1. Introduction
Most of the thermomechanical properties of fiber-reinforced composites are affected by the strength distribution of the constituent fibers. In the past few decades, these composite materials have commonly utilized traditional reinforcing fibers (such as glass and carbon) as reinforcement, but recently, environmental concerns have generated a resurgence of interest in using natural fibers. In particular, plant fibers, such as coir, flax and sisal, are emerging as potential alternatives for man-made fibers in polymer-based composites. It is now widely accepted that natural fibers and biocomposites made from natural sources integrate sustainable, eco-friendly and relatively low-cost industrial products, which can replace the dominance of petroleum-based products in the future [
1].
Presently, bamboo fiber is being considered an important alternative reinforcement to synthetic fibers and has great potential to be used in structural polymer composites [
2]. However, one of the main concerns for industrial application is a much higher variability in the mechanical properties of this natural product. Because the bamboo fibers are extracted by retting followed by mechanical processing, the mechanical properties of the obtained fibers are affected by the natural variability in the plant, by the processing stage and by the damage sustained during processing [
3]. In comparison with synthetic fibers, bamboo fibers have a significantly higher variation in diameter between fibers and within a fiber [
4,
5]. As a consequence, natural fibers exhibit considerable strength variation, which is found to be correlated to fiber diameter and length. In other words, this means that there is no specific value to represent their strength. This leads to the necessity of employing an efficient method for the evaluation of the fiber strength and the prediction of its size dependence.
Knowledge of the mechanical behavior of an advanced fibrous composite requires a comprehensive evaluation of the strength of the fibers. Many attempts have been made for strength prediction in the fibers. The distribution of the failure stress of brittle fibers, such as carbon [
6], ceramic [
7] and jute fibers [
8], is most often characterized using the Weibull statistics. This model is based on the failure of a chain in which the weakest link controls rupture and seems to be well adapted to describe a set of tensile test results carried out at one gauge length [
9]. Existing analytical models for the tensile failure of fibers can be divided into two groups according to the Weibull rule. Models in the first group are based on the two- or three-parameter model, which is called the standard Weibull model or linear-law model. However, there is a growing amount of experimental data suggesting that the standard Weibull distribution shows the inadequacies for describing the experimentally observed fiber strength scatter and the strength dependence on the size [
10]. Models of the second group are based on the power-law model [
8,
11], which is a modified form of the standard Weibull model. Such an analysis introduces a parameter that represents the fiber diameter variations and succeeds in reconciling the mismatch of the fiber strength scatter at a fixed gauge length. However, most attempts have paid attention to studying the within-fiber diameter variation. Very few of the models try to investigate the effect of between-fiber diameter variation on the tensile strength.
Contrary to the previous studies, of primary interest in the present study is implementing the modeling of the statistical distribution of facture strength. Tensile experiments are conducted to estimate the statistical strength properties for the bamboo fibers over a variation of lengths ranging from 20 to 60 mm and diameters ranging from 196.6 to 584.3 μm. The standard Weibull model is modified by incorporating the diameter variation to investigate the effect of fiber length and diameter on the tensile strength. In addition, weak link scaling analysis is conducted to check the validity of the proposed mode for predicting the fracture strength and its size dependence.
2. Experimental Procedure
The bamboo fibers (density of 1.035 g/cm
3) used in this work were delivered from Ban, Ltd., Tokushima, Japan. They were selected from one of the most common bamboo species, known as Moso bamboo. All samples were extracted from the bamboo culms of at least three years of age. The tensile tests were conducted using a WDW3050 computer-controlled universal testing machine (Kexin Testing Instrument Co., Ltd., Changchun, China) that registered the displacement of the clamps and the force applied to the fiber. Each fiber specimen was mounted on a stiff paper frame. The fiber length outside the frame determined the gauge length. To fix the fibers as straightly as possible between the clamps, fiber ends were glued with a double-sided adhesive onto the paper frame in accordance with the preparation procedure described in ASTM (American Society for Testing and Materials) D3379-89 [
12]. Upon clamping of the ends of the frame by the jaws of the testing machine, frame sides were carefully cut in the middle, as shown in
Figure 1.
Figure 1.
Schematic of the fixture set-up for the fiber tensile test.
Figure 1.
Schematic of the fixture set-up for the fiber tensile test.
In order to investigate the influence of a variety of fiber dimensions on the fracture properties, bamboo fibers were used to produce sets of samples at five different gauge lengths with 20, 30, 40, 50 and 60 mm and at five different test diameters with 196.6, 317.3, 398.4, 508.8 and 584.3 μm at a 20-mm gauge length. Indeed, in this batch of samples, fibers were found to normally range from 150 to 600 μm in diameter. Therefore, they were divided into five typical groups that were capable of representing a significant variation in average diameter between fibers. Meanwhile, these grouping diameters were selected based on the consideration that the average diameter at each of the five groups was constant within an accuracy of ±15 μm. At least 20 individual samples were mounted for testing at each of these fiber sizes, because samples broken close to the end had not been considered in the analysis. Consequently, a total of 238 fiber tests were carried out for obtaining 20 qualified specimens per test size. It should be noted that special care was taken during handling in order to avoid the creation of additional defects and changes to flaw distributions.
All static tests were displacement controlled with a cross head speed of 0.5 mm/min, which were performed at ambient conditions. A load cell of 200 N was applied for all fiber specimens. The load-displacement curve was recorded during the test. The cross-sectional area was evaluated from the diameter measured using an optical microscope at five different locations along each sample length. Although the bamboo fiber cross-section has a polygonal shape and fiber thickness varies somewhat along the fiber [
3], the average cross-sectional area, needed to convert the applied loads to stresses, is calculated by assuming that each individual fiber has a constant cross-section and is perfectly circular in shape [
13]. Through image analysis of the fracture profile in
Figure 2, bamboo fiber seems to have a brittle fracture behavior, shaped by a filoselle.
Figure 2.
Fracture profile of a bamboo fiber.
Figure 2.
Fracture profile of a bamboo fiber.
3. Statistical Modeling
Weibull analysis is a widely-used statistical tool for describing the strength behavior of brittle materials, which is based on the assumption that failure at the most critical flaw leads to total failure of the specimen [
14,
15]. Owing to the varying severity of flaws along the volume of the fibers, the strength of fibers is found to be statistically distributed. Therefore, the distribution of fiber strength, σ
f, under tension is generally described by means of the standard Weibull model [
16]:
where
P is the failure probability of the fiber. κ is the Weibull modulus, which defines the variability of the distribution. A lower κ corresponds to a broad distribution of fracture strength and a higher κ to a narrow distribution. σ
0 is the characteristic Weibull strength corresponding to a reference gauge volume
V0.
V is the fiber volume.
By rearranging and taking the natural logarithm of both sides of Equation (1), the following expression is obtained:
For a constant tested volume, Equation (2) is reduced to:
Hence, a plot of X = ln σf vs. should give a straight line if the material strength variability is described by the Weibull distribution. The shape parameter κ and the scale parameter σ0 can be determined by plotting X against Y.
The cumulative probability of failure
P of the fiber can be estimated as median ranks assigned to the measured strength values, σ
fi, at each gauge length and diameter, using the following approximation:
where
n is the number of data points (
i.e., the number of strength measurements performed at the given gauge size).
i is the rank of the
i-th number in the ascending ordered strength data point (
i = 1 corresponds to the smallest and
i =
n corresponds to the largest).
According to the nature of the mathematical function, the average value of
can be calculated with Equation (1):
where Γ is the gamma function.
Equation (1) assumes that the defect density is homogeneously distributed over the volume of the material. In other words, the defect distribution is regarded as a function of the length and diameter of the fiber. If a bamboo fiber has a circular cross-section, the volume of the fiber is
V = (1/4) × π
D2L.
D is the fiber diameter, and
L is the gauge length. Consequently, Equation (1) can be expanded to:
Thus, for the constant fiber diameter, the resulting Weibull distribution is identical to Equation (1), except that volumes are substituted by lengths (
L):
where
L0 is the reference length and
L is the fiber length.
To consider the effect of fiber diameter, for the constant gauge length
L, Equation (6) can be written as:
where
D0 is the reference diameter and
D is the fiber diameter.
Therefore, the average strength based on Equations (7) and (8) is given by, respectively:
However, bamboo fibers are quite different from man-made fibers, especially in their variable geometrical structure, such as between-fiber and within-fiber diameter variation. For example, variations in cross-sectional area along the length of a fiber induce variations in stress that may lead to a different overall fiber strength compared to a constant fiber cross-section with the same average area [
5]. There is some experimental evidence suggesting that the above conventional Weibull model does not always adequately describe the fiber strength and its dependence on the fiber volume for many natural fibers [
8]. As a consequence, a modified form of Equation (1) was suggested by Watson and Smith [
17]:
where γ is the exponential parameter that is a measure of the sensitivity of the strength to the test volume
V. The lower γ, the lower the decrease of strength for increasing volume
V. It is an empirical parameter that is introduced to improve the predictive level of the Weibull model with respect to experimental data. The modified (power-law) model has been justified to have a wider applicability than the standard (linear-law) model [
8]. The reason is that under the framework of the modified model, the equation has taken into account the effect of not only the diameter variation between the fibers, but also the within-fiber diameter variation, especially for natural fibers. The parameters involved in the distribution of Equation (11) can be calculated by the maximum likelihood estimation (MLE) from the strength data.
Therefore, Equations (9) and (10) can be changed as follows:
where the reference length
L0 and reference diameter
D0 are chosen to be 1 mm and 1 μm for mathematical convenience, respectively.
5. Conclusions
In this work, the strength of bamboo fibers was quantified in order to have a better understanding of the behavior of the final composite materials. To obtain a realistic estimate of the fiber properties, a large population of bamboo fibers was individually tested in tension using various gauge dimensions. The fracture strength distribution of bamboo fibers is statistically analyzed using a modified Weibull model to evaluate the effect of different size variables: fiber length and diameter. The obtained results reveal a size effect of fiber strength in both axial and radial directions, in good agreement with experimental observation. The reduction in mechanical properties in bamboo fibers with increasing fiber specimen size is not only caused by the accumulation of defects over the volume of a material, but also by between-fiber and within-fiber diameter variations.
Furthermore, the predicted scaling of fiber strength with both length and diameter has been verified. It is shown that the modified Weibull distribution is much more appropriate than the standard Weibull distribution for the description of the strength distribution of the bamboo fibers at different gauge dimensions, which is attributable to the fact that the modified Weibull distribution can take into account the effect of variations in diameter between fibers and within a fiber.