## 1. Introduction

Composite materials are produced by the combination of primarily two, but sometimes more chemically distinct constituents. One of the constituents (the reinforcement) is embedded in another constituent (the matrix). Synthetic fibers are the most frequently used reinforcing phase with various polymer matrices. However, synthetic fibers are not biodegradable and environmentally friendly, and it is significant challenge to manage the debris from conventional synthetic fiber-reinforced composites. During the last decade, scientists and engineers have been looking for environmentally friendly, biodegradable, and green composites as an alternative. Natural fiber composites (NFC) have the potential to fulfill most of the governing requirements. Natural fibers are extracted from renewable resources, hence their production cost of fiber is comparatively low. NFCs are used in various applications ranging from household to automotive (i.e. door panels, shelves, engine insulation, dashboard, etc.)[1].

However, research on NFC began much later than that on the synthetic fiber-reinforced composites. Hence, there is very little research dealing with jute fiber-reinforced epoxy composites, most of which involves the experimental approach. The influence of various loading conditions for tensile strength and improvement of mechanical properties was studied experimentally[2,3]. Researchers have modified the tensile properties of the jute fiber epoxy composite by introducing the laminated paper method[4]. The literature also describes the chemical enrichment of jute fiber in order to develop the mechanical properties of jute fiber composites[5]. For numerical analysis, the mechanical properties of the composite or its constituents must be required. The effective material properties of the composite can be predicted using both analytical and numerical methods[6]. The various analytical methods can be classified as the mechanics of material (MOM) method, the semi-empirical method, and the elasticity approach[7]. Finite element analysis (FEA) is used in the modern vehicle, airplane, ship and space industries for determining the suitable and safe design of various equipment[8]. The finite element numerical homogenization technique has become favored for predicting the effective elastic properties[9]. In this technique, the reinforcement fiber and matrix are assumed to be homogeneous throughout the composite. The composite can be considered as a combination of unit volumes known as representative volume elements (RVEs). The effective mechanical properties of the jute fiber epoxy have been determined according to this criterion by modeling only one RVE, which then represents the entire model[10]. Finite element modeling can be performed using a range of finite element software including ANSYS, ABAQUS, NASTRAN, LS-DYNA, etc. An FEA for determining the effective mechanical properties of a sisal fiber polystyrene composite had been performed by Adeniyi et. al. using the commercial FEA package ABAQUS[9].

In the present work, the micromechanical modeling of the jute fiber-reinforced epoxy composite was performed. A three-dimensional RVE was developed for the FEA, which was then performed using periodic boundary conditions to investigate the effective material properties of the jute fiber epoxy composite. The elastic properties obtained from the FEA were validated using several analytical methods. The facile commercial FEA software ABAQUS was utilized in this analysis.

## 2. Numerical modeling

The present analysis considers, jute fiber reinforcement embedded in the epoxy matrix phase to generate the jute epoxy reinforced composite. The composite is considered to be void free and homogeneous, such that the constituents are considered to be isotropic. The composite is considered as a uniform distribution of fiber in the matrix, with no imperfections between fiber and matrix[11]. Thus, the composite is a real unidirectional fiber-reinforced composite. A three dimensional RVE is used to predict the mechanical properties which are determined for various volume fractions of fiber.

### 2.1 Constitutive equation

The general 3D constitutive relation for anisotropic material is given by Eq. (1)[9-11].

where σ_{i} and ε_{j} are stress and strain components respectively, and C_{ij} is the material stiffness matrix with 21 independent components known as elastic constants. Based on the independent elastic constant, a composite material can be classified as anisotropic, monoclinic, orthotropic, transversely isotropic or isotropic. In the present investigation, the composite is considered to be transversely isotropic. An orthotropic material is regarded as transversely isotropic when one of its principal planes is the plane of isotropy. The transversely isotropic independent coefficients are reduced to 5, where the nonzero coefficients are 12 in Eq. (1). The reduced form of the 3D constitutive equation is given by Eq. (2)[9].

The effective elastic properties can be calculated when the stiffness tensor C is known. The elastic properties from the stiffness matrix are predicted using the following equations[11].

where *E*_{1}, *E*_{2}, *υ*_{12}, and *G*_{12} are longitudinal elastic modulus, transverse elastic modulus, major Poisson’s ratio, and in-plane shear modulus, respectively.

### 2.2 Finite element analysis (FEA)

FEA is the most widely used numerical method for solving a range of engineering problems[12]. The well-known commercial finite element software ABAQUS was used in the present study. A Python script was also used for the post-processing. The steps used in this study are presented in Fig 1.

### 2.3 Generation of RVE

The jute fiber epoxy composite displays a periodic repetition of the same three-dimensional RVE. Thus the modeling of one RVE, rather than the entire composite is sufficient for the FEA. It is necessary to calculate the true dimension of the RVE. A hexagonal RVE was used in the present analysis, where *a*_{1}, *a*_{2}, and *a*_{3} are the lengths along the respective axes. A hexagonal RVE is shown in Fig 2 for the volume fraction of 40%. The volume fraction can be written as Eq. (4)[11].

where *d _{f}* is the diameter of the jute fiber;

*a*

_{3}=

*a*

_{2}tan(60°), and

*a*

_{2}= 4

*a*

_{1}

### 2.4 Material properties and mesh generation

The effective elastic properties of the unidirectional jute fiber-reinforced epoxy matrix composite were used in investigated in the present work. The jute fiber and epoxy were both considered to be isotropic materials. The mechanical properties of the jute fiber and epoxy resin are presented in Table 1.

The three-dimensional RVE was generated for various volume fraction from 0.1 to 0.9 and the effective mechanical properties of the composites were calculated. The hex(sweep) meshing method was applied with a global mesh size of 0.15 and the C3D8 eight-node linear brick element was employed. The meshed model of hexagonal RVE is presented in Fig 3.

### 2.5 Boundary conditions

The homogenization model is widely used for composite materials because these materials consist of periodic RVE arrays. A periodic boundary condition is used to analyze the periodic RVE representation using three linear perturbation steps. The boundary condition used for this investigation is presented in Table 2.

The unit strain was maintained in all directions of the RVE to determine the longitudinal Young’s modulus, the transverse Young’s modulus, the in-plane Poisson’s ratio and the shear modulus. By applying unit strain, it was possible to calculate the stress, average value of which gave the required components of the constitutive tensor *C*_{ij} [11]. The components can be calculated using Eq (6).

where *V* is the volume of the RVE.

## 3. Analytical modeling

Analytical models are very popular for calculating the effective mechanical properties of the composites. In the current investigation, the FEA results were compared with analytical models.

### 3.1 Mechanics of materials modeling

The mechanics of material (MOM) is based on the following assumptions: (i) the bonding between reinforcement and matrix is perfect, (ii) the diameter of the fiber is uniform, (iii) the fibers are continuous, parallel and possess uniform strength, (iv) the reinforcement and matrix are both elastic, and (v) the composite has no voids[6]. This modeling approach is also known as the mixing method for unidirectional composites. The longitudinal Young’s modulus (*E _{1}*) and the major Poisson’s ratio (

*υ*), the transverse Young’s modulus (

_{12}*E*), and the in-plane shear modulus (

_{2}*G*) can be computed using the following equation derived using MOM approach.

_{12}The longitudinal properties are calculated by the Voigit model and the transverse properties are evaluated using the Reuss model[11].

Thus, the longitudinal properties are given by Eq. (6)

where *V _{f}* and

*V*are the volume fraction of jute fiber and epoxy matrix, respectively;

_{m}*E*, and

_{f}*E*are respective Young’s moduli of the jute fiber and epoxy matrix; and

_{m}*υ*and

_{f}*υ*are respective Poisson’s ratios of the jute fiber and epoxy.

_{m}The transverse properties are given by Eq. (8)

The shear properties are given by Eq. (9)

where *G _{f}* and

*G*are shear moduli of the jute fiber and epoxy matrix, respectively.

_{m}### 3.2 Semi-empirical modeling

Halpin and Tsai interpreted semi-empirical equations by introducing various equations predicted by curve fitting. The longitudinal properties were the same as those obtained by the MOM approach. The transverse properties were improved by introducing one reinforcing factor depending on geometry of the fiber along with, packing and boundary conditions of the fiber and matrix[7].

Here longitudinal properties are given by Eq. (10)

The transverse properties are given by Eq. (12)

where $\eta =\left[\left\{\left({E}_{f}/{E}_{m}\right)-1\right\}/\left\{\left({E}_{f}/{E}_{m}\right)+\xi \right\}\right]$ in which *ξ* is the reinforcing factor. For the circular fiber in a square packing geometry *ξ* = 2 which gives an excellent fit for the transverse properties with elasticity solution.

The shear properties are given by Eq. (13)

where $\eta =\left(\frac{\left({E}_{f}/{E}_{m}\right)-1}{\left({E}_{f}/{E}_{m}\right)+\xi}\right)$ with *ξ*=2

### 3.3 Elasticity Modeling

The elasticity models, also known as composite cylinder assemblage (CCA) models, consider the composites as periodic RVE representation. The CCA model makes the following assumptions: (i) the fibers have circular cross-section, (ii) the fibers are dispersed in a periodic structural sequence, and (iii) a continuous fiber is bonded by a cylindrical matrix[7]. Here the longitudinal properties are given by Eq. (14)

where ${B}_{1}=2{\nu}_{m}^{2}Vf-{\nu}_{m}+{\nu}_{m}{V}_{f}-1-{V}_{f}$ and ${B}_{2}=2{\nu}_{f}^{2}-{V}_{f}{\nu}_{f}-2{V}_{f}{\nu}_{f}^{2}+{V}_{f}+{\nu}_{f}-1$

The transverse properties are given by Eq. (16)

where,

Out of plane shear modulus, *G*_{23} is calculated by solving the following equation.

where ${\eta}_{m}=3-4{\nu}_{m}\hspace{0.17em}\text{and}\hspace{0.17em}{\eta}_{f}=3-4{\nu}_{f}$

Shear Properties:

## 4. Results and discussion

The finite element software ABAQUS was successfully executed for the analysis. The average stresses were calculated from the unit strain by the application displacement boundary conditions. The Python script was used to compute the coefficients of the constitutive tensor *C*_{ij} from the average stresses obtained from FEA. The elastic properties were then investigated using the constitutive tensor. The average stress contours for steps 1-3 are shown in Fig. 4(a-c), respectively.

The results obtained from finite element homogenization model and various other analytical models for the jute fiber epoxy composites were compared. The jute fiber reinforced epoxy composite was considered as a unidirectional composite lamina. The fibers were uniformly distributed and aligned parallel to each other in the longitudinal direction.

It is noted that the predicted finite element results and analytical results for the longitudinal elastic modulus were in agreement for all values of volume fraction as shown in Fig 5. The longitudinal elastic modulus increased at the same rate as the volume fraction of fiber. Fig 6 shows the major Poisson’s ratio for the varying volume fraction of the jute fiber-reinforced epoxy composites. The Poisson’s ratios show a considerable decrease with increasing volume fraction.

The MOM and Halpin-Tsai analytical models slightly over-estimated the Poission’s ratio.

However, the numerical homogenization finite element model showed a good agreement with the elasticity CCA analytical approach. Since the composite was considered to be transversely isotropic, the elastic moduli of both transverse directions were the same. A comparison of transverse Young’s moduli for the different models is presented in Fig 7. The transverse elastic modulus also increased with the fiber volume fraction, although the increase was slower than for the longitudinal modulus. Fig 8 shows the in-plane shear modulus, which also slowly increases with the volume fraction of jute fiber. The finite element homogenization result is closely similar to the Halpin-Tsai and CCA elasticity analytical models.

## 5. Conclusions

In this study, the effective material properties of jute fiber-reinforced epoxy composites were obtained using a novel technique based on the finite element modeling method. The effective material properties obtained by FEA were compared with the analytical results. The longitudinal Young’s modulus, transverse Young’s modulus, and shear modulus all increases as the percentage volume fraction of jute fiber increased. However, the Poisson’s ratio decreased with increasing volume fraction the jute fiber. This finite element technique shows a significant similarity with previously developed analytical methods for the effective prediction of the material properties of jute epoxy composite.