Multi-Scale Probabilistic Analysis for the Mechanical Properties of Plain Weave Carbon / Epoxy Composites Using the Homogenization Technique

: Probabilistic analyses of carbon fabric composites were conducted using the Monte Carlo simulation based on a homogenization technique to evaluate the mechanical properties of composites and their stochastic nature. First, the homogenization analysis was performed for a micro-level structure, which ﬁber and matrix are combined. The e ﬀ ective properties obtained from this analysis were compared with the results from the rule of mixture theory to verify the homogenization analysis. And, tensile tests were conducted to clearly evaluate the result and the reliability was veriﬁed by comparing the results of the tensile tests and homogenization analysis. In addition, the Monte Carlo simulation was performed based on homogenization analyses to consider the uncertainties of the micro-level structure combined of ﬁber and matrix. Next, the results of this simulation were applied to the macro-level structure combined of the tow and matrix to perform the Monte Carlo simulation based on the homogenization technique. Finally, the sensitivity analysis was conducted to identify the e ﬀ ect of constituents of the carbon plain weave composite and the linear correlation of the micro-and macro-level structures combined of the ﬁber / matrix and tow / matrix, respectively. The ﬁndings of this study veriﬁed that the e ﬀ ective properties of the plain weave carbon / epoxy composite and their uncertainties depended on the properties of the carbon ﬁber and epoxy, which are the basic constituents of plain weave carbon / epoxy composites.


Introduction
Carbon fiber reinforced composites are extensively utilized in various industries including aerospace, defense, automotive, and sports, due to their high specific strength and specific stiffness and excellent fatigue resistance. These composites can be classified into unidirectional (UD) and woven composite materials. In particular, woven carbon/epoxy composites feature improved in-plane mechanical properties and impact resistance due to tow structure [1][2][3][4][5]. Consequently, woven carbon/epoxy composites are mainly used in industries instead of the UD composites.
The design of structures using carbon fabric composites requires highly reliable mechanical properties. The homogenization technique is a numerical procedure in which a heterogeneous elastic media is converted into an equivalent model, which is energetically equivalent to the authentic heterogeneous media. For fiber reinforced composites, the homogenization technique assumes the composite structures are composed of a periodic microstructure as shown in Figure 1, and reduces to a unit cell problem [6][7][8]. Hence, the homogenization method is used to estimate the various mechanical properties of composite a unit cell problem [6][7][8]. Hence, the homogenization method is used to estimate the various mechanical properties of composite materials, which takes into consideration their constituents and microstructure [9][10][11]. For example, Babu et al. [12] created four representative volume elements (RVEs) including square and hexagonal arrays that have different fiber orientations in UD composites. Furthermore, they predicted the effective properties by comparing the random sequential adsorption (RSA) algorithm developed in this study with the Halpin-Tsai and Mori-Tanaka homogenization techniques. Qi et al. [13] modeled the RVEs by considering the fiber orientation and ply angle of carbon fiber reinforced plastics (CFRPs), predicted the effective properties based on cross-scale simulations, and compared them with the tensile test results. Gao et al. [14] suggested the Integrated Computational Material Engineering framework implemented by the rule of mixture (ROM) to produce a unidirectional CFRP laminated structure, and compared the tensile tests and 3-point bending tests results with the simulation results. The abovementioned studies proposed various RVE models to predict the effective properties of the fiber level. However, these models have limitations because they do not consider the uncertainties of mechanical properties caused by the structural characteristics of the fiber-matrix composites and various inherent defects [15]. For this, Lee et al. [16] identified the linear correlation of the effective and material properties using the Monte Carlo simulation (MCS) to consider the uncertainties of the properties of fiber and matrix. Kamiński and Kazimierczak [17] analyzed the material properties according to deviations based on the square RVEs including single cylindrical fibers. However, the abovementioned studies only investigated the fiber levels of UD composites with considerably simpler internal structural shapes than those of the fabric composites. Figure 1. The assume structure composites of periodic; square array [16].
Fabric composites are disadvantageous because it is difficult to evaluate the mechanical properties using the conventional homogenization techniques at the fiber level due to the complexity of the internal structure of the constituents. To overcome this, the structural characteristics of fabric composites should be considered by conducting multi-scale analyses to consider the tow level, which fiber and matrix are combined, and the higher level with the tow/matrix combination. Goda et al. [18] predicted the mechanical behavior of different plain weave structures using the discrete homogenization approach for the fabric composites. Chen et al. [19] developed a multi-level homogenization model to determine the effective properties of the woven glass/epoxy composite multilayer printed circuit board. However, the abovementioned studies did not consider the uncertainties of the properties in each constituent at the micro level of fiber/matrix combination (tow) and macro-level of tow/matrix combinations (laminae) in the fabric composite, respectively. Several studies have been conducted to evaluate the effective properties by considering the uncertainties of the mechanical properties of fabric composites. For example, Zhou et al. [20] used the MCS method to consider the uncertainties related to the elastic properties of plain weave composites and evaluated the effective properties of the composite. Hayat et al. [21] developed a multi-scale homogenization technique to predict the elastic behavior of fabric composites. The abovementioned studies analyzed the effective properties of the fabric composites by considering the variability at the macro-level. However, the variability of the mechanical properties of composites comes from the uncertainties of the fiber and matrix at the micro-level. The effective properties of composites should be, therefore, evaluated under the multi-scale condition from the micro-level to the macro-level. Furthermore, these Fabric composites are disadvantageous because it is difficult to evaluate the mechanical properties using the conventional homogenization techniques at the fiber level due to the complexity of the internal structure of the constituents. To overcome this, the structural characteristics of fabric composites should be considered by conducting multi-scale analyses to consider the tow level, which fiber and matrix are combined, and the higher level with the tow/matrix combination. Goda et al. [18] predicted the mechanical behavior of different plain weave structures using the discrete homogenization approach for the fabric composites. Chen et al. [19] developed a multi-level homogenization model to determine the effective properties of the woven glass/epoxy composite multilayer printed circuit board. However, the abovementioned studies did not consider the uncertainties of the properties in each constituent at the micro level of fiber/matrix combination (tow) and macro-level of tow/matrix combinations (laminae) in the fabric composite, respectively. Several studies have been conducted to evaluate the effective properties by considering the uncertainties of the mechanical properties of fabric composites. For example, Zhou et al. [20] used the MCS method to consider the uncertainties related to the elastic properties of plain weave composites and evaluated the effective properties of the composite. Hayat et al. [21] developed a multi-scale homogenization technique to predict the elastic behavior of fabric composites. The abovementioned studies analyzed the effective properties of the fabric composites by considering the variability at the macro-level. However, the variability of the mechanical properties of composites comes from the uncertainties of the fiber and matrix at the micro-level. The effective properties of composites should be, therefore, evaluated under the multi-scale condition from the micro-level to the macro-level. Furthermore, these studies have the limit in that the linear correlation of the properties at the micro-and macro-levels was not analyzed.
In this study, the multi-scale probabilistic analysis on the mechanical properties of plain-weave carbon/epoxy composites using the homogenization technique was performed. First, the effective properties of tow (meso, Figure 2b) were evaluated through homogenization analysis at the micro-level with fiber/matrix combination (Figure 2a). In addition, the homogenization analysis result was verified using the ROM theory and the reliability was verified by comparing the tensile test result and volume fraction measurement with the homogenization analysis results. Next, the probabilistic homogenization analysis was performed using the MCS to consider the uncertainties of the properties of the constituents. Finally, the effective properties of the fabric composites at the macro-level with tow/matrix combinations ( Figure 2c) were obtained by using the mechanical properties results derived from the homogenization analyses at the micro-level. Furthermore, the linear correlation between the basic constituents (fiber and matrix) for effective properties, and the probabilistic analysis result were analyzed.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 3 of 14 studies have the limit in that the linear correlation of the properties at the micro-and macro-levels was not analyzed. In this study, the multi-scale probabilistic analysis on the mechanical properties of plain-weave carbon/epoxy composites using the homogenization technique was performed. First, the effective properties of tow (meso, Figure 2b) were evaluated through homogenization analysis at the microlevel with fiber/matrix combination (Figure 2a). In addition, the homogenization analysis result was verified using the ROM theory and the reliability was verified by comparing the tensile test result and volume fraction measurement with the homogenization analysis results. Next, the probabilistic homogenization analysis was performed using the MCS to consider the uncertainties of the properties of the constituents. Finally, the effective properties of the fabric composites at the macrolevel with tow/matrix combinations ( Figure 2c) were obtained by using the mechanical properties results derived from the homogenization analyses at the micro-level. Furthermore, the linear correlation between the basic constituents (fiber and matrix) for effective properties, and the probabilistic analysis result were analyzed.

Material and Specimen
To verify the homogenization technique, we performed the tensile tests for carbon/epoxy composites from USN 175B (SK Chemicals, Seong-Nam City, Republic of Korea) consisting of TR50S 12K carbon fibers and resin. USN 175B from SK Chemical is a unidirectional lamina, the material properties are thickness of 0.171 mm, weight of 261 g/m 2 , tensile modulus of 240 GPa, fiber density of 1.77 g/cm 3 , and resin density of 1.2 g/cm 3 (Provided by the manufacturer, SK Chemicals, Seong-Nam City, Republic of Korea) [25]. The specimens were manufactured using the Autoclave method for UD prepreg and their volume fraction (VF) was 60.08%. The manufacturing conditions were atmospheric pressure of 0.5 MPa, 3 min at 90 °C, 90 min at 125 °C, and vacuum was applied. The tensile tests were performed on seven specimens in accordance with ASTM D3039 [26]. Test conditions were carried out at a crosshead speed of 2 mm/min under atmospheric conditions at room temperature, and sandpaper was used for the grip of the specimen.

Homogenization Analysis Technique
The experiments for predicting the properties of the carbon composites described in the previous section has disadvantages in terms of the cost and time consumption [27]. Therefore, an analytical

Material and Specimen
To verify the homogenization technique, we performed the tensile tests for carbon/epoxy composites from USN 175B (SK Chemicals, Seong-Nam City, Korea) consisting of TR50S 12K carbon fibers and resin. USN 175B from SK Chemical is a unidirectional lamina, the material properties are thickness of 0.171 mm, weight of 261 g/m 2 , tensile modulus of 240 GPa, fiber density of 1.77 g/cm 3 , and resin density of 1.2 g/cm 3 (Provided by the manufacturer, SK Chemicals, Seong-Nam City, Korea) [25]. The specimens were manufactured using the Autoclave method for UD prepreg and their volume fraction (VF) was 60.08%. The manufacturing conditions were atmospheric pressure of 0.5 MPa, 3 min at 90 • C, 90 min at 125 • C, and vacuum was applied. The tensile tests were performed on seven specimens in accordance with ASTM D3039 [26]. Test conditions were carried out at a crosshead speed of 2 mm/min under atmospheric conditions at room temperature, and sandpaper was used for the grip of the specimen.

Homogenization Analysis Technique
The experiments for predicting the properties of the carbon composites described in the previous section has disadvantages in terms of the cost and time consumption [27]. Therefore, an analytical method is required for predicting the properties of composites. However, it is practically impossible to individually analyze each fiber array by modeling in random order. To address this, the homogenization technique was utilized, which predicts the effective properties by selecting an RVE model based on the Appl. Sci. 2020, 10, 6542 4 of 14 assumption that the unit structures are repeatedly arranged in ideal structure. The RVE model can define a unit cell by a repetitive pattern of carbon fiber and matrix, which typically use square and hexagonal arrays [28,29].
In this study, the unit cells of the unidirectional carbon/epoxy composites under micro-scale were modeled as square arrays to predict the effective properties using the homogenization technique, as shown in Figure 3a. Here, the total volume of unit cell was assumed to be unity, and the unit cells were modeled for carbon fiber volume fractions of 10-70% to predict the mechanical properties of composites with the volume fraction using the ABAQUS [30]. Figure 3b depicts the FE model of the unit cell defined. Element of C3D8R with eight nodes per element and six degrees of freedom per node were used for this model. Here, the total number of elements and nodes was 10,836 and 12,166, respectively. The Mori-Tanaka theory [31,32] was utilized for the homogenization technique to calculate the mechanical behavior. This theory predicts the interaction between the each phase by assuming each inclusion is embedded; reversely, in an infinite matrix, is remotely loaded by the average media strain or stress, respectively. Here, we obtained the mechanical properties of carbon/epoxy composites using the Mori-Tanaka theory implemented in ABAQUS.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 4 of 14 method is required for predicting the properties of composites. However, it is practically impossible to individually analyze each fiber array by modeling in random order. To address this, the homogenization technique was utilized, which predicts the effective properties by selecting an RVE model based on the assumption that the unit structures are repeatedly arranged in ideal structure. The RVE model can define a unit cell by a repetitive pattern of carbon fiber and matrix, which typically use square and hexagonal arrays [28,29]. In this study, the unit cells of the unidirectional carbon/epoxy composites under micro-scale were modeled as square arrays to predict the effective properties using the homogenization technique, as shown in Figure 3a. Here, the total volume of unit cell was assumed to be unity, and the unit cells were modeled for carbon fiber volume fractions of 10-70% to predict the mechanical properties of composites with the volume fraction using the ABAQUS [30]. Figure 3b depicts the FE model of the unit cell defined. Element of C3D8R with eight nodes per element and six degrees of freedom per node were used for this model. Here, the total number of elements and nodes was 10,836 and 12,166, respectively. The Mori-Tanaka theory [31,32] was utilized for the homogenization technique to calculate the mechanical behavior. This theory predicts the interaction between the each phase by assuming each inclusion is embedded; reversely, in an infinite matrix, is remotely loaded by the average media strain or stress, respectively. Here, we obtained the mechanical properties of carbon/epoxy composites using the Mori-Tanaka theory implemented in ABAQUS.  As shown in Figure 4, the repetitive boundary conditions were applied to derive the effective properties for each orientation: E11,tow, E22,tow, E33,tow, G12,tow, G13,tow, G23,tow, v12,tow, v13,tow, and v23,tow. Also, as shown in Table 1, the mechanical properties of the fiber and matrix provided by manufacturer were used.  As shown in Figure 4, the repetitive boundary conditions were applied to derive the effective properties for each orientation: E 11,tow , E 22,tow , E 33,tow , G 12,tow , G 13,tow , G 23,tow , v 12,tow , v 13,tow , and v 23,tow . Also, as shown in Table 1, the mechanical properties of the fiber and matrix provided by manufacturer were used.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 4 of 14 method is required for predicting the properties of composites. However, it is practically impossible to individually analyze each fiber array by modeling in random order. To address this, the homogenization technique was utilized, which predicts the effective properties by selecting an RVE model based on the assumption that the unit structures are repeatedly arranged in ideal structure. The RVE model can define a unit cell by a repetitive pattern of carbon fiber and matrix, which typically use square and hexagonal arrays [28,29]. In this study, the unit cells of the unidirectional carbon/epoxy composites under micro-scale were modeled as square arrays to predict the effective properties using the homogenization technique, as shown in Figure 3a. Here, the total volume of unit cell was assumed to be unity, and the unit cells were modeled for carbon fiber volume fractions of 10-70% to predict the mechanical properties of composites with the volume fraction using the ABAQUS [30]. Figure 3b depicts the FE model of the unit cell defined. Element of C3D8R with eight nodes per element and six degrees of freedom per node were used for this model. Here, the total number of elements and nodes was 10,836 and 12,166, respectively. The Mori-Tanaka theory [31,32] was utilized for the homogenization technique to calculate the mechanical behavior. This theory predicts the interaction between the each phase by assuming each inclusion is embedded; reversely, in an infinite matrix, is remotely loaded by the average media strain or stress, respectively. Here, we obtained the mechanical properties of carbon/epoxy composites using the Mori-Tanaka theory implemented in ABAQUS.  As shown in Figure 4, the repetitive boundary conditions were applied to derive the effective properties for each orientation: E11,tow, E22,tow, E33,tow, G12,tow, G13,tow, G23,tow, v12,tow, v13,tow, and v23,tow. Also, as shown in Table 1, the mechanical properties of the fiber and matrix provided by manufacturer were used.

Homogenization Analysis and Experimental Results
The effective properties of the homogenization analysis are outlined in Table 2 based on the change in volume fraction. To verify the homogenization technique used in this study, the effective properties were calculated using the ROM [33,34], then compared with the homogenization analysis results. Table 3 compares the results of the homogenization analysis and the simple ROM (SROM) for the E 11 tow. From table, the homogenization analysis results are well conformed with the SROM results.  Table 3. Normalized E 11,tow comparison for the rule of mixture and the homogenization analysis result. The aforementioned analysis results should be compared with the experimental results to enable a more detailed evaluation of the homogenization technique. To compare the homogenization analysis results with the experimental results, tensile tests were performed in accordance with the method described in Section 2.1 for unidirectional carbon/epoxy composites (TR50S/Epoxy); the results are shown in Figure 5. Table 4 summarizes the homogenization analysis and experimental results. The error rates of E 11,tow and v 12,tow were 1.13 and 9.09%, respectively. Thus, the homogenization analysis result correlated well with the tensile test results. From these results, the homogenization analysis technique in this study was verified for the carbon/epoxy composites.

Random Variables for the Probabilistic Homogenization Technique at the Micro-Level
The homogenization technique described in Section 2 could evaluate the effective properties at the micro-level in which fibers and matrix are physically combined to form tow. However, probabilistic analysis, rather than deterministic analysis, should be employed because the composites have uncertainties in material properties due to several inherent features [35]. To address this, the probabilistic homogenization analysis based on the MCS was performed using the commercial software ABAQUS and ISIGHT [36]. Table 5 lists the selected random variables. Each random variable was assumed to have a normal distribution consisting of 1000 samples. For the coefficient of variation (COV) of 0.05 was applied. Table 6 shows a portion of the set of random variables calculated using this method.

Effective Properties at the Micro-Level
The probabilistic homogenization analysis was performed by considering the variability of the fiber/matrix due to the inherent features for the 60% volume fraction; the distribution of the effective properties is shown in Figure 6. From figure, the effective properties calculated at the micro-level by probabilistic homogenization analysis follow the normal distribution, which are similar to those of the random variables selected in Section 3.1. This indicates that the variability of the effective

Random Variables for the Probabilistic Homogenization Technique at the Micro-Level
The homogenization technique described in Section 2 could evaluate the effective properties at the micro-level in which fibers and matrix are physically combined to form tow. However, probabilistic analysis, rather than deterministic analysis, should be employed because the composites have uncertainties in material properties due to several inherent features [35]. To address this, the probabilistic homogenization analysis based on the MCS was performed using the commercial software ABAQUS and ISIGHT [36]. Table 5 lists the selected random variables. Each random variable was assumed to have a normal distribution consisting of 1000 samples. For the coefficient of variation (COV) of 0.05 was applied. Table 6 shows a portion of the set of random variables calculated using this method.

Random Variables for the Probabilistic Homogenization Technique at the Micro-Level
The homogenization technique described in Section 2 could evaluate the effective properties at the micro-level in which fibers and matrix are physically combined to form tow. However, probabilistic analysis, rather than deterministic analysis, should be employed because the composites have uncertainties in material properties due to several inherent features [35]. To address this, the probabilistic homogenization analysis based on the MCS was performed using the commercial software ABAQUS and ISIGHT [36]. Table 5 lists the selected random variables. Each random variable was assumed to have a normal distribution consisting of 1000 samples. For the coefficient of variation (COV) of 0.05 was applied. Table 6 shows a portion of the set of random variables calculated using this method.

Effective Properties at the Micro-Level
The probabilistic homogenization analysis was performed by considering the variability of the fiber/matrix due to the inherent features for the 60% volume fraction; the distribution of the effective properties is shown in Figure 6. From figure, the effective properties calculated at the micro-level by probabilistic homogenization analysis follow the normal distribution, which are similar to those of the random variables selected in Section 3.1. This indicates that the variability of the effective properties at the micro-level with fiber and matrix combinations depends on the variability of the fiber and matrix, which are the basic constituents.

Random Variables for the Probabilistic Homogenization Technique at the Micro-Level
The homogenization technique described in Section 2 could evaluate the effective properties at the micro-level in which fibers and matrix are physically combined to form tow. However, probabilistic analysis, rather than deterministic analysis, should be employed because the composites have uncertainties in material properties due to several inherent features [35]. To address this, the probabilistic homogenization analysis based on the MCS was performed using the commercial software ABAQUS and ISIGHT [36]. Table 5 lists the selected random variables. Each random variable was assumed to have a normal distribution consisting of 1000 samples. For the coefficient of variation (COV) of 0.05 was applied. Table 6 shows a portion of the set of random variables calculated using this method.

Effective Properties at the Micro-Level
The probabilistic homogenization analysis was performed by considering the variability of the fiber/matrix due to the inherent features for the 60% volume fraction; the distribution of the effective properties is shown in Figure 6. From figure, the effective properties calculated at the micro-level by probabilistic homogenization analysis follow the normal distribution, which are similar to those of the random variables selected in Section 3.1. This indicates that the variability of the effective properties at the micro-level with fiber and matrix combinations depends on the variability of the fiber and matrix, which are the basic constituents.

Random Variables for the Probabilistic Homogenization Technique at the Micro-Level
The homogenization technique described in Section 2 could evaluate the effective properties a the micro-level in which fibers and matrix are physically combined to form tow. However probabilistic analysis, rather than deterministic analysis, should be employed because the composite have uncertainties in material properties due to several inherent features [35]. To address this, th probabilistic homogenization analysis based on the MCS was performed using the commercia software ABAQUS and ISIGHT [36]. Table 5 lists the selected random variables. Each random variable was assumed to have a normal distribution consisting of 1000 samples. For the coefficient o variation (COV) of 0.05 was applied. Table 6 shows a portion of the set of random variables calculate using this method.

Effective Properties at the Micro-Level
The probabilistic homogenization analysis was performed by considering the variability of th fiber/matrix due to the inherent features for the 60% volume fraction; the distribution of the effectiv properties is shown in Figure 6. From figure, the effective properties calculated at the micro-level b probabilistic homogenization analysis follow the normal distribution, which are similar to those o the random variables selected in Section 3.1. This indicates that the variability of the effectiv properties at the micro-level with fiber and matrix combinations depends on the variability of th fiber and matrix, which are the basic constituents.

Random Variables for the Probabilistic Homogenization Technique at the Micro-Level
The homogenization technique described in Section 2 could evaluate the effective pro the micro-level in which fibers and matrix are physically combined to form tow. probabilistic analysis, rather than deterministic analysis, should be employed because the co have uncertainties in material properties due to several inherent features [35]. To addres probabilistic homogenization analysis based on the MCS was performed using the co software ABAQUS and ISIGHT [36]. Table 5 lists the selected random variables. Each variable was assumed to have a normal distribution consisting of 1000 samples. For the coe variation (COV) of 0.05 was applied. Table 6 shows a portion of the set of random variables c using this method.

Effective Properties at the Micro-Level
The probabilistic homogenization analysis was performed by considering the variabi fiber/matrix due to the inherent features for the 60% volume fraction; the distribution of the properties is shown in Figure 6. From figure, the effective properties calculated at the micr probabilistic homogenization analysis follow the normal distribution, which are similar t the random variables selected in Section 3.1. This indicates that the variability of the properties at the micro-level with fiber and matrix combinations depends on the variabi fiber and matrix, which are the basic constituents.

Random Variables for the Probabilistic Homogenization Technique at the Micro-Leve
The homogenization technique described in Section 2 could evaluate the eff the micro-level in which fibers and matrix are physically combined to for probabilistic analysis, rather than deterministic analysis, should be employed beca have uncertainties in material properties due to several inherent features [35]. T probabilistic homogenization analysis based on the MCS was performed usin software ABAQUS and ISIGHT [36]. Table 5 lists the selected random variab variable was assumed to have a normal distribution consisting of 1000 samples. F variation (COV) of 0.05 was applied. Table 6 shows a portion of the set of random v using this method.

Effective Properties at the Micro-Level
The probabilistic homogenization analysis was performed by considering th fiber/matrix due to the inherent features for the 60% volume fraction; the distribu properties is shown in Figure 6. From figure, the effective properties calculated a probabilistic homogenization analysis follow the normal distribution, which are the random variables selected in Section 3.1. This indicates that the variabili properties at the micro-level with fiber and matrix combinations depends on th fiber and matrix, which are the basic constituents.

Random Variables for the Probabilistic Homogenization Technique at the M
The homogenization technique described in Section 2 could evalua the micro-level in which fibers and matrix are physically combine probabilistic analysis, rather than deterministic analysis, should be empl have uncertainties in material properties due to several inherent featu probabilistic homogenization analysis based on the MCS was perfor software ABAQUS and ISIGHT [36]. Table 5 lists the selected rando variable was assumed to have a normal distribution consisting of 1000 s variation (COV) of 0.05 was applied. Table 6 shows a portion of the set of using this method.

Effective Properties at the Micro-Level
The probabilistic homogenization analysis was performed by cons fiber/matrix due to the inherent features for the 60% volume fraction; th properties is shown in Figure 6. From figure, the effective properties cal probabilistic homogenization analysis follow the normal distribution, w the random variables selected in Section 3.1. This indicates that the properties at the micro-level with fiber and matrix combinations depe fiber and matrix, which are the basic constituents.

Random Variables for the Probabilistic Homogenization Techniq
The homogenization technique described in Section 2 cou the micro-level in which fibers and matrix are physically probabilistic analysis, rather than deterministic analysis, should have uncertainties in material properties due to several inher probabilistic homogenization analysis based on the MCS w software ABAQUS and ISIGHT [36]. Table 5 lists the selec variable was assumed to have a normal distribution consisting variation (COV) of 0.05 was applied. Table 6 shows a portion of using this method.

Effective Properties at the Micro-Level
The probabilistic homogenization analysis was performe fiber/matrix due to the inherent features for the 60% volume fr properties is shown in Figure 6. From figure, the effective prop probabilistic homogenization analysis follow the normal distr the random variables selected in Section 3.1. This indicates properties at the micro-level with fiber and matrix combinati fiber and matrix, which are the basic constituents.

Effective Properties at the Micro-Level
The probabilistic homogenization analysis was performed by considering the variability of the fiber/matrix due to the inherent features for the 60% volume fraction; the distribution of the effective properties is shown in Figure 6. From figure, the effective properties calculated at the micro-level by probabilistic homogenization analysis follow the normal distribution, which are similar to those of the random variables selected in Section 3.1. This indicates that the variability of the effective properties at

Homogenization Analysis Technique at the Macro-Level
Fabric composites have reinforced structures with the weave form of tow and are geometrically more complex than the UD composites. Furthermore, non-homogeneity of the structure and the uncertainties in properties are caused by the unique characteristics of manufacturing and weaving deviations.
The tow, which is an intermediate material of fabric composites, is a combination of UD fiber and matrix. Thus, the properties of tow are also influenced by the properties of the fiber and matrix

Homogenization Analysis Technique at the Macro-Level
Fabric composites have reinforced structures with the weave form of tow and are geometrically more complex than the UD composites. Furthermore, non-homogeneity of the structure and the uncertainties in properties are caused by the unique characteristics of manufacturing and weaving deviations.
The tow, which is an intermediate material of fabric composites, is a combination of UD fiber and matrix. Thus, the properties of tow are also influenced by the properties of the fiber and matrix and their variability, as mentioned in Section 3. Therefore, to effectively evaluate the effective properties at the macro-level of fabric composites, the probabilistic homogenization analysis should be performed by considering the properties and their variability at the micro-level. Therefore, the representative unit cell (RUC) of the fabric composites was selected and the fabric composite was designed according to the literature [37][38][39]. The design geometry is shown in Figure 7a and Table 7. The tow thickness was designed by changing the size according to the volume fractions of 40, 50, 60, and 70%. Figure 7b exhibits the finite element (FE) model for RUC for the fabric composite. C3D10 with five nodes per element and six degrees of freedom per node was used for the FE model. The total numbers of elements and nodes were 267,578 and 375,805, respectively. The Mori-Tanaka theory was used for the homogenization technique. The repetitive boundary condition was applied for the boundary condition which is same condition for the micro-level with fiber/matrix combination. Furthermore, a set of 1000 data derived from the micro-level were applied to the properties of tow and epoxy to perform the homogenization technique based on MCS. and their variability, as mentioned in Section 3. Therefore, to effectively evaluate the effective properties at the macro-level of fabric composites, the probabilistic homogenization analysis should be performed by considering the properties and their variability at the micro-level. Therefore, the representative unit cell (RUC) of the fabric composites was selected and the fabric composite was designed according to the literature [37][38][39]. The design geometry is shown in Figure 7a and Table 7.
The tow thickness was designed by changing the size according to the volume fractions of 40, 50, 60, and 70%. Figure 7b exhibits the finite element (FE) model for RUC for the fabric composite. C3D10 with five nodes per element and six degrees of freedom per node was used for the FE model. The total numbers of elements and nodes were 267,578 and 375,805, respectively. The Mori-Tanaka theory was used for the homogenization technique. The repetitive boundary condition was applied for the boundary condition which is same condition for the micro-level with fiber/matrix combination. Furthermore, a set of 1000 data derived from the micro-level were applied to the properties of tow and epoxy to perform the homogenization technique based on MCS. Table 8 outlines these values.    Figure 8 shows the homogenization analysis results based on MCS for the volume fractions of 40, 50, 60, and 70%. All the effective properties including E11,eq,macro, E33,eq,macro, and G12,eq,macro followed the normal distribution. This trend was maintained, even when the volume fraction changed. However, the average and standard deviations of the stiffness increased with the volume fraction, whereas the opposite was true for the Poisson's ratio. The reason is that as the volume fraction of the tow increases, the stiffness also increases; thus resulting in differences in the standard deviation.

Random Variables for the Probabilistic Homogenization Technique at the Micro-Level
The homogenization technique described in Section 2 could evaluate the effective properties at the micro-level in which fibers and matrix are physically combined to form tow. However, probabilistic analysis, rather than deterministic analysis, should be employed because the composites have uncertainties in material properties due to several inherent features [35]. To address this, the probabilistic homogenization analysis based on the MCS was performed using the commercial software ABAQUS and ISIGHT [36]. Table 5 lists the selected random variables. Each random variable was assumed to have a normal distribution consisting of 1000 samples. For the coefficient of variation (COV) of 0.05 was applied. Table 6 shows a portion of the set of random variables calculated using this method.

Effective Properties at the Micro-Level
The probabilistic homogenization analysis was performed by considering the variability of the fiber/matrix due to the inherent features for the 60% volume fraction; the distribution of the effective properties is shown in Figure 6. From figure, the effective properties calculated at the micro-level by probabilistic homogenization analysis follow the normal distribution, which are similar to those of the random variables selected in Section 3.1. This indicates that the variability of the effective properties at the micro-level with fiber and matrix combinations depends on the variability of the fiber and matrix, which are the basic constituents.

Random Variables for the Probabilistic Homogenization Technique at the Micro-Level
The homogenization technique described in Section 2 could evaluate the effective properties at the micro-level in which fibers and matrix are physically combined to form tow. However, probabilistic analysis, rather than deterministic analysis, should be employed because the composites have uncertainties in material properties due to several inherent features [35]. To address this, the probabilistic homogenization analysis based on the MCS was performed using the commercial software ABAQUS and ISIGHT [36]. Table 5 lists the selected random variables. Each random variable was assumed to have a normal distribution consisting of 1000 samples. For the coefficient of variation (COV) of 0.05 was applied. Table 6 shows a portion of the set of random variables calculated using this method.

Effective Properties at the Micro-Level
The probabilistic homogenization analysis was performed by considering the variability of the fiber/matrix due to the inherent features for the 60% volume fraction; the distribution of the effective properties is shown in Figure 6. From figure, the effective properties calculated at the micro-level by probabilistic homogenization analysis follow the normal distribution, which are similar to those of the random variables selected in Section 3.1. This indicates that the variability of the effective properties at the micro-level with fiber and matrix combinations depends on the variability of the fiber and matrix, which are the basic constituents.

Random Variables for the Probabilistic Homogenization Technique at the Micro-Level
The homogenization technique described in Section 2 could evaluate the effective properties at the micro-level in which fibers and matrix are physically combined to form tow. However, probabilistic analysis, rather than deterministic analysis, should be employed because the composites have uncertainties in material properties due to several inherent features [35]. To address this, the probabilistic homogenization analysis based on the MCS was performed using the commercial software ABAQUS and ISIGHT [36]. Table 5 lists the selected random variables. Each random variable was assumed to have a normal distribution consisting of 1000 samples. For the coefficient of variation (COV) of 0.05 was applied. Table 6 shows a portion of the set of random variables calculated using this method.

Effective Properties at the Micro-Level
The probabilistic homogenization analysis was performed by considering the variability of the fiber/matrix due to the inherent features for the 60% volume fraction; the distribution of the effective properties is shown in Figure 6. From figure, the effective properties calculated at the micro-level by probabilistic homogenization analysis follow the normal distribution, which are similar to those of the random variables selected in Section 3.1. This indicates that the variability of the effective properties at the micro-level with fiber and matrix combinations depends on the variability of the fiber and matrix, which are the basic constituents.

Random Variables for the Probabilistic Homogenization Technique at the Micro-Level
The homogenization technique described in Section 2 could evaluate the effective properties at the micro-level in which fibers and matrix are physically combined to form tow. However, probabilistic analysis, rather than deterministic analysis, should be employed because the composites have uncertainties in material properties due to several inherent features [35]. To address this, the probabilistic homogenization analysis based on the MCS was performed using the commercial software ABAQUS and ISIGHT [36]. Table 5 lists the selected random variables. Each random variable was assumed to have a normal distribution consisting of 1000 samples. For the coefficient of variation (COV) of 0.05 was applied. Table 6 shows a portion of the set of random variables calculated using this method.

Effective Properties at the Micro-Level
The probabilistic homogenization analysis was performed by considering the variability of the fiber/matrix due to the inherent features for the 60% volume fraction; the distribution of the effective properties is shown in Figure 6. From figure, the effective properties calculated at the micro-level by probabilistic homogenization analysis follow the normal distribution, which are similar to those of the random variables selected in Section 3.1. This indicates that the variability of the effective properties at the micro-level with fiber and matrix combinations depends on the variability of the fiber and matrix, which are the basic constituents.

Random Variables for the Probabilistic Homogenization Technique at the Micro-Level
The homogenization technique described in Section 2 could evaluate the effective properties at the micro-level in which fibers and matrix are physically combined to form tow. However, probabilistic analysis, rather than deterministic analysis, should be employed because the composites have uncertainties in material properties due to several inherent features [35]. To address this, the probabilistic homogenization analysis based on the MCS was performed using the commercial software ABAQUS and ISIGHT [36]. Table 5 lists the selected random variables. Each random variable was assumed to have a normal distribution consisting of 1000 samples. For the coefficient of variation (COV) of 0.05 was applied. Table 6 shows a portion of the set of random variables calculated using this method.

Effective Properties at the Micro-Level
The probabilistic homogenization analysis was performed by considering the variability of the fiber/matrix due to the inherent features for the 60% volume fraction; the distribution of the effective properties is shown in Figure 6. From figure, the effective properties calculated at the micro-level by probabilistic homogenization analysis follow the normal distribution, which are similar to those of the random variables selected in Section 3.1. This indicates that the variability of the effective properties at the micro-level with fiber and matrix combinations depends on the variability of the fiber and matrix, which are the basic constituents.

Random Variables for the Probabilistic Homogenization Technique at the Micro-Level
The homogenization technique described in Section 2 could evaluate the effective properties at the micro-level in which fibers and matrix are physically combined to form tow. However, probabilistic analysis, rather than deterministic analysis, should be employed because the composites have uncertainties in material properties due to several inherent features [35]. To address this, the probabilistic homogenization analysis based on the MCS was performed using the commercial software ABAQUS and ISIGHT [36]. Table 5 lists the selected random variables. Each random variable was assumed to have a normal distribution consisting of 1000 samples. For the coefficient of variation (COV) of 0.05 was applied. Table 6 shows a portion of the set of random variables calculated using this method.

Effective Properties at the Micro-Level
The probabilistic homogenization analysis was performed by considering the variability of the fiber/matrix due to the inherent features for the 60% volume fraction; the distribution of the effective properties is shown in Figure 6. From figure, the effective properties calculated at the micro-level by probabilistic homogenization analysis follow the normal distribution, which are similar to those of the random variables selected in Section 3.1. This indicates that the variability of the effective properties at the micro-level with fiber and matrix combinations depends on the variability of the fiber and matrix, which are the basic constituents.

Random Variables for the Probabilistic Homogenization Technique at the Micro-Level
The homogenization technique described in Section 2 could evaluate the effective properties at the micro-level in which fibers and matrix are physically combined to form tow. However, probabilistic analysis, rather than deterministic analysis, should be employed because the composites have uncertainties in material properties due to several inherent features [35]. To address this, the probabilistic homogenization analysis based on the MCS was performed using the commercial software ABAQUS and ISIGHT [36]. Table 5 lists the selected random variables. Each random variable was assumed to have a normal distribution consisting of 1000 samples. For the coefficient of variation (COV) of 0.05 was applied. Table 6 shows a portion of the set of random variables calculated using this method.

Effective Properties at the Micro-Level
The probabilistic homogenization analysis was performed by considering the variability of the fiber/matrix due to the inherent features for the 60% volume fraction; the distribution of the effective properties is shown in Figure 6. From figure, the effective properties calculated at the micro-level by probabilistic homogenization analysis follow the normal distribution, which are similar to those of the random variables selected in Section 3.1. This indicates that the variability of the effective properties at the micro-level with fiber and matrix combinations depends on the variability of the fiber and matrix, which are the basic constituents.

Random Variables for the Probabilistic Homogenization Technique at the Micro-Level
The homogenization technique described in Section 2 could evaluate the effective prop the micro-level in which fibers and matrix are physically combined to form tow. H probabilistic analysis, rather than deterministic analysis, should be employed because the com have uncertainties in material properties due to several inherent features [35]. To address probabilistic homogenization analysis based on the MCS was performed using the com software ABAQUS and ISIGHT [36]. Table 5 lists the selected random variables. Each variable was assumed to have a normal distribution consisting of 1000 samples. For the coeff variation (COV) of 0.05 was applied. Table 6 shows a portion of the set of random variables ca using this method.

Effective Properties at the Micro-Level
The probabilistic homogenization analysis was performed by considering the variabilit fiber/matrix due to the inherent features for the 60% volume fraction; the distribution of the e properties is shown in Figure 6. From figure, the effective properties calculated at the microprobabilistic homogenization analysis follow the normal distribution, which are similar to the random variables selected in Section 3.1. This indicates that the variability of the e properties at the micro-level with fiber and matrix combinations depends on the variabilit fiber and matrix, which are the basic constituents.

Random Variables for the Probabilistic Homogenization Technique at the Micro-Level
The homogenization technique described in Section 2 could evaluate the effec the micro-level in which fibers and matrix are physically combined to form probabilistic analysis, rather than deterministic analysis, should be employed becau have uncertainties in material properties due to several inherent features [35]. To probabilistic homogenization analysis based on the MCS was performed using software ABAQUS and ISIGHT [36]. Table 5 lists the selected random variabl variable was assumed to have a normal distribution consisting of 1000 samples. For variation (COV) of 0.05 was applied. Table 6 shows a portion of the set of random va using this method.

Effective Properties at the Micro-Level
The probabilistic homogenization analysis was performed by considering the fiber/matrix due to the inherent features for the 60% volume fraction; the distributio properties is shown in Figure 6. From figure, the effective properties calculated at t probabilistic homogenization analysis follow the normal distribution, which are s the random variables selected in Section 3.1. This indicates that the variability properties at the micro-level with fiber and matrix combinations depends on the fiber and matrix, which are the basic constituents.

Random Variables for the Probabilistic Homogenization Technique at the Mi
The homogenization technique described in Section 2 could evaluate the micro-level in which fibers and matrix are physically combined probabilistic analysis, rather than deterministic analysis, should be employ have uncertainties in material properties due to several inherent feature probabilistic homogenization analysis based on the MCS was perform software ABAQUS and ISIGHT [36]. Table 5 lists the selected random variable was assumed to have a normal distribution consisting of 1000 sam variation (COV) of 0.05 was applied. Table 6 shows a portion of the set of ra using this method.

Effective Properties at the Micro-Level
The probabilistic homogenization analysis was performed by consid fiber/matrix due to the inherent features for the 60% volume fraction; the d properties is shown in Figure 6. From figure, the effective properties calcu probabilistic homogenization analysis follow the normal distribution, wh the random variables selected in Section 3.1. This indicates that the v properties at the micro-level with fiber and matrix combinations depend fiber and matrix, which are the basic constituents.

Random Variables for the Probabilistic Homogenization Technique
The homogenization technique described in Section 2 could the micro-level in which fibers and matrix are physically c probabilistic analysis, rather than deterministic analysis, should b have uncertainties in material properties due to several inheren probabilistic homogenization analysis based on the MCS was software ABAQUS and ISIGHT [36]. Table 5 lists the selected variable was assumed to have a normal distribution consisting of variation (COV) of 0.05 was applied. Table 6 shows a portion of th using this method.

Effective Properties at the Micro-Level
The probabilistic homogenization analysis was performed b fiber/matrix due to the inherent features for the 60% volume frac properties is shown in Figure 6. From figure, the effective proper probabilistic homogenization analysis follow the normal distrib the random variables selected in Section 3.1. This indicates t properties at the micro-level with fiber and matrix combination fiber and matrix, which are the basic constituents.

Random Variables for the Probabilistic Homogenization T
The homogenization technique described in Section the micro-level in which fibers and matrix are phy probabilistic analysis, rather than deterministic analysis, have uncertainties in material properties due to several probabilistic homogenization analysis based on the M software ABAQUS and ISIGHT [36]. Table 5 lists the variable was assumed to have a normal distribution con variation (COV) of 0.05 was applied. Table 6 shows a por using this method.

Effective Properties at the Micro-Level
The probabilistic homogenization analysis was per fiber/matrix due to the inherent features for the 60% volu properties is shown in Figure 6. From figure, the effectiv probabilistic homogenization analysis follow the norma the random variables selected in Section 3.1. This ind properties at the micro-level with fiber and matrix com fiber and matrix, which are the basic constituents.

Random Variables for the Probabilistic Homogen
The homogenization technique described i the micro-level in which fibers and matrix probabilistic analysis, rather than deterministic a have uncertainties in material properties due t probabilistic homogenization analysis based o software ABAQUS and ISIGHT [36]. Table 5 variable was assumed to have a normal distribu variation (COV) of 0.05 was applied. Table 6 sho using this method.

. Effective Properties at the Micro-Level
The probabilistic homogenization analysis fiber/matrix due to the inherent features for the properties is shown in Figure 6. From figure, th probabilistic homogenization analysis follow th the random variables selected in Section 3.1. properties at the micro-level with fiber and ma fiber and matrix, which are the basic constituen  Figure 8 shows the homogenization analysis results based on MCS for the volume fractions of 40, 50, 60, and 70%. All the effective properties including E 11,eq,macro , E 33,eq,macro , and G 12,eq,macro followed the normal distribution. This trend was maintained, even when the volume fraction changed. However, the average and standard deviations of the stiffness increased with the volume fraction, whereas the opposite was true for the Poisson's ratio. The reason is that as the volume fraction of the tow increases, the stiffness also increases; thus resulting in differences in the standard deviation. However, Poisson's ratio is influenced more strongly by the matrix, thus exhibiting a different trend than observed for the stiffness. Particularly in the case of v 23,eq,macro , the increase in stiffness in the direction of E 33,eq,macro is insignificant; thus, it shows a significant decreasing tendency when the volume fraction increases. To quantify the inherent uncertainties in the properties of composites, the COV was derived from each effective property and compared. Table 9 outlines the results. The COV of the stiffness changed minimally against the volume fraction. However, the COV of v 12,eq,macro showing a ratio of in-plane strain stiffness increased with the volume fraction. In contrast, v 23,tow , which indicates the ratio of out-of-plane strain shows the opposite trend. Therefore, the effects of the design variables on the stiffness and Poisson's ratio of the fabric composite appear differently between the two, and they also have the same out-of-plane strain.

Effective Properties at the Macro-Level
Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 14 However, Poisson's ratio is influenced more strongly by the matrix, thus exhibiting a different trend than observed for the stiffness. Particularly in the case of v23,eq, macro, the increase in stiffness in the direction of E33,eq, macro is insignificant; thus, it shows a significant decreasing tendency when the volume fraction increases. To quantify the inherent uncertainties in the properties of composites, the COV was derived from each effective property and compared. Table 9 outlines the results. The COV of the stiffness changed minimally against the volume fraction. However, the COV of v12,eq, macro showing a ratio of in-plane strain stiffness increased with the volume fraction. In contrast, v23,tow, which indicates the ratio of out-of-plane strain shows the opposite trend. Therefore, the effects of the design variables on the stiffness and Poisson's ratio of the fabric composite appear differently between the two, and they also have the same out-of-plane strain.  (b) Equivalent transverse modulus (E 33,eq,macro ); (c) Equivalent shear modulus (G 12 , eq,macro ); (d) Equivalent shear modulus (G 23,eq,macro ); (e) Equivalent Poisson's ratio (v 12,eq,macro ); (f) Equivalent Poisson's ratio (v 23,eq,macro ). The linear correlations were evaluated to accurately determine the effects of the mechanical properties of the fiber and matrix, i.e., the constituents of fabric composites, on the effective properties at the macro-level. First, the linear correlation of the mechanical properties between the constituents and the effective properties of the micro-level was analyzed using the simple correlation analysis method of statistical method and the results are shown in Figure 9a. Figure 9a shows that E 11,tow has the highest and second highest correlation with E f and E m , respectively. There are also correlations with G m , G f , and v m , but none with v f . In a positive correlation, when the mechanical properties increase, the other effective properties also increase, whereas in a negative correlation, the opposite is true. Furthermore, when there is no correlation, even if the mechanical properties change, the other effective properties remain unchanged. Hence, the results suggest that E 11,tow is under the dominant influence of the elastic modulus of the fiber, whereas E 22,tow and E 33,tow are under the dominant influence of the elastic modulus and Poisson's ratio of the matrix. Furthermore, G 12,tow , G 13,tow , and G 23,tow are under the dominant influence of the shear modulus of elasticity of the matrix, whereas v 12,tow and v 13,tow are under the influences of the Poisson's ratios of the fiber and the matrix. Lastly, v 23,tow is under the dominant influence of the Poisson's ratio of the matrix. Through this analysis, the linear correlation between the mechanical properties of the constituents (carbon fiber and epoxy) and the effective properties of the micro level could be identified. Figure 9b shows the linear correlation between the effective properties of the micro-and macro-levels. For example, E 11,eq, macro exhibits the highest correlations with E 11,tow, E 22,tow, and E 33,tow and the second highest correlation with E m . Furthermore, correlations with other properties excluding these can be also identified. The reason for this is the geometrical shape of the fabric composites of tow with fiber and matrix combination, which was woven in three dimensions.
Since the properties of the macro-level are ultimately determined by the mechanical properties of the constituents, the linear correlation analysis for the effective properties of the macro-level and the mechanical properties of the constituents is also necessary. The results are shown in Figure 9c. For example, E 11,eq,macro and E 22,eq,macro have the highest correlation with E f and the second highest correlation with, E m . Furthermore, the existence of correlations with other properties excluding them could be also identified. By contrast, the shear modulus of elasticity at the macro-level is significantly impacted by the shear modulus of elasticity of epoxy. Finally, the Poisson's ratio of the macro-level is greatly influenced by the Poisson's ratios of epoxy. From the above results, the fabric composites are significantly impacted by the mechanical properties of constituents, and are also influenced by the matrix mechanical properties due to the complex geometric shapes.