Elastic Properties of Jute Fiber Reinforced Polymer Composites with Different Hierarchical Structures

A two-stage micromechanics technique is used to predict the elastic modulus, as well as the major and minor Poisson’s ratio of unidirectional natural fiber (NF) reinforced composites. The actual NF microstructure consists of cellulose, hemicellulose, lignin, lumen, etc., and these constituents and their contributions are neglected in classical models while quantifying their mechanical properties. The present paper addresses the effect of the real microstructure of the natural jute fiber (JF) by applying a micromechanics approach with the Finite Element Method. Six different hierarchically micro-structured JFs are considered to quantify the JF elastic properties in the first level of homogenization. Later, the JF reinforced polypropylene matrix properties are investigated in the second stage by adopting a homogenization approach. Taking into account the different hierarchical structures (HS), the fiber direction modulus (E1), transverse modulus (E2 and E3), in-plane and out-of-plane shear modulus (G12 and G23), and major (ν12, ν13) and minor (ν23, ν21) Poisson’s ratios are estimated for JF and JF reinforced polypropylene composites. The predicted elastic modulus from micromechanics models is validated against the analytical results and experimental predictions. From the present work, it is observed that the HS of NF needs to be considered while addressing the elastic properties of the NF-reinforced composite for their effective design, particularly at a higher volume fraction of NF.


Introduction
Natural fibers (NFs) are also called lignocellulosic fibers, which are extracted from plants, and these fibers contain different proportions of cellulose, hemicelluloses, lignin, and lumen. The main constituent among all of these is cellulose, and the percentage of cellulose is in the range of 50-70% [1]. Microfibrils are formed from these cellulose chains, and these microfibrils aggregate together to form macrofibrils through an amorphous matrix termed lignin and hemicellulose. The hollowness in this structure is called the lumen. Cellulose, hemicellulose, lignin, and lumen are the main constituents of plant-based fibers. Based on the differences in the percentage of contributions in the NF, the hierarchical structure (HS) taken up by the lumen has been found to be 27.2%, 6.8%, and 34.0% for sisal, flax, and jute, respectively. It was noted that the presence of lignin makes the cellulose rigid [28]. These jute fiber reinforced (JFR) composites have a wide range of applications in household, engineering, building structures, door frames, furniture, shopping bags, etc. and these composites can be thermally stable in the range of 250-365 • C [29]. Another important aspect of jute fiber (JF) is the large lumen, which needs to be reflected in the manufacturing process of composite materials. This hollow lumen has a clear influence on the stiffness of the NF composites. These lumen percentages must be considered when designing the properties of NF composites [30]. JFs consist of a high percentage of hollow lumen structures, which could be beneficial for sound energy conversion [31]. However, this lumen will never contribute to the mechanical properties [32] and will remain as it is inside the composites, and the lumen is a large tubular void in the middle of each NF [33,34].
From the above findings, the authors of this paper observed that the identification of lumen and cellulose percentages is the key step in tailoring the properties of NF composites. The lumen represents the hollowness of the NF and the cellulose promotes the strength of the fiber. Most NF studies have been limited to E, i.e., Young's modulus in the longitudinal direction. However, NFR composites are orthotropic in nature, requiring the use of nine elastic constants for effective design of NFR composite structures. This data is not yet available for JFR composites. There are few studies on Poisson's ratio, with more emphasis on elastic modulus and shear modulus. However, Poisson's ratio requires knowledge of the coupling between deformation in the lateral and longitudinal directions of composite materials. While Poisson's ratio does not matter much for regular materials, it does matter a lot for composite materials, which are made of two different materials that work together when loaded. This behavior can be identified by Poisson's ratio. At the same time, the fiber alignment direction also plays a definite role in Poisson's ratio and must be taken into account. Lumen hollowness and its percentage are dependent on the type of NF. This lumen percentage influence on all elastic properties has not been addressed so far. While comparing the experimental and analytical or simulation studies of the properties of NF composites, the experimental results consider all NF constituents, i.e., cellulose, hemicelluloses, lignin, and lumen, while in the analytical or simulation studies, only the technical fiber [12] property will be considered to estimate the overall composite properties. Considering the aforementioned knowledge gaps, the present problem focuses on the estimation of the nine elastic properties of a jute NFR composite considering the six HSs with different lumen percentages using analytical and simulation studies.

Material and Methods
Experimental procedures for estimating the longitudinal modulus of straight natural fiber reinforced composite and their mechanical characterization are well described in the previous article [35]. Using the same procedure, the JFR epoxy composites are prepared and tested. The JFs are treated with NAOH solution and the weight of straight JFs is measured and placed in a mold. The epoxy matrix is poured over the JF according to the fiber weight fraction. The mold is cured for 24 h after which the specimen is removed from the mold. Subsequently, the specimens are cut from the same lamina according to ASTM standards. Five samples are prepared at each weight and tested for longitudinal modulus.
Unidirectional JF is used for the fabrication of NFR polymer composites. Epoxy resin (LY556) and compatible hardener (HY951) are used as a hosting medium. Numerical studies are performed considering the volume fraction, and for conducting the experimental studies, the volume fraction is converted to weight fraction using the density of JFs and epoxy resin. The weight fraction of JF is kept at 12.95% and 36.47% based on the volume fraction of JFs (10% and 30%). Composite specimens are prepared using hand layup technique. Five specimens are prepared for each configuration and tested according to the ASTM D638 standard (Figure 1a). The tensile tests were carried out using the Universal Tensile Testing Machine at the Prasad V. Potluri Siddhartha Institute of Technology, Kanuru, Vijayawada, Andhra Pradesh, India (Figure 1b). Table 1 shows the elastic modulus obtained from experimental tensile tests. epoxy resin. The weight fraction of JF is kept at 12.95% and 36.47% based on the volume fraction of JFs (10% and 30%). Composite specimens are prepared using hand layup technique. Five specimens are prepared for each configuration and tested according to the ASTM D638 standard (Figure 1a). The tensile tests were carried out using the Universal Tensile Testing Machine at the Prasad V. Potluri Siddhartha Institute of Technology, Kanuru, Vijayawada, Andhra Pradesh, India (Figure 1b). Table 1 shows the elastic modulus obtained from experimental tensile tests.

Analytical Studies of Jute Fiber Reinforced Composites
In the plant-based NFs, the strong network of hydrogen bonds between the hydroxyl groups of neighboring chains causes the cellulose to organize in a hierarchical way [36]. Cellulose is the main structural component of plant cell walls [37]. In this work, six types of such structures are taken for analysis and the elastic modulus is estimated. The hierarchical structures (HSs) and the percentage of each constituent are presented in Table 2.

Analytical Studies of Jute Fiber Reinforced Composites
In the plant-based NFs, the strong network of hydrogen bonds between the hydroxyl groups of neighboring chains causes the cellulose to organize in a hierarchical way [36]. Cellulose is the main structural component of plant cell walls [37]. In this work, six types of such structures are taken for analysis and the elastic modulus is estimated. The hierarchical structures (HSs) and the percentage of each constituent are presented in Table 2. These HS structures are designed based on the cellulose percentage i.e., some fibers have maximum cellulose (61%) and some fibers have minimum cellulose (39%). Application of the Micromechanics method to the composite materials to evaluate their elastic properties will start with the selection of the Representative Volume Element (RVE). The NFs in the matrix phase are thought to be straight and spread out evenly.
The space between the JFs and fiber is b f and the thickness is t f . The size of the RVE is represented by l c , b c and t c where l c is the length of the RVE, b c is the width of the RVE, and t c is the thickness of the RVE (Figure 2). The RVE shows the whole lamina of the JF, which can be made by putting the RVEs next to each other over and over again.
These HS structures are designed based on the cellulose percentage i.e., some fibers have maximum cellulose (61%) and some fibers have minimum cellulose (39%). Application of the Micromechanics method to the composite materials to evaluate their elastic properties will start with the selection of the Representative Volume Element (RVE). The NFs in the matrix phase are thought to be straight and spread out evenly.
The space between the JFs and fiber is bf and the thickness is tf. The size of the RVE is represented by lc, bc and tc where lc is the length of the RVE, bc is the width of the RVE, and tc is the thickness of the RVE (Figure 2). The RVE shows the whole lamina of the JF, which can be made by putting the RVEs next to each other over and over again.

Longitudinal Modulus E1
To find out the fiber direction modulus of a JFR polymer matrix composite, an electrical analogy was made. When applying the numerical calculations, the HS of JF is taken into account. The HS of NF includes lumen, lignin, hemicellulose, and cellulose. A lumen in the NF is treated as a hollow member; lignin, hemicellulose, and cellulose are different in terms of geometry and material properties. These fibers are uniformly distributed throughout the matrix material.
All constituents present share the load acting on the RVE. Lumen, lignin, hemicellulose, and cellulose will take the load as shown in Figure 3a.

Longitudinal Modulus E 1
To find out the fiber direction modulus of a JFR polymer matrix composite, an electrical analogy was made. When applying the numerical calculations, the HS of JF is taken into account. The HS of NF includes lumen, lignin, hemicellulose, and cellulose. A lumen in the NF is treated as a hollow member; lignin, hemicellulose, and cellulose are different in terms of geometry and material properties. These fibers are uniformly distributed throughout the matrix material.
All constituents present share the load acting on the RVE. Lumen, lignin, hemicellulose, and cellulose will take the load as shown in Figure 3a.
have maximum cellulose (61%) and some fibers have minimum cellulose (39%). Application of the Micromechanics method to the composite materials to evaluate their elastic properties will start with the selection of the Representative Volume Element (RVE). The NFs in the matrix phase are thought to be straight and spread out evenly.
The space between the JFs and fiber is bf and the thickness is tf. The size of the RVE is represented by lc, bc and tc where lc is the length of the RVE, bc is the width of the RVE, and tc is the thickness of the RVE ( Figure 2). The RVE shows the whole lamina of the JF, which can be made by putting the RVEs next to each other over and over again.

Longitudinal Modulus E1
To find out the fiber direction modulus of a JFR polymer matrix composite, an electrical analogy was made. When applying the numerical calculations, the HS of JF is taken into account. The HS of NF includes lumen, lignin, hemicellulose, and cellulose. A lumen in the NF is treated as a hollow member; lignin, hemicellulose, and cellulose are different in terms of geometry and material properties. These fibers are uniformly distributed throughout the matrix material.
All constituents present share the load acting on the RVE. Lumen, lignin, hemicellulose, and cellulose will take the load as shown in Figure 3a. The forces shared by all the constituents are given in Equation (1). Using the relation between the forces and stresses, Equation (2) is developed. Using Hooke's law, the stress is directly proportional to the strain, as follows: Substituting the Equation (3) in (2) gives: Under the condition of a perfect bond between the constituents of the fiber and the matrix, the strain generated in the RVE is equal to the strain in the fiber and the strain developed in the matrix.

Transverse Modulus
This modulus is obtained from the RVE subjected to transverse loading as shown in Figure 3b,c.
The transverse elongation under the applied load is equal to the transverse extension generated in all constituents, such as fiber and matrix. Again, JF is considered with lumen, lignin, hemicellulose, and cellulose, considering all the constituents the total elongation is represented as in Equation (8) thus: Replacing the deformation with the strain, the Equation (9) can be obtained. Using the strain in the above is modified as: The transverse strain is obtained by rearranging the equation, the Equation (10) is obtained. Finally the transverse strain in terms of strain of the each constituent and volume fraction of respective constituent the Equation (11) is obtained.
To get the ε2Φ Using the relation between the strain and stress in terms of modulus in the respective directions, the Equation (11) becomes: After applying the assumption of Equation (13), the Equation (12) becomes: Materials 2022, 15, 7032 The same analogy is applied to calculate the G 12 as presented in Equation (15).
Substituting the corresponding values of the fiber constituents' matrix elastic modulus and their percentage in Equations (7), (14) and (15), the longitudinal modulus and transverse modulus and shear modulus of the JFR composite will be estimated, respectively.

First Stage of Homogenization
Further, using the micromechanics and finite element method, the nine elastic properties of the JFR composites were estimated. The work is carried out in two stages. In the first stage, only JF properties were determined by considering different HSs. Each HS contains different constituents such as cellulose, hemicelluloses, lignin, and lumen in different proportions. This stage is considered the first stage of homogenization. In the second step, the properties of the JFR polypropylene composite are estimated using the Finite Element Based Software ANSYS 19.2. To ensure that the simulation models are accurate, the results of the FE models are checked against the analytical results. Figure 4a shows the cross section of a unidirectional NFR composite, which is illustrated to understand the HS. A fiber bundle can be seen in Figure 4b and the uniform distribution of each fiber in the fiber bundle is idealized to be spread regularly, and the analysis of one fiber is enough to estimate the fiber bundle properties in Figure 4c,d. σ2Φ = σ2α = σ2β = σ2γ = σ2δ = σm (13) After applying the assumption of Equation (13), the Equation (12) becomes: The same analogy is applied to calculate the G12 as presented in Equation (15).
Substituting the corresponding values of the fiber constituents' matrix elastic modulus and their percentage in Equations (7), (14) and (15), the longitudinal modulus and transverse modulus and shear modulus of the JFR composite will be estimated, respectively.

First Stage of Homogenization
Further, using the micromechanics and finite element method, the nine elastic properties of the JFR composites were estimated. The work is carried out in two stages. In the first stage, only JF properties were determined by considering different HSs. Each HS contains different constituents such as cellulose, hemicelluloses, lignin, and lumen in different proportions. This stage is considered the first stage of homogenization. In the second step, the properties of the JFR polypropylene composite are estimated using the Finite Element Based Software ANSYS 19.2. To ensure that the simulation models are accurate, the results of the FE models are checked against the analytical results. Figure 4a shows the cross section of a unidirectional NFR composite, which is illustrated to understand the HS. A fiber bundle can be seen in Figure 4b and the uniform distribution of each fiber in the fiber bundle is idealized to be spread regularly, and the analysis of one fiber is enough to estimate the fiber bundle properties in Figure 4c,d. The unit cell contains lumen, cellulose, and matrix phase, which are obtained by selecting a fiber from the bundle. In this case study, it is divided into two phases. In the first stage of homogenization, the JF properties are estimated by including cellulose, lumen, lignin, hemicelluloses, and later, using the properties of the JF with all its constituents, the fiber reinforced matrix properties are estimated. These homogenization concepts are used to understand the potential of electrical systems [38]. Similarly, a transverse thermal conductivity model was recently proposed [39] considering the hollow portion of the NF (lumen), and the remaining portion of the fiber is treated as cellulose. In this work, along with the lumen percentage, the lignin, cellulose, and hemicellulose percentages are also reflected in the RVE to estimate the natural properties of JF. The JF contains between 61-71% cellulose, a large amount of hemicelluloses (14-20%), lignin (12-13%), and pectin (0.2%), as cited in Ref. [40]. The Young's modulus of each constituent of the fiber is provided in Table 3. Table 3. Constituent properties of Jute Fiber [36]. The properties of the RVE can be estimated by making the RVEs in a square array look like they are perfect and setting the appropriate boundary conditions. The size of the RVE is determined based on the volume fraction of the fiber constituents. For the HS-1 model, the cellulose percentage is 61%, the hemicellulose percentage is 14%, the lignin is 12%, and the lumen is 13%. Based on these percentages, the radius of each constituent is calculated. For this structure, the square RVE size is 10 × 10 nm 2 , and the diameter of the lumen is calculated by equating the percentage of lumen to the size of the RVE, which is the area of the lumen. The lumen is treated as a hollow circle in the square RVE. The radius of the lumen is calculated according to the volume fraction. For example, the volume fraction of lumen in the total volume of the RVE is 13% for the HS-1 model. However, the cross-sectional areas are important in this calculation. The thicknesses of all the constituents are the same in the RVE. Hence, the areas of the constituents represent the volume fractions of the constituents. For the fixed RVE size (10 × 10 nm 2 ) and fixed lumen percentage (13% for the HS-1 model), the radius of the lumen is calculated by dividing the lumen area (π/4·d lu 2 ) to the total RVE area (10 × 10 nm 2 ) and equating the outcome to 13% (lumen percentage for the HS-1 model) where d lu is the diameter of the lumen. Similarly, the remaining constituents' dimensions are also estimated. The cellulose area is obtained by subtracting the lumen, lignin, and hemicellulose areas from the RVE. The FE models corresponding to HS-1 and HS-6 are presented in Figure 5. Using the geometrical data listed in Table 4 and the properties of the constituents (Table 3), a FE model is generated for all the considered structures (as given in Table 2) to estimate the elastic properties of JF under all possible loading applications [38].  Using the geometrical data listed in Table 4 and the properties of the constituents (Table 3), a FE model is generated for all the considered structures (as given in Table 2) to estimate the elastic properties of JF under all possible loading applications [38]. The possible loading cases are longitudinal loading, in-plane transverse loading, out-of-plane transverse loading, in-plane shear, and out-of-plane shear loading. A solid 186 element has been used to describe the model generated for the analysis. This solid 186 is defined by 20 nodes, and each node possesses three directional freedoms, i.e., in the X, Y, and Z directions [41]. Converged FE models are used for the analysis. One-eighth of the RVE is modelled for the analysis in terms of symmetry from the perspectives of loading, geometry, and boundary conditions. Before finding the required properties of the FE model, the model needs to ensure that the selected unit cell should reflect the total behavior of the selected material. For that, the nodes corresponding to the X = 0, Y = 0, and Z = 0 areas are arrested to move in the X, Y, and Z directions, respectively. Multipoint constraints are applied to the corresponding nodes of the FE model in the positive directions [41,42]. The longitudinal modulus is obtained by applying uniform pressure parallel to the fiber (Z axis) and, using Hooks' law, the longitudinal modulus is obtained (Figure 3a). The transverse modulus is obtained by applying load in the X and Y directions of the FE model, respectively (Figure 3b,c). The in plane shear modulus is calculated using models loaded in the XZ plane, and the out-of-plane modulus is obtained by applying load in the XY plane. The major Poisson's ratio is calculated by dividing −ε 2 /ε 1 where ε 1 is the longitudinal strain ε 2 is the lateral strain of the composite material.

Second Stage of Homogenization
The final JFR composites are evaluated by considering the JF properties, which are obtained by using the methodology proposed in Section 4.1. Considering six hierarchical structural models and their properties, the final JFR composites are estimated and presented in Section 5.2. The concentration of lumen percentage differs between the six HSs.
Using the properties of JF from the first stage of homogenization, the fiber-reinforced polypropylene composite is estimated. The second stage of homogenization is used to Figure out how different HS structures affect the final properties of the composite.

Validation of Simulation Studies
The FE models are validated by comparing the results with experimental results [43,44]. The experimental results are available for 10 and 30% volume fraction (12.95% and 36.45% weight fraction). Using the method proposed in Sections 3 and 4, the longitudinal modulus is predicted and compared to the experimental and analytical results (Table 5).  Figure 6 shows the variation of fiber directional or longitudinal modulus (E 1 ), transverse modulus (E 2 and E 3 ), in-plane (G 12 ) and out-of-plane shear modulus (G 23 ). Among all the moduli, E 1 is more than E 2 or E 3 , G 12 , and G 23 . A declining trend is observed in all the moduli except G 12 from HS-1 to HS-6. Changing the HS from HS-1 to HS-6 decreases the E 1 from 80.78 to 54.12 Gpa. The possible reason for the decrease is the increase of lumen (hollowness) in the fiber. Increasing the lumen decreases the cellulose percentage, and cellulose is the main load-bearing element of the fiber [29]. About 33% of E 1 is decreased by changing the HS from HS-1 to HS-6. The reason for the decrement is an increase in the lumen percentage from HS-1 to HS-6. Compared to E 1 , the transverse modulus of E 2 and E 3 is affected more due to lumen percentage. As a result, 42.78% of E 2 decreased from HS-1 to HS-6. A different scenario is observed in shear modulus. The in-plane shear modulus is increases from HS-1 to HS-6, which means that the lumen percentage is not affected by the in-plane modulus and the contribution of cellulose is dominated by the decrease caused by the lumen. As a result, the G 12 improves by 44.4% from HS-1 to HS-6 models. G 23 become less bright as the lumen percentage goes up, just as lumen does with E 1 and E 2 .

21.330
21.42 22.62 6.04% Figure 6 shows the variation of fiber directional or longitudinal modulus (E1), transverse modulus (E2 and E3), in-plane (G12) and out-of-plane shear modulus (G23). Among all the moduli, E1 is more than E2 or E3, G12, and G23. A declining trend is observed in all the moduli except G12 from HS-1 to HS-6. Changing the HS from HS-1 to HS-6 decreases the E1 from 80.78 to 54.12 GPa. The possible reason for the decrease is the increase of lumen (hollowness) in the fiber. Increasing the lumen decreases the cellulose percentage, and cellulose is the main load-bearing element of the fiber [29]. About 33% of E1 is decreased by changing the HS from HS-1 to HS-6. The reason for the decrement is an increase in the lumen percentage from HS-1 to HS-6. Compared to E1, the transverse modulus of E2 and E3 is affected more due to lumen percentage. As a result, 42.78% of E2 decreased from HS-1 to HS-6. A different scenario is observed in shear modulus. The in-plane shear modulus is increases from HS-1 to HS-6, which means that the lumen percentage is not affected by the in-plane modulus and the contribution of cellulose is dominated by the decrease caused by the lumen. As a result, the G12 improves by 44.4% from HS-1 to HS-6 models. G23 become less bright as the lumen percentage goes up, just as lumen does with E1 and E2. The ratio of lateral strain to the longitudinal strain of the composite material gives the Poisson's ratio. Composite materials have two types of Poisson's ratios. Major Poisson's ratio (ν 12 or ν 13 ) minor Poisson's ratio (ν 23 and ν 21 ). For transversely isotropic materials such as (E 2 = E 3 ), the magnitude of the major Poisson's ratio ν 12 or ν 13 is the same and minor Poisson's ratio ν 21 and ν 23 are same. However, the presence of lumen inside the JF makes the difference between the major Poisson's ratio ν 12 and ν 13 and the minor Poisson's ratio ν 21 and ν 23 .

Simulation Results of JF Using Micromechanics Approach (First Stage of Homogenization)
The major Poisson's ratios ν 12 and ν 13 increase from HS-1 to HS-6 due to an increase in lumen percentage (Figure 7). This response is caused by excessive deformation in longitudinal loading due to lumen in the transverse loading. The minor Poisson's ratios ν 23 , ν 32 decrease from HS-1 to HS-6. In the transverse loading, excessive deformation in the longitudinal loading due to lumen is the reason for this response. N 31 and ν 32 are the same, and the magnitude is very small, and the changes in these properties are constant from HS-1 to HS-6. From the whole of Figure 7, it is observed that the ν 12 , ν 13 magnitudes are the same. Moreover, the ν 23 and ν 32 magnitudes are also the same.
gitudinal loading due to lumen in the transverse loading. The minor Poisson's ra ν32 decrease from HS-1 to HS-6. In the transverse loading, excessive deformatio longitudinal loading due to lumen is the reason for this response. ν31 and ν32 are th and the magnitude is very small, and the changes in these properties are consta HS-1 to HS-6. From the whole of Figure 7, it is observed that the ν12, ν13 magnitu the same. Moreover, the ν23 and ν32 magnitudes are also the same.    Figure 8a [42]. The lumen behavior differs for the HS finite element model when subjected to longitudinal, in-plane transverse, and out-of-plane transverse directions. Figure 8b,c show the FE deformation contours in X, Y, and Z directions for the FE model under in-plane transverse and out-of-plane transverse loading (X and Y-directions). The JF with lumen behaved differently in the in-plane transverse directions than a transverse isotropic material (Figure 8b). Figure 8a shows the FE contours of the HS-1 model subjected to directional fiber loading.
the Poisson's ratio. Composite materials have two types of Poisson's ratios. Major Poisson's ratio (ν12 or ν13) minor Poisson's ratio (ν23 and ν21). For transversely isotropic materials such as (E2 = E3), the magnitude of the major Poisson's ratio ν12 or ν13 is the same and minor Poisson's ratio ν21 and ν23 are same. However, the presence of lumen inside the JF makes the difference between the major Poisson's ratio ν12 and ν13 and the minor Poisson's ratio ν21 and ν23.
The major Poisson's ratios ν12 and ν13 increase from HS-1 to HS-6 due to an increase in lumen percentage (Figure 7). This response is caused by excessive deformation in longitudinal loading due to lumen in the transverse loading. The minor Poisson's ratios ν23, ν32 decrease from HS-1 to HS-6. In the transverse loading, excessive deformation in the longitudinal loading due to lumen is the reason for this response. ν31 and ν32 are the same, and the magnitude is very small, and the changes in these properties are constant from HS-1 to HS-6. From the whole of Figure 7, it is observed that the ν12, ν13 magnitudes are the same. Moreover, the ν23 and ν32 magnitudes are also the same.    Figure 8a shows the FE contours of the HS-1 model subjected to directional fiber loading.

Simulation Results of JFR Polypropylene Using Micromechanics Approach (Second Stage of Homogenization)
In this section, the elastic properties of JFR polymer composites are presented by conducting analytical and simulating studies. Jute, a NF with lumen, lignin, hemicellulose, and cellulose was considered for the study. The homogenized properties of JF with different HS are measured. Six HSs models were considered, and each structure is different based on the lumen percentage. The homogenized properties of six hierarchically structured JFs were further used in the second level of homogenization to quantify the JFR polymer matrix composite. Figure 9 depicts the FE models at 0.1 and 0.6 volume fractions, as well as the homogenized JF and polypropylene matrix representations.

Simulation Results of JFR Polypropylene Using Micromechanics Approach (Second Stage of Homogenization)
In this section, the elastic properties of JFR polymer composites are presented by conducting analytical and simulating studies. Jute, a NF with lumen, lignin, hemicellulose, and cellulose was considered for the study. The homogenized properties of JF with different HS are measured. Six HSs models were considered, and each structure is different based on the lumen percentage. The homogenized properties of six hierarchically structured JFs were further used in the second level of homogenization to quantify the JFR polymer matrix composite. Figure 9 depicts the FE models at 0.1 and 0.6 volume fractions, as well as the homogenized JF and polypropylene matrix representations. The longitudinal modulus (E1) of the JFR polymer composite is presented in Figure  10. The longitudinal modulus decreases at all volume fractions of JF in all Hs models considered for the study, from HS-1 to HS-6. The decrease in E1 is greater at higher volume fractions than at lower volume fractions of JF. Lumen is the primary parameter that de- The longitudinal modulus (E 1 ) of the JFR polymer composite is presented in Figure 10. The longitudinal modulus decreases at all volume fractions of JF in all Hs models considered for the study, from HS-1 to HS-6. The decrease in E 1 is greater at higher volume fractions than at lower volume fractions of JF. Lumen is the primary parameter that determines the property of the HS model. The modulus E 1 decreases as the lumen percentage increases from HS-1 to HS-6. Analytical results are also compared with numerical results and good agreement is found. In the authors' previous studies, it was found that the synthetic fiber reinforced composite longitudinal modulus (E 1 ) is not affected by deboning defects and moisture defects [24,41], but the behavior of the NF is different when compared to natural fiber. The HS of the NF has a considerable influence on the E 1 . From HS-1 to HS-6, the percentage of lumen has increased as a result; the cellulose percentage has decreased as a result, the E 1 has decreased from HS-1 to HS-6. Not all the JFs will show the same HS, and selecting the fibers with a high cellulose percentage or low lumen percentage is desirable to achieve the high longitudinal modulus. Perfect alignments between the analytical and FE results are observed at every volume fraction of JF. Transverse Modulus (E2) also decreased from HS-1 to HS-6 and at a higher volume fraction of JF; the property loss is high, whereas at lower volume fraction of JF the effect of HS is negligible (Figure 11). The main changing parameter of HS is the percentage of lumen in the structure, it increases from HS-1 to HS-6, and the increase of the percentage of lumen decreases the contribution of cellulose in the fiber as a result of the decrease of the modulus. The simulation results are compared with the analytical results, and both results are in good agreement. Transverse Modulus (E 2 ) also decreased from HS-1 to HS-6 and at a higher volume fraction of JF; the property loss is high, whereas at lower volume fraction of JF the effect of HS is negligible (Figure 11). The main changing parameter of HS is the percentage of lumen in the structure, it increases from HS-1 to HS-6, and the increase of the percentage of lumen decreases the contribution of cellulose in the fiber as a result of the decrease of the modulus. The simulation results are compared with the analytical results, and both results are in good agreement.
Transverse Modulus (E2) also decreased from HS-1 to HS-6 and at a higher volume fraction of JF; the property loss is high, whereas at lower volume fraction of JF the effect of HS is negligible (Figure 11). The main changing parameter of HS is the percentage of lumen in the structure, it increases from HS-1 to HS-6, and the increase of the percentage of lumen decreases the contribution of cellulose in the fiber as a result of the decrease of the modulus. The simulation results are compared with the analytical results, and both results are in good agreement. The in-plane shear modulus (G 12 ) is not changed with HS ( Figure 12). The magnitude of G 12 magnitude increases with increasing fiber percentage; however, the influence of HS is negligible on this property. This means that increasing lumen percentage or decreasing cellulose is not affected by final G 12 . The in-plane shear modulus (G12) is not changed with HS ( Figure 12). The magnitude of G12 magnitude increases with increasing fiber percentage; however, the influence of HS is negligible on this property. This means that increasing lumen percentage or decreasing cellulose is not affected by final G12.   The out-of-plane shear modulus G23 is influenced by the type of HS of JF, especially at volume fractions of 0.6, 0.5, 0.4, and 0.3 of JF (Figure 13). At lower volume fractions of JF, i.e., 0.1 and 0.2, no such changes are observed in the G23.  Figure 14 shows the major Poisson's ratios ν12 and ν13 of a JFR polymer composite. ν12 is estimated from the transverse and longitudinal strain of the RVE. The longitudinal strain is obtained by applying the load in the direction (1) of the fiber, under the same load, the RVE will experience transverse strain (ε2), and then the ratio of the transverse strain (ε2) to the longitudinal strain (ε1) will be the major Poisson's ratio (ν12). Similarly, ν13 is obtained by dividing (ε3) and (ε1) of the JFR composite of the same RVE.
Compared with synthetic fiber composites, ν12 and ν13 are not the same for HS-1, HS-2 and HS-3 structured composites. The highest percentage of cellulose is responsible for this deviation in HS-1, HS-2 and HS-3 model structures. The largest lumen content giving the same response in transverse directions (2 and 3). Unlike ν12 and ν13 of JF, the major  Figure 14 shows the major Poisson's ratios ν 12 and ν 13 of a JFR polymer composite. ν 12 is estimated from the transverse and longitudinal strain of the RVE. The longitudinal strain is obtained by applying the load in the direction (1) of the fiber, under the same load, the RVE will experience transverse strain (ε 2 ), and then the ratio of the transverse strain (ε 2 ) to the longitudinal strain (ε 1 ) will be the major Poisson's ratio (ν 12 ). Similarly, ν 13 is obtained by dividing (ε 3 ) and (ε 1 ) of the JFR composite of the same RVE. Poisson's ratio of JFR polypropylene composite ( Figure 7) shows a clear variation up to HS-3; after that, the ν12 becomes the same as the ν13. This behavior is only due to the matrix phase. The minor Poisson's ratio ν21 and ν23 are presented in Figure 15. Compared to ν21, the magnitude of ν23 is very high. The reason for the high magnitude is the increased response of the lumen of the JF. Compared with synthetic fiber composites, ν 12 and ν 13 are not the same for HS-1, HS-2 and HS-3 structured composites. The highest percentage of cellulose is responsible for this deviation in HS-1, HS-2 and HS-3 model structures. The largest lumen content giving the same response in transverse directions (2 and 3). Unlike ν 12 and ν 13 of JF, the major Poisson's ratio of JFR polypropylene composite (Figure 7) shows a clear variation up to HS-3; after that, the ν 12 becomes the same as the ν 13 . This behavior is only due to the matrix phase.
The minor Poisson's ratio ν21 and ν23 are presented in Figure 15. Compared to ν 21 , the magnitude of ν 23 is very high. The reason for the high magnitude is the increased response of the lumen of the JF. The minor Poisson's ratio ν21 and ν23 are presented in Figure 15. Compared to ν21, the magnitude of ν23 is very high. The reason for the high magnitude is the increased response of the lumen of the JF. Currently, jute fiber is utilized in a variety of industries, such as textiles, vehicles, and even some load-bearing applications. In the automotive industry, bio-polymers and advanced composites made from jute are used for manufacturing parts like cup holders, trunk liners, and door panels [45][46][47]. Currently, jute fiber is utilized in a variety of industries, such as textiles, vehicles, and even some load-bearing applications. In the automotive industry, bio-polymers and advanced composites made from jute are used for manufacturing parts like cup holders, trunk liners, and door panels [45][46][47].

Conclusions
Natural fiber reinforced composites manufactured with jute fibers (JFs) and polypropylene matrix are analyzed considering the hierarchical structure (HS) of jute. Different HSs are considered based on different percentages of JF constituents, such as cellulose, lignin, hemicellulose, and lumen.
The type of HS of JF was found to influence the elastic properties of jute fiber reinforced (JFR) polyester composites. Analytical and micromechanics-based finite element models are used to generate the results. A perfect alignment is observed between the analytical and simulation results.
Increasing the lumen percentage from 13 to 35% in the JF from HS-1 to HS-6, decreases the longitudinal modulus (E 1 ) of the JF from 83.75 to 54.12 GPa. The transverse modulus (E 2 and E 3 ) of the same fiber decreased from 12.70 to 6.90 GPa. The in-plane shear modulus G 12 increased from 3.46 to 4.19 GPa, while the out-of-plane shear modulus G 23 decreased again from 7.12 to 3.69 GPa. The lumen and cellulose percentages of JF are not significantly influenced by E 1 and E 2 and the out-of-plane shear modulus (G 23 ) at a lower volume fraction of JF in the polypropylene matrix. From both analytical and numerical simulation models, it is found that the influence of HS on in-plane shear modulus is negligible.
The major Poisson's ratio (ν 12 and ν 13 ) of JF for different HS models is the same. However, for JFR polyester composites, there was a clear difference in ν 12 and ν 13 for HS-1, HS-2, and HS-3 models. This is only due to the role of the polypropylene matrix. For JF with different HS models, the magnitude of minor Poisson's ratio ν 21 is much smaller than ν 23 . The same trend continued for JFR composites due to more elongation of the lumen under transverse loading.  ∆γ Transverse deformation of Hemi cellulose ∆δ Transverse deformation of Cellulose ∆m Transverse deformation of matrix ε 2 Φ Transverse Strain of RVE ε 2 α Transverse strain of Lumen ε 2 β Transverse strain of Lignin ε 2 γ Transverse strain of Hemi cellulose ε 2 δ Transverse strain of Cellulose ε 2 m Transverse strain of matrix ε 2 Φ Transverse Strain of RVE