Keywords

1 Introduction

In order to reduce the environmental pollution produced by synthetic materials, interest in renewable and bio-sustainable materials has grown in recent years [1,2,3]. Nowadays, it has become common the use of composite materials reinforced with glass, carbon and Kevlar fibers in industrial applications. However, the impact on the environment have made researchers to focus their studies on natural fibers. Composite materials reinforced with natural fibers can be found in nature, a clear example is wood composed of cellulose (fiber) and lignin (resin) [4]. The mechanical properties of these materials have been used by ancient cultures (6000 BC) in order to reinforce ceramics, mummies and decorative objects. However, their potential was underestimated for other applications. Composite materials with natural fibers have several advantages compared to synthetic materials such as biodegradability, low density, low energy consumption for their manufacture, eco friendly, non-abrasive, economical, renewable, excellent mechanical properties and can be found in the nature, moreover they are abundant [5, 6]. From the point of view of the mechanical properties, these materials present anisotropy which is a direct dependence of the properties with the orientation state of the reinforcing fibers. In this case, two situations can be found, the first when the fibers are perfectly aligned and the second, when they are randomly distributed. Due to their simplicity and low computational cost [7], the ellipses method from photo micrographs is widely used to evaluate the orientation of short fibers. Vélez et al. [8] showed that the error associated with this method is minimal in the case of rigid short fibers. Some works have been developed with computational tools such as ModFlow, Ansys or SolidWorks that simulate the behavior of the material through finite element analysis to predict the fiber orientation. The main disadvantage of this analysis is that it does not include properties of natural reinforcements (lignocellulosic) in the database [5]. A big boost for the study of reinforcing fibers of composite materials is the work accomplished by Advani et al. [9], this technique reduces considerably the computational cost of the process by calculating the second and fourth order moments in this way, the orientation tensors are completely defined. Subsequently, Neves et al. [10] studied the injection process of polycarbonate reinforced with short glass fibers at 10% of concentration in volume determining the effect exerted by the orientation of the fibers on the mechanical properties of the material. A comparison of results with a commercial software is also shown. A new method proposed by Kim and Lee [11] presents a simple non-destructive method using X-ray images, the orientation is defined based on the intensity of the image. Abdennadher [7] evaluates the fiber orientation considering the fiber morphology by using a scanning electron microscopy (SEM). In this case specialized equipment is required. Recently, a non-destructive μCT technique with X-ray (Micro Computed Tomography) has been introduced to analyze the compounds microstructure [12, 13]. The disadvantages associated with these methods lie in the high costs and limitation in using certain natural fibers. This paper aims to validate an efficient, low cost and minimal user intervention tool for characterization of short fiber orientation tensor in polymeric compounds as well as its morphological properties. The manuscript has been organized as follows: Sect. 2 describes the theoretical basis of the methods used in this work. Section 3 presents the methodology used and finally, in Sect. 4 the results and conclusions of this work are presented.

2 Theoretical Foundation

2.1 Fiber Orientation and Tensors

Fiber orientation of composite materials allow to predict the material behavior in micromechanical terms. The composite is stiffer and stronger in the direction of greater orientation of the fibers and weaker and more flexible in the direction of lesser orientation [14]. The first works developed in this field are based on Jeffery’s model [15], who proposed the use of the probability distribution of orientation function (ODF) in non-Newtonian fluids. The motion equation is based on the volume control approach and proposes a resolution technique which provides accurate responses, however the analysis can last several days [6]. After that, a spherical harmonic approach was introduced, the method can be adapted to a finite element software for flow simulation inside the molding cavity. The results are as precise as the results using Bay’s technique, however the memory requirements make difficult its use in industrial applications. Considering previous works, Advani y Tucker III [9] changed the ODF movement focus and left the control volume analysis to adopt a new point of view focused on the moments of orientation distribution [14]. This technique is known as orientation tensors which allows to obtain concise answers in few seconds additionally, it is suitable for industrial applications.

2.2 Orientation Tensors

Tensional notation is an effective way of representing the orientation state of short fiber reinforced composites. It contains all the necessary information to represent the compound behavior with less computational effort than the required when using statistical orientation distribution. The orientation tensors can be defined by forming didactic products of the vector p and then integrating the product of the tensors with the distribution function in all possible directions. The study carried out by Advani and Tucker III [9], shows the effectiveness in using the second and fourth order tensors for the fiber orientation estimation the same that are determined by Eqs. (1) and (2).

$$ a_{ij} = \oint {p_{i} p_{j} \psi \left( p \right) dp} $$
(1)
$$ a_{ijkl} = \oint {p_{i} p_{j} p_{k} p_{l} \psi \left( p \right) dp} $$
(2)

The second-order orientation tensor with a suitable closure approximation can precisely predict the distribution of the fibers of the material [9, 16]. The tensors are calculated from the average orientation data obtained from each fiber (among N individual fibers) and are represented by Eq. (3):

$$ a_{ij} = \frac{{\mathop \sum \nolimits_{n = 1}^{N} P_{i} P_{j} F_{n} }}{{\mathop \sum \nolimits_{n = 1}^{N} F_{n} }} $$
(3)

where \( F_{n} \) is a weighting function of the nth fiber. In total, there are nine tensors distributed in a matrix of 3 × 3 where the sum of the components of the main diagonal is the unit [10]. If the tensors are symmetrical, it is enough to calculate the value of six tensors of the whole matrix, Eqs. (4)–(7):

$$ a_{11} = \frac{1}{N}\mathop \sum \limits_{n = 1}^{N} cos^{2} \theta_{n} , $$
(4)
$$ a_{12} = a_{21} = \frac{1}{N}\mathop \sum \limits_{n = 1}^{N} sen^{2} \theta_{n} cos^{2} \phi_{n} sen\phi_{n} , $$
(5)
$$ a_{13} = a_{31} = \frac{1}{N}\mathop \sum \limits_{n = 1}^{N} sen\theta_{n} cos\theta_{n} cos\phi_{n} , $$
(6)
$$ a_{23} = a_{32} = \frac{1}{N}\mathop \sum \limits_{n = 1}^{N} sen\theta_{n} cos\phi_{n} sen\phi_{n} . $$
(7)

where N is the total number of fibers existing in the image, \( \theta_{n} \) and \( \phi_{n} \) are the orientation angles in the plane and in the space of each fiber respectively. The \( a_{11} \) value represents the fiber orientation in relation to the flow direction, constituting in the reference direction tensor [17], as follows: \( a_{11} < 0.35 \) in relation to the flow direction, constituting in the reference direction tensor; \( a_{11} > 0.7 \) orientation considered parallel to the direction of flow and \( 0.5 < a_{11} < 0.6 \) a random orientation. In the case of two-dimensional analysis with symmetric tensors, it is enough to calculate the value of three tensors \( \left( {a_{11} , a_{12} , a_{22} } \right) \). Thus, the preferred angle is represented by Eq. (8):

$$ tan2\alpha = \frac{{2a_{12} }}{{a_{11} - a_{22} }} $$
(8)

The summary of the fibers orientation can be shown through the orientation of an ellipsoid, where the autovectors of the 3 × 3 matrix delivers information related to the direction and the eigenvalues give the anisotropic grade of the material.

3 Methodology

3.1 Materials

The images used in this work were provided by the research group GiMaT (Research Group on New Materials and Transformation Processes) of the Politécnica Salesiana University, Cuenca-Ecuador. In order to obtain the composite, guadua Angustifolia kunth (GAK) was used as natural reinforcing material and homopolymer polypropylene H-306 as matrix. During the elaboration process, superficial treatments were carried out to guarantee the good adhesion of the components. Subsequently, the injection material was performed as a modeling method. Normally, the variation in the orientation or breakage of the fibers occurs in this step. Microscopic images were captured using light microscopy with a BX51M microscope and an Olympus DP 72 digital camera with 5x magnification. In order to obtain the complete field of analysis, a combined image acquisition tool called MIA was used, this tool allows individual captures of adjacent positions and their subsequent combination to form only one image of the entire field of analysis. Two samples were fabricated at 30% of fiber concentration by weight. From each of the samples nine images were obtained to know the direction of flow during the injection process. Figure 1 shows the areas and depth of the specimen from which the images were obtained.

Fig. 1.
figure 1

Zones and depths of image acquisition. Zone 1 next to the flow inlet and Zone 3 the farthest. Depth (1) 1 mm from the surface, (0) at the center of the specimen and (−1) at 3 mm from the surface.

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3.2 Pre-processing

The pre-processing corresponds to all the tools and methods applied prior to the analysis of the fibers. Initially it was necessary to determine the value of the scale which is the number of one-dimensional pixels representing a defined value, usually 1 mm. Since the top edges of the images were not horizontal and the area of analysis does not have a standard value, it was necessary to make inclination adjustments and crop the image.

To eliminate noise or particulate matter, some filtering alternatives were evaluated. Since it is an analysis of morphological characteristics specifically, methods such as erosion or dilation that could modify the shape of the fibers were not considered as valid tools for this work. The application of reconstruction by erosion shows a remarkable improvement in the results, since through erosion the fibers are filtered by the size and what remains acts as a marker for the reconstruction. The selected structuring element was size 2 diamond. Figure 2 shows the difference between the evaluated methods. These results were decisive to select the filtering method.

Fig. 2.
figure 2

Result of filtering by the techniques: (a) Original, (b) Erosion, (c) Dilation, (d) reconstruction by erosion.

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Subsequently, the thresholding process, which consists in determining the value of the grayscale that will serve as a separation between what will be considered fiber (white) and matrix (black) was performed. The proposed methodology is based on an automatic thresholding process using Otsu’s method which chooses the threshold value to minimize the intraclass variance of the thresholded black and white pixels. The manual thresholding was used only for comparison between the algorithm and the information from commercial software.

4 Processing

The image processing started with feature extraction. In this step, each of the objects presents in the image were separated and the ellipses method was applied. This method consists of determining the adjusted and inscribed ellipse to the object, using the fourth spatial moments. The angle formed between the major and the horizontal axis determines the fiber orientation. Figure 3 displays examples the method applied. In order to define the fiber length, the Feret diameter was calculated. This diameter represents the distance between two parallels lines that are tangential to the projection contour. The fiber width corresponds to the perpendicular line at the object orientation which was traced from the centroid.

Fig. 3.
figure 3

Ellipses method application

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As the analysis is performed on fibers, it is necessary to establish geometric parameters that differentiate fibers from particles. The main constraint corresponds to the aspect ratio which is the ratio between the length and width of the fibers. The minimum value depends on the composition of each material. For this particular case, the minimum aspect ratio was 2.5 [18]. Another restriction corresponds to the maximum limit of fiber length defined in 80 μm according to characteristics of elaboration of the test piece. These data may vary depending on the forming process of the material.

The resulting objects from this filtering meet pre-established conditions and were used for the tensor analysis. Using Eqs. (4)–(7), previously described, the second-order orientation tensor was calculated, where \( a_{11} \) indicates the fibers orientation tendency. If this value is greater than 0.7 it means that the fibers follow the direction of the injection flow. The higher this value, the fibers will be more aligned to the flow direction. Likewise, the preferred angle delivers the tilt value of all fibers in the image using the eigenvalues of the tensor matrix.

4.1 Validation

Initially, three standard images with fiber distributions in the ranges: −10 to 10, 80 to 100 and 35 to 145, covering all possible orientations of the fibers were used. The geometric characteristics of these images were known beforehand. Once the systematic error was obtained and the correct behavior of the algorithm was verified, the images of the composite were analyzed. The obtained results were compared with the Olympus Stream Essentials ® software analysis, found in [18].

5 Results

5.1 Algorithm Validation in Pattern Images

Table 1 shows the comparison between the actual and calculated data of each image. Section (a) corresponds to fibers in the range −10 to 10°; section (b) has fiber information in the range 80 to 100° and section (c) corresponds to fibers in the range 35 to 145°.

Table 1. Comparison of the pattern image results
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As can be seen in the fourth column (Difference), the absolute error for all cases was minimal. The algorithm accurately identified the number of objects existing in the image. The values of the orientation tensors had an error rate of 1.9% in \( \varvec{a}_{12} \) and the most important tensor \( \varvec{a}_{11} \), presented a difference of 0.05%. The results also showed that the average of the fibers orientation had an error of 2%. In terms of length and width, the error was 3.77%. It is important to note that the obtained error values were lower than those found in the literature specified in [18].

5.2 Analysis of PP-GAK Composite at 30% Weight Concentration

Table 2 summarizes the percentages of similarity for the test specimens A and B with 30% concentration by weight, considering as baseline the results from [18]. According to the obtained data, the difference between the two methods is low, especially in the \( a_{11} \) (98,25%), preferential angle (96,40%) and overage orientation (96,09%), which are important parameters to determine the micromechanical behavior of the material. It is worth notice that the main element of the tensor orientation is higher than 0.8 for each value of z. A certain asymmetry was also observed across the thickness. Generally highly oriented zones are those regions in which the polymer matrix solidifies first, with slight variations due to the forces produced by front advance of the flow, as it was reported by Fajardo et al. [19], using a commercial software.

Table 2. PP-GAK 30% percentage of similarity summary
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6 Conclusions

An effective method for the semi-automatic determination of morphological and geometric properties of natural fibers reinforcing biocomposites was developed.

Of all the evaluated aspects, \( a_{11} \) is the most important value since it determines the general fiber orientations of the image additionally it is used to predict the micromechanical behavior of the material. The similarity rate in relation to the data obtained with commercial software was 98,25%, and with the standard image 99,8%, which guarantees a good performance of the proposed method.

The pattern image analysis allowed to evaluate the performance of the developed algorithm and to predict the system error interval.

Pre-processing techniques are decisive in determining geometric characteristics, especially in small objects and are suitable for precision applications as the presented in this work. The application of reconstruction by erosion was considered adequate since it eliminates the noise present in the image keeping the original shape of the objects. One of the causes of the difference of values corresponds to the lack of definition of edges that in small objects has greater influence.

All evaluated aspects in this images of PP-GAK composite show a similarity rate higher than 90%, the values corresponding to \( a_{11} \) have a similarity of 98% and the average preferred orientation angle presents a similarity higher than 96%.

The similarity rate could have been even higher if the analysis area used in the base work had been known. According to a report with graphs of the area used, issued by the SSE the space and inclination of the original images was assumed.

The proposed method optimizes the analysis time in comparison to the traditional process which requires at least two stages of analysis with different softwares, one to extract the geometric attributes of the reinforcement material and the other for the post-processing generated by the tensor of orientation.