concept to predict fatigue life under such conditions. These models help to capture the anisotropic effects of fiber orientation [14,15], which is critical for accurate fatigue life estimation in materials such as PPGF40.

Moreover, recent advances have demonstrated the potential of multi-scale modeling approaches to reduce testing requirements and improve fatigue life predictions for fiber-reinforced composites. Tang et al. [16] developed a multi-scale fatigue model for chopped carbon fiber-reinforced composites under tension-tension loading, showing how fiber orientation and microstructural variability influence fatigue life. In a complementary study, they analyzed [17] the effect of fiber orientation distributions on constant fatigue life diagrams for sheet molding compounds (SMC), highlighting the importance of integrating fiber orientation data into fatigue modeling strategies. These studies underscore the value of microstructure-informed modeling for accurate fatigue predictions, motivating the present work on short glass fiber-reinforced thermoplastics.

Given the extensive research conducted on the fatigue behavior of composite fiber materials in recent decades, a final model category has been proposed.

This model describes the deterioration of the composite material directly through specific damage mechanisms, such as transverse cracking of the matrix or delamination size.

These models correlate one or more measured damage variables to a mechanical property of the material, thus representing a quantitative progression of the current state of damage. They are often referred to as mechanistic or micromechanical models [18].

Therefore, fatigue life prediction models can be divided into three categories: empirical fatigue models, residual stiffness or residual strength phenomenology models, and micromechanical or mechanistic models. The objective of these models is to predict fatigue life. Each model uses its criterion to determine the moment of final failure and, consequently, the fatigue life.

Fatigue life prediction models utilize information from S-N curves or Goodman diagrams and introduce a failure criterion that determines fatigue life. Regardless of the characterization of the S-N curve and the fatigue behavior of the material, Sendeckyj [6] recommends considering the following assumptions:

* The behavior of the S-N curve can be described by a deterministic equation.

* Static strength is not related to service life or end-of-life residual strength.

The fundamental principle underlying residual strength models is the existence of a clear failure criterion. This criterion states that failure will occur when the applied stress reaches the level of residual strength [19]. In the residual stiffness approach, fatigue failure is defined as the degradation of the modulus to a critical level, which is determined through experimentation.

The relationship between damage accumulation models and fatigue life prediction methodologies is a close one, as the fatigue life of a given material can be predicted by establishing a failure criterion based on the accumulated damage. The critical damage variable for each type of damage is identified through empirical investigation.

In this context, this article focuses on the micromechanical modeling of the monotonic behavior of PPGF40, taking damage into account. The modeling approach is defined based on experimental results and
associated multi-scale analyses. The micromechanical model used, how damage is considered, and how the non-linear behavior of the matrix is linearized using the secant moduli method are exhaustively described in this paper. In addition, our proposed hybrid method is novel in that it couples micromechanical, phenomenological, and analytical models, allowing for a more accurate and robust estimation of fatigue damage. A key experimental result guiding this approach is the direct relationship between fatigue life and the initial decrease in stiffness, serving as an early indicator of damage progression. By incorporating this critical insight, our method enhances predictive capability, offering a significant improvement over existing models.

## 2. Description of material and experimental methods

### 2.1. Material

The material used, PPGF40, is composed of a polypropylene base reinforced with 40 % short glass fibers by weight. The manufacturing process involves mixing polypropylene granules with short glass fibers, followed by injection molding to produce rectangular plates measuring 400 × 320 × 3 mm³, following ASTM D638 standards for specimen dimensions. Tensile test specimens were prepared using water-jet cutting from the injection-molded plates. A total of 3 plates were fabricated, from which 15 specimens were extracted for testing. These specimens were cut from the central region of the plates to ensure a generally uniform microstructure. To investigate the material's anisotropy caused by fiber orientation, samples were cut in three directions: 0° (longitudinal), 90° (transverse), and 45° relative to the injection direction. Fig. 1 illustrates the specimen geometry, designed to achieve uniform stress distribution throughout the gauge section.

### 2.2. Methods of mechanical characterization

#### 2.2.1. Loading-unloading

To examine the development of macroscopic damage and plasticity under quasi-static conditions, load-unload tests were performed at different maximum stresses using an MTS servo-hydraulic machine. The tests were performed at various maximum stresses and a displacement speed of 2 mm/min, equivalent to a strain rate of 4 × 10⁻⁴ s⁻¹. For each test, the sample undergoes eight to ten load-unload cycles with a gradually increasing stress range of approximately 10 % of the ultimate tensile stress (Su), as shown in Fig. 2. This procedure enables the plotting of stress-strain curves for the three tested orientations (0°, 45°, and 90°). The residual strain at zero stress provides an assessment of macroscopic apparent residual plastic deformation. The Young's modulus is determined by calculating the slope of the curve in the quasi-linear region of each cycle. The evolution of the Young's modulus throughout the test serves as an indicator of macroscopic damage.

#### 2.2.2. Fatigue test

Fatigue tests were carried out on an MTS 830 hydraulic fatigue machine (MTS Systems Corporation, Eden Prairie, MN, USA), specifically designed for load-controlled fatigue testing. The machine was equipped with a 10 kN load cell, and tension-tension load-controlled fatigue tests were conducted at varying maximum loading levels (Fmax). The minimum applied force (Fmin) was always set at 10 % of the

maximum applied force. A force-controlled sinusoidal signal was applied with a constant amplitude and a fixed load ratio R of 0.1. The main test frequency was set at 10 Hz, corresponding approximately to a deformation rate of 0.25 s⁻¹. Fatigue tests are preceded by an initial cycle of load-unload-reload at the applied fatigue stress to determine the first stiffness decrease (E₁/E₂). This initial loss of stiffness can also be determined through monotonic load-unload tests. Our fatigue machine typically reaches the setpoint, i.e. the desired amplitude, after 22 cycles, which corresponds to the stabilization period, as shown in Fig. 3. The stabilization process duration depends on several factors, such as machine power, material resistance, and applied frequency. The fatigue behavior of the three tested orientations (0°, 90°, and 45°) was studied at a frequency of 10 Hz by analyzing the evolution of fatigue stiffness loss (Eₙ), apparent plastic deformation, and self-heating throughout the tests. The dynamic modulus for each cycle (Eₙ) was calculated using the peak-to-peak method.

<figure><img src="image_2.png" /><figcaption>Fig. 2. Experimental methodology for load-unload tests.</figcaption></figure>

## 3. Presentation of the hybrid approach

Various models are available for predicting fatigue life, which can be categorized into analytical, phenomenological, and micromechanical approaches. Analytical S-N approaches aim to establish a connection between applied stress and the number of cycles to failure. Phenomenological approaches focus on the macroscopic damage suffered by the material, which results in a loss of stiffness, strength, or both.

<figure><img src="image_3.png" /><figcaption>Fig. 3. Cycle count required to stabilize setpoint.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 1. Fatigue specimen geometry.</figcaption></figure>
Micromechanical or progressive approaches, based on observations of damage mechanisms, propose establishing a micro damage variable directly linked to the material's failure. In a multiscale approach, micromechanical methods are used to link the experimentally identified damage variable to a macroscopic property of the material, typically stiffness. This approach was initially proposed by Highsmith and Reifsnider [20] and has since been widely adopted by other authors, including Reifsnider and Gao [21], Jain et al. [22,23], Le Pen et al. [24], Guo et al. [25,26], Fitoussi et al. [27,28], and Derrien et al. [29]. Therefore, this approach involves coupling phenomenological and micromechanical methodologies. By understanding the microstructure of the material and the primary damage mechanism(s) that lead to failure, the material laws of the composite in question can be determined.

The analysis of damage mechanisms confirmed that decohesion of the fiber-matrix interface leads to the creation of microcracks and their subsequent coalescence, resulting in the formation of a macrocrack and material failure. Jendi et al. [30] proposed this damage scenario to model the monotonic behavior of composite materials. The authors incorporated this scenario into a Mori-Tanaka homogenization approach, establishing a quadratic failure criterion at the fiber-matrix interface associated with a Weibull probabilistic law.

The approach to damage in this study is based on micromechanics. The homogenization model gradually introduces several families of heterogeneity, including undamaged fibers that fully contribute to the material's stiffness, microcracks generated by the activation of the statistical interfacial failure criterion, and debonded fibers that still partially contribute to the material's stiffness. This work proposes using the approach employed by Jendli et al. [31] to predict the monotonic behavior of PPGF 40. However, in this case, the behavior of the thermoplastic matrix is defined step by step through a linearization procedure. Unlike the polyester matrices studied by Jendli et al. [30], whose behavior is linear and brittle, the polypropylene matrix exhibits nonlinear behavior due to plasticity. Therefore, linearization of the latter is crucial for the homogenization approach.

A mean-field approach with a secant formulation, initially proposed by Hutchison and subsequently adopted by Berveiller and Zaoui [32], is suggested as a linearization method. This approach describes the matrix's behavior as a succession of elastic states during loading, making it suitable for the iterative homogenization proposed.

The analysis of fatigue damage mechanisms has shown that the first fatigue phase (for $n < 1000$ in our case) is crucial in the development of damage mechanisms, as demonstrated by shirinbayan et al. [33,34]. It has been observed that most of the damage occurs for several cycles equal to 10% of the lifespan. During this initial phase, a significant loss of rigidity is evident. Therefore, the lifespan is determined by the very early stages of damage. This proposal aims to examine the correlation between the initial loss of rigidity due to fatigue, denoted as $E_1/E_0$, and the fatigue lifespan. To achieve this, the evolution of the macroscopic damage parameter at the first cycle, represented as $D_1 = 1 - E_1/E_0$, is plotted against $N_r$ for various amplitudes and three orientations (0°, 45°, and 90°) at a frequency of 10 Hz, as illustrated in Fig. 4.

On this curve, the relationship between the damage parameter $D_1$ and $N_r$ can be considered unique, regardless of the fiber orientation.

$$D_1 = a \times N_r \qquad (1)$$

The $D_1$ versus $N_r$ curve can be considered a 'master' curve that provides fatigue life for a given amplitude, regardless of microstructure. This is achieved through knowledge of the stiffness reduction caused by a cycle applied at that amplitude. This approach is similar to that proposed by Jain et al. (23), which associates the macroscopic damage state under quasi-static load with that reached at a certain point on the S-N curve. The methodology proposed by the authors predicts S-N curves for other composite microstructures based on the 'master curve' corresponding to the established S-N curve for a specific reference microstructure. This method has already been validated and published in a

<figure><img src="image_5.png" /><figcaption>Fig. 4. $D_1$ as a function of $N_r$ for the three study orientations.</figcaption></figure>

previous study on SMC composites. It combines and leverages the three approaches of micromechanics, phenomenology, and empiricism. Therefore, predicting the evolution of stiffness reduction for various stress values of a given microstructure allows for easy deduction of the associated Wöhler curve. In other words, simulating the behavior under monotonic loading for a given microstructure provides access to the fatigue life through the 'master curve'. The following paragraph explains the micromechanical model used for this purpose.

## 4. Micromechanical modeling

The micromechanical model utilizes the Mori-Tanaka approach to derive the macroscopic properties of the equivalent homogeneous material (MHE) from a known volume fraction (VER) of the heterogeneous material. This approach is based on the microstructure of the VER and the properties of the different phases, allowing for the derivation of the stiffness tensor of the equivalent homogeneous material and the mean stress and strain fields for each phase. The model determines the fraction of debonded fibers for each increment of applied load using a local damage criterion at the fiber-matrix interface. To account for interfacial damage, the model progressively introduces microcrack density, represented by zero stiffness heterogeneities in a penny shape. The local criterion also results in a reduction in the volume fraction of active fibers that contribute to the composite's stiffness. The observed fiber-matrix interface damage is reproduced step by step, including undamaged fibers and a proportion of debonded fibers. The matrix behavior is linearized using a secant approach type. A detailed description of this micromechanical approach is provided below.

The equations provided enable the calculation of composite stiffness and mean stress fields in various phases. The stresses at the interface between the fibers and matrix can be derived as described below. The model assumes isotropic properties for both the matrix and fibers. Input data for the model includes volume fraction, average length, and orientation distribution of the reinforcements. The composite stiffness tensor can be calculated using the following expression:

$$C^{\text{comp}} = [I + f(Q)(I + f(S - I)Q)]^{-1} \qquad (2)$$

Where $\langle \rangle$ denotes the average, $C^m$ and $S$ represent the stiffness tensor of the matrix and the Eshelby tensor, respectively. $Q$ is a tensor known as 'pseudo-localization', which is defined for each fiber orientation family '0'.

$$Q^{\theta} = ((C^m - C^{\theta})(S^{\theta} - I) - C^{\theta})^{-1}(C^{\theta} - C^m) \qquad (3)$$

$C^\theta$ represents the stiffness tensor of the reinforcement families oriented at $\theta$° (refer to Fig. 5). It is important to note that in relatively thin
<figure><img src="image_6.png" /><figcaption>Fig. 5. Fiber-matrix interface constraints [35].</figcaption></figure>

composites, all fibers are assumed to be contained in the plane of the sheet. As a result, the representation can be simplified by considering that each fiber family is only characterized by its orientation in the θ plane.

For a macroscopic applied stress σimp, the tensor of average stresses experienced by a family of fibers oriented at θ is given by:

$$ \sigma^{\theta} = (I + (S^{\theta} - I))(I + f(S - I)Q)C^{m-1}\sigma^{imp} \quad (4) $$

By considering the continuity condition of normal stress at the interface between the fiber and matrix, we can deduce its expression. The normal stress at the interface takes the following form depending on the average stress in the fiber.

$$ \sigma_n = \vec{T}^{-} * \vec{n}^{-} \quad (5) $$

where $\vec{T}^{-}$ is the stress vector at the fiber-matrix interface point with a normal vector $\vec{n}^{-}$

The normal and shear interfacial stresses, σn and τ, at each point on the fiber-matrix interface (Fig. 6) are calculated using the average stress in the fibers for each orientation. It is important to note that the interfacial stresses σn and τ are calculated at each point on the interface defined by the angle φ for each orientation θ.

Fitoussi et al. [36] proposed a fracture criterion for the fiber-matrix interface based on the calculation of local normal and tangential stress fields. They then proposed a local debonding criterion at the interface, which is of the Coulomb type and couples the normal and tangential stresses through a linear function as shown in the equation:

$$ \sigma + \beta\tau = R \quad (6) $$

The variable R represents the strength of the interface, while β is a coupling parameter. Jendli et al. [31] proposed a second formulation, which is quadratic and expressed in the following form:

$$ \left(\frac{\sigma}{\sigma_0}\right)^2 + \left(\frac{\tau}{\tau_0}\right)^2 = 1 \quad (7) $$

The parameters σ0 and τ0 correspond to the normal and tangential stresses at the interface failure, respectively. To account for the statistical aspect of interface failure associated with local variations in microstructure, we can define a Weibull-type probability of interface

failure at each fiber-matrix interface point defined by the fiber orientations θ and φ, as shown in Fig. 4.

$$ (\theta, \varphi) = 1 - \exp\left(-\left(\left(\frac{\sigma}{\sigma_0}\right)^2 + \left(\frac{\tau}{\tau_0}\right)^2\right)^m\right) \quad (8) $$

where m is a statistical parameter related to the microstructure of the material. σ and τ are the normal and shear stresses respectively, which depend on the applied macroscopic stress, the volume fraction, the fiber orientation considered and the elastic properties of both the matrix and the fibers.

Thus, for each load increment, the local stresses at the interface are evaluated and the proportion of fibers affected by interface damage is assessed for each fiber orientation.

At the local scale, when a fiber experiences interface damage, two effects may occur, the presence of a crack around the fiber, and a reduction in the fiber's contribution to the composite's overall stiffness.

Micromechanical modelling can represent the first effect by introducing a plate-like ellipsoidal heterogeneity, which has a shape similar to that of a coin or 'penny shape'. The second effect can be modelled by reducing the volume fraction of fibers in the considered orientation family.

Therefore, once the interfacial break probability is calculated for each fiber family, the damaged state at each calculation step n can be defined as a microstructure containing:

- Good fibers (undamaged at their interface) (denoted as $f_n^{nd}$)

- Active fibers (denoted as $f_n^{act}$) that still contribute to the stiffness of the composite, including two populations: undamaged fibers (denoted as $f_n^{nd}$) and a residual fraction of fibers that have undergone interface damage but still partially contribute to the reinforcement of the composite. Their contribution is defined by a reduction coefficient "k" applied to the number of fibers debonded at their interface [36].

- Microcracks (denoted as $f_n^{mc}$).

However, at each calculation step n, the determination of the local interface fracture probabilities allows the local damage state to be described by the following equations:

$$ f_n^{nd} = (1 - P_r^n) + f_{n-1}^{nd} \quad (9) $$

$$ f_n^{act} = \left( k \sum_{i=1}^{n} P_r^i \right) + f_n^{nd} \quad (10) $$

$$ f_n^{mc} = (h \cdot P_r^n f_{n-1}^{nd}) + f_{n-1}^{mc} \quad (11) $$

The reduction parameter 'k' was evaluated through finite element calculations on a representative cell that contained a partially debonded fiber. The value of 'k' falls between 0.3 and 0.5. Additionally, 'h' represents the ratio of the volume of the introduced 'penny shape' with zero stiffness (representing an interfacial microcrack) to the volume of the debonded fiber. This parameter is evaluated geometrically.

This micromechanical description of the damage state can be

<figure><img src="image_7.png" /><figcaption>Fig. 6. Stresses exerted on the fiber as a function of its orientation, θ (°), concerning the loading direction.</figcaption></figure>
realized independently of the applied load: monotonic (quasi-static or dynamic) or fatigue.

The local stresses at the fiber-matrix interface involved in the damage criterion (Eq (12)) also depend on the properties of the matrix. Since the matrix exhibits non-linear behavior, it is necessary to account for it incrementally. An incremental approach using secant moduli was proposed by Qiu & Weng [37] in the 1990s. It is advantageous to describe the plasticity (nonlinearity) of the matrix as a series of elastic states during loading of the composite (Fig. 7). Thus, the homogenization model (here Mori-Tanaka) provides the stress state of the matrix at each calculation step. Qiu & Weng [37] suggest using the generalized Ludwik's law of behavior for combined stresses.

This simple law is represented by the following equation in terms of Von Mises effective stress and strain:

$$ \sigma = \sigma_y + c \cdot (\epsilon_p)^n \quad (12) $$

Where $\epsilon_p$ is the plastic strain in terms of Von Mises, $\sigma_y$ is the yield stress, and c and n are directly identified from the stress-strain curves performed on the matrix alone.

The secant modulus of the matrix $E_s^i$, as well as the Poisson's ratio $\nu_s^i$, are defined elsewhere by the following equations:

$$ E_i^s = \frac{1}{\frac{1}{G} + \frac{\epsilon_p}{\sigma_y + h(\epsilon_p)^n}} \quad (13) $$

$$ \nu_i^s = \frac{1}{2} - \left(\frac{1}{2} - \nu_0\right) \frac{E_i^s}{E_0} \quad (14) $$

The linearization method is schematized below:

At each step of monotonic loading, the stress and strain states of the matrix are determined, and secant moduli are calculated to redistribute local stresses. The non-linearity of the matrix influences the local stresses at the fiber-matrix interface, resulting in a strong coupling between damage and plasticity at the local scale. It is important to note that the secant modulus method is only valid when unloading is not simulated, which is the case in our application.

## 5. Different steps of the micromechanical model

Let 'i' represent the calculation step. Based on this increment, the model can be organized into the following steps:

* For $i = 1$, consider undamaged material and elastic matrix.

Homogenize and estimate the overall material stiffness tensor from the properties of the different constituents using the mean field approach proposed by Mori and Tanaka, coupled with Eshelby's equivalent inclusion model. This section presents the estimation of the local stress field around the inclusions, identifying the normal and tangential components of the interfacial stress.

<figure><img src="image_8.png" /><figcaption>Fig. 7. The linearization method of matrix behavior.</figcaption></figure>

Additionally, it includes the calculation of the fracture criterion at the fiber-matrix interface, as well as the quantification of the volume fraction of debonded fibers, microcracks, and active fibers.

The calculation of $E_s$ is also included.

* For $i > 1$, the damaged material and matrix are represented by $E_s$.

- The first homogenization is performed between the matrix and microcracks.

- The second homogenization step should consider the fibers and calculate local stresses at the fiber-matrix interface and in the matrix. The fracture criterion at the fiber-matrix interface should be calculated, and the volume fraction of debonded fibers and microcracks should be quantified.

- Finally, calculate $E_s$.

For $i > 1$, homogenization should be performed in two steps, and this process should continue until the last increment is reached. The following paragraph presents the results of the simulation of the monotonic tensile behavior. These results are then utilized to predict the S-N curves for various reinforcement orientations.

## 6. Results and discussions

### 6.1. Modelling plasticity alone

The secant modulus evolution of the matrix is evaluated based on the tensile curve obtained at a speed of 4.10–4 s-1, which corresponds to the load-unload-reload tests performed before each fatigue test. The Mori and Tanaka model is then integrated, and a simulation of the tensile curve is performed for each orientation (refer to Fig. 8). These simulations do not initially simulate any damage to highlight the effect of matrix non-linearity.

The orientation of the matrix significantly affects its plasticity. Specifically, at 90° (T-direction), the orientation reflects the material's non-linear behavior, while at 0° (longitudinal direction), damage is responsible for the non-linearity.

### 6.2. Modeling of damage at the fiber-matrix interface

The damage criterion parameters ($\sigma_0$, $\tau_0$, m, h, and k) were determined using an inverse method based on the results at 0 and 90°, as detailed in Ayari's work [38]. Subsequently, they were validated for the 45° orientation. Plasticity is also considered in these simulations using the secant modulus method. Fig. 9 presents a comparison of the simulated stiffness reduction with the experimental results.

We observe a satisfactory agreement between the experimental results and those derived from the model. Therefore, the model allows us to predict the loss of stiffness independently of the microstructure.

### 6.3. Fatigue life prediction

In Fig. 10, the first measured loss of stiffness during the initial cycle has been plotted as a function of fatigue life (N$_r$) in passive voice. A direct relationship has been established between damage caused by monotonic loading and fatigue life under cyclic loading. This relationship appears to be independent of microstructure. It is worth noting that similar observations have been made for other materials, such as several families of SMC and PA66GF30.

The approach for predicting S-N curves used a hybrid method. Interfacial decohesion, the dominant phenomenon under monotonic loading and fatigue, causes the material's macroscopic damage.

The methodology for predicting fatigue life, as shown in Fig. 10, involves the following steps:

Firstly, identify the damaging elastoplastic model by simulating curves that illustrate the evolution of macroscopic damage as a function of applied stress (D, $\sigma$) for the chosen microstructure. Secondly, model
<figure><img src="image_9.png" /><figcaption>Fig. 8. Simulation results of the traction curves of PPGF40 taking into account the plasticity of PP.</figcaption></figure>

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 9. Results of the micromechanical simulation of damage under monotonic tensile loading.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 10. Diagram of the hybrid methodology predicting fatigue life [33].</figcaption></figure>

the monotonic loading tensile test for any other microstructure.

The value of macroscopic damage $D_s$ can be read for a stress $\sigma = S$ on the modeled curve ($D_s, \sigma$).

- The number of cycles to failure $N_r$ can be determined directly from the curve ($D_1, N_r$). Therefore, record the pair of values ($S, N_r$).

Repeat these steps for other stress values $S$ to establish the S-N curve for the chosen microstructure.

The methodology has been validated by comparing it with experimental results of S-N curves for the PPGF40 at orientations of 0°, 45°, and 90°. Furthermore, S-N curves have been modeled for orientations of 15°, 30°, and 60°, as shown in Fig. 11.

## 6.4. Fatigue failure criterion

Calculating a fatigue failure criterion is essential for any structural analysis. This discussion will focus on the Tsai-Wu failure criterion, which is easily implementable and pre-implemented in commercial computational codes. The mathematical equation for the Tsai-Wu criterion for 2D loading is as follows:

$$F_1\sigma_{11} + F_2\sigma_{22} + F_{11}\sigma_{11}^2 + F_{22}\sigma_{22}^2 + F_{66}\sigma_{12}^2 + 2F_{12}\sigma_{11}\sigma_{22} < 1 \quad (15)$$

$F_1, F_2, F_{11}, F_{22}, F_{66}$, and $F_{12}$ are material parameters that require evaluation based on the number of cycles. For simplicity, we assume equal rupture stresses in tension and compression. The Tsai-Wu coefficients are defined as follows:

$$F_1 = F_2 = 0, F_{11} = -\frac{1}{\bar{X}^2}, F_{22} = -\frac{1}{\bar{Y}^2}, F_{66} = -\frac{1}{S^2} \quad (16)$$

Where $X, Y$ are respectively the tensile failure stresses in direction 1, 2 and S the shear failure stress in the plane. In the case of a tensile stress $\sigma_X$ applied in the x direction, the Tsai-Wu criterion leads to the following expression

$$\sigma_X < \left[ \frac{1}{\frac{\sigma_X^4}{x^2} + \frac{\sigma_Y^4}{y^2} + c^2 S^2 \left( \frac{1}{S^2} + 2F_{12} \right)} \right]^{1/2} \quad (17)$$

The symbols c and s represent cos($\theta$) and sin($\theta$) respectively, where $\theta$ is the angle between the main direction of the fibers and the local axis of the material. X and Y correspond to the values of $\sigma_X$ for $\theta = 0°$, and $\theta = 90°$. Therefore, for a given number of cycles to failure $N_r$, these values can be directly read from the S-N curves of the different orientations presented in Fig. 10.

The values of S and $F_{12}$ need to be determined. This can be achieved by plotting the $\sigma_X$ curves against $\theta$ at different numbers of cycles to rupture $N_r$, as shown in Fig. 11.

The values of $\sigma_X$ can be directly deduced from Fig. 10, which represents the different S-N curves. The value of S is initially determined by assuming that $F_{12}$ is zero, following the Tsai-Hill criterion. Subsequently, the value of $F_{12}$ is determined using the Tsai-Wu criterion by fitting the curves presented in Fig. 12.
<figure><img src="image_13.png" /><figcaption>Fig. 11. Experiment-simulation comparison on S-N curves for all orientations.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Fig. 12. Evolution of $\sigma_x$ depending on the orientation ($\theta$) for different $N_r$.</figcaption></figure>

Once the four parameters are identified for the various targeted numbers of cycles, the Tsai-Wu criterion can be plotted as a function of the number of cycles $N_r$, as shown in Fig. 13.

7. Conclusion

This work demonstrates that fatigue life in PPGF40 can be reliably

<figure><img src="image_15.png" /><figcaption>Fig. 13. Tsai-Wu criterion for different numbers of cycles.</figcaption></figure>
predicted by linking early stiffness degradation—specifically, the initial loss of modulus—to long-term cyclic performance via a hybrid, multi-scale framework. Through extensive experiments varying fiber orientation, loading frequency, strain rate, and thermal conditions, we confirmed that fiber–matrix interfacial decohesion is the dominant damage mechanism initiating fatigue, and that the proposed Mori–Tanaka-based micromechanical model, augmented by a statistical local failure criterion and stepwise matrix plasticity via a secant modulus formulation, can reproduce both stiffness loss and S–N curves across multiple orientations with commendable accuracy. The calibrated Tsai–Wu fatigue failure criterion derived from these simulations provides a practical design tool for composite structures, significantly reducing the need for exhaustive fatigue testing. Beyond the specific case of PPGF40, this methodology is transferable: it can be adapted to other microstructures and integrated with process-simulation tools to capture fiber orientation or length distributions and thereby optimize the chain from manufacturing process, through actual microstructure, to fatigue performance. In future work, the framework will be extended to address variable-amplitude loading, environmental effects (temperature, humidity), and mixed mode fatigue, in order to reinforce its industrial relevance and applicability.

## Consent to participate

Not applicable.

## Consent to publish

Not applicable.

## Availability of data and materials

The authors declare that the data and the materials of this study are available within the article.

## CRediT authorship contribution statement

**Mohammadali Shirinbayan:** Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Project administration, Methodology, Formal analysis. **Samia Nouira:** Writing – review & editing, Writing – original draft, Visualization, Validation, Formal analysis. **Jihed Zghal:** Writing – review & editing, Writing – original draft, Visualization, Supervision, Formal analysis. **Joseph Fitoussi:** Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Project administration.

## Declaration of competing interest

The authors declare that they have no conflicts of interest.

## Data availability

No data was used for the research described in the article.

## References

[1] Flexural fatigue and failure probability analysis of polypropylene-glass hybrid fibres reinforced epoxy composite laminates - Marwa [Internet]. [cited 2024 Apr 18]. Available from, in: A. Abd El-baky, Mohamed A. Attia, Madeha Kamel (Eds.), 2018.

[2] M. Eftekhari, A. Fatemi, Creep-fatigue interaction and thermo-mechanical fatigue behaviors of thermoplastics and their composites, Int. J. Fatigue 91 (2016) 136–148.

[3] S. Nouira, M. Shirinbayan, J. Peixinho, K. Benfriha, T. Hassine, J. Fitoussi, Effect of processing conditions on morphology and mechanical damage in glass-reinforced polypropylene composite, Polym. Compos. 45 (9) (2024) 8251–8263.

[4] J. Sommer, M. Hajikazemi, I. De Baere, W. Van Paepegem, Fatigue behaviour of thermoplastic glass/polypropylene composite cross-ply laminates: an experimental

study with in-situ damage observations and numerical validation, Compos. Part B Eng. 252 (2023) 110494.

[5] Fong J.T. What Is Fatigue Damage? In: Damage in Composite Materials: Basic Mechanisms, Accumulation, Tolerance, and Characterization [Internet]. ASTM International, 1982. p. 243–66. Available from: https://store.astm.org/stp34330s.html.

[6] Sendeckyj G. Chapter 10 - Life Prediction for Resin-Matrix Composite Materials. In: Reifsnider KL, Eds., Composite Materials Series [Internet]. Elsevier; 1991 [cited 2024 Mar 11]. p. 431–83. [Fatigue of Composite Materials; vol. 4]. Available from: https://www.sciencedirect.com/science/article/pii/B9780444705075500147.

[7] M. Eftekhari, A. Fatemi, On the strengthening effect of increasing cycling frequency on fatigue behavior of some polymers and their composites: experiments and modeling, Int. J. Fatigue 87 (2016) 153–166.

[8] J. Choi, H. Lee, H. Lee, N. Kim, A Methodology to Predict the Fatigue Life under Multi-Axial Loading of Carbon Fiber-Reinforced Polymer Composites Considering Anisotropic Mechanical Behavior, Materials. (Basel) 16 (5) (2023) 1952.

[9] A.I. Selmy, N.A. Azab, M.A Abd El-baky, Flexural fatigue characteristics of two different types of glass fiber/epoxy polymeric composite laminates with statistical analysis, Compos Part B Eng 45 (1) (2013) 518–527.

[10] Influence of nanoparticles addition on the fatigue failure behavior of metal matrix composites: comprehensive review - ScienceDirect [Internet]. 2024. Available from: https://www.sciencedirect.com/science/article/pii/S1350630723007057.

[11] A. Ince, G. Glinka, A modification of Morrow and Smith-Watson-Topper mean stress correction models, Fatigue Fract. Eng. Mater. Struct. 34 (11) (2011) 854–867.

[12] P.P. Gillis, Manson-Coffin fatigue, Acta Metall 14 (12) (1966) 1673–1676.

[13] R. Branco, J.D. Costa, F.V. Antunes, Fatigue behaviour and life prediction of lateral notched round bars under bending-torsion loading, Eng. Fract. Mech. 119 (2014) 66–84.

[14] I. Feki, M. Shirinbayan, S. Nouira, E. Heripre, R. Tie Bi, J. Maeso, et al., Multi-scale experimental investigation of porosity-induced damage effects in filament-wound carbon fiber reinforced epoxy composites used in hydrogen storage tanks, Polym. Compos. (2024) 29121 pc.

[15] M.F. Arif, N. Saintier, F. Meraghni, J. Fitoussi, Y. Chemisky, G. Robert, Multiscale fatigue damage characterization in short glass fiber reinforced polyamide-66, Compos Part B Eng 61 (2014) 55–65.

[16] H. Tang, G. Zhou, Z. Chen, L. Huang, K. Avery, Y. Li, et al., Fatigue behavior analysis and multi-scale modelling of chopped carbon fiber chip-reinforced composites under tension-tension loading condition, Compos. Struct. 215 (2019) 85–97.

[17] H. Tang, Z. Chen, G. Zhou, X. Sun, Y. Li, L. Huang, et al., Effect of fiber orientation distribution on constant fatigue life diagram of chopped carbon fiber chip-reinforced Sheet Molding Compound (SMC) composite, Int. J. Fatigue 125 (2019) 394–405.

[18] E.K. Gamstedt, L.A. Berglund, T. Peijs, Fatigue mechanisms in unidirectional glass-fibre-reinforced polypropylene, Compos. Sci. Technol. 59 (5) (1999) 759–768.

[19] J.R. Schaff, B.D. Davidson, Life Prediction Methodology for Composite Structures. Part II—Spectrum Fatigue [Internet]. [cited 2024 Mar 11]. Available from: https://journals.sagepub.com/doi/10.1177/002199839703100203, 1997.

[20] Reifsnider K., Highsmith A., Reifsnider K. Stiffness-Reduction Mechanisms in Composite Laminates. 1982;103–17.

[21] K.L. Reifsnider, Z. Gao, A micromechanics model for composites under fatigue loading, Int. J. Fatigue 13 (2) (1991) 149–156.

[22] A. Jain, J.M. Veas, S. Straesser, W. Van Paepegem, I. Verpoest, S.V. Lomov, The Master SN curve approach – A hybrid multi-scale fatigue simulation of short fiber reinforced composites, Comos. Part Appl. Sci. Manuf. 91 (2016) 510–518.

[23] A. Jain, W. Van Paepegem, I. Verpoest, S.V. Lomov, A feasibility study of the Master SN curve approach for short fiber reinforced composites, Int. J. Fatigue 91 (2016) 264–274.

[24] E. Le Pen, D. Baptiste, G. Hug, Multi-scale fatigue behaviour modelling of Al–Al2O3 short fibre composites, Int. J. Fatigue 24 (2) (2002) 205–214.

[25] G. Guo, J. Fitoussi, D. Baptiste, N. Sicot, C. Wolff, Modelling of Damage Behaviour of a Short-Fiber Reinforced Composite Structure by the Finite Element Analysis Using a Micro-Macro Law, Int. J. Damage. Mech. 6 (3) (1997) 278–299. Jul 1.

[26] L. Zhang, Z. Li, H. Zhang, Z. Liu, P. Zhu, Fatigue failure mechanism analysis and life prediction of short fiber reinforced polymer composites under tension-tension loading, Int. J. Fatigue 160 (2022) 106880.

[27] J. Fitoussi, G. Guo, D. Baptiste, Determination of a tridimensional failure criterion at the fibre/matrix interface of an organic-matrix/discontinuous-reinforcement composite, Compos. Sci. Technol. 56 (7) (1996) 755–760.

[28] J. Fitoussi, G. Guo, D. Baptiste, A statistical micromechanical model of anisotropic damage for S.M.C. composites, Compos. Sci. Technol. 58 (5) (1998) 759–763.

[29] K. Derrien, J. Fitoussi, G. Guo, D. Baptiste, Prediction of the effective damage properties and failure properties of nonlinear anisotropic discontinuous reinforced composites, Comput. Methods Appl. Mech. Eng. 185 (2) (2000) 93–107.

[30] Z. Jendli, J. Fitoussi, F. Meraghni, D. Baptiste, Anisotropic strain rate effects on the fibre–matrix interface decohesion in sheet moulding compound composites, Comos. Sci. Technol. 65 (3) (2005) 387–393.

[31] Z. Jendli, F. Meraghni, J. Fitoussi, D. Baptiste, Multi-scales modelling of dynamic behaviour for discontinuous fibre SMC composites, Compos. Sci. Technol. 69 (1) (2009) 97–103.

[32] M. Berveiller, A. Zaoui, An extension of the self-consistent scheme to plastically-flowing polycrystals, J. Mech. Phys. Solids. 26 (5) (1978) 325–344.

[33] M. Shirinbayan, S. Nouira, M.A. Imaddahen, J. Fitoussi, Microstructure-sensitive investigation on the plastic deformation and damage initiation of fiber-reinforced polypropylene composite, Compos. Part B Eng. 286 (2024) 111790.
[34] M. Shirinbayan, I. Feki, S. Nouira, R.T. Bi, J. Maeso, C. Thomas, et al., Multi-scale damage analysis of filament-wound carbon fiber-reinforced epoxy composites for hydrogen storage tanks under high strain rates, Polym. Compos. (2024) 29273 pc.

[35] H. Tang, S. Zhang, L. He, Z. Yang, T. Liu, 3D printing of high-stiffness and high-strength glass fiber reinforced PEEK composites by selective laser sintering, Compos Part Appl Sci Manuf 187 (2024) 108470.

[36] J. Fitoussi, S. Nouira, K. Benfriha, M.A. Laribi, A. Kallel, R.T. Bi, et al., Investigation of manufacturing process effects on microstructure and fatigue prediction in

composite automotive tailgate design, Int. J. Adv. Manuf. Technol. 130 (9) (2024) 4295-4310.

[37] Y.P. Qiu, G.J. Weng, The influence of inclusion shape on the overall elastoplastic behavior of a two-phase isotropic composite, Int. J. Solids. Struct. 27 (12) (1991) 1537-1550.

[38] H. Ayari, M. Shirinbayan, A. Imaddahen, S. Tamboura, H. Ben Daly, A. Tcharkhtchi, et al., Micromechanical modelling of dynamic behavior of advanced sheet molding compound (A-SMC) composite, Appl. Compos. Mater. 27 (3) (2020) 321-335.