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Abstract

Fatigue analysis and life prediction are the key and challenging issues in the mechanics research of composite material. In this paper, based on the progressive damage analysis method and fatigue cumulative damage theory, a three-dimensional fatigue cumulative damage numerical model is constructed by introducing the orthotropic constitutive equation, 3D-Hashin fatigue criterion, and material performance degradation scheme. Two loading conditions of tension–tension fatigue and compression–compression fatigue of Carbon Fiber composite laminates were investigated. In the tension–tension fatigue experiments, when the cycle is $10^{6}$ times, the experiment and the numerical simulation all not completely destroyed. However, progressive degradation of stiffness and strength was observed in certain elements, accompanied by some internal damage. On the other hand, the experimental and predicted values for compression–compression fatigue were within a triple error band. Then the cumulative damage evolution law of each laminate under fatigue load was analyzed. Compression–compression fatigue damage initiated from the center of the hole and expanded to both sides of the composite laminate, and its fatigue life increased with the increase of the compression–compression fatigue load. Considering the degradation of stiffness in composite materials under fatigue loading, which leads to frequency attenuation in the structure, this paper established a connection between the fatigue state and frequency to obtain the frequency attenuation curve under fatigue loading. The relationship between frequency and fatigue life was revealed, allowing for the prediction of remaining fatigue life in practical engineering based on the rate of inherent frequency reduction.

Keywords

composite materials fatigue life prediction cumulative damage frequency attenuation

Introduction

In comparison to traditional metallic materials, composite materials exhibit favorable fatigue resistance. However, composite materials possess anisotropic properties and exhibit diverse ply configurations, which leads to a more complex failure mechanism of fatigue cumulative damage. Fatigue damage in composite materials, such as fiber fracture, interfacial debonding, and matrix degradation, mutually induce and couple with each other, posing challenges in assessing fatigue life.^1,2 At present, a small allowable value is usually given in the design of composite structural parts, and the use is relatively conservative, which fails to give full play to the weight-reducing performance of composite materials. In recent aircraft designs, composite structural parts are transitioning from secondary load-bearing parts to primary load-carrying parts, and their fatigue life and durability problems have become more prominent.

Research on composite material fatigue life can be categorized into three types: fatigue life models, phenomenological models for residual strength/residual stiffness, and progressive damage models for fatigue.^3,4 Despite the distinct characteristics of fatigue performance between composite materials and metals, fatigue behavior of composite materials can still be described using S-N curves. The variables of the model mainly include: stress level, stress mean value, stress ratio, laminate stiffness, and lay-up method.

Hashin⁵ established a criterion that can identify fatigue damage, and this criterion can better identify two modes of fiber fatigue and matrix fatigue. Reifsunide⁶ adopted the Mori-Tanaka method, and the stress field of the composite material was calculated using the average stress formula, and then the corresponding fatigue criterion was constructed, which can identify the interface and the performance of the corresponding components from the perspective of mesomechanics. Gathercole⁷ systematically studied the fatigue problem of composite materials and constructed a corresponding fatigue life model. The model introduced the fatigue stress amplitude and stress mean value, and finally gave the expression of the equal-life model, and related parameters explained. Philippidis et al.⁸ established a fatigue criterion capable of identifying fatigue failure under multiaxial loading conditions, which bears resemblance to the Tsai-Wu criterion under static loading. Kawai⁹ investigated axial and off-axis experiments on the off-axis fatigue life of plain fabric composite laminates at room temperature and 100°C. The stress ratio was 0.1, and the S-N curve of two test conditions was obtained. Seyhan¹⁰ conducted tension–tension fatigue tests on bidirectional woven E-glass fiber composite materials and established a mathematical model for probabilistic fatigue life distribution using the Weibull distribution. The tests were performed under three different stress levels and a stress ratio of 0.1, and the probability of fatigue failure at various stress levels was calculated based on the probabilistic statistical model. Yang¹¹ investigated the vibration fatigue behavior of aeroengine blades through experiments. The vibration mode diagram of the composite material blade is measured to obtain the region with the largest vibration stress distribution, which is also the weakest part of fatigue. The fatigue damage is monitored according to the change of the natural frequency of the component. Guo¹² conducted experimental research on the tension–tension fatigue characteristics of adhesive-bonded composite structures and obtained a fatigue prediction model by fitting the S-N curve. The experimental results demonstrated a nonlinear decrease in adhesive stiffness with an increasing number of fatigue cycles.

In the realm of residual strength and residual stiffness modeling, Broutman and Sahu¹³ developed a linear residual strength model, which postulates that the material’s residual strength exhibits a linear decrease with the accumulation of fatigue cycles in the structural load. Schaff and Davidson¹⁴ further refined and optimized the aforementioned linear model. For variable-amplitude fatigue loads, Yao and Himmel^15,16 devised a fatigue life prediction method based on the theories of strength degradation and cumulative damage. Stojkovic¹⁷ proposed a two-parameter strength degradation model under equal-amplitude spectral loading, which requires a small experiment to fit the parameters of the model. Philippidis and Vassilopoulos¹⁸ found that the modulus E(n) of the laminate has a linear relationship with the number of loading cycles according to the fatigue experiments of composite materials. Liu¹⁹ established a two-parameter model of stiffness degradation, which offers improved characterization of the three stages of fatigue damage evolution.

In the domain of progressive damage analysis models, the characterization of fatigue progressive damage differs from that of residual strength and residual stiffness models. It involves the introduction of damage variables to simulate the comprehensive process of damage accumulation in composite materials subjected to fatigue loads. Shokrieh²⁰ et al. established the progressive fatigue cumulative damage theory of composite materials. This model can be divided into three major sections: fatigue failure assessment, material performance degradation, and stress analysis. The stress analysis module primarily relies on finite element software, such as utilizing three-dimensional solid elements in Abaqus for stress analysis. The fatigue failure assessment module introduces seven fatigue failure modes through subroutines: fiber compression and tension failure, interlayer compression and tension failure, matrix–fiber shear failure, matrix compression, and tension failure. Adam²¹ et al. proposed a residual strength and residual stiffness model applicable to any stress state and stress ratio for individual unidirectional plies. Beheshty²² et al. normalized the fatigue test data to establish an equivalent model of fatigue life, which is applicable to any stress level and stress ratio. Wang²³ conducted extensive static strength and fatigue experiments on unidirectional laminates and fitted model parameters for fatigue life calculation based on the experimental results. Feng²⁴ employed a mesoscale unit cell model of three-dimensional braided composites and defined fiber bundle and matrix damage within the Umat subroutine of Abaqus. The results show that the fatigue damage first appears at the contact surface between the fiber bundle and the matrix in the unit cell, and the damage begins to spread from the surface and inside of the fiber bundle, and the damage mode is in good agreement with the experiment.

Regarding the fatigue monitoring of composite materials using frequency-based techniques, Abo-Elkhier²⁵ et al. conducted bending experiments on glass fiber laminates while simultaneously performing modal tests. The findings revealed a correlation between the changes in modal parameters and the fatigue state of composite structures, thereby providing novel approaches for composite material evaluation. By combining modal parameters such as natural frequency and damping ratio with fatigue life, fitting curves were derived, enabling the monitoring of structural fatigue life through modal parameter variations. Liao²⁶ et al. obtained the frequency attenuation curve of structural components under fatigue loads by initially establishing a fatigue calculation model and conducting modal analysis, thereby deriving the stiffness values at specific cycle counts. Subsequently, an artificial neural network was introduced to predict the fatigue life of glass fiber composites. Angela²⁷ et al. analyzed the material performance degradation and delamination damage of composite materials under fatigue load, and found that matrix damage and strength reduction will lead to the spread of lamellar damage. Angela²⁸ et al. discussed the development of the Smart Time XB method and predicts the evolution of delamination damage in composite structures. Angela²⁹ et al. verified the correctness of the model by introducing only cycle hopping strategy and taking composite laminates as the research object. Malekinejad³⁰ et al. summarized the failure behavior of composites, including the properties of joints and the material mechanics behavior of bonding. Bhowmik³¹ et al. summarized the influence of loading conditions, machinery, and wear on fatigue behavior of natural fiber polymer composites.

Considering the complexity and importance of fatigue damage and life prediction of composite materials, it is of great significance to stud of composite materials under fatigue loads for the reliability design of composite structural parts. This paper first establishes a numerical model that can predict the stiffness and strength degradation process of composite materials under fatigue loads and evaluate the fatigue life. Furthermore, various failure modes of composite materials under fatigue loads are comprehensively considered. The fatigue life of composite materials under tension-tension fatigue and compression–compression fatigue conditions is studied, and the mechanical properties and failure laws under fatigue loads are revealed. Finally, the stiffness attenuation and frequency of the material under fatigue load are combined to obtain the frequency attenuation curve under fatigue load, which is of great significance for the health monitoring of structures in actual engineering and the prediction of remaining fatigue life through the decline rate of natural frequencies.

Theory of fatigue life of composite materials

Fatigue failure criterion

With the degradation of unit stiffness and strength, various damages will occur inside the composite material, so it is necessary to define a suitable failure criterion to identify fatigue damage. Considering the various failure modes that may occur in composite materials under fatigue loads, the fatigue judgment criteria³² for the failure modes selected in this paper are as follows:

Fiber compression fatigue failure criteria ( $σ_{11}$ <0):

{(\frac{σ_{11}}{X_{C} (n, σ, k)})}^{2}

(1)

Fiber tensile fatigue failure criteria ( $σ_{11} \geq$ 0):

{(\frac{σ_{11}}{X_{T} (n, σ, k)})}^{2} + {(\frac{σ_{12}}{S_{12} (n, σ, k)})}^{2} + {(\frac{σ_{13}}{S_{13} (n, σ, k)})}^{2} \geq 1

(2)

Matrix-fiber shear fatigue failure criterion ( $σ_{11}$ <0):

{(\frac{σ_{11}}{x_{C} (n, σ, k)})}^{2} + {(\frac{σ_{12}}{S_{12} (n, σ, k)})}^{2} + {(\frac{σ_{13}}{S_{13} (n, σ, k)})}^{2} \geq 1

(3)

Matrix tensile fatigue failure criterion ( $σ_{22} \geq$ 0):

{(\frac{σ_{22}}{Y_{T} (n, σ, k)})}^{2} + {(\frac{σ_{12}}{S_{12} (n, σ, k)})}^{2} + {(\frac{σ_{23}}{S_{23} (n, σ, k)})}^{2} \geq 1

(4)

Matrix extrusion fatigue failure criterion ( $σ_{22}$ <0):

{(\frac{σ_{22}}{Y_{C} (n, σ, k)})}^{2} + {(\frac{σ_{12}}{S_{12} (n, σ, k)})}^{2} + {(\frac{σ_{23}}{S_{23} (n, σ, k)})}^{2} \geq 1

(5)

Delamination Tensile Fatigue Failure Criterion ( $σ_{33} \geq$ 0):

{(\frac{σ_{33}}{z_{T} (n, σ, k)})}^{2} + {(\frac{σ_{13}}{S_{13} (n, σ, k)})}^{2} + {(\frac{σ_{23}}{S_{23} (n, σ, k)})}^{2} \geq 1

(6)

Delamination Compressive Fatigue Failure Criterion ( $σ_{33} <$ 0):

{(\frac{σ_{33}}{z_{C} (n, σ, k)})}^{2} + {(\frac{σ_{13}}{S_{13} (n, σ, k)})}^{2} + {(\frac{σ_{23}}{S_{23} (n, σ, k)})}^{2} \geq 1

(7)

where

σ_{i j}

represents the stress tensor at the material integration point,

X_{T} (n, σ, k)

and

Y_{T} (n, σ, k)

represent the longitudinal and transverse residual tensile strength of the single-layer plate under uniaxial fatigue.

X_{C} (n, σ, k)

and

Y_{C} (n, σ, k)

represent the residual compressive strength of the single-layer plate in the longitudinal and transverse directions under uniaxial fatigue.

Z_{T} (n, σ, k)

and

Z_{C} (n, σ, k)

represent the residual tensile strength and residual compressive strength of the single-layer plate in the normal direction under uniaxial fatigue.

S_{12} (n, σ, k)

S_{13} (n, σ, k)

S_{23} (n, σ, k)

represent shear residual fatigue strength of the laminate under uniaxial shear fatigue load, k is the stress ratio, and n is the number of cycles.

When a composite material component fails, the number of cycles corresponding to the material is the ultimate fatigue life of the composite material. Considering that the fiber is the main load-bearing part of the composite material structure, when the fiber damaged unit extends to the edges of both sides of the composite laminate along the vertical load direction, it is determined that the overall fatigue failure of the composite material occurs.^33,34

Residual strength and residual stiffness model

Under the fatigue load, the stiffness and strength of the composite material will gradually degrade, and the expression of the residual stiffness degradation model²⁰ is as follows:

E (n, σ, k) = {[1 - {(\frac{\log n - \log 0.25}{\log N_{f} - \log 0.25})}^{λ}]}^{1 / γ} (E_{0} - \frac{σ}{ε_{f}}) + \frac{σ}{ε_{f}}

(8)

where

E (n, σ, k)

represents the remaining stiffness;

E_{0}

is the initial static stiffness; n is the number of applied cycles;

σ

is the magnitude of applied maximum stress;

N_{f}

represents the fatigue life at

σ

, and k represents the stress ratio;

ε_{f}

is the average strain to failure; γ, λ are the curve fitting parameters obtained from the experiment.

The expression of the residual strength degradation model¹⁶ is as follows:

R (n, σ, k) = {[1 - {(\frac{\log n - \log 0.25}{\log N_{f} - \log 0.25})}^{β}]}^{1 / α} (R_{0} - σ) + σ

(9)

where

R (n, σ, k)

is the residual strength;

R_{0}

is the static strength; α, β are the experimental fitting parameters.

When the fatigue damage continues to accumulate, unit damage occurs inside the composite laminate structure. It is necessary to perform sudden drop treatment on the internal units of the composite structure.^35,36 The stiffness degradation scheme in this paper is shown in Table 1:

Table 1.

Sudden drop scheme of composite material stiffness.

Failure mode	Sudden drop stiffness	Degradation coefficient
Fiber tensile mode	$E_{11}$ 、 $E_{22}$ 、 $E_{33}$ 、 $G_{12}$ 、 $G_{13}$ 、 $G_{23}$ 、 $v_{12}$ 、 $v_{13}$ 、 $v_{23}$	0.1
Fiber compress mode	$E_{11}$ 、 $E_{22}$ 、 $E_{33}$ 、 $G_{12}$ 、 $G_{13}$ 、 $G_{23}$ 、 $v_{12}$ 、 $v_{13}$ 、 $v_{23}$	0.1
Matrix tensile mode	$E_{22}$ 、 $G_{12}$ 、 $G_{23}$ 、 $v_{12}$ 、 $v_{23}$	0.2
Matrix compress mode	$E_{22}$ 、 $G_{12}$ 、 $G_{23}$ 、 $v_{12}$ 、 $v_{23}$	0.2
Matrix–fiber shear mode	$G_{12}$ 、 $v_{12}$	0
Delamination tensile mode	$E_{33}$ 、 $G_{13}$ 、 $G_{23}$ 、 $v_{13}$ 、 $v_{23}$	0
Delamination compress mode	$E_{33}$ 、 $G_{13}$ 、 $G_{23}$ 、 $v_{13}$ 、 $v_{23}$	0

Equal-life model

In the general research of life model, if the fatigue performance of composite materials is described for each stress state and stress ratio, a large number of experiments need to be carried out. In order to make the fatigue model have better applicability, the researchers introduced the life prediction model. The constant life model combined with the residual strength and residual stiffness models can avoid the limitation of using special stress ratios in the fatigue life analysis process.

The equal-life model established by Beheshty²² can predict the fatigue life of unidirectional plates, and its longitudinal expression is as follows:

u = \frac{\ln (a / f)}{\ln [(1 - m) (c + m)]} = A + B \lg N_{f}

(10)

where f, A, and B are fitting coefficients, and f = 1.06, which is obtained by Beheshty.²²

The definition of the parameters in the equivalent life expression is as follows:

{\begin{cases} S t r e s s a m p l i t u d e ༚ σ_{a} \\ S t r e s s m e a n ༚ σ_{m} \end{cases} \begin{array}{l} = (σ_{\max} - σ_{\min}) / 2 \\ = (σ_{\max} + σ_{\min}) / 2 \end{array}

(11)

{\begin{cases} C o e f f i c i e n t - a ༚ a = \frac{σ_{a}}{σ_{t}} \\ C o e f f i c i e n t - m ༚ m = \frac{σ_{m}}{σ_{t}} \\ C o e f f i c i e n t - c ༚ c = \frac{σ_{c}}{σ_{t}} \end{cases}

(12)

where σ_{α}

and

σ_{m}

represent stress amplitude and average stress,

σ_{t}

and

σ_{c}

correspond to tensile strength and compressive strength, respectively.

For the life model in the transverse direction of the matrix, literature^37–39 has improved the initial model, and the expression of the improved matrix equal-life model is as follows:

\hat{u} = \frac{\ln (a / f)}{\ln [((σ_{m} / Y_{C}) + 1) ((Y_{T} / Y_{C}) - (σ_{m} / Y_{C}))]}

(13)

For the constant life model in the shear direction, the longitudinal and transverse constant life models have the same effect on the fatigue life in the positive and negative directions for the working conditions subjected to shear fatigue loads. Thus it is not necessary to define the ratio of compressive strength to tensile strength in equation (10). Therefore, for unidirectional laminates, in the fatigue life model under shear fatigue load c = 1, the modified expression is as follows:

u = \lg (\frac{\ln (z / f)}{\ln [(1 - m) (1 + m)]}) = A + B \lg N_{f}

(14)

where z in the formula is calculated according to the following formula:

z = \frac{σ_{a}}{σ_{s h e a r}}

(15)

where

σ_{s h e a r}

is the static shear strength.

Fatigue finite element model and modal analysis

The fatigue and modal calculation process of composite materials is shown in Figure 1. When the progressive degradation of strength and stiffness under fatigue load is not considered, the static strength of composite materials is obtained. First, according to the static strength calculation results, different stress levels are selected to determine the fatigue load of the model, and then on the basis of the static strength prediction. An equal-life model and a progressive degradation model of stiffness and strength of the composite material under fatigue load are introduced, and the number of fatigue cycles is continuously increased. In order to analyze the structural behavior of the component subjected to fatigue loading, this study initially determines the stiffness value of the element at a specific cycle number. Subsequently, the obtained element stiffness value, which replaces the initial material state, is incorporated into the frequency-life relationship under cyclic fatigue loading. Considering the substantial number of fatigue calculations typically involved, encompassing tens of thousands of cycles, and to ensure computational accuracy, a fatigue cycle assumption is adopted. In each fatigue cycle, only the damage and stiffness degradation occurring under the maximum fatigue load are taken into account. Thus, a two-step analysis process is established, where the first step pertains to the application of the maximum fatigue load, while the second step involves the fatigue load. Consequently, each incremental step represents a specific number of cycles.

Figure 1.

Flowchart of composite material fatigue life prediction.

In the calculation of the tensile strength of the composite material, L = 152.4 mm, B = 76.2 mm, width W = 38.1 mm, and the aperture is 6.35 mm. The layering sequence is ${[0 / \pm 45 / 90]}_{3 s}$ , the thickness of a single layer is 0.146 mm, and the thickness H is 3.504 mm. In the prediction of compressive strength, L = 139.7 mm, W = 38.1 mm, and the hole diameter is 6.35 mm. The layering sequence is ${[0 / \pm 45 / 90]}_{4 s}$ , and the thickness H is 4.672 mm. The strength and fatigue life prediction model of composite materials is shown in Figure 2. The boundary conditions for static strength prediction are fixed support boundary conditions on the left, displacement loads are applied to the right end, and uniform pressure is applied to the right end of the fatigue load boundary conditions. In this paper, the calculated ultimate tensile strength of the composite material is 453 MPa, while the compressive ultimate strength is 354 MPa. The experimental values of tensile and compressive strength given in Ref. 40 are 60ksi (413.68 MPa) and 50 ksi (344.73 MPa), respectively. The tensile strength prediction error in this paper is 9.5%, while the compression strength prediction error is 2.7%.

Figure 2.

The finite element model of composite material strength and fatigue prediction.

Under different stress levels, this paper analyzes the tension-tension fatigue when the stress ratio k = 0.1, and the compression–compression fatigue when the stress ratio k = 10. The simulation results are compared with those in the literature.⁴⁰ The composite material selected in this paper is AS4/3501-6, which is used in the strength and fatigue life prediction, and its performance parameters are shown in Table 2. After testing and verification, the strength parameters fitted by fatigue experiments were corrected, and the relevant data are shown in Table 3.³⁹

Table 2.

Mechanical property parameters of AS4/3501-6 composite materials.⁴¹

Characteristic parameters of composite materials
$E_{11}$	147 GPa	$X_{T}$	2004 MPa
$E_{22}$	9 GPa	$X_{C}$	1197 MPa
$E_{33}$	9 GPa	$Y_{T}$	53 MPa
$v_{12}$	0.3	$Y_{C}$	204 MPa
$v_{13}$	0.3	$Z_{T}$	53 MPa
$v_{23}$	0.42	$Z_{C}$	204 MPa
$G_{12}$	5 GPa	$S_{12}$	137 MPa
$G_{13}$	5 GPa	$S_{13}$	137 MPa
$G_{23}$	3 GPa	$S_{23}$	42 MPa

Table 3.

Composite material AS4/3501-6 fatigue fitting data table.

	$λ$	$γ$	$ε_{f}$	$α$	$β$	A	B
Longitudinal tensile	14.57	0.3024	0.0136	0.473	10.03	1.3689	0.1097
Longitudinal compress	14.57	0.3024	0.0136	0.025	49.06	1.3689	0.1097
Transverse tensile	14.77	0.1155	0.0068	0.1255	9.628	0.999	0.096
Transverse compress	14.77	0.1155	0.0068	0.0011	67.63	0.999	0.096
In plane shear	0.70	11	0.101	0.16	9.11	0.0999	0.186
Out of plane shear	0.70	11	0.101	0.2	12	0.299	0.111

Fatigue failure modes and degradation

Damage evolution of tension-tension fatigue failure

Under the fatigue load, the composite fiber and matrix failure modes are shown in Figures 3–5. The composite tensile parts did not fail at the cycle of $10^{6}$ , at the 70% stress level, a small amount of fiber damage began to appear in the 0° ply. At the 80% stress level, the fatigue damage propagated a little, but not to both sides of the composite laminate. A certain amount of matrix damage occurs in the 90° ply at 40% stress level, and the matrix damage of the 90° ply expands to both sides of the composite laminate as the stress level increases. When the stress level reaches 70%, the accumulated damage almost fills the circular hole area of the composite material. At the same time, the matrix damage of the 45° and −45° plies starts from the vicinity of the round hole and spreads to both sides of the composite laminate, and there is almost no matrix tensile damage in the 0° ply. Since the stiffness and strength values in the matrix direction are much smaller than those in the fiber direction, even if a large area of matrix damage occurs, the composite structure does not fail completely, and there is no fiber compression or matrix compression damage in each ply unit of the composite material.

Figure 3.

The tensile fatigue damage of fibers in each layer.

Figure 4.

The tensile fatigue damage of each laminated matrix.

Figure 5.

The fiber compression fatigue damage of each layer.

Considering Figure 6 that the strength and stiffness of the composite material will undergo gradual degradation under fatigue load, the degradation contours of the stiffness cycles of E22 and G12 under $10^{6}$ cycles are shown in Figures 7 and 8. The degradation trend is consistent with the failure mode of the matrix and fibers. The fatigue damage is initiated from the vicinity of the round hole and spreads to both sides of the composite laminate. For elements with fatigue failure, the degradation is performed according to the stiffness drop criterion. For elements that have not experienced fatigue failure, progressive degradation is performed. For example, when the matrix tensile damage occurs in the 90° ply, its stiffness degenerates to about 1800 MPa, and for the element where fiber and matrix damage occurs, the stiffness value degenerates to 180 MPa.

Figure 6.

The Matrix compression fatigue damage of each layer.

Figure 7.

E22 damage degradation nephogram of $10^{6}$ fatigue cycles of each laminate.

Figure 8.

G12 damage degradation nephogram of $10^{6}$ fatigue cycles of each laminate.

In order to reveal the progressive degradation process of the stiffness of the composite material under different cycles, the strength value is normalized. Figure 9 shows the Yt performance degradation process of 90° laminates at 40% stress level and different cycles. The strength degradation starts near the hole and expands to both sides of the composite laminate. Figure 10 shows the stiffness degradation process of the 90° ply at 50% stress level under different cycles. The matrix damage occurred in the ply, and its stiffness value degraded to 1000 MPa, and the stiffness value of the unfailed unit underwent gradual degradation.

Figure 9.

Yt degradation of 90° laminate under different cycle times at 40% stress level.

Figure 10.

G12 Degradation of 90° ply under different cycles at 50% stress level.

To quantify and visualize the level of damage a measure of the relative reduction in the strength parameter due to damage $D_{P}$ , is calculated using equation (16).

D_{P} = 1 - \frac{P_{r}}{P_{initial}}

(16)

Compression–compression fatigue damage and evolution process

In this paper, 80% stress level was used to show the unit failure statistics of various failure modes in the area near the hole. First, the units of all layers near the hole are counted, and then the failure ratio of various damage units in the round hole area is obtained according to the damage mode. It can be found that the units with fatigue damage under fatigue loads are mainly matrix compression damage, followed by matrix–fiber shear damage units, and the proportion of fiber compression damage is about 20%. Considering the matrix–fiber shear fatigue criterion includes the strength degradation items of S12 and S13, the fatigue failure elements are more than fiber compression damage elements. Because the composite structure is finally crushed under the fatigue load, a small number of delaminated compression fatigue damage elements appear in Figure 11.

Figure 11.

Damage ratio of various failure modes in the hole area.

In order to study the damage evolution of composites under compressive-compressive fatigue loading, the failure modes of different plies were analyzed. Figure 12 shows the fiber compression damage and its evolution law of the 0° ply under fatigue load. There is almost no fiber tensile damage in the 0° ply under 60% stress level. With the increase of the stress level, the fiber compression damage starts near the hole and spreads to both sides of the composite laminate. At stress levels of 80% and 90%, when n/N = 0.5, a small number of fiber damage units begin to extend to the edge, the laminate structure still has a certain bearing capacity and has not been completely destroyed. When a large number of continuous fiber damage units extend to both sides of the composite laminate, fatigue damage occurs, and this cycle is the fatigue life of the composite structure.

Figure 12.

The fiber compression damage and its evolution law of the 0° ply.

Figure 13 shows the matrix compression fatigue cumulative damage of different laminates at the 80% stress level. It can be seen that no matrix fatigue damage occurs in the 0° laminate, while the matrix damage unit of other laminates extends to both sides of the laminate. Observing Figures 12–16, it can be seen that the fiber compression damage of the 0° ply near the hole extends to both sides of the sample, which is the final fatigue failure, and a certain amount of interlaminar compression units appear in the and-45° ply. Considering that there are some differences in the final fatigue failure modes of composite laminates under different stress levels, the 0° ply units with ultimate fatigue failure at the 70% stress level are slightly lower than the units at the 80% and 90% stress levels, and interlaminar compression damage units appear. At the same time, considering the applied compressive fatigue load, the stress state did not enter the tensile failure mode or did not meet the fatigue tensile failure criterion, and no fiber tensile damage and matrix tensile damage occurred in each laminate. When fatigue failure occurs, no fiber compression damage occurs in the 90° ply.

Figure 13.

Matrix compression fatigue cumulative damage at the 80% stress level.

Figure 14.

Matrix compression damage under different cycles at 90% stress level.

Figure 15.

Cumulative damage and evolution of 45° ply matrix–fiber shear fatigue.

Figure 16.

Cumulative damage and evolution of 45° ply delamination compression fatigue.

In order to reveal the cumulative damage process of the matrix compression fatigue, the −45° laminate under the 90% stress level is selected for discussion. Under high stress level, a certain amount of damage appears at the beginning, and as the number of cycles increases, the stiffness and strength of the unit are degraded. When n/N = 0.5, a small amount of matrix damage unit extends to both sides of the plate. When the structure fails, a large number of matrix compression elements expand to both sides of the plate.

According to Figure 15, it can be seen that there is no matrix-fiber shear damage at 60% and 70% stress levels, and only a few units of the −45° ply are degraded. The units of fiber shear degradation are sparser than those of fiber compression damage. Observing Figure 16, it can be seen that at the 60% stress level, no fatigue failure occurred after

10^{6}

cycles, and no interlaminar compression damage was observed. At the 70% stress level, a certain amount of interlaminar compression units appear when the structure undergoes fatigue failure. At the stress level of 80% and 90%, when n/N = 0.5, a small number of damage units appear on both sides, and there are crushing conditions on both sides under high stress levels, and more interlaminar compression damage units appear in Table 4.

Table 4.

Comparison of compression–compression fatigue life prediction of composite materials and experimental results in literature.⁴⁰

Stress level(%)	Test value	Test mean	Predictive value	Test log life	Predicted log life	error
90	27	55	52	1.74	1.71	1.72%
	69
	69
80	1326	3376	2100	3.53	3.32	5.95%
	4947
	3854
70	30,547	33,636	72,000	4.53	4.86	−7.28%
	38,999
	31,363
60	1000000	1,000,000	1,000,000	$6$	$6$	/
	1000000
	1000000

Comparing the composite material compression–compression fatigue S-N curve in Figure 17, it can be found that the correlation between composite material simulation and experimental curve fitting is high, and the overall error is less. Therefore, the fitted S-N curve can be used to evaluate fatigue problems under different stress levels.

Figure 17.

Compression–compression fatigue S-N curves of composite materials.

Figure 18 shows the triple error band of the average value of the compression–compression fatigue of composite materials. The three experimental values and predicted values of each group under the three stress levels are all within the triple error band, which validate the reliability of the fatigue life prediction calculation model established in this paper.

Figure 18.

Triple error band for compression–compression fatigue of composite materials.

Modal analysis at a specific number of cycles

The stiffness of the material will decrease under fatigue loading, which will change the frequency of the structure. In this paper, the stiffness value at a specific cycle is derived, and the relationship between fatigue life and frequency attenuation is established Figures 19 and 20. Firstly, the fatigue life of composite laminates is evaluated, and then the Umat subroutine is used to derive the stiffness value of the unit under the specified number of cycles. At the same time, the python script is used to make each unit have independent material parameters, and an inp file is generated. The stiffness values for the specified number of cycles are read, and the initial modes are replaced to calculate the stiffness values for the model material. Finally run the inp file to obtain the relationship between frequency attenuation and life under different cycle numbers. When performing modal analysis on composite laminates, the left end is a fixed boundary, and its first six frequencies and mode shapes are shown in Figure 21. By normalizing the frequency and life, the curve of fatigue life and frequency attenuation can be obtained.

Figure 19.

Flowchart of modal calculation for composite laminates.

Figure 20.

Calculation model for modal analysis of composite laminates.

Figure 21.

Modes and vibration shapes of composite laminates.

Tensile–tensile fatigue frequency attenuation curve

Figure 22 (a)-(e) shows the frequency attenuation under different stress levels (80%, 70%, 60%, 50%, 40% stress level respectively), it is evident that an increase in stress level leads to a more pronounced frequency attenuation. Notably, the higher-order mode exhibits greater sensitivity to fatigue damage in composite materials. In the context of the tensile test specimens subjected to fatigue loading, which showed no signs of fatigue damage, the frequency decay curve indicates that the residual frequency reaches a stable state after a certain number of cycles.

Figure 22.

Frequency attenuation diagram under different stress levels.

Compression–compression fatigue frequency attenuation curve

Figure 23 are the frequency attenuation life curves of composite materials under compression–compression fatigue loading. At 60% stress level, the structure has no fatigue damage and its frequency attenuation is small, while at 80% stress level, the structure has fatigue damage. The frequency attenuation is more obvious, and the second-order mode is more sensitive to fatigue damage at the 60% stress level.

Figure 23.

Frequency attenuation diagram under different stress levels.

Conclusions

In this paper, based on the progressive damage analysis method and fatigue cumulative damage theory, a finite element model for fatigue cumulative damage and life prediction of composite materials is constructed. The tensile–tension fatigue and compression–compression fatigue damage characteristics of composite materials were studied, the damage evolution process of different laminates of composite materials under fatigue load was analyzed, and the relationship between fatigue life and frequency attenuation of composite materials was revealed. The fatigue life of the structure can be indirectly predicted by detecting the natural frequency of the structure, which is of great value to engineering applications. The conclusions are as follows:

(1) The finite element model of fatigue cumulative damage and life prediction of composite materials has good accuracy, and can analyze the problems of composite material tension-tension fatigue and compression–compression fatigue life accurately. The simulation results of tension–tension fatigue life prediction are in good agreement with the experiment. In the life prediction of compression–compression fatigue, the prediction results under four stress levels are all within the triple error band.

(2) In the calculation of tensile–tension fatigue life of composite materials, under the five fatigue loads, no fatigue damage occurs in the experimental and simulation results, which shows that the composite laminate has good fatigue performance. However, a certain amount of fiber and matrix failure occurred inside the composite, along with a progressive degradation of stiffness and strength.

(3) In the process of tension–tension fatigue life prediction, the tensile fatigue cumulative damage of the 90° laminate matrix expands from the center of the hole to the two sides of the laminate. At low stress levels, the matrix tensile damage did not extend to both sides of the board, and at high stress levels, the matrix tensile damage of the 90° ply laminate extended to the entire ply area.

(4) The natural frequency of a structure decays as the stress level increases. For compression–compression fatigue, at the 80% stress level, the frequency of each order decreases more rapidly with the accumulation of fatigue cycles. Considering that fatigue damage mainly occurs near the round hole, when it is close to its life, the fifth order frequency attenuates by about 0.35. In actual engineering, the fatigue life of the structure can be predicted by monitoring the frequency of the composite material structure.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Natural Science Foundation of Chongqing (Grant No. CSTB2023NSCQ-MSX0802), the Science and Technology Research Program of Chongqing Municipal Education Commission (Grant No. KJQN202201105, KJQN202201113).

ORCID iDs

Yu Yang

Zijun Zheng

Jiaru Shao

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Study on fatigue damage and frequency attenuation characteristics of carbon fiber composite

Abstract

Keywords

Introduction

Theory of fatigue life of composite materials

Fatigue failure criterion

Residual strength and residual stiffness model

Equal-life model

Fatigue finite element model and modal analysis

Fatigue failure modes and degradation

Damage evolution of tension-tension fatigue failure

Compression–compression fatigue damage and evolution process

Modal analysis at a specific number of cycles

Tensile–tensile fatigue frequency attenuation curve

Compression–compression fatigue frequency attenuation curve

Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iDs

References