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Fatigue life prediction method of carbon fiber-reinforced composites

  • Jiamei Lai , Yousheng Xia , Zhichao Huang EMAIL logo , Bangxiong Liu , Mingzhi Mo and Jiren Yu
Published/Copyright: May 30, 2024
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e-Polymers
From the journal e-Polymers Volume 24 Issue 1

Abstract

The use of composite laminates is characterized by problems such as poor inter-layer bonding and susceptibility of material properties to fatigue cracking, which seriously threaten structural safety. Research on fatigue damage characteristics and fatigue life prediction of fiber-reinforced composites can help to solve such problems. Carbon fiber-reinforced epoxy resin matrix composite laminates are taken as the object of this study. By analyzing the fatigue failure process and the fatigue failure micromorphology of the specimen, the primary damage forms and fatigue damage characteristics of its fatigue failure were obtained. The fatigue failure process of fiber-reinforced composites was simulated using finite element analysis software ABAQUS and its UMAT subroutine function. The tensile–tensile fatigue damage characteristics and failure mechanism of fiber-reinforced composites were studied, and the fatigue life of the composites was predicted. The feasibility of this life prediction method was verified by comparing it with experimentally obtained damage processes and fatigue lives. This intuitive and reliable life prediction method has good research potential for predicting the fatigue limit of fiber-reinforced composites.

Keywords: carbon fiber-reinforced composites; fatigue failure characteristics; fatigue life models; fatigue life prediction

1 Introduction

Carbon fiber-reinforced composites are high-performance composites that use carbon fiber as a reinforcing material to strengthen the matrix of resins, rubber, and ceramics (1,2,3). With low specific gravity, high specific modulus and strength, and good designability, carbon fiber composites have a wide range of applications in many fields, including aerospace vehicles, civil engineering and construction, and sporting goods (4,5,6). With the deepening of technology and the development of applications, these areas are setting higher standards for the performance of composite materials, which includes the requirement for fatigue life (7,8,9). For example, many components on aircraft are subjected to high-frequency, low-amplitude cyclic loads of 108–1012 cycles. In engineering practice, fatigue cracking leads to stiffness degradation and reduced load-carrying capacity, threatening component safety (10). In addition, aerospace vehicles operate in harsh environments and under complex and variable loading conditions. The degradation of composite materials under fatigue loading is one of the reasons for more damage in the early years of service (11).

Composites face serious fatigue problems in their applications. Understanding the fatigue damage characteristics and mechanisms of composites and predicting the fatigue limit and fatigue life of composite components can help ensure the safety and reliability of composite components in use. This has important practical implications. With the in-depth research on the fatigue performance of composite materials, more and more scholars have proposed different types of fatigue life prediction models for the fatigue damage characteristics of fiber composites. With the in-depth research on the fatigue performance of composite materials, more and more scholars have proposed different types of fatigue life prediction models for the fatigue damage characteristics of fiber composites. Yao and Himmel (12) constructed and demonstrated a cumulative fatigue damage model for predicting the life of fiber-reinforced composites under variable-amplitude fatigue loading, which represents the damage state of composite laminates through loss of residual strength based on the strength decay phenomenon. Ahmadzadeh et al. (13) evaluated the effect of different layups of quasi-isotropic carbon fiber composites on the development of fatigue damage and proposed a stiffness-based progressive damage model. Zhang et al. (14) developed a fatigue life prediction model by conducting tensile–tension cyclic loading tests and monitoring the fatigue damage during the tests, taking into account the fracture morphology and damage accumulation of the composite material. The results showed that the method could better predict the fatigue life of composite laminates under tensile–tension loading. Feng et al. (15) conducted tensile fatigue tests on composite laminates at different fatigue stress levels, analyzed and compared the specimens’ static and fatigue failure modes, and finally proposed a new S–N curve model with good accuracy with the experimental results. Frydrych et al. (16) prepared multi-axial warp-knitted fabric/matrix-reinforced composite samples using vacuum-assisted resin transfer molding. They were subjected to tensile–tensile fatigue cycles at different load levels, and equivalent residual strength and stiffness degradation were obtained. Ramakrishnan and Mallick (17) conducted tests using butterfly-shaped Arcan specimens and statistical analysis using the Weibull distribution, and they concluded that the tensile strength of the material decreases with increasing shear stress. Yudhanto et al. (18) sewed laminates with different seam densities using Vectran threaded sewing needles and performed experiments with unstitched specimens, showing that the fatigue life of the specimens with a seam density of 3 mm × 3 mm was slightly better than that of the unstitched as well as the seam density of 6 mm × 6 mm. The improvement in fatigue performance and the delay in the onset of stiffness degradation for specimens with a stitch density of 3 mm × 3 mm were mainly due to the stitching process effectively impeding edge delamination. Quantitative analysis of the damage at different cyclic and stress levels was then carried out, and the results showed that stitch density mainly influenced the growth rate of delamination. Mall et al. (19) used a four-beam satin carbon fiber/epoxy composite prepared by the heat vacuum assisted resin transfer molding process and conducted a comparative study of fatigue experiments on specimens with and without notches. The experiments showed that nothing did not affect the fatigue life. Tang et al. (20) proposed a modified degradation law that could predict cumulative fatigue damage at two different displacement amplitude loads and different crack opening displacement ranges, and the predicted results were in good agreement with the experimental results. Huang et al. (21) used stable temperature rise data and a modified thermal imaging method to determine the fatigue limit, and using parametric calibration and normalized failure threshold stiffness calculations, the entire S–N curve could be obtained in a short time. Whitworth (22) evaluated the residual strength degradation of graphite/epoxy composite laminates and proposed a model relating residual strength to the number of fatigue cycles and maximum stress. Liu and Mahadevan (23) discussed the feasibility of using probabilistic models to describe the fatigue behavior of composites and quantify the uncertainty in fatigue life prediction. Safaei (24) developed a finite element method model based on Galerkin’s method to analyze the elastic stress field of a flat plate subjected to axial loading. The cell method was used to model the composite material, as special boundary conditions were applied to the model. The validity of the model was verified by calculating the displacement, strain, and stress comparisons of the junctions by MATLAB software. Lian and Yao (25) proposed a new stiffness degradation model and then derived a strength degradation model and coupled it with the proposed stiffness degradation model, thus predicting the fatigue behavior and fatigue life of E-glass/epoxy composite laminates. Huang et al. (21) proposed a new constant-amplitude life prediction model that better describes the longitudinal fatigue behavior of unidirectional laminates under large stress ratios and has good accuracy in either tension–compression or compression–compression modes. Aoki et al. (26) simulated intra-ply and inter-ply damage in composite laminates using a continuum damage mechanics model and an adhesion cohesion model and observed the progressive evolution of damage in laminates during fatigue by evaluating both types of damage.

Composites are mainly exposed to complex fatigue loads in the form of tension–tension in practical engineering applications. The fatigue failure microstructure of the specimens was analyzed using tensile and tensile–tensile fatigue tests for carbon fiber-reinforced epoxy resin matrix composite laminates. The fatigue failure process of the composites was simulated using finite element analysis techniques and the subroutine function of ABAQUS. The tensile–tensile fatigue damage characteristics of the composites and the failure mechanisms of the composites were investigated, and the prediction on the fatigue life of the composites was made.

2 Experiment

2.1 Preparation of specimens

The carbon fiber-reinforced epoxy resin matrix composite for the experiment was prepared using the vacuum assisted resin transfer molding (VARTM) composite molding process as shown in Figure 1(a). The CF12-300 unidirectional carbon fiber sheet (Xiamen Weman Material Technology Co., Ltd.) was used to prepare the composite laminates. The carbon fiber was laid up with a stacking sequence of [0/90]4. The resin material was epoxy resin R688. The curing agent was amine H3268 (Xiamen Weman Material Technology Co., Ltd.). The main processes for specimen manufacturing are as follows: a clean, flat piece of tempered glass served as the mold. First, the cut carbon fiber sheets were stacked flat and placed in the middle of the mold in the layup sequence. To make the carbon fiber laminate easy to demold after molding, the carbon fiber material was made to be wrapped by the release sheet. It was covered with a flow-through mesh, which guided the resin to cover and saturate the carbon fiber layup quickly. Finally, the mold was closed through a vacuum bag and sealing tape. After the vacuum mold was prepared, the resin inlet was closed. The air outlet was connected to the vacuum pump, and the air pressure inside the vacuum mold was pumped to −0.3 MPa by the vacuum pump. The air extraction port was closed, and the vacuum mold was left to stand for 10 min to see whether there are any air leaks in the vacuum mold. After the vacuum mold was made sure that there were no leaks, the epoxy resin was proportioned with the hardener in a ratio of 5:1 and stirred for about 8 min. Through the pumping force of a vacuum pump, the epoxy resin mixture was transported into the vacuum mold, and the epoxy resin mixture completely saturated the carbon fibers. Subsequently, the vacuum mold was kept sealed. The vacuum mold was removed after 20 h of curing at room temperature, and the carbon fiber laminate was manufactured. The carbon fiber volume fraction of the manufactured carbon fiber boards was 58%. According to American society of testing materials (ASTM) D3039 standard (27), the specimen shapes for this test were all of the same straight type. The specific parameters of the specimens are shown in Figure 1. First, the carbon fiber laminate was cut by the waterjet into straight strips of specimens. Subsequently, 6061 aluminum alloy with a thickness of 2.08 mm was pasted on both ends of the specimen as a reinforcing sheet.

Figure 1 Specimen fabrication process, shape, and dimensions of the specimen are expressed as follows: (a) diagram of the VARTM process and (b) main view and top view of the specimen.
Figure 1

Specimen fabrication process, shape, and dimensions of the specimen are expressed as follows: (a) diagram of the VARTM process and (b) main view and top view of the specimen.

2.2 Static tensile testing and fatigue testing

According to ASTM D3479 standard (28), static tensile test was performed on the ETM105D microcomputer-controlled electronic universal test machine (Shenzhen Wance Equipment Co., Ltd, China). The maximum test force was 100 kN. The length of the experimental specimen was 250.0 mm, the width was 25.0 mm, the average thickness was 2.5 mm, and the cross-sectional area was 62.5 mm². The specimen was clamped at both ends with equal length and parallel to the direction of movement of the fixture during the experiment. To ensure that the specimen was approximately static in tension, it was necessary to maintain a device force loading rate of 2 mm·min−1. The first breaking load of the stretching process was obtained using a force transducer and was recorded as the maximum stretching force of the specimen. The experiments were all carried out at a room temperature of 25℃. To obtain sufficient valid experimental data, a minimum of eight specimens were used for each set of experiments. The tensile strengths obtained from the experiments are shown in Table 1.

Table 1

Average of ultimate tensile strength of specimens

Load (kN) 60.89 62.66 65.74 61.93 57.01 Average value 61.65
UTS (MPa) 934.58 961.75 1,009.03 950.55 875.03 Average value 946.19

Tensile–tensile fatigue test for carbon fiber composites was performed according to ASTM D3039 standard (27). The maximum and minimum stresses of the test were tensile stresses to ensure that the specimen was in a tensile state at all times during the loading process. The experimental equipment was the INSTRON 8800 electro-hydraulic servo fatigue test system (Instron, USA). The loading waveform was a sine wave, and the loading frequency was 10 Hz. The loading frequency was precisely controlled so that the frequency variation did not exceed 0.5 Hz, the maximum and minimum load ratio was 0.1, and the load variation did not exceed 5 kN. The loaded fatigue load increased linearly to the set maximum load value within 2 s during the experiment. Subsequently, the fatigue load was varied according to the loading waveform and load amplitude. The fatigue test process was protected and limited by load and position limit values. Experiments were carried out at three average values of the maximum load, 55%, 60%, and 70% of the static tensile strength, with each load level repeated at least four times. The center line of the specimen was aligned with the alignment center line of the upper and lower clamps when the specimen was clamped. If the specimen failed under this load condition or if the number of cycles had exceeded 106, it was decided that the specimens had reached their fatigue limit. At the end of the experiment, the damage was checked and recorded, and the specimen was protected for further fracture analysis. The number of fatigue limit cycles obtained experimentally for different load levels is shown in Table 2.

Table 2

Number of fatigue cycle loading of the specimen

Load level Experiment 1 Experiment 2 Experiment 3 Experiment 4 Experiment 5 Experiment 6 Experiment 7 Experiment 8
70% UTS 12,690 27,123 29,724 42,538 15,324 23,445 9,712 17,668
60% UTS 45,661 387,273 149,071 199,342 238,823 165,233 267,474 324,562
55% UTS 74,801 897,441 196,095 707,418 176,655 642,173 246,687 654,483

2.3 Result and analysis

2.3.1 Process of fatigue failure

During the fatigue test loading of the specimens, these specimens all showed obvious fiber breaks, matrix cracks, and delamination damage, which are closely related to the composition of the laminate. At the same time, after analyzing the damage development throughout the fatigue loading process, it can be found that the damage of fiber-reinforced composites usually shows stage characteristics. Its fatigue process can be divided into three stages: the initial stage, the middle stage of development, and the final stage (Figure 2).

Figure 2 (a) Static tensile test and (b) fatigue tests.
Figure 2

(a) Static tensile test and (b) fatigue tests.

At the initial stage of fatigue loading (phase from 0 loading to 10% of the maximum number of cycles), when the fatigue load is rapidly and linearly increased from 0 to a standard static load value, the laminate is accompanied by many crisp fracture sounds, indicating that the initial damage within the laminate occurred before cyclic loading. The earliest damage to the laminate occurs at the edge of the laminate after cyclic loading has started. Figure 3(b) shows that many tiny intra-laminar matrix cracks first appear at the edges. These cracks expand as the load cycle increases, with the matrix cracks gradually developing toward the middle of the laminate until the cracks reach the fibers. As shown in Figure 3(c) and (d), some of the fine fibers then break and pull out of the laminate, while the rate of crack expansion increases. Cracking is mainly in the direction of fiber orientation, but cracks also develop slowly toward the width of the specimen. The initial damage is mainly in the form of intra-layer cracks and localized fine fiber fractures, which are slight and generally slow to develop.

Figure 3 At the initial stage of fatigue cyclic loading, the specimens showed some initial damage. The damage in the initial stage of the specimen was mainly in the form of intra-layer cracks and local fine fiber fracture, and the damage was relatively mild. The main manifestations are as follows: (a) 70% UTS, N: 543 the condition of the specimen without damage, (b) 70% UTS, N: 1,357 specimens with localized fine fiber breakage, (c) 70% UTS, N: 1,899 matrix crack extension of laminate specimens, and (d) 70% UTS, N: 2,713 increased fiber breakage of laminate specimens. In the middle of the fatigue load cycle loading, the matrix crack within the layer accelerates to both ends of the specimen. Since the inter-layer cracks expand faster than the intra-layer cracks, it makes the inter-layer damage gradually become the main form of damage. The main manifestations are as follows: (e) 70% UTS, N: 6,780 matrix crack extension of laminate specimens, (f) 70% UTS, N: 10,849 increased matrix crack expansion in laminate specimens, (g) 70% UTS, N: 18,986 inter-laminar crack expansion in laminate specimens, and (h) 70% UTS, N: 20,342 increased inter-laminar crack expansion in laminate specimens. In the final stage of the fatigue load cycle loading, the inter-laminar cracks of the laminate specimens tend to saturate and the delamination damage is very severe. When the sound of fiber breakage becomes more frequent, a large number of fibers are pulled off and blown to various places, and finally, the specimen as a whole suddenly fails. The main manifestations are as follows: (i) 70% UTS, N: 21,427 laminate specimens show a significant delamination damage, (j) 70% UTS, N: 22,240 increased delamination damage of laminate specimens, (k) 70% UTS, N: 25,766 the laminate specimens showed obvious fiber breakage, and (l) 70% UTS, N: 27,123 the fiber breakage of the laminate specimens increased leading to complete failure.
Figure 3 At the initial stage of fatigue cyclic loading, the specimens showed some initial damage. The damage in the initial stage of the specimen was mainly in the form of intra-layer cracks and local fine fiber fracture, and the damage was relatively mild. The main manifestations are as follows: (a) 70% UTS, N: 543 the condition of the specimen without damage, (b) 70% UTS, N: 1,357 specimens with localized fine fiber breakage, (c) 70% UTS, N: 1,899 matrix crack extension of laminate specimens, and (d) 70% UTS, N: 2,713 increased fiber breakage of laminate specimens. In the middle of the fatigue load cycle loading, the matrix crack within the layer accelerates to both ends of the specimen. Since the inter-layer cracks expand faster than the intra-layer cracks, it makes the inter-layer damage gradually become the main form of damage. The main manifestations are as follows: (e) 70% UTS, N: 6,780 matrix crack extension of laminate specimens, (f) 70% UTS, N: 10,849 increased matrix crack expansion in laminate specimens, (g) 70% UTS, N: 18,986 inter-laminar crack expansion in laminate specimens, and (h) 70% UTS, N: 20,342 increased inter-laminar crack expansion in laminate specimens. In the final stage of the fatigue load cycle loading, the inter-laminar cracks of the laminate specimens tend to saturate and the delamination damage is very severe. When the sound of fiber breakage becomes more frequent, a large number of fibers are pulled off and blown to various places, and finally, the specimen as a whole suddenly fails. The main manifestations are as follows: (i) 70% UTS, N: 21,427 laminate specimens show a significant delamination damage, (j) 70% UTS, N: 22,240 increased delamination damage of laminate specimens, (k) 70% UTS, N: 25,766 the laminate specimens showed obvious fiber breakage, and (l) 70% UTS, N: 27,123 the fiber breakage of the laminate specimens increased leading to complete failure.
Figure 3

At the initial stage of fatigue cyclic loading, the specimens showed some initial damage. The damage in the initial stage of the specimen was mainly in the form of intra-layer cracks and local fine fiber fracture, and the damage was relatively mild. The main manifestations are as follows: (a) 70% UTS, N: 543 the condition of the specimen without damage, (b) 70% UTS, N: 1,357 specimens with localized fine fiber breakage, (c) 70% UTS, N: 1,899 matrix crack extension of laminate specimens, and (d) 70% UTS, N: 2,713 increased fiber breakage of laminate specimens. In the middle of the fatigue load cycle loading, the matrix crack within the layer accelerates to both ends of the specimen. Since the inter-layer cracks expand faster than the intra-layer cracks, it makes the inter-layer damage gradually become the main form of damage. The main manifestations are as follows: (e) 70% UTS, N: 6,780 matrix crack extension of laminate specimens, (f) 70% UTS, N: 10,849 increased matrix crack expansion in laminate specimens, (g) 70% UTS, N: 18,986 inter-laminar crack expansion in laminate specimens, and (h) 70% UTS, N: 20,342 increased inter-laminar crack expansion in laminate specimens. In the final stage of the fatigue load cycle loading, the inter-laminar cracks of the laminate specimens tend to saturate and the delamination damage is very severe. When the sound of fiber breakage becomes more frequent, a large number of fibers are pulled off and blown to various places, and finally, the specimen as a whole suddenly fails. The main manifestations are as follows: (i) 70% UTS, N: 21,427 laminate specimens show a significant delamination damage, (j) 70% UTS, N: 22,240 increased delamination damage of laminate specimens, (k) 70% UTS, N: 25,766 the laminate specimens showed obvious fiber breakage, and (l) 70% UTS, N: 27,123 the fiber breakage of the laminate specimens increased leading to complete failure.

As shown in Figure 3(e) and (f), as the cyclic loading continues (at 10–75% of the maximum number of cycles), the matrix cracks on the outermost 0° direction of the laminate continue to grow and become more pronounced. When the crack in the layer accelerates to the ends of the specimen, small cracks also appear between the laminate layers and expand toward the ends of the specimen, growing faster than the cracks within the layers. As shown in Figure 3(c) as well as (d), at this time, inter-layer damage gradually becomes the main form of damage, which is also an important symbol of fatigue loading damage development to the middle stage. In the thickness direction of the laminate, debonding between the different layers causes delamination initially at one end of the specimen in the thickness direction, with new delamination damage continuously occurring in the direction of the other end. Compared to the specimen in the stage of Figure 3(b), the mid-fatigue stage shows more forms of damage and considerably more damage. At the same time, the damage develops more rapidly, with inter-laminar cracks gradually extending to the ends of the specimen. However, there are fewer fracture sounds in the laminate, and the duration of the entire development phase is longer.

In the final stages of fatigue loading (between 75% and 100% of the maximum number of cycles), the inter-laminar cracks extend to the ends of the laminate and approach saturation, at which point the delamination damage is very severe. As shown in Figure 3(i) and (j), the specimens show the development of cracks within the layer, with more and more fibers in the tensile specimen breaking and pulling away from the layer. This is accompanied by a progressively more frequent fiber breakage sound, with a portion of the fractured carbon fibers breaking away from the original laminate and moving out of the specimen by friction. As shown in Figure 3(k) and (l), as the load was cyclically loaded, more fibers in the specimen split away from the matrix and the layup and were continuously stretched. This is accompanied by more frequent sounds of fiber breakage in the specimen and, finally, the sudden failure of the specimen as a whole. With a loud bursting sound, many fibers are pulled off and blown to various places. The final stage’s duration is shorter than the first two stages of fatigue loading. The damage to the specimen is already very evident after the development of the two steps, and the laminate properties decay severely so that the final damage occurs very quickly.

2.3.2 Micro-morphological analysis

In order to validate the subsequently proposed finite element model for fatigue life prediction along with the visible damage conditions in Section 2.3.1, optical microscopy and scanning electron microscopy were used to observe the failed specimens. The optical microscopic images were taken using an Axio Scope A1 Zeiss metallurgical microscope, and the scanning electron micrographs were taken using a Hitachi SU8010 cold-field scanning electron microscope.

Analysis of the fiber fracture appearance after fiber breakage shows that Figure 4(a) shows the fine fiber breakage and (b) shows that the carbon fibers in the middle layer of the laminate have moved after breakage. Through the aforementioned phenomenon, it can be judged that after the delamination of the layer, the fibers shifted spatially, leaving the original layer, and brittle fracture occurred after reaching the load-bearing limit. Figure 4(c) shows the expansion of inter-laminar cracks in laminates, and it can be seen that after the cracks are generated in the matrix, on the one hand, the cracks will expand in the same direction as the fiber orientation, leading to further delamination damage between the fibers and the matrix. On the other hand, cracks can penetrate through the substrate in the entire thickness direction, leading to the crack extension to another fiber-to-substrate contact surface and triggering further delamination damage. Figure 4(d) shows the surface specimen of the 90° layer after delamination failure, in which the silver-white color is the carbon fiber and the black color is the resin matrix. It can be seen that only a few scattered resins are adhered to the fiber surface, indicating that the interfacial cracks between the carbon fibers and the epoxy resin will be extended over a large area, which will lead to the aggravation of the interfacial debonding damage. Figure 4(e) shows the fiber fracture surface of the cracked side of the laminate surface, and (f) shows the fiber fracture morphology after fatigue fracture. The fiber fracture surfaces are relatively neat but have a certain step shape. Fracture of the fibers indicates that the load is transmitted in the direction of the thickness of the specimen. The first fibers to break are the surface fibers, followed by the inner fibers.

Figure 4 Optical microscope was used to analyze the fiber fracture morphology of laminates after fracture failure. The main manifestations are as follows: (a) 70% UTS, N: 6,780 the section at the beginning of the fiber break, (b) 70% UTS, N: 6,780 fiber movement, (c) 70% UTS, N: 20,342 inter-layer crack expansion, (d) 70% UTS, N: 20,342 interface debonding, (e) 70% UTS, N: 25,766 fiber breaks at cracks, and (f) 70% UTS, N: 25,766 fiber fracture pattern.
Figure 4

Optical microscope was used to analyze the fiber fracture morphology of laminates after fracture failure. The main manifestations are as follows: (a) 70% UTS, N: 6,780 the section at the beginning of the fiber break, (b) 70% UTS, N: 6,780 fiber movement, (c) 70% UTS, N: 20,342 inter-layer crack expansion, (d) 70% UTS, N: 20,342 interface debonding, (e) 70% UTS, N: 25,766 fiber breaks at cracks, and (f) 70% UTS, N: 25,766 fiber fracture pattern.

The surface of the carbon fibers is relatively smooth, as can be seen from the fibers and epoxy matrix samples within the layup layer obtained by electron scanning microscopy in Figure 5. As the untreated carbon fibers are usually inert (29), there is only a tiny amount of resin matrix on the surface of the fibers after debonding, reflecting the poor bonding of the fibers to the epoxy resin matrix. As a result, the interfacial properties are often unsatisfactory when used as a reinforcement with an epoxy resin matrix to form a composite. By analyzing Figure 5(b), it can be speculated that when the same layer of the composite material is loaded, there is a large gap between the strength of the fibers and the epoxy resin, which makes the deformation capacity of the epoxy matrix and the carbon fibers different and the deformation difference is more prominent. In addition, the interfacial bonding between the fibers and the epoxy resin is poor, which, on the one hand, causes the debonding of the fibers and the resin and, on the other hand, leads to fractures and cracks in the matrix. In addition, the fatigue specimens in Figure 5(b) and (c) show a different failure profile for the epoxy matrix compared to the tensile specimens in Figure 5(a). The specimens that fractured in tension had a more pronounced brittle fracture pattern with a wavy undulating surface. The specimens that failed in fatigue after cyclic loading had a less brittle matrix and a more subdued surface. Compared to the matrix in (b), the matrix in (c) similarly shows many split-independent matrix miniparticles after a greater number of cycles, but the surface of the matrix in (c) is less undulating. Analyzing this phenomenon with the load level and the fatigue loading process, it is concluded that producing such small particles of the finely ground matrix is related to the rate of crack expansion at lower load levels.

Figure 5 Use a scanning electron microscope for observation. The matrix of tensile fracture of the specimen is brittle, and its surface is undulating. However, after cyclic loading, the matrix fracture is less brittle and the surface is more peaceful. The main manifestations are as follows: (a) static tensile specimen morphology, (b) the specimen interface morphology after 27,123 cycles of fatigue loading, and (c) the specimen interface morphology after cyclic loading 149,071 cycles under fatigue load.
Figure 5

Use a scanning electron microscope for observation. The matrix of tensile fracture of the specimen is brittle, and its surface is undulating. However, after cyclic loading, the matrix fracture is less brittle and the surface is more peaceful. The main manifestations are as follows: (a) static tensile specimen morphology, (b) the specimen interface morphology after 27,123 cycles of fatigue loading, and (c) the specimen interface morphology after cyclic loading 149,071 cycles under fatigue load.

3 Fatigue life prediction methods

3.1 Fatigue life prediction theory

When the finite element software ABAQUS is used to establish the model and load the fatigue loading, the material properties of the carbon fiber laminate are degraded, fatigue damage is judged, and the stiffness degradation needs to have a definite judgment standard (30). The UMAT subroutine was used to make additions to the constitutive relationships of the carbon fiber laminates, thus continuously updating the stress relationships of the model. Shokrieh’s (31) guidelines for determining fatigue damage were applied to determine the occurrence of different forms of fatigue damage. The strength and stiffness degradation equations were used to obtain the fatigue residual strength and stiffness of the material composite. Naderi’s (32) method of sudden drop in properties was used for the failure determination of different material properties. Ultimately, carbon fiber laminate fatigue life was predicted.

3.1.1 Stress analysis

The mechanical properties of fiber composites are characterized by anisotropy and inhomogeneity, making the mechanical properties behave differently in different directions. Since the composite monolayer consists of two parts, the matrix and the fibers, the monolayer is considered to be homogeneous and orthotropically anisotropic when analyzing the mechanical properties of the material. Therefore, according to the mechanical theory of composites, the intrinsic relationship of anisotropic composites (33) can be expressed as follows:

(1) σ 1 σ 2 σ 3 τ 23 τ 31 τ 12 = C 11 C 12 C 13 C 14 C 15 C 16 C 21 C 22 C 23 C 24 C 25 C 26 C 31 C 32 C 33 C 34 C 35 C 36 C 41 C 42 C 43 C 44 C 45 C 46 C 51 C 52 C 53 C 54 C 55 C 56 C 61 C 62 C 63 C 64 C 65 C 66 ε 1 ε 2 ε 3 γ 23 γ 31 γ 12

where C ij is the stiffness coefficient and is an intrinsic property of the material and where C 12 = C 21, C 13 = C 31, and C 23 = C 32. Each stiffness coefficient expression can be expressed by the modulus of elasticity, shear modulus, and Poisson’s ratio of the material as follows (33):

(2) C 11 = 1 − ν 23 ν 32 E 2 E 3 ∆ C 12 = ν 12 + ν 13 ν 32 E 2 E 3 ∆ C 13 = ν 13 + ν 12 ν 23 E 2 E 3 ∆ C 22 = 1 − ν 13 ν 31 E 1 E 3 ∆ C 23 = ν 23 + ν 21 ν 13 E 1 E 3 ∆ C 33 = 1 − ν 12 ν 21 E 1 E 2 ∆ C 44 = G 23 C 55 = G 31 C 66 = G 12 ∆ = 1 − ν 12 ν 21 − ν 23 ν 32 − ν 13 ν 31 − 2 ν 12 ν 23 ν 31 E 1 E 2 E 3

where E 1, E 2, and E 3 are the moduli of elasticity of the material in the three axial directions, G ij is the in-plane shear modulus of the material, ν ij is the Poisson’s ratio of the material, and ∆ is the value of the third-order principal subequation that corresponds to the determinant in the flexibility matrix of the material.

From the aforementioned equation, the stress–strain relationship of a material can be expressed by the elastic modulus, shear modulus, and Poisson’s ratio. Under the effect of fatigue loading, the stress state of the composite also changes. As the number of fatigue cycles increases, the loading capacity of composites usually shows a tendency to degrade. Therefore, when applying finite element analysis, it is necessary to adjust the stiffness matrix to the degraded stiffness continuously and to update the stresses in the composite using the degraded stiffness matrix.

3.1.2 Guidelines for determining fatigue damage

Shokrieh (31) determined the fiber and matrix damage of the material based on Hashin’s criterion (34,35). A failure criterion applied to the fatigue damage mode of carbon fiber composite laminates was developed using residual strength and stiffness (35). The five primary forms of damage during fatigue were formulated as fiber tensile damage, fiber compression damage, base fiber shear, matrix cracking, and matrix extrusion. When performing finite element analysis, what kind of damage occurs in the element body is evaluated according to the corresponding judgment basis.

  1. The formula for determining the fiber tensile damage is as follows:

    (3) σ xx X T ( n , σ , k ) 2 + σ xy S xy ( n , σ , k ) 2 + σ xz S xz ( n , σ , k ) 2 ≥ 1

  2. The formula for fiber compression damage determination is as follows:

    (4) σ xx X C ( n , σ , k ) 2 ≥ 1

  3. The formula for determining the base fiber shear damage is as follows:

    (5) σ xx X C ( n , σ , k ) 2 + σ xy S xy ( n , σ , k ) 2 + σ xz S xz ( n , σ , k ) 2 ≥ 1

  4. The formula for making a determination of matrix cracking damage is as follows:

    (6) σ yy Y T ( n , σ , k ) 2 + σ xy S xy ( n , σ , k ) 2 + σ yz S yz ( n , σ , k ) 2 ≥ 1

  5. The formula for deciding on matrix crush damage is as follows:

(7) σ yy X C ( n , σ , k ) 2 + σ xy S xy ( n , σ , k ) 2 + σ yz S yz ( n , σ , k ) 2 ≥ 1

where n is the number of cycles; σ is the stress; k is the stress ratio; σ xx , σ yy , and σ zz are the three directional principal stresses of each element; σ ij is the remaining stress component of each element; X T ( n , σ , k ) is the residual longitudinal strength of a single-layer laminate after a certain number of cycles under tensile load; X C ( n , σ , k ) is the residual longitudinal strength in the case of compressive load, and so on; Y T ( n , σ , k ) and Z T ( n , σ , k ) are the transverse and normal strengths; S xy ( n , σ , k ) is the shear strength of a single ply under shear fatigue loading in the xy-direction; and S xz ( n , σ , k ) and S yz ( n , σ , k ) are similar. For any element in the model, if the stress component satisfies one of the aforementioned equations, it means that the corresponding damage occurs in that element.

3.1.3 Strength and stiffness degradation

The macroscopic expression of material properties can be expressed in strength and stiffness. The progressive degradation law of material properties in this study is based on the stiffness degradation formula and strength degradation formula of composites during fatigue derived by Wang (36) and Yang et al. (37).

(8) E ( n , σ , R ) E ( 0 ) = 1 − 1 − σ c 1 σ U 1 c 2 n N f a 1 + b 1 σ

(9) S ( n , σ , R ) S ( 0 ) = 1 − 1 − σ σ U n N f a 2 + b 2 σ

where E ( n , σ , R ) is the residual fatigue stiffness; E ( 0 ) is the initial stiffness; S ( 0 ) is the residual fatigue strength; N f is the fatigue life at the maximum fatigue stress level at a specified stress ratio; and a 1 , b 1 , c 1 , a 2 , and b 2 are the constants, which are obtained by fitting the experimental data.

3.1.4 Methods of performance degradation

When a material is subjected to cyclic loading and its properties, there is then a rapid and sudden drop in the stiffness and strength of the laminate. When using finite element analysis (FEA) for fatigue life prediction, a unit can be considered to have decayed in stiffness and strength. When it reaches a value judged by the Shokrieh and Lessard (31) damage criterion, the unit is deemed to have suffered some form of damage, and a sudden drop in performance can be applied to the unit. The material performance degradation is often to a very small value, indicating that the unit has failed and is no longer involved in the subsequent damage reduction. Naderi and Maligno (32) concluded that there are several failure scenarios for composites under cyclic loading, namely, fiber tensile or compression fracture, matrix tensile or compression fracture, delamination damage, and matrix shear failure. The fatigue life of composites was predicted based on these failure scenarios, and the results showed that the predictions were in good agreement with the experimental results, proving the reliability of the method. Therefore, this study refers to and adopts the approach of the fatigue damage accumulation model proposed by Naderi and Maligno (32) for different sudden decreases in performance under various forms of damage and concludes that:

  1. When fibers are stretched or compressed to the point of failure, the damage has progressed to the final stage. The laminate can no longer carry the load, and the laminate has been completely damaged. The stiffness, Poisson’s ratio, and strength of the laminate are all degraded to an approximate value of 0, which, in this study, is taken to be 0.0001. No other form of damage is judged after that:

    (10) [ E xx , E yy , E xy , ν xy ] = [ 0 , 0 , 0 , 0 ]

    (11) [ X T , X C , Y T , Y C , S xy ] = [ 0 , 0 , 0 , 0 , 0 ]

  2. As the matrix does not bear the main load, matrix damage is not devastating, so when only matrix damage occurs to the material, only a portion of the properties in the direction of the matrix are degraded. Other forms of damage to the specimen still need to be judged. (a) When the matrix is subjected to tensile or compression-induced failure, the material’s shear modulus, shear strength, and shear Poisson’s ratio drop abruptly to a value close to zero:

    (12) [ E xx , E yy , E xy , ν xy ] = [ E xx , 0 , E xy , 0 ]

    (13) [ X T , X C , Y T , Y C , S xy ] = [ X T , X C , 0 , Y C , S xy ]

  3. When shear failure of the fibers and matrix occurs, the material is no longer subjected to in-plane shear loads. This causes the shear stiffness, Poisson’s ratio, and strength to drop abruptly to a value close to zero. However, the fibers and matrix can still withstand the load, so it is necessary to continue to determine other forms of damage

(14) [ E xx , E yy , E xy , ν xy ] = [ E xx , E yy , 0 , 0 ]

(15) [ X T , X C , Y T , Y C , S xy ] = [ X T , X C , Y T , Y C , 0 ]

3.1.5 Fatigue life calculation

Failure of a material occurs when the strength and stiffness of the material drop to a very small value. This means that the material has reached the end of its life and is no longer able to withstand the load. Shokrieh and Lessard (31) optimized the life prediction model based on the life model of Beheshty and Harris (38). It enabled the fatigue life of the material to be predicted by determining the strength, the stress ratio for fatigue loading, and the load value. The fatigue life prediction model for carbon fiber unidirectional laminates in this study is as follows:

(16) u = ln a f ln [ ( 1 − m ) ( c + m ) ] = A + B lg N f

where a = σ a/σ t, m = σ m/σ t, c = σ c/σ t, σ a = (σ max − σ min)/2, σ m = (σ max + σ min)/2, u is the fatigue life of the specimen, σ t is the tensile strength, σ c is the compressive strength, σ max is the maximum fatigue load, σ min is the minimum fatigue load, A and B are obtained by fitting experimental data, and N f represents the fatigue life.

The fatigue performance of carbon fiber composites with [0°/90°]4 layups in tension–tension was tested in the aforementioned study, resulting in the fatigue life of the laminate material at different load levels at a stress ratio of 0.1. Based on the model’s predictive capability for different stress ratios and to build a model to match the Shokrieh and Lessard (31) life prediction model, the predictive capability of the model is thus extended to fatigue loading at different stress ratios. The lg(N)–U relationship was established based on the experimentally obtained S–N relationship. The correlation coefficient was obtained as r = 0.938 by applying the following correlation coefficient formula to its correlation, indicating a high degree of correlation. Thus, the values of A and B in the life prediction model were determined by fitting the lg(N)–U relationship to 0.375 and 0.9039, respectively, resulting in the following fatigue life calculation formula:

(17) u = ln a 1.06 ln [ ( 1 − m ) ( 1 + m ) ] = 0.375 + 0.9039 lg N f

3.2 Finite element analysis

3.2.1 UMAT subroutine

The finite element simulation flow is shown in Figure 5, and the main program imports the relevant variables. The current strain is calculated, and the coefficients in the stiffness matrix are updated. Through the stress–strain constitutive relation formula, the stress at this point is calculated and used as the updated stress value. For elements with different laying directions, the software can solve for the principal stresses of the elements themselves. Define the initial damage of the material as 0, and use the updated stresses and strengths for the failure criterion determination. If less than one, the failure mode is considered not to have occurred; if greater than or equal to 1, the failure is judged to have happened. Five state variables were set, each representing fiber tensile fracture, fiber compression, matrix tensile, matrix compression, and matrix–fiber shear damage of the laminate specimens. When such a failure occurs, the value of the state variable of the species is set equal to 1, and a performance degradation is applied to the laminate specimen. Refer to the method in Chapter III. Element with sudden performance degradation will no longer incur a performance degradation.

If fatigue damage destructive failure is not produced, progressive performance degradation will continue. The fatigue life N of the elements in different principal directions was first calculated based on the stress values of each element in each main direction and the fatigue life prediction equations obtained from the unidirectional laminate tests in Chapter III. Subsequently, by substituting the fatigue life N into the stiffness degradation and strength degradation equations introduced in Chapter 3, the strength and stiffness of the element after the degradation of its performance at n cycles can be obtained. Finally, the abrupt and gradual degradation of material properties is stored in the state variables for state variable updating. These relevant variables were used as initial values for the next incremental step in the fatigue analysis. The layer is assumed to be considered damaged when the fiber element failure extends to both specimen ends.

The simulation flowchart is shown in Figure 6.

Figure 6 Numerical modeling process.
Figure 6

Numerical modeling process.

3.2.2 Finite element analysis setup

Based on the specimen dimensions, a 3D model was created on ABAQUS FEA software. As shown in Figure 7, the S4R shell element was selected, the mesh was divided using the free division method, and the advancing front algorithm was selected. The number of elements in the model is 7,632, and the number of nodes is 30,528. Each model ply is set with the same material properties, i.e., all are given the same single-ply performance parameters. Only the fiber orientation of the different layer differs, and the fiber orientation angle is defined by establishing a local coordinate system. The fiber stacking sequence is [0°/90°]4, the same as the experimental laminate layup sequence. To simplify the model, reduce the number of calculations and save computational costs. The load acting on the laminate changes from moment to moment during each cycle. The calculation efficiency will be greatly reduced if the damage performance is determined for each loading instant or a single cycle (Table 3).

Figure 7 Finite element analysis model of the laminate specimen. The right end of the specimen model was fully cemented, and a uniform load was applied to the other end.
Figure 7

Finite element analysis model of the laminate specimen. The right end of the specimen model was fully cemented, and a uniform load was applied to the other end.

Table 3

Material parameters of the CF1200-6300/R668 unidirectional plate

Unidirectional lamina
Density (kg·m−3) 1,760
Young’s modulus (GPa) E 11 = 123; E 22 = E 33 = 10.1; G 12 = G 13 = 4.6; G 23 = 3.082
Poisson’s ratio μ 12 = μ 13 = 0.28; μ 23 = 0.21;
X T = 2,260; X C = 1,370
Strength (MPa) Y T = 51; Y C = 130
S 12 = 68; S 13 = S 23 = 40
Fracture energy (N·mm−1) G n c = 0.27 ; G s c = 0.49

This finite element analysis is based on two assumptions as follows:

  1. Assuming that fatigue damage occurs only at the time of maximum stress during a cycle of tensile–tensile fatigue loading.

  2. A fixed cycle increment Δn is set for each analysis step, and a laminate degradation and failure analysis is performed after each increase in the specified number of cycles.

Therefore, the simulation analysis step and the loading method have been divided into two parts for easy calculation. First, linear loading is taken to increase the load to the fatigue cycle load level, and this step does not fix the incremental step size. The fatigue cyclic load was then set to a sinusoidal loading form, at which point the incremental step size was fixed. The fatigue loading method in the simulation is a sine wave loading method. In addition, different state variables can be used to describe the damaged state of a laminate specimen. A separate state variable can be used to indicate that at least one of the different forms of damage described previously has occurred to the laminate. Five state variables are set to describe the different forms of damage to the laminate specimen by setting five state variables: fiber tensile fracture, fiber compression, matrix tension, matrix compression, and matrix–fiber shear damage. A state variable of 1 indicates that this form of damage has occurred, while a state variable of 0 indicates that no damage has occurred. Based on the material property degradation criterion and the sudden drop method in this study, the simulation suggests that the specimen reaches the failure point when the fiber tensile or compressive damage of the laminate specimen extends across the entire width of the specimen.

3.2.3 Simulate result and analysis

Different state variables are used to represent different forms of damage and levels of damage during model failure. Figure 8(a)–(h) shows the development of the overall damage accrued during fatigue loading of the 90° layer on the surface and the 90° layer inside the model, respectively. The red blocks represent the danger zones, i.e., where fiber tensile damage has occurred. Blue blocks indicate that no damage has occurred. As can be seen from the diagram, the carbon fibers on the surface appear to be damaged more quickly than the carbon fibers inside when they are both oriented at 90° layup. And their primary areas of tensile damage are approximately the same. From these two aspects, it can be shown that fiber tensile damage is transmitted in the direction from outside to inside.

Figure 8 Development of different stages of fiber tensile damage in the model, where N is the number of cycles: (a)–(d) are the damage situations of the ply 6 layer and (e)–(h) are the damage situations of ply 8 layers.
Figure 8

Development of different stages of fiber tensile damage in the model, where N is the number of cycles: (a)–(d) are the damage situations of the ply 6 layer and (e)–(h) are the damage situations of ply 8 layers.

Figure 9(a)–(h) shows the four forms of damage occurring to the surface 0° layer in the model for fatigue lives of 18,500 and 50,000 cycles: fiber stretching, fiber compression, matrix stretching, and matrix compression. The figure shows that there is little difference in the severity of the matrix damage and fiber damage in the 0° layer. The fiber damage is primarily aggravated by extension from the edge to the interior along the original damage, while the matrix damage gradually develops into a blocky regional damage. This is similar to the damage development observed in fatigue tests. Figure 9(i) and (j) shows the form of damage that occurs in the 90° layer on the model’s surface. The damage to the substrate is more concentrated and severe in the 90° layer compared to the 0° layer. This phenomenon indicates that the loads taken by the substrate in a 90° layer are more complex than in a 0° layer.

Figure 9 Damage of ply1 layer of laminated board under different cycles, where N = 18,500: (a) fiber tensile damage situation, (b) fiber compression damage situation, (c) matrix tensile damage situation, and (d) matrix compression damage situation, where N=50,000: (e) fiber tensile damage situation, (f) fiber compression damage situation, (g) matrix tensile damage situation, and (h) matrix compression damage situation. When the number of cyclic loading cycles N = 50,000, the tensile damage of the model fibers is as follows: damage situation in ply 8 layer of laminated board, (i) fiber tensile damage situation, and (j) matrix tensile damage situation.
Figure 9

Damage of ply1 layer of laminated board under different cycles, where N = 18,500: (a) fiber tensile damage situation, (b) fiber compression damage situation, (c) matrix tensile damage situation, and (d) matrix compression damage situation, where N=50,000: (e) fiber tensile damage situation, (f) fiber compression damage situation, (g) matrix tensile damage situation, and (h) matrix compression damage situation. When the number of cyclic loading cycles N = 50,000, the tensile damage of the model fibers is as follows: damage situation in ply 8 layer of laminated board, (i) fiber tensile damage situation, and (j) matrix tensile damage situation.

Figure 10(a)–(d) shows the form of damage that occurs in the 90° layer on the model’s surface. The damage to the substrate is more concentrated and severe in the 90° layer compared to the 0° layer. This phenomenon indicates that the substrate in the 90° layer carries more load than in the 0° layer. As can be seen, after damage occurs to the surface layer of the laminate model by loading, the damage is transferred to the next ply. Due to the poor load-carrying capacity of the 90° ply, when damage occurs to ply1, the damage is transferred to ply2. The stresses in the ply model layer are redistributed, resulting in damage in some areas of ply2. The absence of damage to ply7 is because the ply suffered severe matrix stretching and matrix fiber shear before damage to the fibers occurred. As shown in Figure 10(e) and (f), this resulted in a low-stress level being transferred to the fibers and no damage to the fibers. In addition, it can be seen from the figure that the fiber damage in ply1 and ply3 is roughly distributed longitudinally and the damage in ply8 is distributed transversely. This is related to the orientation of the fibers in the layup.

Figure 10 When the number of cyclic loading cycles N = 17,500, the tensile damage of the model fibers is as follows: (a) the laying angle of ply 1 layer is 0°, (b) the laying angle of ply 2 layer is 90°, (c) the laying angle of ply 7 layer is 0°, and (d) the laying angle of ply 8 layer is 90°. When the fatigue load loading frequency is N = 17,500, the damage situation of ply 7 is as follows: (e) matrix tensile damage situation and (f) shear damage of matrix and fiber.
Figure 10

When the number of cyclic loading cycles N = 17,500, the tensile damage of the model fibers is as follows: (a) the laying angle of ply 1 layer is 0°, (b) the laying angle of ply 2 layer is 90°, (c) the laying angle of ply 7 layer is 0°, and (d) the laying angle of ply 8 layer is 90°. When the fatigue load loading frequency is N = 17,500, the damage situation of ply 7 is as follows: (e) matrix tensile damage situation and (f) shear damage of matrix and fiber.

Figure 11 shows the stiffness degradation of different layups of the element in the edge region of the model. It can also be seen from the graph that all eight curves show a slow downward trend before any significant stiffness degradation occurs. However, shortly after the fatigue cycle begins, the stiffness of ply7 and ply5 within the model first degrades. A degradation followed this in the stiffness of ply3 in the interior. As the cycle progresses, the stiffness of ply1 degrades to zero, and the stiffness of all the 0° layer in the laminate is reduced to a minimum when the number of cycles reaches approximately 75,000. The stiffness of the 90° layer in the model degrades abruptly only after the 0° layer has failed. Figure 12 shows that the order of stiffness degradation for each ply is: ply7, ply5, ply3, ply1, ply8, ply6, ply2, and ply4. However, the 90° layer in the laminate model suffers a sudden drop in stiffness in the opposite order to the 0° layer. First, the outer layer is subjected to a sudden drop. Then, the innermost layer breaks down. It is noteworthy that ply6 and ply2 were damaged earlier by ply6 and later by ply2. Analysis suggests that this was due to the lower stresses transferred from ply1 to ply2. Although ply6 is closer to the interior, the stresses transmitted by ply7 are higher. The reason for the sudden drop in the stiffness of the 0° layer in the first place can be analyzed in terms of other damage occurring in the layer. As can be seen in Figure 10, ply7 has suffered severe matrix tensile damage. In addition, the stiffness of the layer decreases in an approximately linear fashion following the onset of performance degradation. The rate of degradation falls as it drops closer to 0. When all eight stiffness curves drop to 0, the stiffness of each laminate degrades to 0. It is considered that at this point, the laminate as a whole undergoes destructive damage, loses its load-bearing capacity, and reaches its fatigue limit.

Figure 11 Stiffness degradation of different layers in fatigue damage models.
Figure 11

Stiffness degradation of different layers in fatigue damage models.

Figure 12 Shortly after fatigue cyclic loading, the different layers inside the model gradually began to degrade. The order of stiffness degradation of each layer is: ply 7, ply 5, ply 3, ply 1, ply 8, ply 6, ply 2, and ply 4.
Figure 12

Shortly after fatigue cyclic loading, the different layers inside the model gradually began to degrade. The order of stiffness degradation of each layer is: ply 7, ply 5, ply 3, ply 1, ply 8, ply 6, ply 2, and ply 4.

3.2.4 Reliability verification

To verify the accuracy and reliability of the fatigue life prediction method established in this study for carbon fiber-reinforced epoxy resin matrix composites, the results are obtained from the fatigue damage development process and fatigue damage form of the composite material in the simulation study using the UMAT subroutine and finite element analysis. The results are compared with the results obtained from the aforementioned experimental investigation of the fatigue damage characteristics, damage development process, and damage form of the composite material obtained from the analysis of the tensile–tensile fatigue process and microscopic morphology of the failed specimen. The final analysis of the fatigue damage development, fatigue damage forms, and fatigue life of carbon fiber composite laminates is presented. Figure 13 compares the experimental and simulation results in the development of tensile–tensile fatigue damage in composite materials.

Figure 13 To prove the accuracy and reliability of the fatigue life prediction method for carbon fiber-reinforced epoxy resin matrix composites established in this study, the results obtained from this finite element model calculation are compared with those obtained from the previous experiments. There are several state comparisons as follows: (a) experimental results at the beginning of cyclic loading of fatigue loads, (b) simulation results of the initial period of cyclic loading of fatigue loads, (c) experimental results of cyclic loading of fatigue loads at mid-term, (d) simulation results for the middle period of cyclic loading of fatigue loads, (e) experimental results at the late stage of cyclic loading of fatigue loads, and (f) simulation results of the late fatigue load cyclic loading.
Figure 13

To prove the accuracy and reliability of the fatigue life prediction method for carbon fiber-reinforced epoxy resin matrix composites established in this study, the results obtained from this finite element model calculation are compared with those obtained from the previous experiments. There are several state comparisons as follows: (a) experimental results at the beginning of cyclic loading of fatigue loads, (b) simulation results of the initial period of cyclic loading of fatigue loads, (c) experimental results of cyclic loading of fatigue loads at mid-term, (d) simulation results for the middle period of cyclic loading of fatigue loads, (e) experimental results at the late stage of cyclic loading of fatigue loads, and (f) simulation results of the late fatigue load cyclic loading.

As seen from Figure 13(a) and (b), the material starts to break down at the edge of the layer and near the ends of the specimen. The simulation results are similar to the experimental results. However, comparing the results from the middle stage of damage development shows that the matrix damage appears earlier in the simulation than the obvious matrix damage in the experiment. This may be since inter-laminar bonding forces and inter-laminar cracking were not considered in the simulation, resulting in a higher load being applied to the substrate in the simulation. This is shown by the damage appearing earlier in the simulations for the laminates. As can be seen from Figure 13(c) and (d), the fiber fracture locations in the simulations and experiments are also similar. Both appear at one end of the specimen rather than in the middle of the specimen. The shape of the fracture in the simulations is also similar to the experimental results, with an undulating and uneven pattern. This may be related to the location of the earliest damage in the early stages of damage. As can be seen from Figure 13(e) and (f), the experimental results show that the laminate shows severe matrix and inter-laminar damage as well as fiber breakage as the damage progresses to a later stage until the laminate fails completely. The red squares in the simulation results also indicate that the material has reached its fatigue damage limit at this point and is in a state of complete failure. Overall, the damage form, location of damage initiation, and progression of the laminate model are similar to the experimental results. However, the inability to account for the generation and development of inter-laminar damage allows the matrix damage in the laminate model to appear earlier than the experimental matrix damage. It is also more severe in the early and middle stages of damage development.

Based on the results of FEA, it is generally accepted that when the stiffness of all layer is reduced to a minimum, the model as a whole is destroyed and the fatigue life limit is reached. According to this hypothesis, the degradation results of the stiffness under fatigue cyclic loading in the finite element analysis are counted separately for each of the three cases. Table 4 shows the statistics and comparisons for these three cases. A comparison of the simulation results with the experimental results shows that the error in the simulation is within 35%, which is higher than the prediction accuracy of some of the life models in the industry. This demonstrates the reliability of the life model and life prediction method.

Table 4

Comparison of life prediction results under three load levels

Stress level (MPa) Experiment Simulation Deviation (%)
520.4 468,938 342,500 26.96
567 150,484 101,500 32.55
662 23,015 27,500 19.45

4 Conclusion

  1. As the number of fatigue cycle weeks increases, the damage development of the specimen shows a three-stage trend of initial, intermediate, and final stages. This trend coincides with the stiffness development of the material. In the middle and end stages of specimen damage, inter-layer cracking becomes gradually the main form of damage. The three phases differ in the form of damage and the rate of damage expansion. The middle stage has the longest duration, and inter-laminar damage is the main form of damage. The final stage of inter-layer cracking tends to be saturated, many fibers break, and the specimen is explosively damaged quickly.

  2. By analyzing the microscopic morphology of the fatigue failure specimens, it was found that the fiber fracture was, indeed, the result of stress transfer in the thickness direction. As cracks extend between the matrix and the fibers, debonding and delamination damage are aggravated. Extensive cracks can block stress transfer, and stress concentrations at the crack front can lead to fiber fracture.

  3. The results obtained from the simulation are closer to the experiments in terms of the damage initiation location, the damage extension process, and the damage form of the laminate. The fatigue life values obtained from the simulations are within a small error range from the experimental results, indicating that the fatigue life prediction method for this composite material has a certain degree of reliability. This intuitive and somewhat reliable life prediction method has good research potential for predicting the fatigue limit of fiber-reinforced composites.

  1. Funding information: This work was supported by China National Natural Fund (13008906).

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results and approved the final version of the manuscript. Jiamei Lai, Yousheng Xia and Zhichao Huang designed the experiments and Bangxiong Liu carried them out. Mingzhi Mo developed the model code and performed the simulations. Jiren Yu prepared the manuscript with contributions from all co-authors.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Data availability statement: The data that support the findings of this study are available from the corresponding author upon a reasonable request.

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Received: 2023-09-22
Revised: 2023-11-11
Accepted: 2023-11-17
Published Online: 2024-05-30

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/epoly-2023-0150
Keywords for this article
carbon fiber-reinforced composites; fatigue failure characteristics; fatigue life models; fatigue life prediction
Creative Commons
BY 4.0

Articles in the same Issue

  1. Research Articles
  2. Flame-retardant thermoelectric responsive coating based on poly(3,4-ethylenedioxythiphene) modified metal–organic frameworks
  3. Highly stretchable, durable, and reversibly thermochromic wrapped yarns induced by Joule heating: With an emphasis on parametric study of elastane drafts
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  74. Development of smart core–shell nanoparticles-based sensors for diagnostics of salivary alpha-amylase in biomedical and forensics
  75. Thermoplastic-polymer matrix composite of banana/betel nut husk fiber reinforcement: Physico-mechanical properties evaluation
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