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ISSN : 1598-6721(Print)
ISSN : 2288-0771(Online)
The Korean Society of Manufacturing Process Engineers Vol.18 No.12 pp.28-37
DOI : https://doi.org/10.14775/ksmpe.2019.18.12.028

# Effective Material Properties of Composite Materials by Using a Numerical Homogenization Approach

Anik Das Anto*, Hee Keun Cho**#
*Department of Precision Mechanical Engineering, ANDONG NATIONAL UNIVERSITY
**Department of Mechanical Engineering Education, ANDONG NATIONAL UNIVERSITY
Corresponding Author : marklee1@hanmail.net Tel: +82-54-820-5677, Fax: +82-54-820-7655
09/11/2019 02/12/2019 08/12/2019

## Abstract

Due to their flexible tailoring qualities, composites have become fascinating materials for structural engineers. While the research area of fiber-reinforced composite materials was previously limited to synthetic materials, natural fibers have recently become the primary research focus as the best alternative to artificial fibers. The natural fibers are eco-friendly and relatively cheaper than synthetic fibers. The main concern of current research into natural fiber-reinforced composites is the prediction and enhancement of the effective material properties. In the present work, finite element analysis is used with a numerical homogenization approach to determine the effective material properties of jute fiber-reinforced epoxy composites with various volume fractions of fiber. The finite element analysis results for the jute fiber-reinforced epoxy composite are then compared with several well-known analytical models.

# 균질화 접근법을 통한 복합재의 유효물성치 계산

아 닉 다스 안토*, 조 희근**#
*안동대학교 정밀기계공학과
**안동대학교 기계교육과

## 초록

This is an Open-Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

## 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].

${ σ 1 σ 2 σ 3 σ 4 σ 5 σ 6 } = [ C 11 C 12 C 13 C 14 C 15 C 16 C 12 C 22 C 23 C 24 C 25 C 26 C 13 C 23 C 33 C 34 C 35 C 36 C 14 C 24 C 34 C 44 C 45 C 46 C 15 C 25 C 35 C 45 C 55 C 56 C 16 C 26 C 36 C 46 C 56 C 66 ] { ε 1 ε 2 ε 3 ε 4 ε 5 ε 6 }$
(1)

where σi and εj are stress and strain components respectively, and Cij 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].

${ σ 1 σ 2 σ 3 σ 4 σ 5 σ 6 } = [ C 11 C 12 C 13 0 0 0 C 12 C 22 C 23 0 0 0 C 13 C 23 C 22 0 0 C 36 0 0 0 ( C 22 − C 23 ) / 2 0 0 0 0 0 0 C 66 0 0 0 0 0 0 0 ] { ε 1 ε 2 ε 3 ε 4 ε 5 ε 6 }$
(2)

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].

$E 1 = C 11 − 2 C 12 2 C 22 + C 23 E 2 = [ C 11 ( C 22 + C 23 ) − 2 C 12 2 ] ( C 22 − C 23 ) ( C 11 C 22 − C 12 2 ) ν 12 = C 12 C 22 + C 23 G 12 = C 66$
(3)

where E1, E2, υ12, and G12 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 a1, a2, and a3 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].

$V f = 2 a 1 ( π / 4 ) d f 2 a 1 a 2 a 3$
(4)

where df is the diameter of the jute fiber; a3 = a2tan(60°), and a2 = 4a1

### 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 Cij [11]. The components can be calculated using Eq (6).

$C i j = σ ¯ i = 1 V ∫ V σ i ( x 1 , x 2 , x 3 ) d V w i t h ε j 0 = 1$
(5)

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 (E1) and the major Poisson’s ratio (υ12), the transverse Young’s modulus (E2), and the in-plane shear modulus (G12) can be computed using the following equation derived using MOM approach.

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)

$E 1 = E f V f + E m V m$
(6)

$ν 12 = ν f V f + ν m V m$
(7)

where Vf and Vm are the volume fraction of jute fiber and epoxy matrix, respectively; Ef, and Em are respective Young’s moduli of the jute fiber and epoxy matrix; and υf and υm are respective Poisson’s ratios of the jute fiber and epoxy.

The transverse properties are given by Eq. (8)

$1 E 2 = V f E f + V m E m$
(8)

The shear properties are given by Eq. (9)

$1 G 12 = 1 G f + V m G m$
(9)

where Gf and Gm are shear moduli of the jute fiber and epoxy matrix, respectively.

### 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)

$E 1 = E f V f + E m V m$
(10)

$ν 12 = ν f V f + ν m V m$
(11)

The transverse properties are given by Eq. (12)

$E 2 E m = 1 + ξ η V f 1 − η V f$
(12)

where $η = [ { ( E f / E m ) − 1 } / { ( E f / E m ) + ξ } ]$ 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)

$G 12 G m = 1 + ξ η V f 1 − η V f$
(13)

where $η = ( ( E f / E m ) − 1 ( E f / E m ) + ξ )$ 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)

$E 1 = E f V f + E m V m − 2 E m E f V f ( ν f − ν m ) 2 ( 1 − V f ) E f B 1 + E m B 2$
(14)

$ν 12 = ν f V f + ν m V m + V f V m ( ν f − ν m ) ( 2 E f ν m 2 + ν m E f − E f + E m − E m ν f − 2 E m ν f 2 ) E f B 1 + E m B 2$
(15)

where $B 1 = 2 ν m 2 V f − ν m + ν m V f − 1 − V f$ and $B 2 = 2 ν f 2 − V f ν f − 2 V f ν f 2 + V f + ν f − 1$

The transverse properties are given by Eq. (16)

$E 2 = 2 ( 1 + ν 23 ) G 23$
(16)

$ν 23 = K * − m G 23 K * + m G 23$
(17)

where,

$m = 1 + 4 K * ν 12 2 E 1$
(18)

$K * = K m ( K f + G m ) V m + K f ( K m + G m ) V f ( K f + G m ) V m + ( K m + G m ) V f$
(19)

$K f = E f 2 ( 1 + ν f ) ( 1 − 2 ν f )$
(20)

$K m = E m 2 ( 1 + ν m ) ( 1 − 2 ν m )$
(21)

Out of plane shear modulus, G23 is calculated by solving the following equation.

$A ( G 23 G m ) + 2 B ( G 23 G m ) + C = 0$
(22)

$A = 3 V f ( 1 − V f ) 2 ( G f G m − 1 ) ( G f G m + η f ) + [ G f G m η m + η f η m − ( G f G m η m − η f ) V f 3 ] × [ V f η m ( G f G m − 1 ) − ( G f G m η m + 1 ) ]$
(23)

$B = − 3 V f ( 1 − V f ) 2 ( G f G m − 1 ) ( G f G m + η f ) + 1 2 [ η m G f G m + ( G f G m − 1 ) V f + 1 ] × [ ( η m − 1 ) ( G f G m + η f ) − 2 ( G f G m η m − η f ) V f 3 ] + V f 2 ( η m + 1 ) ( G f G m − 1 ) × [ G f G m + η f + ( G f G m η m − η f ) V f 3 ]$
(24)

$C = 3 V f ( 1 − V f ) 2 ( G f G m − 1 ) ( G f G m + η f ) + [ η m G f G m + ( G f G m − 1 ) V f + 1 ] [ G f G m + η f + ( G f G m η m − η f ) V f 3 ]$
(25)

where $η m = 3 − 4 ν m and η f = 3 − 4 ν f$

Shear Properties:

$G 12 = G m [ G f ( 1 + V f ) + G m ( 1 − V f ) G f ( 1 − V f ) + G m ( 1 + V f ) ]$
(26)

## 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 Cij 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.

## Acknowledgement

This work was supported by a Research Grant of Andong National University (2019)

## Figure

Steps for solve the numerical homogenization problem
Three dimensional hexagonal RVE with circular fiber
Meshed model of hexagonal RVE
Average stresses of RVE
Comparison of FE and analytical longitudinal Young’s modulus
Comparison of FE and analytical in-plane Poisson’s ratio
Comparison of FE and analytical transverse Young’s modulus
Comparison of FE and analytical in-plane shear modulus

## Table

Mechanical properties of jute fiber and epoxy[13,14]
Periodic boundary conditions for maintaining unit strain

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