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Abstract

Geometric reconstruction is an important precondition for the computational micromechanics analysis of chopped fiber reinforced composites. When fiber aspect ratio increases, the maximum fiber volume fraction in reconstruction model reduces rapidly because of jamming limit, which greatly limits the application of reconstruction methods. A novel algorithm is proposed to significantly increase the fiber volume fraction in the reconstruction model of the chopped fiber reinforced composites with large fiber aspect ratio. The algorithm is made up of two stages. At the first stage, fibers are packed into the sublayers of initial filling space to preliminarily design fiber orientation distribution. The unidirectional arrangement of fibers is adopted to achieve high fiber volume fraction. At the second stage, a new multi-step fiber shaking strategy is used to introduce randomness into reconstruction model. The high fiber volume fraction over 30% is achieved within the wide range of fiber aspect ratio from 50 to 200 while the results of the existing methods are not more than 10%, showing the remarkable increase of the fiber volume fraction under large fiber aspect ratio. The proposed algorithm is verified by the statistical results of the four representative microstructural characteristics from reconstruction model and realistic material.

Keywords

Chopped fiber reinforced composites geometric reconstruction algorithm fiber volume fraction fiber aspect ratio microstructural characteristics

Introduction

Computational micromechanics based finite element method is regarded as the most promising method for research on composite material [1]. The first step of finite element method is to reconstruct the microstructures of composites, which is called geometric reconstruction. For reliable analysis result, the fibers in reconstruction model should distribute similarly as in realistic material, which depends on reconstruction algorithms.

Injection molded chopped fiber reinforced composites are widely used industrial materials [2 –4] which show the significant advantages in cost, productivity and fabricability over conventional continuous fiber reinforced composites. For chopped fiber reinforced composites, there are two classes of strategies for geometric reconstruction. The first one is random sequential absorption (RSA) based packing algorithm. Fibers are randomly and sequentially filled into a space and penetrated fibers will be abandoned. This process will iteratively continue until target fiber volume fraction or maximum iteration number has been reached [5]. Pan [1] adopts curved fibers and divides RVE space into fiber-rich sublayers and matrix-rich sublayers, achieving 35.1% volume fraction with the fiber aspect ratio of 20 as a result. Kari et al. [6] reduces fiber aspect ratio in condition that volume fraction can hardly be increased to improve filling efficiency. Tian [7] proposes a fiber growth method and realizes 22.5% volume fraction within the range of fiber aspect ratio from 5 to 15. Besides, an improved RSA algorithm has been proposed by Tian [8] to efficiently generate RVE with specified particle orientation. Li [9] adds statistical orientation information in reconstruction and analyzes the relationship between the number and the orientation error of filled fibers. The fiber volume fraction of RSA based algorithm is mainly limited by fiber aspect ratio. The present fiber volume fraction of reconstruction model can not satisfy the analysis requirement of realistic composite material.

The second one is Monte Carlo based packing algorithm. Fibers are randomly put into a large box and shaken to avoid penetration. Then the box is squashed so that fiber volume fraction could be increased. Shaking and squashing processes are alternately conducted until an expected fiber volume fraction is reached [10,11]. Dirrenberger et al. [12] introduce Poisson law into Monte Carlo process to build the model of random fiber microstructure. Schneider [13] achieves a high fiber volume fraction by the local close arrangement of fibers using Sequential addition and migration (SAM) algorithm and gradient descent method, but the microstructure of the reconstruction model is obviously different from realistic material. Altendorf and Jeulin [14] reconstruct Monto Carlo simulation scheme based densely-packed fiber system. Fibers are bent too much so that systematic errors could be introduced in finite element analysis for cylindrical fibers such as chopped glass fiber. Monte Carlo based packing algorithm utilizes close arrangement or bending fibers to improve fiber volume fraction, reducing reconstruction accuracy and increasing application limitations. Due to Monte Carlo based random shaking process, fiber orientation distribution can not be controlled according to the microstructure of realistic material.

In the existing studies of reconstruction algorithm, the phenomenon that fibers can hardly be filled into a model when fiber volume fraction reaches a certain value is called “jamming”. The maximum fiber volume fraction currently achievable is the jamming limit. The theoretical and experimental curves of the jamming limit with respect to fiber aspect ratio are proposed by Evans and Gibson [15] and are confirmed by many studies [1,6,7,16 –18]. For instance, the jamming limit of the composites with fiber aspect ratio equaling to 70 is not more than 10% according to Evans’ curve. However, the fiber volume fraction in realistic composite products with the same fiber aspect ratio could approach to 20% or higher values. Therefore, the jamming limit makes a great challenge on the application of geometric reconstruction algorithm to the computational micromechanics analysis of the chopped fiber reinforced composites with high fiber volume fraction and large fiber aspect ratio.

In this article, a novel algorithm is proposed to significantly increase the fiber volume fraction of geometric reconstruction model. The injection molding process of chopped fiber reinforced composites gives an inspiration. Before injection, fibers are unidirectionally distributed in granule which is the prefabricated material of injection molded composites. During manufacturing process, granules are melted, stirred and injected into injection molding machine. Fibers suffer sophisticated mechanical and fluidic actions which introduce the randomness of fiber distribution in material. Considering aforementioned process, a two-stage algorithm is proposed to build geometric reconstruction model. Firstly, fibers are unidirectionally arranged in filling space to efficiently improve fiber volume fraction. Several sublayers are adopted to preliminarily design fiber orientation distribution. Then, fibers are shaken by a multi-step strategy to introduce randomness into reconstruction model. By the proposed algorithm, one can build the geometric reconstruction model with much higher fiber volume fraction than the existing methods.

The remainder of this paper is organized as follows. Firstly, three basic definitions of fiber distributions are given. Secondly, the detailed algorithm mechanism is illustrated including the method of fiber initial filling and the multi-step shaking strategy. The comparison of the algorithm and the existing methods is conducted and analyzed. Then, the algorithm is verified based on the four microstructural characteristics of chopped fiber reinforced composites. The discussions and conclusions of this study are given in the last part.

Basic definitions

Microstructural characteristics have important influence on material properties [19 –21] and some basic definitions of the characteristics of chopped fiber reinforced composite material are given in this section, including fiber orientation distribution, spatial distance of fibers and fiber length distribution.

Fiber orientation tensor

In Advani and Tucker’s research [22], two angles $θ$ and $ϕ$ are used to describe the orientation of a fiber as illustrated in Figure 1. The unit vector of a single fiber $P = (P_{1}, P_{2}, P_{3})$ can be written as equation (1):

\begin{array}{l} P_{1} = \sin θ \cos ϕ \\ P_{2} = \sin θ \sin ϕ \\ P_{3} = \cos θ \end{array}

(1)

Figure 1.

The definition of a single fiber P in three-dimensional space.

The fiber orientation distribution at a point in macroscale can be described by a probability distribution function, $ψ (P)$ . A second order orientation tensor $a_{i j}$ is defined based on $ψ (P)$ :

a_{i j} = \int p_{i} p_{j} ψ (P) d P i, j = 1, 2, 3

(2)

Eigen values and eigen vectors are obtained by eigen-decomposition of orientation tensor [23]:

S_{ij} = [\begin{matrix} S_{11} & S_{12} & S_{13} \\ S_{21} & S_{22} & S_{23} \\ S_{31} & S_{32} & S_{33} \end{matrix}] = [\begin{matrix} a_{1} & 0 & 0 \\ 0 & a_{2} & 0 \\ 0 & 0 & a_{3} \end{matrix}] \cdot [\begin{matrix} S_{1} & S_{1} & S_{1} \end{matrix}]

(3)

where the eigen vectors

S_{1}

S_{2}

and

S_{3}

represent three main fiber distribution orientations and the eigen values

a_{1}

a_{2}

and

a_{3}

are the probabilities of the fiber orientation in the direction of corresponding eigen vectors. According to the rule, the larger is the footnote of eigen value, the smaller is the eigen value. Thus, the size relationship of eigen values is

1 \geq a_{1} \geq a_{2} \geq a_{3} \geq 0

. From the normalization condition:

a_{1} + a_{2} + a_{3} = 1

(4)

Spatial distance of fibers

The fibers in injection molded composites could be represented by the segments in three-dimensional space. As shown in Figure 2(a), define two segments as AB and $CD$ . Points $P$ and $Q$ are on the two segments respectively and can be described by two parameters $s$ and $t$ as follows:

\begin{array}{l} P = A + s (B - A), s = \frac{A P}{A B} \\ Q = C + t (D - C), t = \frac{C Q}{C D} \end{array}

(5)

Figure 2.

The four situations of the distance between two segments. (a) $0 \leq s, t \leq 1$ . (b) $(0 \leq s \leq 1) & (t < 0 | t > 1)$ . (c) $(s < 0 | s > 1) & (0 \leq t \leq 1)$ . (d) $(s < 0 | s > 1) & (t < 0 | t > 1)$ .

The squared distance between $P$ and $Q$ is defined as:

f (s, t) = {| P - Q |}^{2}

(6)

where

s

and

t

are the arguments of

f

and can be obtained by taking the partial derivatives:

\begin{array}{l} \frac{\partial f (s, t)}{\partial s} = 0 \\ \frac{\partial f (s, t)}{\partial t} = 0 \end{array}

(7)

There are four situations of the distance between two spatial segments according to the range of $s$ and $t$ as shown in Figure 2. For different situations, the distance between two segments can be calculated as following equation:

Dis = {\begin{array}{l} | PQ |, while 0 \leq s, t \leq 1 \\ \min (| C, AB |, | D, AB |), while (0 \leq s \leq 1) & (t < 0 | t > 1) \\ \min (| A, CD |, | B, CD |), while (s < 0 | s > 1) & (0 \leq t \leq 1) \\ \min (| A, CD |, | B, CD |, | C, AB |, | D, AB |), while (s < 0 | s > 1) & (t < 0 | t > 1) \end{array}

(8)

Fiber length distribution

During injection molding process, part of fibers will be broken and fiber lengths obey skewed distribution. The fiber length distribution in the material with 20% fiber volume fraction is obtained from burning off the matrix. The burn-off temperature is 400 °C (752 °F) and the duration of burning is 3 hours. Over one thousand fibers are measured and the average length of fibers is around 800 μm. The fiber length histogram is shown in Figure 3. Use the Gamma distribution in equation (9) to fit the fiber length distribution and regenerate fiber lengths:

f (l, β, α) = \frac{β^{α}}{Γ (α)} l^{α - 1} e^{- β l}, α > 0, β > 0

(9)

where

α

and

β

are shape parameter and scale parameter, respectively.

Figure 3.

The fiber length histogram of composites with V_f = 20%.

The scheme of the algorithm

In this section, a novel algorithm is proposed to build the geometric reconstruction model of the chopped fiber composites with high fiber volume fraction and large fiber aspect ratio. The reconstruction process of the algorithm consists of two stages. In the first stage, fibers are unidirectionally arranged in initial filling space. Fiber orientation state is preliminarily designed by making use of sublayers. In the second stage, a multi-step strategy is used to shake fibers and introduce randomness into reconstruction model. In the end, the maximum fiber volume fraction of the proposed algorithm is compared with the performance of the existing methods and the results are analyzed and discussed.

Initial filling

In injection molded composites, fibers mainly lie in the injection flow plane with slight deflection angles out of the plane [24 –26]. Thus, the component $a_{3}$ of the orientation tensor is quite small (less than 0.1) so that the out of plane angles of the fibers can be neglected [1,27]. Considering the aforementioned studies and the convenience of modeling in the initial filling stage, $a_{3}$ is assumed to be zero and equation (4) can be simplified as:

a_{1} + a_{2} = 1

(10)

In equation (10), the value of $a_{2}$ depends on the value of $a_{1}$ . In this way, fiber orientation state can be represented by the principle eigen value $a_{1}$ of orientation tensor.

Stapleton et al.’s study [28] indicates that fiber volume fraction can achieve up to 90.7% by unidirectionally filling, which can improve maximum fiber volume fraction significantly and is adopted in the first stage of initial filling. During initial filling process, fibers are arranged in one sublayer in the direction of $S_{1}$ or in another sublayer in the direction of $S_{2}$ as shown in the schematic diagram in Figure 4. To ensure the expected fiber distribution state that the probabilities of fiber orientation in direction of $S_{1}$ and $S_{2}$ are respectively $a_{1}$ and $a_{2}$ , the number of fibers in two directions should satisfy the ratio $a_{1} / a_{2}$ . Considering equation (10), the ratio can be expressed as $a_{1} / (1 - a_{1})$ by the independent value $a_{1}$ . $V_{f}^{1}$ and $V_{f}^{2}$ are the fiber volume fractions of sublayers in the direction of $S_{1}$ and $S_{2}$ , respectively. Each sublayer is divided into many units and each unit contains as least one fiber. $x_{1}$ and $x_{2}$ are the unit sizes of the sublayers. The transverse section shape of the unit cell in the direction $S_{1}$ is square, therefore, the thickness of the sublayer is $x_{1}$ . To ensure that the fibers in sublayer in the direction of $S_{2}$ gain the same space for shaking in the next stage, the thickness of the sublayer in the direction of $S_{2}$ is set to be the same as the thickness of the sublayer in the direction of $S_{1}$ . $V_{f}^{1}$ and $V_{f}^{2}$ are determined by the target fiber volume fraction $V_{f}^{t}$ and the principle eigen value $a_{1}$ .

Figure 4.

The schematic diagram of initial filling process. (a) Two sublayers when a₁<0.6. (b) Three sublayers when a₁≥0.6.

When $a_{1} < 0.6$ , the probabilities of fibers distributing in the two directions are close to each other and the filling space is separated into 2 sublayers as shown in Figure 4(a). The related parameters are calculated by following formulae:

\begin{array}{l} V_{f}^{1} = 2 V_{f}^{t} a_{1} \\ V_{f}^{2} = 2 V_{f}^{t} (1 - a_{1}) \\ x_{1} = \sqrt{\frac{π r^{2}}{V_{f}^{1}}} \\ x_{2} = \frac{a_{1}}{1 - a_{1}} x_{1} \end{array}

(11)

where

r

is fiber radius.

When $a_{1} \geq 0.6$ , the probability of fiber distributing in $S_{1}$ is much higher than that in $S_{2}$ . To avoid the significant difference of fiber volume fraction between sublayers, the filling space are separated into 3 sublayers as shown in Figure 4(b). The fiber orientations in top and bottom sublayers are consistent with $S_{1}$ and the fiber orientation in the middle sublayer is consistent with $S_{2}$ . The related parameters are given:

\begin{array}{l} V_{f}^{1} = 1.5 V_{f}^{t} a_{1} \\ V_{f}^{2} = 3 V_{f}^{t} (1 - a_{1}) \\ x_{1} = \sqrt{\frac{π r^{2}}{V_{f}^{1}}} \\ x_{2} = \frac{a_{1}}{2 (1 - a_{1})} x_{1} \end{array}

(12)

Gitman et al. [29] proposes the existence of wall-effect in the process of heterogeneous material reconstruction which refers to the inability of inclusions to penetrate through RVE boundaries. Wall-effect will lead to the change of material properties near RVE boundaries and weaken the representativeness of RVE. To avoid Wall-effect, Harper et al. [30] makes shaking region twice as large as RVE space to let fibers go through RVE boundaries freely. This study adopts Harper’s method to deal with Wall-effect.

In initial filling process, the fiber content near filling boundaries is generally lower than that in central region. This phenomenon is defined as the nonuniformity of fiber content in this study. To avoid the nonuniformity of fiber content, the size of initial filling region is twice as large as the size of shaking region. In this way, the fiber content within shaking region is uniform. Figure 5 shows an initial filling result of the chopped fiber reinforced composites with the fiber average aspect ratio $R = 50$ , the principle eigen value $a_{1} = 0.8$ and the target fiber volume fraction $V_{f}^{t} = 20 %$ . Commercial software Matlab is used to perform calculations and draw Figure 5.

Figure 5.

The initial filling result of the material with $R = 50$ , $a_{1} = 0.8$ and $V_{f}^{t} = 20 %$ .

A multi-step shaking strategy

During injection molding process, fibers experience sophisticated fluidic and mechanical effects, which lead to the nonuniform distribution of fibers in composite material. Imitating these effects, a multi-step shaking strategy is proposed in this part.

The shaking process is divided into three steps. The first step is global shaking and the state of fibers in shaking region will be tried to updated in this step. Use midpoint coordinates and two angles $(x_{c}, y_{c}, z_{c}, θ, ϕ)$ to describe the fiber state including location and orientation. A shaking step $(Δ x_{c}, Δ y_{c}, Δ z_{c}, Δ θ, Δ ϕ)$ is generated in an appropriate range and the fiber state is updated by the linear superposition of its initial state and the small shaking steps. The mathematical description of fiber state update is given below:

{({x^{'}}_{c}, {y^{'}}_{c}, {z^{'}}_{c}, θ^{'}, ϕ^{'})}^{T} = {(x_{c}, y_{c}, z_{c}, θ, ϕ)}^{T} + {(Δ x_{c}, Δ y_{c}, Δ z_{c}, Δ θ, Δ ϕ)}^{T}

(13)

where the primed variables describe the fiber state after shaking. The descriptions of the components of the shaking step are given in following equation:

\begin{array}{l} Δ x_{c} = Δ y_{c} = \frac{α \cdot x_{1} \cdot r}{N_{\max}} \\ Δ z_{c} = \frac{β \cdot H \cdot r}{N_{\max}} \\ Δ θ = \frac{γ_{1} \cdot θ_{0} \cdot r}{N_{\max}} \\ Δ ϕ = \frac{γ_{2} \cdot ϕ_{0} \cdot r}{N_{\max}} \end{array}

(14)

where

x_{1}

is the unit size introduced in Initial filling section and H is the thickness of shaking region.

ϕ_{0}

equals to

45^{\circ}

and

θ_{0}

is defined by

arcsin (H / L_{f})

N_{\max}

is the maximum number of iterations.

r

obeys the normal distribution

N (0, \frac{1}{3})

, by which randomness is introduced into fiber distribution.

α, β, γ_{1}

and

γ_{2}

are the coefficients of the components to control fiber distribution. The values of these coefficients are obtained from trial and error. Some recommended value ranges are listed in Table 1. For fiber orientation distribution,

γ_{1}

and

γ_{2}

influence the concentration degree of probability density function. The small values of the parameters result to concentrated distributions while the large values result to gentle distributions. It should be note that parameters of global shaking play decisive roles in fiber distribution and should be preferentially considered while parameters in local shaking and breaking & shaking mainly affect the computational cost.

Table 1.

The recommended value ranges of coefficients in equation (14).

Step	$α$	$β$	$γ_{1}$	$γ_{2}$
Global shaking	[3, 5]	[2, 3]	[1, 2]	[1, 2]
Local shaking	[0.5, 2]	[0.1, 1]	[0.1, 1]	[0.1, 1]
Breaking and shaking	[0.1, 1]	[0.1, 1]	[0.1, 1]	[0.1, 1]

Once the state of a fiber is updated, the penetration tests are conducted between this fiber with the other fibers that have finished global shaking. If it passes penetration test, then it will be added into the fiber set of globally shaken fibers. On the contrary, if it fails to pass penetration test, a new shaking step will be generated for next attempt. A maximum attempt number is set for global shaking step and if a fiber fails to update its state within the specified number, it will be recorded in a predefined set to perform the local shaking in the second step.

In the second step called local shaking, the fibers shaken in the first step are fixed and more computing resources will be allocated to the fibers failing to update their state in the first step. For a fiber to be dealt with, a new shaking step is generated and its state is updated according to equations (13) and (14). If the new fiber passes penetration test, it will be added into the fiber set of locally shaken fibers. Otherwise, this process will be attempted continuously until the fiber passes penetration test or the maximum attempting number is reached. If it fails to update fiber state within the specified number, the fiber will be called a stubborn fiber and its length will be broken in the third step.

In the third step, stubborn fibers are broken and shaken. For a stubborn fiber, its aspect ratio and other surrounding fibers limit its shaking and state updating. Changing the state of surrounding fibers may create chain reactions and lead to more penetrated fibers, making the problem even more difficult. Decreasing its aspect ratio, however, is a feasible and practical way. During injection molding process, some of fibers are broken because of mechanical interactions. Inspired by this process, a stubborn fiber is broken so that its aspect ratio is reduced and its state is easier to be updated in this situation. The breakage is supposed to happen at the middle point of the fiber and two new fibers are obtained. Use the scheme in the second step to update the state of these two fibers and those pass penetration test will be added into fiber set of repositioned broken fibers. If a broken fiber still penetrates other fixed fibers, it will be deleted instead of being broken again, for breaking many times will lead to quite short fibers and influences the fiber length distribution in reconstruction model.

Each time before updating the state of a fiber, it is necessary to conduct the penetration tests between the fiber and other fibers. To make a physical sense, all the distances should be larger than fiber diameter. To avoid the distortion of finite element meshes, the minimum distance between surfaces of two fibers is specified to be 5% fiber radius $r_{f}$ [7,30]. In this study, the distance between the center lines of two fibers should satisfy following equation:

Dis (Fibe r_{1}, Fibe r_{2}) \geq MinDis = (d_{f} + 5 % r_{f})

(15)

As shown in the flowchart of the algorithm in Figure 6, the sets of globally shaken fibers, locally shaken fibers and repositioned broken fibers are recorded and these fibers will participate in the next iteration until the maximum number of iterations is reached and a geometric reconstruction model is obtained as a result.

Figure 6.

The flowchart of the proposed algorithm.

Implementation

The implementation of the proposed algorithm is given in this section. The random fiber orientation distribution ( $a_{1} = 0.5$ ) is adopted and the fiber aspect ratio ranges from 10 to 200. Perform 5 reconstruction models for each fiber aspect ratio. The target fiber volume fraction $V_{f}^{t}$ is set to be 40%, which is large enough to test the maximum fiber volume fraction $V_{f}^{\max}$ under the predefined conditions. The results of the proposed algorithm and the existing methods [1,7,16 –18] as well as Evans’ jamming limit curves [15] are plotted in Figure 7. Some necessary discussions are given.

Figure 7.

The maximum fiber volume fraction by the proposed algorithm versus the existing methods.

In Figure 7, the results of existing methods are divided into two classes. One includes methods for the reconstruction of 3D random fiber orientation distribution [7,17,18]. The other includes methods for the reconstruction of flow-plane random and slightly out of plane fiber orientation distribution (2.5D random fiber distribution) [1,16]. Results of these two classes of methods are respectively plotted in blue and red color. By the comparison of Evans’ jamming limit curves and the interpolation curve by this study, it is obvious that the maximum fiber volume fraction of methods for 2.5D random fiber distribution is higher than that for the 3D random fiber distribution in a wide range of fiber aspect ratio. The difference results from the low space utilization rate of 3D random fiber distribution. Actually, 2.5D random fiber distribution with higher space utilization rate is much more similar to microstructures of many practical chopped fiber reinforced composites, such as injection molded and compression molded chopped fiber reinforced composites.

For results of 3D random fiber orientation distribution, $V_{f}^{\max}$ decreases obviously with the increase of fiber aspect ratio in Figure 8. When fiber aspect ratio exceeds 50, $V_{f}^{\max}$ is less than 10. The facts demonstrated that 3D random fiber orientation distribution severely limits high fiber volume fraction results. However, the downtrend of the results of 2.5D random fiber distribution is not significant. Most of these methods face the reconstruction of fibers with small fiber aspect ratio. For example, when fiber aspect ratio is 10, $V_{f}^{\max}$ by Pan [1] and Duschlbauer et al. [16] are 13.5% and 21% respectively while $V_{f}^{\max}$ by the proposed algorithm is more than 40%, twice as much as the existing methods, which demonstrates the breakthrough of the proposed algorithm on high fiber volume fraction.

Figure 8.

The reconstruction models with different fiber aspect ratios. (a) $a_{1} = 0.5$ , $r = 10$ , $V_{f}^{\max} = 40.28 %$ . (b) $a_{1} = 0.5$ , $r = 50$ , $V_{f}^{\max} = 35.12 %$ . (c) $a_{1} = 0.5$ , $r = 100$ , $V_{f}^{\max} = 33.42 %$ . (d) $a_{1} = 0.5$ , $r = 200$ , $V_{f}^{\max} = 30.68 %$ .

Verification of the algorithm

To verify the proposed algorithm, a high-resolution 3D micro computed tomography (μCT) system Xradia 520 Versa is applied to observe the microstructure of injection molded chopped fiber reinforced composites. The observed composite material is composed of E-glass fiber and polypropylene. The fiber volume fraction of samples ranges from 10% to 30% and the sample size is 2.5 mm × 2.5 mm × 3.5 mm. The exposure time of 1 s and the photon energy of 50 keV are adopted. The voxel edge size of 2.5 μm is obtained. A 3D image of composites and the μCT system are shown in Figure 9.

Figure 9.

A 3D image of material and the μCT equipment. (a) A 3D image of injection molded composites. (b) The μCT system Xradia 520 Versa.

The studies of characterization of fiber reinforced composites [8,9,18,31 –34] are referred to and four indexes, including fiber orientation distribution, fiber nearest-neighbor distance distribution, fiber length distribution and fiber volume fraction are chosen to quantitatively describe the microstructural characteristics. The fiber distributions of reconstruction model and realistic material are compared and the quantitative evaluation is given based on the concept of entropy.

Fiber orientation distribution

Fiber orientation distribution has a significant influence on the properties of chopped fiber reinforced composites [21]. Adopt the angles $θ$ and $ϕ$ introduced in Fiber orientation tensor section to describe fiber spatial orientation. For chopped fiber reinforced composites, the statistical distribution of fiber angles can be described by probability density function $ψ (θ, ϕ)$ [22]. There is no difference between $(θ, ϕ)$ and $(π - θ, π + ϕ)$ for the orientation description of one fiber so that the fiber orientation probability density function (PDF) $ψ (θ, ϕ)$ should satisfy following equation：

ψ (θ, ϕ) = ψ (π - θ, ϕ + π)

(16)

which can be expressed in another way:

ψ (P) = ψ (- P)

(17)

The sum of fiber distribution probability in all directions should equal to 1:

\int_{θ = 0}^{π} \int_{ϕ = 0}^{2 π} ψ (θ, ϕ) \sin θ d θ d ϕ = \int ψ (P) d P = 1

(18)

For the material with the fiber volume fraction $V_{f} = 20 %$ and the average fiber length $L = 0.8 mm$ ( $R = 50$ ), its joint probability density distribution of fiber angles $θ$ and $ϕ$ are shown in Figure 10(a) as the results of 3D image processing. $θ$ presents a symmetric unimodal distribution centered at 90 degrees and it decreases at a rapid rate from the center to the sides, indicating that the probability of fiber distribution along the thickness of material is very low and fibers mainly distribute in the injection flow plane. $ϕ$ presents a multimodal distribution taking a period of 180 degrees which results from the symmetry of fiber orientation probability density function illustrated as equations (16) and (17). The distribution center of $ϕ$ depends on the relationship between cutting direction of material sample and the direction of injection flow. When the two directions are consistent with each other, the distribution center is 180 degrees and when there is angle $α$ between two directions, the distribution center will offset $α$ relative to 180 degrees.

Figure 10.

The joint PDF of the fiber angles $θ$ and $ϕ$ of realistic material and reconstruction model. (a) The joint PDF of realistic material. (b) The joint PDF of reconstruction model.

The reconstruction target is made to be $a_{1} = 0.84$ which is the same as the principle eigen value of material sample. Make the lengths of initially filled fibers obey the distribution in Figure 3. A reconstruction model is built by the proposed algorithm and the joint PDF of fiber angles $θ$ and $ϕ$ is plotted in Figure 10(b) compared with that of the realistic material. The two distributions are similar to each other. Furthermore, quantitative comparisons are made.

KL divergence, also called relative entropy, is an effective measurement index to quantify the mismatch between two PDF curves [35]. Define two PDFs $p (x)$ and $q (x)$ . KL divergence can be expressed as:

K L (p | | q) = \int p (x) \log \frac{p (x)}{q (x)} d x

(19)

Considering that fiber orientation distribution is joint probability density distribution of angles $θ$ and $ϕ$ , KL divergence of realistic material and reconstruction model can be calculated by following equation:

K L (ψ_{real} | | ψ_{model}) = \iint ψ_{real} (θ, ϕ) \log \frac{ψ_{real} (θ, ϕ)}{ψ_{model} (θ, ϕ)} d θ d ϕ

(20)

where

ψ_{real} (θ, ϕ)

and

ψ_{model} (θ, ϕ)

are the joint probability density distributions of the microstructural characteristics of realistic material and reconstruction model, respectively. As KL divergency is the absolute error between the cross entropy and the information entropy, a normalized index IS is used:

I S = \frac{K L (ψ_{real} | | ψ_{model})}{\iint ψ_{real} \log ψ_{model}}

(21)

IS measures the difference of microstructural characteristics between realistic material and reconstruction model and is adopted in this article.

6 samples with different fiber volume fractions are tested, each containing over 100 fibers. For each sample, reconstruct material microstructure for 5 times and calculate the IS values. The trend of the IS values changing with fiber volume fraction is shown in Figure 11. The IS values show a slow upward trend with the increase of fiber volume fraction. The errors between two PDFs are quite small, indicating that the fiber orientation distribution of reconstruction model is accurate, which is of great importance to further finite element analysis.

Figure 11.

The IS of fiber orientation distribution with variable $V_{f}$ .

Fiber nearest-neighbor distance distribution

In material space, define the distance of arbitrary fiber $q$ to the referenced fiber $p$ as $Z_{p q}$ . Then, the normalized nearest-neighbor distance of the referenced fiber $p$ are expressed in [20] as the following equation:

d_{m} = \min_{p \neq q} | Z_{p q} | / D

(22)

where

D

is fiber diameter. It is naturally satisfied that

d_{m} > 1

, otherwise penetration will happen. On the other hand,

d_{m}

will gradually reduce to 1 when fiber volume fraction keeps increasing. In this condition,

d_{m}

represents the degree of fiber concentration around referenced fiber

p

. Hojo et al. [36] and Pan [1] indicate that the higher the degree of fiber aggregation, the more serious the stress concentration effect on fiber-matrix interface, which affects the overall failure properties of composites. In this part, the error of fiber nearest-neighbor distance distribution is studied and given out.

For the material with the fiber volume fraction $V_{f} = 20 %$ , the principle eigen value $a_{1} = 0.84$ and the average fiber length $L = 0.8 m m$ ( $R = 50$ ), the fiber nearest-neighbor distance distribution is shown in Figure 12, comparing with that of reconstruction model. The two distributions agree well with each other and the deviation may come from the minimum gap distance between two fiber surfaces (set as 5% $r_{f}$ ) in penetration test. The IS values are illustrated in Figure 13. The results show the small error of the fiber nearest-neighbor distance distribution which plays an important part on the stress distribution results of finite element analysis.

Figure 12.

The fiber nearest-neighbor distance distributions of realistic material and reconstruction model.

Figure 13.

The IS of fiber nearest-neighbor distance distribution with variable $V_{f}$ .

Fiber length distribution

It is proved by Thomason’s study [19] that fiber length plays an important part on the strength of chopped fiber reinforced composites. Bowyer and Bader [37] gives the formula of fiber critical length to explain its influence on the failure mode of the composites:

l_{c} = \frac{r_{f} σ_{t}}{τ}

(23)

where

σ_{t}

is the tensile strength of fiber and

τ

is the interfacial shear strength. Make

l_{f}

the average fiber length. When

l_{f} < l_{c}

, the maximum binding force between fiber and matrix is smaller than its ultimate load and fibers are more likely to be pulled out from matrix. When

l_{f} \geq l_{c}

, the maximum binding force will exceed fiber ultimate load, fibers are more possible to be broken with the increase of load [38,39]. In realistic material, mixture failure mode usually happens because fiber lengths vary within a wide range.

The PDF of fiber lengths in the realistic material with $V_{f} = 20 %$ and the reconstruction model are plotted and compared in Figure 14. The lengths of the fibers in reconstruction model are regenerated by the fiber length distribution of realistic material so that the two PDFs match well with each other. The IS values in Figure 15 also show the high accuracy of the fiber length distribution in reconstruction models.

Figure 14.

The fiber length distributions of realistic material and reconstruction model.

Figure 15.

The IS of fiber length distribution with variable $V_{f}$ .

Fiber volume fraction

Fiber volume fraction mainly influences the elastic properties of chopped fiber reinforced composites [40]. Two groups of tests are conducted to evaluate the errors of the fiber volume fractions in reconstruction models.

Make the fiber aspect ratios be 50 and 100, respectively. Fix the fiber orientation state $a_{1}$ at 0.5. The target fiber volume fraction is gradually increased and reconstruct 5 times for each volume fraction. The errors of each group are shown in Figure 16. Under an ideal condition, there should be a linear relationship between the target and the achieved fiber volume fractions as the dash line in Figure 16. After the target fiber volume fraction reaches 15%, the achieved fiber volume fraction begins to deviate from the linear relationship and tends to be a stable value. The decrease of fiber volume fraction results from the removing of penetrated fibers. Essentially, a maximum fiber volume fraction is a balance between the fiber shaking process and the deletion of the fibers failing to pass penetration tests. Besides, the maximum fiber volume fraction with $R = 100$ is smaller than that with $R = 50$ because longer fibers make shaking process more difficult and reduce filling efficiency. On the whole, the errors are small when the target fiber volume fraction is not more than 15%. If a higher volume fraction is intended, compensation should be made for the target volume fraction.

Figure 16.

The errors between achieved $V_{f}^{\max}$ and $V_{f}^{t}$ .

Discussion and conclusion

Discussion

A novel algorithm is proposed in this article to significantly increase the maximum fiber volume fraction of the reconstruction model for the chopped fiber reinforced composites with large fiber aspect ratio. The reconstruction process of the algorithm is divided into two stages. In the first stage, fibers are unidirectionally arranged in initial filling space. Fiber orientation state can be preliminarily designed using sublayers. In the second stage, fibers are adequately shaken and partly broken imitating the sophisticated flow and mechanical actions during injection process. In this way, not only the maximum fiber volume fraction is significantly increased but also one can design the microstructural characteristics of reconstruction model. The proposed algorithm breaks through the jamming limit during the geometric reconstruction process of chopped fiber reinforced composites. The comparisons with the existing methods show that the results of the proposed algorithm are better in the wide range of fiber aspect ratio from 10 to 200. The advantage of the algorithm is much more significant when the fiber aspect ratio is larger than 50, in which condition the result of this article is higher than 30% while the results of the existing methods are less than 10%. The high fiber volume fraction and the large fiber aspect ratio of the reconstruction model by the proposed algorithm can satisfy the requirements of majority materials. Four representative characteristics are chosen to quantitatively evaluate the accuracy of the algorithm based on the concept of entropy.

Despite of the aforementioned advantages, some limitations of the proposed algorithm are given for an objective perspective. Firstly, Filling of curved fibers is beyond the capability of the proposed algorithm. Another limitation results from the assumption that the component of orientation tensor $a_{3}$ is zero. On one hand, the assumption may overestimate the degree of fiber in-plane orientation leading to errors of predicted mechanical properties by subsequent finite element analysis. On the other hand, the influence of out of plane orientation $a_{3}$ should not by neglected for some features in practical structures, such as complicated geometry, manufacturing defects (weld line), some particular positions (injection gate) and a few special injection molding process (co-injection molding) [26,41 –44]. Some studies will be carried out to deal with more practical features in our future works. However, in general condition, the proposed algorithm is still effective and practical for the geometrical reconstruction of chopped fiber reinforced composites.

Conclusion

The main contributions of this article are as follows:

A novel algorithm is proposed to increase the maximum fiber volume fraction of the reconstruction model of the chopped fiber reinforced composites with large fiber aspect ratio. The proposed algorithm is proved to be better than the existing methods in the wide range of fiber aspect ratio for 10 to 200. When the fiber aspect ratio is larger than 50, the more significant advantage is shown that the maximum fiber volume fraction is higher than 30% compared to 10% by the existing methods.

The fiber orientation distribution and the fiber length distribution of reconstruction model can be predesigned and controlled by the proposed algorithm, which provides more practical results for subsequent studies.

The statistical descriptions of four representative microstructural characteristics, including fiber orientation distribution, fiber nearest-neighbor distance distribution, fiber length distribution and fiber volume fraction, are used to verify the proposed algorithm by comparing the statistical results of reconstruction models and micro-CT images of realistic material.

The proposed algorithm is quite effective in increasing the fiber volume fraction of the reconstruction model with large fiber aspect ratio, enhancing the applicability of reconstruction model to the computational micromechanics analysis of practical composites products. The composites with thin shell such as injection molded chopped fiber composites and sheet molded compounds chopped fiber composites are recommended to be reconstructed by the proposed algorithm.

Footnotes

Declaration of conflicting interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research is supported by the Key National Natural Science Foundation of China (Grant No.U1864211) and the National Natural Science Foundation of China (Grant No.11772191).

ORCID iD

Ping Zhu

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A novel algorithm for significantly increasing the fiber volume fraction in the reconstruction model with large fiber aspect ratio

Abstract

Keywords

Introduction

Basic definitions

Fiber orientation tensor

Spatial distance of fibers

Fiber length distribution

The scheme of the algorithm

Initial filling

A multi-step shaking strategy

Implementation

Verification of the algorithm

Fiber orientation distribution

Fiber nearest-neighbor distance distribution

Fiber length distribution

Fiber volume fraction

Discussion and conclusion

Discussion

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References