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Abstract

As aviation and space technology advances rapidly, fiber-reinforced polymer (FRP) composites are finding growing applications under extreme service conditions, particularly under high-thermal environments. Assessing the fatigue resistance of FRP composites across different temperature ranges has become a major concern. By applying the Force-Heat Equivalent Energy Density (FHEED) principle and damage failure analysis, a temperature and damage dependent fatigue strength model (TDDFS) was established for FRP composites, grounded in physical mechanisms. The TDDFS model considers the comprehensive influence of temperatures, evolution of constituent properties, load level, layup configurations, especially the damage evolution with temperature on the fatigue strength. The well comparisons between model predictions and experimental data demonstrate the reliability and reasonability of TDDFS model. The study also analyzed the fatigue performance and temperature-related damage evolution of FRP composites. These insights contribute to a reliable theoretical approach for TDDFS prediction and deepen the understanding of fatigue degradation and damage mechanisms in FRP materials at elevated temperatures.

Graphical Abstract

Keywords

FRP composites temperature and damage fatigue strength modeling and analysis

Highlights

(1) The physics-based temperature dependent fatigue strength model of FRP composites is developed.

(2) The model considers the effects of temperature, constituents’ properties and damage evolution.

(3) The TDDFS of FRP composites can be well predicted by the model.

(4) This study provides the understanding on fatigue analysis and strength prediction method.

Introduction

Fiber-reinforced polymer (FRP) composites have obtained significant attention in aerospace, defense, and transportation industries due to their exceptional properties, including high specific mechanical properties, strong designability, and superior processability.^1–9 The mechanical properties and fatigue behavior of FRP composites are predominantly governed by the type of reinforcing fibers, while the toughness of the matrix resin significantly influences their fatigue crack propagation resistance and damage tolerance characteristics.^10–14 With the escalating service temperature requirements in these advanced engineering applications, investigating the thermomechanical properties of FRP composites has evolved into a crucial scientific challenge.^15–18 Notably, fatigue strength, a pivotal indicator of material durability, exhibits pronounced sensitivity to thermal conditions and progressive damage accumulation. The comprehensive investigations into the fatigue response at different temperatures and damage conditions are significant for evaluating the durability of FRP composites. Particularly, establishing a robust quantitative theoretical framework to characterize the synergistic effects of thermal degradation and microstructural damage on fatigue strength has become a fundamental scientific challenge, driving substantial research in materials science and mechanical engineering.^19–21

Extensive research has been devoted to understanding the thermomechanical characteristics and damage evolution in FRP composites. Monte et al.²² and Sonsino et al.²³ systematically studied the axial fatigue performance of short glass fiber-reinforced polyamide 6.6 across temperature gradients. Their experimental findings revealed a temperature-induced degradation of fatigue strength while demonstrating near-constant S-N curve slopes regardless of thermal conditions. Loos et al.²⁴ investigated the fatigue strength of carbon nanotube/polyurethane composites, which demonstrated significant enhancement in fatigue resistance across both high-cycle and low-cycle regimes upon CNT incorporation. Ren et al.^25,26 performed the research regarding the fatigue damage modes of single-walled carbon nanotubes reinforced epoxy (CNT/epoxy) composites, and used S-N curves to describe the fatigue behavior of nanocomposites, which suggests that CNT bridging serves as the primary fatigue resistance mechanism. Guo et al.^27,28 proposed a stress-normalized method for fatigue life prediction that accounts for the effect of temperature on fatigue damage, and enables life prediction under high-temperature and variable-amplitude loading. Furthermore, by considering the microstructure and ply thickness of multidirectional laminates, they developed a fatigue life prediction model suitable for high-temperature conditions where compression-dominated fatigue failure prevails. Ramakrishnan et al.²⁹ proposed a multi-scale stiffness-based damage model that quantifies composites degradation through constituents failure mechanisms, and establishing explicit correlations between damage accumulation and mechanical property deterioration. These investigations have significantly advanced the understanding of the fatigue properties evolution in FRP composites under combined thermal and mechanical damage, thereby establishing a critical foundation for physics-based modeling frameworks. Moreover, there is an urgent need to develop a temperature-dependent fatigue strength prediction model capable of estimating thermal fatigue performance based on fundamental material parameters. However, under coupled thermomechanical loading, the fatigue damage progression follows a sequential mechanism involving matrix cracking, crack bridging, interfacial failure between fiber and matrix, and ultimately fiber breakage. Currently, the existing models fail to account for the synergistic effects of real-time damage states and their temperature-dependent evolution on the fatigue resistance of FRP composites.

Consequently, the present investigation focuses on characterizing the fatigue performance and damage evolution of FRP composites under thermal variations, with the goal of developing a mechanistic model for fatigue strength prediction under multi-stage damage accumulation at elevated temperatures. To this end, we propose a TDDFS model by integrating the FHEED principle with established damage progression laws. In particular, this study innovatively introduces the FHEED principle into the field of fatigue failure of FRP composites at different temperatures. By establishing the equivalent relationship between heat energy and strain energy, a novel TDDFS theoretical model can be developed. The model systematically incorporates the synergistic effects of temperature, cyclic loading parameters, layup configurations, and crucially, the temperature-dependent damage evolution. The validation was conducted through experimental data and predictions from the TDDFS model, complemented by parametric analyses to quantify the sensitivity of fatigue properties to critical damage and temperature variables. This model can quantitatively characterize the entire process of fatigue damage evolution without relying on any empirical fitting parameters. Moreover, compared with the empirical model proposed by Jen et al.,^30,31 the presented model has a stronger physical foundation and greater universality. This study establishes a robust theoretical framework for fatigue strength prediction and damage tolerance assessment of FRP composites in elevated temperature environments.

Theoretical derivations

The damage evolution in FRP composites under varying temperatures involves a progressive and intricate process, characterized by the cumulative development of micro-scale damage mechanisms, including interfacial debonding, matrix microcracking, and fiber breakage.^21,32 Exploring the fatigue damage behavior and theoretical characterizing the temperature-dependent fatigue strength of FRP composites under thermo-mechanical fatigue load are the key issue for safety use and reliability evaluation.

Extending the FHEED principle established in our prior research,^33–36 a novel theoretical framework was developed to quantitatively characterize the temperature-dependent fatigue behavior of FRP composites, representing the innovative theoretical model of its kind. According to this principle, the underlying assumption is that a critical energy storage value exists for a given FRP composites, i.e. the fatigue failure of composites is associated with a fixed maximum energy storage value. Meanwhile, since the failure of FRP composites is mainly a result of the combined effect of strain energy and heat energy, the critical energy storage value consists of the critical strain energy and heat energy of the composites. In order to clarify and deepen the modeling understanding on the fatigue strength, the schematic illustration of energy equivalence principle of FRP composites under fatigue load is depicted in Figure 1. Therefore, the governing equation can be derived as follows:

W_{Total} = W_{s} (T) + α \cdot W_{T} (T)

(1)

where

W_{Total}

represents the critical volumetric energy density at which FRP composites undergo fatigue failure, in units of J/m³.

T

represents the experimental temperature (in Kelvin) for the composite specimens.

W_{s} (T)

is the critical strain energy density of FRP composites, in units of J/m³.

W_{T} (T)

is the heat energy per unit volume of FRP composites, in units of J/m³.

α

proposes an energy partition coefficient to quantify the differential contributions of strain energy and heat energy to FRP composite fatigue failure.

Figure 1.

Schematic illustration of energy equivalence principle and damage accumulation of FRP composites under fatigue load.

It is assumed that the density of the composite material is independent of temperature due to the weak temperature sensitivity. The heat energy density of composites at temperature T is given by:

W_{T} (T) = \int_{0}^{T} ρ C_{p c} (T) d T

(2)

where

ρ

is the density of the composite materials, which remains effectively constant across the temperature range owing to its minimal thermal variation, in units of Kg/m³.

C_{p c} (T)

represents the constant pressure p, temperature-dependent specific heat capacity of the composites, expressible as,³⁷ in units of J kg⁻¹ K⁻¹:

C_{p c} (T) = C_{p f} (T) \cdot M_{f} + C_{p m} (T) \cdot M_{m}

(3)

where

M_{f}

and

M_{m}

correspond to the mass fractions of the fiber reinforcement and polymer matrix. The subscripts

c

f

and

m

denote the composites, fiber and matrix.

The influence of temperature on the critical strain energy density at failure of FRP composites, denoted by $W_{s} (T)$ , is composed of the critical strain energy densities of the fiber and matrix at temperature T. Therefore, we obtain the expression for $W_{s} (T)$ :

W_{s} (T) = W_{s}^{m} (T) + W_{s}^{f} (T)

(4)

The critical strain energy densities of fiber and matrix at temperature

T

are denoted by

W_{s}^{f} (T)

and

W_{s}^{m} (T)

, respectively.

A primary objective is to derive the expressions for the critical strain energy density of the fiber and matrix components. When the service temperature is lower than the glass transition temperature, it can be assumed that the matrix approximately exhibits linear elastic behavior within the analyzed temperature range. Building on prior work, we derive the relationship between the thermally influenced internal stress distribution in constituent materials and the external load applied to the composite, yielding the following formulation³⁸:

σ_{w}^{f} (T) = \frac{E_{f} (T) \cdot σ_{w} (T)}{V_{f} E_{f} (T) + (1 - V_{f}) E_{m} (T)}

(5)

σ_{w}^{m} (T) = \frac{E_{m} (T) \cdot σ_{w} (T)}{V_{f} E_{f} (T) + (1 - V_{f}) E_{m} (T)}

(6)

Where

σ_{w}^{f} (T)

represents the average stress in the fiber, while

σ_{w}^{m} (T)

represents that in the matrix, both evaluated at temperature

T

, in units of MPa.

E_{f} (T)

represents the Young’s modulus of fiber, while

E_{m} (T)

represents that of the matrix, both evaluated at temperature

T

, in units of MPa.

V_{f}

denotes the volume fraction of fiber in the composite. The fatigue load applied to FRP composites at temperature

T

is denoted by

σ_{w} (T)

To account for the damage mechanisms including interfacial degradation, matrix cracking, and fiber breakage in fatigue strength prediction, the damage parameters were integrated into the average stress equations of constituents. Building on the mechanics of continuous damage, the intrinsic damage in composites model³⁹ was further generalized to accommodate temperature-dependent behavior:

σ_{c} (T) = E_{c} (T) ε_{c} (T) (1 - D (T))

(7)

where

σ_{c} (T)

and

ε_{c} (T)

are the stress and strain of composites at temperature

T

, respectively.

E_{c} (T)

is the Young’s modulus of composites which can be calculated by the law of mixtures (

E_{c} (T) = V_{f} E_{f} (T) + (1 - V_{f}) E_{m} (T)

D (T)

represents the temperature dependent damage parameter, with a value range of 0 to 1. FRP composites demonstrate complex thermomechanical damage progression. To simplify the description of the damage extension process, a damage parameter is introduced, which can take into account the effect of damages such as degradation of interfacial properties, the influence of matrix microcracking and fiber breakage on composite fatigue resistance. In this regard, Farahani proposed a fatigue degradation criterion^29,40:

\begin{array}{l} D (T) = {(1 - \frac{E_{f} (T) V_{f} \cos θ}{E_{c} (T)}) (1 - f^{*}) \frac{\ln (N (T) + 1)}{\ln (n N_{f} (T))}} + {(1 - \frac{E_{f} (T) V_{f} \cos θ}{E_{c} (T)}) f^{*} (\frac{N (T)}{n N_{f} (T)})} \\ + {\frac{E_{f} (T) V_{f} \cos θ}{E_{c} (T)} (1 - \frac{σ_{\max} (T) (1 - R)}{2 σ_{ult} (T)}) \frac{\ln (1 - \frac{N (T)}{n N_{f} (T)})}{\ln (\frac{1}{n N_{f} (T)})}} \end{array}

(8)

where

f^{*}

is the friction coefficient introduced by the fiber with

0 \leq f^{*} \leq 1

N (T)

is the number of cycles at temperature

T

N_{f} (T)

is the fatigue life at temperature

T

σ_{\max} (T)

is the maximum applied fatigue stress at temperature

T

σ_{u l t} (T)

is ultimate tensile stress at temperature

T

, in units of MPa.

R

is stress ratio (

σ_{\min} / σ_{\max}

n

is the assumed percentage of run-out or drop in stiffness.

θ

is the angle between the fiber and the load direction.

Assuming a fiber direction of $θ = 0^{\circ}$ gives,

\begin{array}{l} D (T) = {\frac{E_{m} (T) V_{m}}{E_{c} (T)} (1 - f^{*}) \frac{\ln (N (T) + 1)}{\ln (n N_{f} (T))}} + {\frac{E_{m} (T) V_{m}}{E_{c} (T)} f^{*} (\frac{N (T)}{n N_{f} (T)})} \\ + {\frac{E_{f} (T) V_{f}}{E_{c} (T)} (1 - \frac{σ_{\max} (T) (1 - R)}{2 σ_{ult} (T)}) \frac{\ln (1 - \frac{N (T)}{n N_{f} (T)})}{\ln (\frac{1}{n N_{f} (T)})}} \end{array}

(9)

Equation (5) and Equation (6) denote the mean stresses in the fiber and matrix of FRP composites under external load without considering damage effects. Combining the definition of damage with the introduction of corresponding damage parameters, the temperature-dependent mean stresses of the fiber ( $σ_{w D}^{f} (T)$ ) and matrix ( $σ_{w D}^{m} (T)$ ) containing the damage growth are derived, respectively:

σ_{w D}^{f} (T) = \frac{E_{f} (T) \cdot σ_{w} (T) \cdot (1 - D (T))}{V_{f} E_{f} (T) + (1 - V_{f}) E_{m} (T)}

(10)

σ_{w D}^{m} (T) = \frac{E_{m} (T) \cdot σ_{w} (T) \cdot (1 - D (T))}{V_{f} E_{f} (T) + (1 - V_{f}) E_{m} (T)}

(11)

The mean stress formulas for fiber and matrix varying with temperature and damage were established. The fiber and matrix can be considered to have reached the ultimate bearing state when the ultimate fatigue load ( $σ_{w} (T)$ ) on the FRP composite is applied. Below the glass transition temperature, the nonlinear behavior of the stress-strain relationship of the polymer shows a slightly weaker behavior, but still approximately follows linear elasticity. The derivation of critical strain energy density is performed for both fiber and matrix components at temperature $T$ , yielding the following results:

W_{s}^{f} (T) = \frac{E_{f}^{2} (T) \cdot σ_{w}^{2} (T) \cdot V_{f} \cdot {(1 - D (T))}^{2}}{2 E_{f} (T) \cdot {[V_{f} E_{f} (T) + (1 - V_{f}) E_{m} (T)]}^{2}}

(12)

W_{s}^{m} (T) = \frac{E_{m}^{2} (T) \cdot σ_{w}^{2} (T) \cdot V_{m} \cdot {(1 - D (T))}^{2}}{2 E_{m} (T) \cdot {[V_{f} E_{f} (T) + (1 - V_{f}) E_{m} (T)]}^{2}}

(13)

The critical strain energy and heat energy versus temperature for FRP composites taking into account the damage effects are given, combining equations (1), (2), and (4), and $W_{Total}$ at any reference temperature $T_{0}$ is expressed as:

W_{Total} = W_{s}^{f} (T_{0}) + W_{s}^{m} (T_{0}) + α \cdot \int_{0}^{T_{0}} ρ C_{p c} (T) d T

(14)

When the experimental temperature reaches the melting point of the polymer matrix, the composite loses its load-bearing capacity. Consequently, the strain energy equals zero, and the

W_{Total}

T_{melting}

is obtained:

W_{Total} = α \cdot \int_{0}^{T_{melting}} ρ C_{p c} (T) d T

(15)

Based on equations (14) and (15), we obtain the ratio coefficients $α$ :

α = \frac{W_{s}^{f} (T_{0}) + W_{s}^{m} (T_{0})}{\int_{T_{0}}^{T_{melting}} ρ C_{p c} (T) d T}

(16)

Subsequently, combing equations (1), (4), (14), and (16), at any temperature $T$ , the critical strain energy of FRP composites can be expressed as:

W_{s}^{f} (T) + W_{s}^{m} (T) = [W_{s}^{f} (T_{0}) + W_{s}^{m} (T_{0})] \cdot \frac{\int_{T}^{T_{melting}} C_{p c} (T) d T}{\int_{T_{0}}^{T_{melting}} C_{p c} (T) d T}

(17)

Substituting equations (8), (12), and (13) into equation (17) to derive $σ_{w} (T)$ , the temperature dependent fatigue strength model for FRP composites taking into account the damage growth was developed:

\begin{array}{l} σ_{w} (T) = {[\frac{V_{f} E_{f} (T) + (1 - V_{f}) E_{m} (T)}{V_{f} E_{f} (T_{0}) + (1 - V_{f}) E_{m} (T_{0})} (1 - \frac{\int_{T_{0}}^{T} C_{p c} (T) d T}{\int_{T_{0}}^{T_{melting}} C_{p c} (T) d T})]}^{\frac{1}{2}} \\ \times [\frac{\frac{E_{f} (T_{0}) V_{f}}{E_{c} (T_{0})} (1 - f^{*}) \frac{\ln (N (T_{0}) + 1)}{\ln (n N_{f} (T_{0}))} - \frac{E_{m} (T_{0}) V_{m}}{E_{c} (T_{0})} f^{*} (\frac{N (T_{0})}{n N_{f} (T_{0})}) - \frac{E_{f} (T_{0}) V_{f}}{E_{c} (T_{0})} (1 - \frac{σ_{\max} (T_{0}) (1 - R)}{2 σ_{u l t} (T_{0})}) \frac{\ln (1 - \frac{N (T_{0})}{n N_{f} (T_{0})})}{\ln (\frac{1}{n N_{f} (T_{0})})}}{\frac{E_{f} (T) V_{f}}{E_{c} (T)} (1 - f^{*}) \frac{\ln (N (T) + 1)}{\ln (n N_{f} (T))} - \frac{E_{m} (T) V_{m}}{E_{c} (T)} f^{*} (\frac{N (T)}{n N_{f} (T)}) - \frac{E_{f} (T) V_{f}}{E_{c} (T)} (1 - \frac{σ_{\max} (T) (1 - R)}{2 σ_{u l t} (T)}) \frac{\ln (1 - \frac{N (T)}{n N_{f} (T)})}{\ln (\frac{1}{n N_{f} (T)})}}] σ_{w} (T_{0}) \end{array}

(18)

where

C_{p c} (T)

is formulated by equation (3).

The TDDFS model (equation (18)) provides the theoretical framework for predicting the fatigue strength of unidirectional FRP composites at different temperatures and different damage levels. The model incorporates the synergistic influence of thermal and mechanical loading conditions, with particular emphasis on temperature dependent damage evolution and its impact on the fatigue performance of FRP composites. It should be noted that the present model is established under the premise of a constant ambient temperature T, and is applicable for predicting the fatigue strength of FRP composites under this condition. The potential influence of material self-heating during cyclic loading was not considered in the model. Moreover, the model can predict the temperature-dependent fatigue strength of composites under constant load but varying fiber content, requiring only a limited set of commonly available material parameters from standard handbooks or literature. The detailed analysis regarding the various factors affecting the fatigue properties of FRP composites can be conducted by employing the model, and it contributes to enhanced comprehension on the fatigue strength degradation and damage evolution of FRP composites at elevated temperatures.

Considering the relationship between damage and layup, a temperature dependent damage parameter is introduced $D (T)$ ⁴²:

\begin{array}{l} D (T) = {(1 - \frac{F (T)}{E_{c} (T)}) (1 - f^{*}) \frac{\ln (N (T) + 1)}{\ln (n (T) N_{f} (T))}} + {(1 - \frac{F (T)}{E_{c} (T)}) f^{*} (\frac{N (T)}{n (T) N_{f} (T)})} \\ + {\frac{F (T)}{E_{c} (T)} (1 - \frac{σ_{\max} (T) (1 - R)}{2 σ_{u l t} (T)}) \frac{\ln (1 - \frac{N (T)}{n (T) N_{f} (T)})}{\ln (\frac{1}{n (T) N_{f} (T)})}} \end{array}

(19)

where the term

F (T)

in equation (19) varies based on the composite laminate layup. The

F (T)

parameter requires external determination as it originates from the analysis of composite laminates. Meanwhile, the modeling framework in equation (18) does not account for ply-layer effects. In order to quantitatively characterize the fatigue strength of orthogonal woven composites, the corresponding

F (T)

parameter was adopted.

Specifically, the rule of mixtures can be applied to orthogonal woven composites, resulting in:

F (T) = K_{f} E_{f} (T) V_{f}

(20)

Note that $V_{f}^{*} = K_{f} V_{f}$ represents the volume fraction of fibers oriented parallel to the loading direction.

Substituting equations (12), (13), and (19) into equation (17) yields $σ_{w} (T)$ , and then the temperature dependent fatigue strength model for FRP composites with different lay-ups, taking into account damage growth:

\begin{array}{l} σ_{w} (T) = {[\frac{V_{f} E_{f} (T) + (1 - V_{f}) E_{m} (T)}{V_{f} E_{f} (T_{0}) + (1 - V_{f}) E_{m} (T_{0})} (1 - \frac{\int_{T_{0}}^{T} C_{p c} (T) d T}{\int_{T_{0}}^{T_{melting}} C_{p c} (T) d T})]}^{\frac{1}{2}} \\ \times [\frac{1 - (\frac{F (T_{0}) \cdot (1 - f^{*})}{E_{c} (T_{0})} \cdot \frac{\ln (N (T_{0}) + 1)}{\ln (n (T_{0}) N_{f} (T_{0}))}) - (1 - \frac{F (T_{0})}{E_{c} (T_{0})}) (\frac{f^{*} \cdot N (T_{0})}{n (T_{0}) N_{f} (T_{0})}) - \frac{F (T_{0})}{E_{c} (T_{0})} (1 - \frac{σ_{\max} (T_{0}) (1 - R)}{2 σ_{u l t} (T_{0})}) \frac{\ln (1 - \frac{N (T_{0})}{n (T_{0}) N_{f} (T_{0})})}{\ln (\frac{1}{n (T_{0}) N_{f} (T_{0})})}}{1 - (\frac{F (T) \cdot (1 - f^{*})}{E_{c} (T)} \cdot \frac{\ln (N (T) + 1)}{\ln (n (T) N_{f} (T))}) - (1 - \frac{F (T)}{E_{c} (T)}) (\frac{f^{*} \cdot N (T)}{n (T) N_{f} (T)}) - \frac{F (T)}{E_{c} (T)} (1 - \frac{σ_{\max} (T) (1 - R)}{2 σ_{u l t} (T)}) \frac{\ln (1 - \frac{N (T)}{n (T) N_{f} (T)})}{\ln (\frac{1}{n (T) N_{f} (T)})}}] σ_{w} (T_{0}) \end{array}

(21)

The TDDFS model (equation (21)) was developed to predict the fatigue strength of orthogonal woven composites under varying temperatures and progressive damage conditions. A fundamental assumption of this model is that the composites are at a constant ambient temperature T, and the model does not incorporate self-heating effects during cyclic loading. Based on this premise, this model explicitly considers the combined effects of temperature, load level, especially the damage evolution with temperature on the fatigue strength. This study establishes a theoretical framework to elucidate the temperature dependent damage progression and its dominant role in composite fatigue degradation. The proposed model utilizes standard material properties readily available in technical databases, ensuring broad applicability. In this study, the TDDFS model supplies a theoretical prediction method for fatigue strength at different temperatures, and enables a systematic analysis of significant influencing factors about fatigue properties evolution in orthogonal woven composites, thereby advancing the understanding of their strength degradation and damage accumulation in different temperature environments.

Results and discussion

Fatigue strength prediction of polymer composites at different temperatures

The TDDFS model (equation (18)) of FRP composites, developed in the above section, aims to predict the fatigue strength at different temperatures and multi-factor conditions. To verify the scientific validity and precision of the developed model, this section compares the experimental fatigue strength with the predictions of TDDFS model across varying fiber contents, load levels, and temperatures for systematically assessing their agreement.

CNTs-GNP and epoxy composites with different fiber contents and load levels

The TDDFS model (equation (18)) was used to predict the temperature-dependent fatigue strength of three nanocomposites including 10% CNTs/epoxy, 10% Graphene nanoplatelets/epoxy (GNP/epoxy), and 9%GNP-1%CNTs/epoxy. The material parameters used in this model are sourced from published literature and material handbooks. Table 1 lists the parameters required for the model to calculate CNTs-GNP and epoxy composites. As shown in Figure 2, the predicted fatigue strengths for these composites agree well with the experimental results. Figures 3 and 4 reveal that the model predictions exceed experimental values at elevated temperatures, primarily because the current formulation neglects both the temperature-dependent degradation of CNT/Graphene Young’s modulus and high-temperature interfacial bond weakening. Overall, the discrepancy between predicted and experimental values falls within an acceptable margin of error, and thus the TDDFS model (equation (18)) can reasonably predict the fatigue strength of FRP composites under various temperatures, different load levels and number of cycles. It is noteworthy that the fatigue strength of 9%GNP-1%CNTs/epoxy composites is higher than that of 10% CNTs/epoxy and 10% GNP/epoxy composites. This enhancement stems from CNTs bridging adjacent GNP to form a three-dimensional nanofiller network, which suppresses GNP agglomeration and improves the mechanical performance. Furthermore, the study confirms a gradual decline in fatigue strength for CNT/GNP reinforced epoxy composites with increasing temperature, which is consistent with the temperature-fatigue relationships.

Table 1.

Material parameters of 10% CNTs/epoxy, 10%GNP/epoxy and 9% GNP-1%CNTs/epoxy.

Material parameters	10%CNTs/epoxy	10%GNP/epoxy	9%GNP-1%CNTs/epoxy
$V_{f}$ ³⁰	0.1	0.1	0.01CNTs/0.09GNP
$V_{m}$ ³⁰	0.9	0.9	0.9
$T_{0}$ ³⁰	245K
$E_{f}$ ^42,43	CNTs:245K-325K:1112.82GPa GNP:245K-325K:1000GPa
$E_{m}$ ³⁰	245K:3.38GPa, 275K:3.03GPa, 295K:3.27GPa, 325K:2.8GPa
$N = N_{f}$ ³⁰	$λ$ = 60%	$λ$ = 57.5%	$λ$ = 60%
$λ = \frac{σ_{\max}}{σ_{u l t}}$	245K:448841	275K:314330	245K:444633
	275K:166549	295K:436494	275K:576066
	295K:107018	325K:701563	295K:50147
	325K:1000000	$λ$ = 60%	325K:124922
	$λ$ = 62.5%	275K:165947	$λ$ = 65%
	245K:125387	295K:141794	245K:262364
	275K:58374	325K:124484	275K:243229
	295K:49107	$λ$ = 65%	295K:15890
	$λ$ = 65%	275K:5710	325K:100218
	245K:96073	295K:2449	$λ$ = 70%
	275K:10804	325K:37871	245K:32127
	295K:17633		275K:44227
	325K:29022		325K:13339
$T_{melting}$	458K⁴⁴
n	1.67⁴⁰
$f^{*}$	0.1(weak),0.3(medium),0.5(strong)^29,45
$R$	0.1³⁰
Specific heat capacity^46,47	CNTs/GNP:Cp(T) = 0.054 × $10^{- 3}$ T+20.769(J kg⁻¹ K⁻¹)
Specific heat capacity^46,47	Epoxy resin:Cp(T) = 2.15 T + 1167(J kg⁻¹ K⁻¹)

Figure 2.

TDDFS model predictions and experimental data for 10% CNTs/epoxy at (a) 60% load level, (b) 62.5% load level, (c) 65% load level.

Figure 3.

TDDFS model predictions and experimental data for 10% GNP/epoxy at (a) 57.5% load level, (b) 60% load level, (c) 65% load level.

Figure 4.

TDDFS model predictions and experimental data for 9% GNP-1%CNTs/epoxy at (a) 60% load level, (b) 65% load level, (c) 70% load level.

CNTs and epoxy composites with different fiber contents and load levels

The TDDFS model (equation (18)) predicted the fatigue strength of 0.5% CNTs/epoxy and 1% CNTs/epoxy composites at varying temperatures. All material parameters employed in this model are derived from the open literature and material handbooks. Table 2 lists the parameters required for the model to calculate CNTs and epoxy composites. Notably, the fatigue strength of 1% CNTs/epoxy is lower than that of 0.5% CNTs/epoxy because excessive CNT incorporation induces the stress concentration at agglomerate location, thereby reducing overall fatigue resistance. Furthermore, the experimental results in this study confirm that the fatigue strength of CNTs/epoxy composites gradually decreases with rising temperature. As shown in Figures 5 and 6, the TDDFS model predictions (equation (18)) for both 0.5% and 1% CNTs/epoxy composites as a function of temperature exhibit reasonable agreement with the experimental data. But, at high temperatures, the predicted values of the model (equation (18)) were slightly higher than the experimental results. This discrepancy arises because the model assumes the temperature dependent Young’s modulus for CNTs, and does not quantify the interfacial properties degradation, which causes the overestimate of model predictions at high temperatures. Nevertheless, the overall error between predictions and experimental data remains within acceptable limits. Thus, the proposed model can reliably predict the fatigue strength of polymer composites across a wide temperature range under varying load levels and cycle counts.

Table 2.

Material parameters of 0.5% CNTs/epoxy and 1% CNTs/epoxy.

Material parameters	0.5%CNTs-epoxy resin	1%CNTs-epoxy resin
$V_{f}$ ³¹	0.005	0.01
$V_{m}$ ³¹	0.995	0.99
$T_{0}$ ³¹	248K 248 K
$E_{f}$ ⁴³	248K-313 K:1112.82 GPa
$E_{m}$ ³¹	248K:3.345 GPa, 273K:3.05 GPa
	298K:3.223 GPa, 313K:2.988 GPa
$N = N_{f}$ ³¹	$λ$ = 40%	$λ$ = 35%
$λ = \frac{σ_{\max}}{σ_{u l t}}$	248K:1000000 273K:1000000	248K:1000000
	298K:432819	298K:119159
	313K:90184	313K:118706
	$λ$ = 45%	$λ$ = 40%
	248K:75680	248K:102463
	273K:129593	273K:1000000
	298K:102736	298K:54714
	313K:46276	313K:31042
	$λ$ = 50%	$λ$ = 45%
	248K:19470	248K:35405
	273K:20994	273K:72581
	298K:43984	298K:16492
	313K:15666	313K:16566
	$λ$ = 55%	$λ$ = 50%
	248K:8121	248K:12823
	273K:11619	273K:49479
	298K:13614	298K:8937
	313K:3406	313K:7381
$T_{melting}$	458K⁴⁴
n	1.67⁴⁰
$f^{*}$	0.1(weak),0.3(medium),0.5(strong)^29,45
$R$	0.1³¹
Specific heat capacity^46,47	CNTs:C_p(T) = 0.054 × $10^{- 3}$ T + 20.769(J kg⁻¹ K⁻¹)
Specific heat capacity^46,47	Epoxy resin:C_p(T) = 2.15 T + 1167(J kg⁻¹ K⁻¹)

Figure 5.

TDDFS model predictions and experimental data for 0.5% CNTs/epoxy at (a) 40% load level, (b) 45% load level, (c) 50% load level, and (d) 55% load level.

Figure 6.

TDDFS model predictions and experimental data for 1% CNTs/epoxy at (a) 35% load level, (b) 40% load level, (c) 45% load level, and (d) 50% load level.

AS-4 fiber and PEEK composites with different layups and loading levels

The fatigue behavior of AS-4/PEEK composites (ICI Fiberite Co., USA) with various stacking sequences was predicted across temperature ranges using the TDDFS formulation (equation (21)). The material parameters applied in this model are obtained from publicly available literature and material handbooks. Table 3 lists the parameters required for the model to calculate AS-4 fiber and PEEK composites. Due to the difficulty in directly measuring fiber/matrix interfacial strength parameters under different ply orientations, an inverse analysis method based on characteristic temperature points was employed to determine the interfacial properties. The results indicate that cross-ply AS-4/PEEK composites exhibit higher fatigue strength than quasi-isotropic counterparts. This difference stems from the cross-ply layup’s ability to efficiently disperse and resist multidirectional stress, thereby enhancing overall fatigue resistance. In contrast, the quasi-isotropic layups, while maintaining balanced strength across multiple directions, are prone to stress concentration in specific orientations, ultimately reducing fatigue strength. Additionally, the study reveals a gradual decline in fatigue strength of AS-4/PEEK composites with rising temperature. As evidenced in Figure 7, the close agreement between the TDDFS model predictions (equation (21)) and experimental data confirms the model’s validity and accuracy.

Table 3.

Material parameters of AS-4/PEEK composites (from ICI Fiberite Co. (USA)).

Material parameters	AS-4/PEEK
Plies/laminates	[0/90]_4S	[0/+45/90/−45]_2S
$V_{f}$	0.61⁴⁹
$V_{m}$	0.39⁴⁹
$T_{0}$	298K⁴⁸
$E_{f}$	234 GPa^49,50
F $(T_{0})$ ⁴¹	298K–423 K:71370 MPa	298K–423 K:35685 MPa
F $(T)$ ⁴¹	298K–423 K:71370 MPa	298K–423 K:35685 MPa
$E_{c}$ ²⁹	298K:77680 MPa	298K:54503 MPa
	348K:76384 MPa	348K:52085 MPa
	373K:75293 MPa 398K:73112 MPa	373K:51461 MPa 398K:50135 MPa
	423K:71500 MPa	423K:47950 MPa
$N = N_{f}$ ⁵¹	298K–423 K:1000000	298K–423 K:1000000
$λ = \frac{σ_{\max}}{σ_{u l t}}$ ⁵¹	298K:0.53	298K:0.66
	348K:0.43	348K:0.52
	373K:0.45	373K:0.53
	398K:0.41	398K:0.5
	423K:0.41	423K:0.5
$T_{melting}$	616K⁴⁸
n(T)⁵¹	298K:1.02	298K:1.01
	348K:1.05	348K:1.15
	373K:1.1	373K:1.12
	398K:1.13	398K:1.2
	423K:1.2	423K:1.23
$f^{*}$	0.236	0.5
$R$	0.1⁵²
Specific heat capacity^47,52	AS-4 fiber: C_p(T) = 0.084 + 38.911× $10^{- 3}$ T-1.464 × $10^{5}$ T⁻²-17.364 × $10^{- 6}$ T²(J kg⁻¹ K⁻¹)
Specific heat capacity^47,52	PEEK:Cp(T) = 4.75T-0.73(J kg⁻¹ K⁻¹)

Figure 7.

TDDFS model predictions and experimental data for AS-4/PEEK at (a) Cross-ply [0/90]_4S, (b) Quasi-isotropic [0/+ 45/90/−45]_2S.

Analysis of influencing factors of FRP composite materials

This study develops a TDDFS theoretical model based on the FHEED principle, which establishes the equivalence between strain energy and heat energy. By incorporating a damage parameter into the critical strain energy equations for both fiber and matrix, the model accounts for damage effects, including matrix microcracking and fiber breakage, on fatigue strength. Furthermore, this section analyzes key factors influencing the fatigue strength of FRP composites and their temperature-dependent variations.

The influence of temperature on the degree of damage and fatigue strength

Building upon the proposed TDDFS model, we systematically examined temperature effects on both damage parameters and fatigue strength in FRP composites, employing AS-4/PEEK cross-ply laminates as a representative case study. As shown in Figure 8, the fatigue strength exhibits a gradual decline with rising temperature, whereas the damage parameters increase.²⁹ This trend can be attributed to the temperature-induced degradation of material properties, which reduces the load-bearing capacity of composites. Meanwhile, the high-temperature properties degradation of fiber and matrix promotes premature failure in the composites, reflecting an elevated damage state. Consequently, temperature elevation directly diminishes the fatigue strength of the composites while simultaneously amplifying their damage degree, and then a dual mechanism that synergistically accelerates the fatigue strength reduction.

Figure 8.

Temperature-dependent evolution of damage accumulation and fatigue resistance.

The influence of interface bonding performance on damage at different temperatures

The cross-sectional analysis characterizes the relationship between fiber/matrix bonding properties and damage degree of composites at varying temperatures, as shown in Figure 9. The results demonstrate that at a fixed temperature, the damage parameters escalate as the decrease of interfacial parameter f ^*, which can be interpreted that the fiber/matrix bonding transitions from strong to weak caused the increasingly damage degree of FRP composites. Moreover, under the identical bonding condition, the damage parameters rise with increasing temperature. These analysis results suggest that weakened fiber/matrix bonding and elevated temperatures exacerbates microscopic damage mechanisms, including matrix cracking, fiber fracture, and interfacial defects. In particular, at elevated temperatures, FRP composites with inferior fiber/matrix bonding will achieve the significant damage degree, which further leads to the decreasing fatigue strength.

Figure 9.

Temperature dependent variation of damage level with fiber/matrix bonding properties.

The influence of young's modulus at different temperatures on fatigue strength

Figure 10 demonstrates that fatigue strength exhibits distinct temperature-dependent correlations: a positive relationship with matrix Young’s modulus versus a negative relationship with fiber Young’s modulus. This contrast is particularly pronounced in cross-ply FRP composites, where fiber modulus typically exceeds matrix modulus by an order of magnitude. Continuously increasing the fiber Young’s modulus causes its high-stiffness, and exacerbate the stress concentration. Further, It will reduce the interlaminar shear strength, and accelerate the crack propagation while lowering the fracture toughness, which leads to the decrease of temperature dependent fatigue strength of FRP composites under fatigue load. Meanwhile, the detrimental effect of increasing fiber Young’s modulus on fatigue strength becomes progressively more pronounced with rising temperature. In addition, the Young’s modulus level of the polymer matrix is relatively low, and increasing its modulus can significantly improve the mechanical properties, which is beneficial to the strength enhancement of FRP composites under fatigue load. Moreover, the enhancement of fatigue strength diminishes progressively with rising temperature. This can be interpreted that the lower matrix Young’s modulus at elevated temperatures has a negative effect on the fatigue strength. Consequently, this concludes that moderately increasing the matrix Young’s modulus enhances the fatigue resistance of FRP composites, whereas high level of the fiber Young’s modulus detrimentally impacts the fatigue performance.

Figure 10.

The influence of Young’s modulus of fibers and matrix on the fatigue strength of FRP composites at different temperatures.

Conclusions

In this study, we developed a TDDFS model for FRP composites based on the FHEED principle and damage failure analysis. The model enables accurate prediction of the fatigue strength of FRP composites under varying temperatures and damage levels. The combined effects of layup configurations, temperature, constituents’ properties evolution, and thermal induced damage variation on fatigue strength were systematically investigated and quantified in the model. The comparisons between model predictions and experimental data demonstrate that the TDDFS theoretical model reliably captures the temperature dependent fatigue behavior of FRP composites.

In addition, the analysis of factors affecting the fatigue properties of FRP composites was conducted, and providing the following insights: (1) As temperature increases, the damage degree escalates, and resulting in a decline in fatigue strength; (2) At a constant temperature, weakened bonding properties at the fiber/matrix interface promotes the damage accumulation, thereby reducing the fatigue strength; (3) While the increase of matrix Young’s modulus enhances the fatigue strength at lower temperatures, this beneficial effect diminishes with rising temperature. Conversely, excessively high Young’s modulus of the fiber induces the stress concentration phenomena, ultimately degrading the fatigue performance. This research develops a mechanistic theoretical framework for predicting temperature-dependent fatigue behavior in FRP composites, integrating damage progression physics with strength degradation mechanisms. The model provides both predictive capability and fundamental understanding of composite performance under thermo-mechanical fatigue conditions.

At the current stage, the validation and application scope of this model is explicitly confined to the fatigue strength prediction for unidirectional or orthogonally woven composites under constant ambient temperature. Accordingly, the model is not applicable to the following scenarios that have not been validated: (1) unconventional 3D woven or other complex braided composites; (2) structures with stress concentration effects such as holes, notches, or joints; (3) laminates subject to hygrothermal aging or moisture absorption effects; and (4) loading conditions with very high stress ratios.

Due to current experimental limitations, the test data of fatigue strength of FRP composites under high temperature conditions remain relatively scarce. Nevertheless, a systematic validation of the model was conducted within the temperature range covered by the available experimental data. The results demonstrate that the model predictions are in good agreement with the experimental measurements. It should be noted, however, that when the model is applied to predict the fatigue strength at elevated temperatures beyond the validation range, the predictions may tend to be higher than actual values. This overestimation is primarily attributed to the fact that the current model does not fully account for key factors such as the modulus reduction of CNTs/GNPs and the softening of interfacial strength at high temperatures. Therefore, the established model is explicitly applicable only to fatigue strength predictions within the temperature range from room temperature up to the glass transition temperature. The future work will focus on two main aspects: first, obtaining additional experimental data over a broader high-temperature range to further enhance the model’s applicability and generalizability. Afterwards, improving the physical foundation of the model under high-temperature conditions by incorporating mechanisms such as interfacial property degradation, thereby increasing its predictive accuracy and generalization ability.

Footnotes

Declaration of Conflicting Interests

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

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (Nos. 12202086, 12572238, 12302271); Natural Science Foundation of Chongqing, China (No. CSTB2025NSCQ-GPX0853); Science and Technology Research Program of Chongqing Municipal Education Commission (No. KJZD-K202500709); Opening Project of Key Laboratory of Testing Technology for Manufacturing Process, Southwest University of Science and Technology (No. 24kfzk03); Research and Innovation Program for Graduate Student in Chongqing (No. 2025S0097).

ORCID iD

Ying Li

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.*

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Theoretical modeling and influencing factor analysis for the temperature and damage dependent fatigue strength of fiber-reinforced polymer composites

Abstract

Keywords

Highlights

Introduction

Theoretical derivations

Results and discussion

Fatigue strength prediction of polymer composites at different temperatures

CNTs-GNP and epoxy composites with different fiber contents and load levels

CNTs and epoxy composites with different fiber contents and load levels

AS-4 fiber and PEEK composites with different layups and loading levels

Analysis of influencing factors of FRP composite materials

The influence of temperature on the degree of damage and fatigue strength

The influence of interface bonding performance on damage at different temperatures

The influence of young's modulus at different temperatures on fatigue strength

Conclusions

Footnotes

Declaration of Conflicting Interests

Funding

ORCID iD

Data Availability Statement

Appendix

References