A rapid method for determining Weibull parameters of carbon fiber strength

Abstract

This study proposes a new rapid measurement method for determining the Weibull parameters of carbon fiber strength distribution based on tensile testing of carbon fiber bundles. Unlike traditional fiber bundle tests, which require both stress and strain measurements, this method only requires testing the tensile strength of the fiber bundles. The process involves three key steps: First, a formula derived from the Weibull distribution function was used to establish the relationship between the Weibull parameters and the tensile strength of fiber bundles of different lengths. Next, fiber bundles of varying lengths were selected for tensile strength testing. Finally, the Weibull parameters were calculated by substituting the test results into the formula. This method offers several advantages: First, compared to single fiber testing, this method eliminates the need to test a large number of individual fiber strength values, requiring only a few bundles of different lengths to be tested. The experimental equipment is both reliable and affordable. Second, unlike the traditional fiber bundle test, which involves testing both tensile strength and the $F\sim \epsilon$ (or $\sigma \sim \epsilon$ ) curve (including strain measurements), this method focuses solely on tensile strength. As a result, the testing equipment is simpler to operate, and time costs are reduced. Third, the data processing is straightforward, requiring only a few data points to be inputted into the formula. The Weibull parameters of fiber strength obtained through this method have been validated by multiple experiments conducted on fiber bundles of different lengths, confirming its accuracy and reliability.

Introduction

Fiber is the main load-bearing phase of composite materials, and its strength determines the macroscopic mechanical performance of the continuous-fiber-reinforced composite. At the microscale, the single fiber strength distribution at different lengths plays an essential role in the micromechanical study of the fiber–matrix interface stress transfer.^1–4 Therefore, it is necessary to accurately determine the strength of a single fiber through experimentation.

The manufacturing process of fibers may introduce defects in their microstructure. As a result, it is difficult to fully characterize the tensile strength of these fibers using an average value. Instead, it should be characterized by a probability distribution. ⁵ Both theoretical and experimental results have shown that there is a significant randomness in the strength of single fibers. The strength of brittle fiber is often described in line with the weakest-link theory^6,7 where it is assumed that a given volume of material fails at the point of most detrimental flaw. Based on the weakest-link theory, the random distribution of defects can be expressed as a Weibull formula at a macro level^8–13 As confirmed experimentally by Coleman, ¹⁴ the strength of single fibers conforms to this Weibull distribution. Since then, it has been widely applied in engineering, especially for the two-parameter Weibull model.^15–23 The two-parameter Weibull distribution model is expressed as the following function

$$P\left(L,\sigma \right)=1-\exp \left[-\frac{L}{{\delta }_0}{\left(\frac{\sigma }{{\sigma }_0}\right)}^m\right]$$

(1)

where $P\left(L,\sigma \right)$ represents the failure probability of single fiber of length $L$ under an applied stress that is no greater than $\sigma$ ; ${\sigma }_0$ and $m$ denotes the scale parameter and shape parameter of the Weibull distribution, respectively; ${\delta }_0$ indicates the reference characteristic length. The shape parameter m, also known as Weibull modulus, is used to characterize the uniformity and reliability of the material. The larger the value of m, the greater the uniformity and reliability of the material is. Therefore, Weibull modulus m is commonly used to draw comparison in the strength distribution of different materials.

In majority of studies,^24–37 these two parameters, ${\sigma }_0$ and $m$ are determined by conducting experiments. It is summarized from the relevant literature that there are three main testing methods involved: (1) Single fiber tensile test method^24–32; (2) single fiber fragmentation test method^33–36; and (3) dry fiber bundle tensile test method. ³⁷

The single fiber tensile test is the most common method used to obtain the Weibull parameters required for the distribution of fiber strength. Through this test, the stiffness and strength of individual fibers can be determined directly. Notably, each fiber must be manually selected from a bundle and then mounted onto two plastic tabs, with each tab holding the fiber at one end, as shown in Figure 1. A fitted straight line is drawn by inputting the test data into equation (2), where $b=\ln \left(L/{\delta }_0\right)$ .

$$\ln \ln \left(1/\left(1-P\right)\right)=m\hspace{0.17em}·\hspace{0.17em}\ln \hspace{0.17em}\mathrm{\sigma }-m·\hspace{0.17em}\ln \hspace{0.17em}{\sigma }_0+b$$

(2)

Figure 1.

Schematic diagram of single fiber tensile testing fixture.

The slope of the line is indicated by the Weibull shape parameter $m$ , and the intercept of the line represents the Weibull scale parameter. As demonstrated by Mesquita ²⁴ through a large dataset derived from experiment, a large number (over ∼200) of the tests is required for the Weibull parameters of fiber strength to be accurately determined. The samplings of ∼50 tests can cause Weibull modulus to deviate by ± 1. There are two disadvantages to this testing method. On one hand, it is a lengthy process to test a large number of specimens for ensuring the accuracy. On the other hand, some weaker fibers are more likely to break during the process of manual selection, but these fibers may affect the tested probability distribution.

Single fiber fragmentation test (SFFT) is recognized as an indirect testing method. It is usually performed to measure the fiber with a shorter gauge length, typically less than 1 mm. In this method, one single fiber is embedded in a dogbone specimen of epoxy resin after curing as shown in Figure 2. The transparent epoxy allows the fiber breakage in the specimen to be observed in situ. The fragmentation test is carried out by applying tensile load to the epoxy dogbone at a low rate using a mini tensile tester mounted on an optical microscope stage. As the load increases, the number of breaks increases. The mean fragment length is used to determine the parameters of Weibull strength distribution at short gauge lengths. Ganesh ³⁶ demonstrated that a different population of defects becomes dominant at shorter gauge lengths. In the SFFT, loading and microscope observation are carried out simultaneously, which makes the testing process relatively complex and prone to significant errors.

Figure 2.

Schematic diagram of single fiber fragmentation test.

Dry fiber bundle tensile test represents another indirect testing method. Instead of single fibers, fiber bundles were used for testing Weibull parameters. The tensile force-strain $\left(F-\epsilon \right)$ curve^15,21,22,37 of fiber bundle was tested by experiment as seen in Figure 3. The Weibull parameters were obtained from this $F-\epsilon$ curve. Compared to the above two methods, this dry fiber bundle tensile test method strikes a balance between time consumption and test accuracy. Equation (3) shows the rationales of fiber bundle test. The tensile fracture force $F_{\mathit{max}}$ and fracture strain ${\epsilon }_b$ can be determined from the tensile $F-\epsilon$ curve. Then, the Weibull parameters can be obtained using equation (3).

$$m=1/\ln \left(\frac{{AN}_0{E}_f{\epsilon }_b}{F_{\mathit{max}}}\right)$$

(3)

Figure 3.

Schematic diagram of fiber bundle tensile test.

The advantage of the dry fiber bundle test is that a single fiber bundle experiment is sufficient to determine the distribution of strength values. Meanwhile, it is simple to process the data. Conversely, the disadvantage of this method is that the strain must be tested to determine the fracture strain ${\epsilon }_b$ . However, the strain testing of fiber bundles is complex. Strain gauges may induce vibration deviations during the gradual fracture of some fibers, which causes the measured strain to deviate from the actual value. When using a gripping-type strain gauge for testing, the grips are in direct contact with the fiber specimen. The fracture of each fiber exerts some impact on the grips. During the tensile testing of the fiber bundle, the tensile load gradually increases. When the load is relatively small, no fiber breaks, and the strain measurement is highly accurate. As the load increases, fibers gradually fracture, and the fracture locations exhibit a random distribution. As a result, some fracture points of the fibers are closer to the strain gauge grips. The energy released during the fracture generates a significant impact on the grips, which may sometimes cause the grips to shift. At this point, the measured strain is no longer accurate, and therefore using the stress-strain method to test the fiber bundle can introduce errors. Very few studies have addressed the difficulties and improvements in measuring the strain of fiber bundles under tension using a gripping-type strain gauge. Some scholars have avoided using a strain gauge to measure the deformation of the specimen, instead directly using the tensile displacement of the testing machine to substitute the specimen’s deformation. This introduces the deformation of the testing machine, leading to another source of error. The numerous defects of directly measuring the stress-strain of the fiber bundle mentioned above are difficult to resolve, prompting us to seek a new testing method to avoid these issues.

Therefore, a new dry fiber bundle testing method, which is different from the traditional way of fiber bundle test, is proposed in this study. It only requires the strength of the fiber bundle to be tested. The Weibull parameters can be determined through the length and fracture force of fiber bundle, which removes the requirement of testing the strain value. Therefore, the test process is simplified, and errors are reduced.

Theoretical model

The fiber bundle model is illustrated in Figure 4. In this model, $N_0$ parallel fibers of length $L$ is fixated at both ends. The cross-section area of a single fiber is denoted as A. The fiber bundle is subjected to a tensile force $F$ parallel to the fiber direction as seen in Figure 4.

Figure 4.

Fiber bundle model for tensile testing.

The assumptions of the fiber bundle model are as follows ³⁷ :

(1) The distribution of single fiber strength under tension follows the two-parameter Weibull distribution as given in equation (1).

(2) The relationship between applied stress, denoted as $\sigma$ , and strain, denoted as $\epsilon$ , for a single fiber follows Hooke’s law up to fracture. $E_f$ represents the elastic modulus of the fiber.

$$\sigma =E_f\epsilon$$

(4)

From equations (1) and (4), it can be obtained that

$$P\left(\epsilon \right)=1-\exp \left[-\frac{L}{{\delta }_0}{\left(\frac{\epsilon }{{\epsilon }_0}\right)}^m\right]$$

(5)

where $P\left(\epsilon \right)$ represents the failure probability of a single fiber under strain no greater than $\epsilon$ , and ${\epsilon }_0$ represents the scale parameter of Weibull distribution for strain. ${\epsilon }_0={\sigma }_0/E_f$ .

• (3) The interaction between the fibers is negligible. As the tensile load increases, some fibers break. When $n=N_{0}P\left(\epsilon \right)$ fibers break, the load carried by them is instantaneously distributed equally among the remaining $N=N_0-n=N_0\left[1-P\left(\epsilon \right)\right]$ fibers, and the stress can be expressed as:

$$F=E_f\epsilon A\left(N_0-n\right)={E_f\epsilon AN}_0\left[1-P\left(\epsilon \right)\right]$$

(6)

Substituting equation (5) into equation (6), the force $F$ can be expressed as:

$$F=E_f\epsilon A{N}_0\hspace{0.17em}\exp \left[-\frac{L}{{\delta }_0}{\left(\frac{\epsilon }{{\epsilon }_0}\right)}^m\right]$$

(7)

when $N_0$ is large enough, equation (7) is a continuous and differentiable equation. According to equation (7), the $F-\epsilon$ curve for a 3K carbon fiber bundle is shown in Figure 5. $F_{\mathit{max}}$ and ${\epsilon }_b$ at the highest point of the curve represent the breaking force and breaking strain, respectively. After reaching the point $F_{\mathit{max}}$ , the load gradually decreases to zero.

Figure 5.

Theoretical $F-\epsilon$ curve for a carbon fiber bundle, where $E_f=225GPa$ , $D_f=7\mu m$ , $N_0=3000$ , $L={\delta }_0=50mm$ , $m=1.837$ , and ${\epsilon }_0=0.0116$ .

According to equation (7), the force $F$ reaches $F_{\mathit{max}}$ when $dF/d\epsilon =0$ ( ${\epsilon =\epsilon }_b$ ).

$$\frac{dF}{d\epsilon }=E_{f}A{N}_0\hspace{0.17em}\exp \left[-\frac{L}{{\delta }_0}{\left(\frac{\epsilon }{{\epsilon }_0}\right)}^m\right]\left[1-m\frac{L}{{\delta }_0}{\left(\frac{\epsilon }{{\epsilon }_0}\right)}^m\right]=0$$

(8)

From equation (7) and equation (8), the $F_{\mathit{max}}$ and ${\epsilon }_b$ can be obtained as follows: where $e=2.718$ .

$${\epsilon }_b=\frac{{\sigma }_0}{E_f}{\left(\frac{{\delta }_0}{L}\frac{1}{m}\right)}^{\frac{1}{m}}$$

(9)

$$F_{\mathit{max}}={AN}_0{\sigma }_0{\left(\frac{{\delta }_0}{L}\frac{1}{me}\right)}^{1/m}$$

(10)

According to equation (10), different fiber bundle lengths lead to different breaking forces. Therefore, two breaking forces $F_{\mathit{max}}$ can be obtained when two fiber bundles of different lengths ( $L_1,L_2$ ) are selected and substituted into equation (10).

When

$$When\hspace{0.17em}L=L_1,F_{\max \backslash \_1}={AN}_0{\sigma }_0{\left(\frac{{\delta }_0}{L_1}\frac{1}{me}\right)}^{1/m}$$

(11)

$$When\hspace{0.17em}L=L_2,F_{\max \_2}={AN}_0{\sigma }_0{\left(\frac{{\delta }_0}{L_2}\frac{1}{me}\right)}^{1/m}$$

(12)

From equation (11) and (12), the Weibull shape parameter $m$ can be obtained and expressed as equation (13), while the Weibull scale parameter ${\sigma }_0$ (characteristic length is ${\delta }_0$ ) can be obtained and expressed as equation (14).

$$m=\frac{\mathit{ln}\left(L_2/L_1\right)}{\mathit{ln}\left(F_{\max \hspace{0.17em}\_1}/F_{\max \hspace{0.17em}\_2}\right)}$$

(13)

$${\sigma }_0=\frac{F_{\max \hspace{0.17em}\_1}}{{AN}_0}{\left(\frac{{\delta }_0}{L_1}\frac{1}{me}\right)}^{-1/m}$$

(14)

Equation (13) and equation (14) demonstrate the principle of this testing method. There is no need to test the strain of the fiber bundle. This method requires a test conducted only on the fracture forces of two different bundle lengths. Thus, it is easily operable and the testing error is reduced.

Experiment

Experiment and results

Based on the above theory, 3K fiber bundles of varying lengths, produced by Zhongfu Shenying (SYT45-3K carbon fiber), were selected for tensile testing. Weibull parameters are obtainable by substituting bundle length and fracture force into equation (13) and equation (14). The equipment used to conduct tensile test is shown in Figure 6. Through a wire clamp and a tension sensor with a capacity of 1 kN, it captures the maximum tension force in real time without measuring the strain during test.

Figure 6.

Fiber bundle tensile testing machine (with wire fixtures).

The tensile strength of 3K carbon fiber bundle, with a length of $L_1$ = 50 mm and $L_2$ = 300 mm, was tested on the tensile experiment as shown in Figure 7.

Figure 7.

Tensile test of two fiber bundles with different length ( $L_1$ = 50 mm, $L_2$ = 300 mm).

Figure 8 shows the force–time curve of a sample with a fiber bundle length of $L_1$ = 50 mm and $L_2$ = 300 mm during the tensile test. After the maximum value was reached, the tensile load decreased gradually to zero. The fracture force $F_{\max \hspace{0.17em}\_1}$ and $F_{\max \hspace{0.17em}\_2}$ were determined automatically by software through the force–time curve.

$$m=\frac{\mathit{ln}\left(L_2/L_1\right)}{\mathit{ln}\left(F_{\max \hspace{0.17em}\_1}/F_{\max \hspace{0.17em}\_2}\right)}=\frac{\mathit{ln}\left(6.000\right)}{\mathit{ln}\left(2.652\right)}=1.837$$

(15)

$${\sigma }_0\left({\delta }_0=50mm\right)=\frac{F_{\max \hspace{0.17em}\_1}}{{AN}_0}{\left(\frac{{\delta }_0}{L_1}\frac{1}{me}\right)}^{-1/m}=2.536\text{GPa}$$

(16)

Figure 8.

Experimental curve of tensile force–time for two carbon fiber bundles of different lengths.

Table 1 shows the result of average test strength for five bundles with the same length. As shown in equation (15) and equation (16), the Weibull parameters ( $m=1.837$ , ${\sigma }_0=2.54\text{GPa}$ ) are obtained by substituting these test results into equation (13) and equation (14). Compared with other types of carbon fiber materials, for example, Baxevanakis’s data ³³ ( $m=6.7$ , ${\sigma }_0=2.66\text{GPa}$ ), and CHI’s data ³⁷ ( $m=4.8$ , ${\sigma }_0=5.85\text{GPa}$ ) by single fiber tensile testing method, the carbon fiber in the present work ( $m=1.837$ , ${\sigma }_0=2.54\text{GPa}$ ) exhibits significant strength dispersion.

Table 1.

Testing results for different length bundle.

Fiber bundle length (mm)	$F_{\mathit{max}}$ (N)	Average $F_{\mathit{max}}$ (N)	Standard deviation $\sigma$ (N)	Variation coefficient
$L_s$ = 50	109	122.0	10.7	8.8%
	116
	121
	127
	137
$L_l$ = 300	41	46.0	4.2	9.1%
	44
	45
	48
	52

Figure 9 shows the photo of the fiber bundle after it has broken. Each fiber breaks at different locations, which explains the difficulty to test the strain using a strain gauge when progressive fracture occurs. Due to the impact of fracture energy on the strain gauge, it causes a shift of the strain testing device, resulting in significant errors.

Figure 9.

Fracture photos of carbon fiber bundle under tensile load.

Table 2 lists the comparative test results between the new dry fiber bundle testing method and the traditional single fiber testing method for the same carbon fiber model. The test data of the traditional single fiber tensile strength method are sourced from Reference 38. Comparison of the test data indicates that the Weibull parameters from both methods are in close agreement, with differences within approximately 10%. The differences in test outcomes primarily arise from the distinct quality characteristics of different carbon fiber types manufactured by various suppliers. In addition, different testing methods may also bring certain differences. The traditional single filament tensile testing method is subject to too many influencing factors, including the influence of the number of single filaments on statistical results, whether samples damaged by single filaments during the selection process are included in statistics, and the influence of fixtures on single filament strength. Consequently, comparative analysis using the same testing method can yield more reliable and meaningful performance evaluations.

Table 2.

Comparison of two testing methods for the same carbon fiber material.

Test sample	Shape parameter $\mathit{m}$		Scale parameter ${\mathit{\sigma }}_{\mathbf{0}}$ (GPa)
	Single fiber tensile test	The new dry fiber bundle test	Single fiber tensile test	The new dry fiber bundle test
Toray T300 B-3K-40 B	6.48	6.11	2.51	2.76

Discussions

Equation (10) can be converted into equation (17) which is simpler, where $K={AN}_0{\sigma }_0{\left({\delta }_0/me\right)}^{1/m}$ . The strength of the fiber bundle declines with the increase of length.

$$F_{\mathit{max}}={AN}_0{\sigma }_0{\left(\frac{{\delta }_0}{L}\frac{1}{me}\right)}^{1/m}=K{L}^{-\left(\frac{1}{m}\right)}$$

(17)

According to the Weibull parameters obtained from the test as described in the previous section, the relationship between bundle length $L$ and fracture force $F_{\mathit{max}}$ was shown in Figure 10. In order to validate this testing method, the strength of carbon fiber bundles (SYT45-3K) as obtained from theoretical analysis and experimental test was compared. The bundles of six different lengths (50 mm, 100 mm, 150 mm, 200 mm, 250 mm, and 300 mm) were chosen to test their strength. The corresponding strength values are 122 GPa, 85 GPa, 73 GPa, 65 GPa, 53 GPa, and 46 GPa, respectively. The coefficient of variation is around 9%. As seen from Figure 10, the results of theoretical analysis and experimental test are clearly comparable within a length range of less than 0.3 m. When the fiber length is greater than 0.3 m, the strength value changes less and equation (17) is no longer accurate. As the fiber length of the test samples exceeds 0.3 m, the error between the test results and the theoretical predictions gradually exceeds 10%. One possible explanation for these situations is the presence of multiple potential failure modes in the material being tested. This leads to the inaccuracy of the two-parameter Weibull model (as shown in equation 1) in describing the fiber strength distribution, while the three-parameter Weibull model may provide a more accurate representation. ¹⁶ This aspect will be further studied in the future. The third parameter ${\sigma }_t$ is defined as a threshold stress below which the failure probability is zero.

$$P\left(L,\sigma \right)=1-\exp \left[-\frac{L}{{\delta }_0}{\left(\frac{\sigma -{\sigma }_t}{{\sigma }_0}\right)}^m\right]$$

(18)

Figure 10.

Comparison between theoretical analysis by equation (17) and experimental strength of carbon fiber bundles with different lengths.

The method presented in this paper for obtaining Weibull parameters from fiber bundle testing requires only a few tests. The fiber bundle contains a large number of fibers (e.g., N = 3000), which exhibit relatively stable statistical properties at the macroscopic level. In contrast, the method of obtaining Weibull parameters through testing a large number of single fiber strengths and conducting statistical analysis requires a sample size of N >80, which is a considerable amount of work. The operator’s experience and skill play a critical role in the testing accuracy. Any individual fibers selected from the fiber bundle that fall below the limit strength will not survive, ultimately affecting the accuracy of the analysis. ¹⁶

Conclusions

A new rapid method for fiber bundle tensile testing was proposed in this study for measuring Weibull parameters. Compared with the traditional fiber bundle test method, it eliminates the need for strain measurement. The testing equipment is easy to operate and reduces time costs. Additionally, errors can be mitigated during the test. There is no need to test strain, the stiffness of the testing machine does not need to be very high, and the structure can be designed to be relatively small and lightweight.

This method was validated through tensile tests conducted on bundles with different lengths. The strength predicted theoretically is generally consistent with the results of multiple experimental tests. As shown by both theoretical and experimental analyses, this testing method is applicable for measuring Weibull parameters of other brittle fibers.