Document Type : Research Paper
Authors
^{1} Department of Nanoscience and Microsystem Engineering, University of New Mexico, Albuquerque, NM, USA
^{2} Department of Mechanical Engineering, University of New Mexico, Albuquerque, NM, USA
^{3} Department of Nuclear Engineering, University of New Mexico, Albuquerque, NM, USA
Abstract
Keywords
Main Subjects
Material Mechanical Properties Prediction of Layer with Multidirectional Fiber Ply Reinforced Polymer Matrix Composites
M.M. Hassan^{ a*}, A.B. Siddique^{b}, B. Bohanon^{c}
^{a} Department of Nanoscience and Microsystem Engineering, University of New Mexico, Albuquerque, NM, USA
^{b} Department of Mechanical Engineering, University of New Mexico, Albuquerque, NM, USA
^{c} Department of Nuclear Engineering, University of New Mexico, Albuquerque, NM, USA
KEYWORDS 

ABSTRACT 
Reinforced polymer; Multidirectional fiber; Mechanical properties; TsaiWu failure criteria; Finite element analysis. 
Carbon fiber reinforced plastics (CFRP) are now being used in primary structures of airplanes, ships, and automotive engineering and for applications that demand sustained high reliability and strength during longterm operations. The present work provides a threedimensional (3D) model that has been established for the simulation of the multidirectional carbon fiber reinforced plastic (CFRP) composite layer which enables understanding of the mechanical properties at failure using finite element simulation software ANSYS. The matrix is considered isotropic and elasticplastic. Moreover, the carbon fiber is considered transversely isotropic and linear elastic for the finite element modeling. The maximum stressstrain criterion is used to determine the failure of fiber and matrix of composite layer for both analytical and modeling analysis. The modeling result shows that the elastic modulus in fiber orientation is significantly higher. Numerical and analytical results of nonlinear mechanical stress and the distribution on the multilayer show good agreement with experimental results. 
Fiberreinforced polymer composites (FRP) are the most vastly used composite materials actually and research on FRP composites has been increasingly developed due to their ample structural potential [1]. FRP composites have steadily gained various applications in different fields. The modern concept and technology of fiberreinforced polymer composites have accompanied ample change with the clear concept of composite laminates [2]. As advanced engineering materials, Composite laminates have many applications primarily as components in aircraft, ships, cars, rail vehicles, power plants, civil engineering structures, robots, prosthetic devices, sports equipment, etc. [35]. The reasons behind this huge demand are mainly due to the high strengthtoweight ratio, adjustable mechanical properties, and fatigue resistance [6].
The elastic properties of composite layers depend on multifarious factors like the configuration of the laminates, materials used, and the production method. Laminated material with the required strength and stiffness characteristics with respect to specific design conditions can be determined by a different arrangement of layers and fiber directions [7]. The mechanical and elastic properties of composites are not fixed until the final structure is manufactured. These characteristics could be achieved by employing standardized tests or with numerical methods [810]. Due to the expenses and inconveniences in experimental methods, it is more common to estimate these properties by numerical calculations.
The elastic properties of composites are determined by the mechanical characteristics of the constituent materials. In order to predict and characterize composite behavior, different analytical and numerical techniques have been applied [11][12][13]. Analytical solutions do not always provide good results due to the complexity and the number of equations arising from complex geometries, loading conditions, and material characteristics [1416]. Extracting effective elastic properties is the main requirement in studying the mechanics of composite materials.
The tensile properties of fiberreinforced composites polymer are affected by several factors, such as fibermatrix interaction, volume fraction, aspect ratio (diameter to length of fiber), fiber orientation, and rules of the mixture model [1719]. Rehan et al. [20] investigated the fiber orientation effects of multidirectional composite laminates with special stacking sequences leading to isolating the orientation parameter at the crack interface. The stacking structures also influenced on mechanical properties of carbon fiber/epoxy IM7/9377 laminates and carbon/ glass interlayer hybrid composite was investigated by Roy et al. [21] and Weili et al. [22] respectively. They found that the mechanical properties as tensile and compressive moduli increase for hybrid composite with the increase of fiber concentration. Daniel et al. [23] proposed a failure theory to predict the lamina strength under multiaxial stress on the strain rate effect. The axial and radial stress was calculated as a function of axial distance in an elastic fiber surrounded by a plastic matrix by the finite element method and the results demonstrated that the axial stress increased linearly from the end towards the center; the radial stress was more evenly distributed [24]. Xi Deng et al. [25] proposed a new threedimensional micromechanical model for the analysis of multidirectional fiber of reinforced polymer and showed a good agreement with the experimental results. The local fiber deformations and microdebonding between matrix and fibers are caused by void influence on composite tensile strength investigated by Olivier et al. [26]. Another researcher Nirbhay et al.[27] investigated the strength of composite with various stacking sequences of carbon fiber and found that the carbon fiber with crossply lamination is stiffer than the angle ply glass fiber lamination.
If the fibers are parallel with the direction of loading, the optimum stiffness and strength can be achieved, On the other hand, when the fibers are perpendicular to the direction of applied load, then the strength of the composite is comparatively low. However, fibers can vary in different orientations as 90°, 45°, or 30° angles to accommodate for the direction of the applied loads to give better strength. This article aims to predict the material property of a layer with multidirectional fiber ply reinforced polymer matrix composites. A composite layer consists of multiple numbers of ply that have been used to perform different tests (tensile, expansion, or shear) using the finite element analysis software ANSYS. Some predefined material properties were taken and simulation was carried out with the specific number of plies and their regular alternating orientation. The deviation between numerical results and modeling results of elastic modulus and share modulus were calculated.
Fibers are randomly oriented in composite material. Material properties are varied for different arrangements of the fibers. The stressstrain equation for a transversely isotropic material in the elastic region follows Hook’s law and the relationship can be written as [28].
(1) 


where, and are the stress in x, y, and zdirection. and are the strain in x, y, and zdirection. The zaxis is considered the fiber direction and the xy plane is the transverse basal plane. The share modulus of x, y and z directions is . If the normal stress acts in the xdirection, the elastic modulus is defined as
(2) 
Composite laminate has individual layers consisting of unilateral plies with the same or regularly alternating orientation as shown in Fig. 1. The layers can also be applied from thermosetting or thermoplastic polymers, metals, and fabric or have a spatial 3D reinforced structure.
Fig. 1. Graphical representation of the composite layer 
If the layer consists of m longitudinal plies oriented at 0° with the thickness (i) (i=1,2,3,…,m) and n transverse plies with 90° with the thickness (j) (j=1,2,3,…,n) are made of the same composite materials. Next, stresses and form the plane stress state in the global coordinate frame and can be transformed in terms of stresses in the principal coordinates of the plies as the equation given below whereas the total thickness is
The and are the longitudinal and transverse thickness of ply respectively.
(3) 
Similarly, the Stresses in the principal coordinates of the plies are related to the corresponding strains given below.
(4) 
The stresses acting on a lamina in the local axis are of two types i.e., the normal and shear stress. So the lamina will fail if each of the local axes' normal or shear stresses is equal to or exceeds the corresponding ultimate strengths of the unidirectional lamina [29]. When these components are higher than the corresponding tension or compression yield strength, then the failure occurs. The and symbols represent the normal stress corresponding to compression and tension respectively.
(5) 
The most common failure criterion used to predict the failure of composite materials is the maximum stress criterion which is a linear, stressbased, and failure mode dependent criterion without stress interaction [30]. The failure index of a lamina is given by equation No. 4 using the maximum stress failure theory.
The lamina strains are resolved in the local axes. A lamina will fail if these normal or shear strains are equal to or exceed the corresponding ultimate strains of the unidirectional lamina. The and symbols represent the normal strain corresponding to compression and tension respectively.
(6) 
The failure index of a lamina is given by equation no.7 using the maximum strain failure theory.
(7) 
Finite element analysis is one of the most powerful tools for the design and analysis of any complex problem. It involves the approximation of a continuous function by a discrete model and the deformation of physical solids whereas, the whole body of the model is divided into several smaller parts considered as elements that are interconnected with the nodes. The steps for the finite element modeling are shown in Fig. 2. The commercial FEA ANSYS software (Canonsburg, Pennsylvania) was used for finite element modeling.
PLANE183, an Axisymmetric 8node element type was used as carbon fiber. The quadratic displacement behavior was exhibited by SOLID186, a homogeneous structural Solid is a higherorder threedimensional 20node solid element. The element was defined by 20 nodes with three degrees of freedom (x, y and z) per each one. The elements had nonlinear capabilities, such as large deflection, large strain, and elasticplastic stressstrain of the material constitutive. The finite element model dimensions orders were several hundred times smaller than real specimens due to computation time limitations and finer element requirements for better accuracy. The length and width of each layer were 5mm and 5 mm. The thickness of the individual layer was 0.5 mm. The total number of layers/ply was 20 due to the limitation of computing time of the ANSYS software version. The fibers were in the X direction in the odd layers and for the case of even layers, the fibers were in the Y direction. The angle between the two plies was 90ᵒ. In order to carry the various internal and external loads applied, fibers can be positioned in any direction within the tube. The proper element size and density were determined by convergence study of the tensile test so that the model in Fig. 3 is meshindependent. The matrix's total number of elements is 6000 and the number of nodes is 28,060. The element meshing was very important to improve the accuracy of the FEM shown in Fig. 4. Applying the boundary conditions was the most crucial part of the simulation. Boundary conditions virtually and mathematically simulate the theoretical test conditions. It helped to understand the basics and actual working of the tests. All boundary conditions have been applied to the solid model for the tensile test by setting the X, Y and Z plane fixed on one side and making the strain on another side 1.
Fig. 2. Flow chart for finite element modeling 
Fig. 3. (Left) FEA 3D model of 20 layers and (Right) Mesh of the model (Isometric View) 
Fig. 4. (Left) Element meshing of the 3D model, and (Right) Boundary condition applied on the Meshing of the model (Isometric View) 
The volume fraction of the fiber and matrix in composite material plays an important role in shaping its properties. The fiber attributes almost 80% to 90% of the properties of the composite material again it also depends on the volume fraction of the fiber too. A regular threedimensional long fiber in a matrix was used to describe the volume fraction of the single layer of the composite as shown in Fig. 5. The mechanical properties of carbon fiber and epoxy are given in Table 1 [31]. The nominal diameter of the carbon fiber was 5–10 µm and it was composed mostly of carbon atoms. The total number of nodes and elements was 19,822 and 3,870 were used for a single fiber. From the volume fraction, the sides of the square i.e. length and width of the matrix were calculated using the following relation as w_=sqrt(pi_/v_frctn)*r_ where w_ is the width of a square matrix of the representative element, v_frctn is the volume fraction of fiber and r_ is the radius of the fiber. After the tensile test through the modeling, the stress and young’s modulus in the axial direction of Z was calculated at 137 GPa with a strain of 0.01%. The maximum Von Mises stress is found at 2.5 GPa. The Von Mises stress is used to determine whether an isotropic and ductile metal will yield a complex loading condition or not. In addition, the Poisson’s ratio of composite material for straininduced in x and y direction because of the force in zdirection i. e. ν_{12}= ν_{x}/ ν_{z} and ν_{13}= ν_{y} / ν_{z}, where Z_{1} is fiber direction and X_{2} & Y_{3} are transverse directions. Finally, the numerical calculation of the elastic modulus, E_{1} = E_{f}*v_{f }+ E_{m}*v_{m}; (f and m denote fiber and matrix) is 155.77GPa which validates the simulation results in the volume fraction of 0.62. The elastic properties of fiber content in singleply are shown in Fig. 6 with different volume fractions.
(a) 
(b) 
(c) 
Fig. 5. Unit cell of square array fiber packing geometry in 3D (a) for onefiber (b) Von Mises stress distribution and (c) FEA mesh both fiber and matrix (Front view) 
Table 1. Geometrical characteristics of the specimens
Materials 
Elastic Moduli 
Poisson’s Ratio 
Share Moduli 


(GPa) 
(GPa) 
(GPa) 

Carbon fiber 
252 
22.30 
0.35 
0.30 
0.027 
8.29 
22.1 

Epoxy 
2.03 
2.03 
0.4 
0.4 
0.4 
0.725 
0.725 



Fig. 6. Materials properties of a single layer of carbon fiber with reinforced polymer matrix composites extracted from modeling: (a) Young moduli, E_{x} and E_{z}, and share moduli, G_{XY, }and G_{ZX. }and (b) Poisson’s ratio, . 


Table 2. Comparison of material properties of Carbon fiber reinforced (carbon fiber of 54.7%) by modeling and experiment 


CF 54.7% Vol 
Elastic Moduli 
Poisson’s Ratio 
Share Moduli 


(GPa) 
(GPa) 
(GPa) 

Modeling 
137.5 
8.9 
0.372 
0.345 
0.020 
3.56 
12.9 

Exp 
139.42 
9.34 
0.330 
 
 
 
9.29 

The uniaxial tensile tests were conducted on 20 layers of the composite of 0.5 mm thickness along x, y and z directions for extracting the mechanical properties i. e. young modulus of E1, E_{2}, and E_{3} and Poisson’s ratio of ν_{12}, ν_{21}, ν_{23}, ν_{32}, ν_{31} and ν_{13}. The shear modulus, Gzx, of carbon fibers' shear crosssections is higher than shearing along with carbon fiber longitudinal direction (Gxy). However, as carbon fiber volume fraction increases, the Poisson’s ratios decrease, which indicates less contraction in the transverse directions under tensile loads. At the fibermatrix interface with less fiber content, the stress concentration is greater [32]. The variation of fiber concentration is more significant in higher stress regions of the composite layer. In addition, when the fiber concentration is increased, the stress concentration at the fiber decreased and the shear stress at the fibermatrix interface became more intense. Therefore, the ratio between the matrix and fiber concentration is important, whereas the interfacial stress is reducing the premature interfacial failure and increases the mechanical properties. The modeling predicted elastic characteristics of the composite are compared with the experimental results [33] and presented in Table 2 which shows a good agreement.
In this study, a 20layer 3D composite with 3065 vol% carbon fiber was investigated. The modeling results show that Young’s modulus in fiberoriented directions and shear modulus is significantly higher than that of the matrix. The shear modulus of shearing fiber crosssections is even much higher than the shearing along with fiber longitudinal direction. The carbon fiber volume fraction can enhance structural stability by lowering the composites' Poisson ratios. In this investigation, the modeling results were in good agreement with the experimental data. Several phenomena such as fiber sliding and fragmentation (which can affect the debonding mechanism) have not been included in the model. Also, timedependent effects have been neglected for the modeling work. The MoriTanaka model is an effective field theory based on Eshelby's elasticity method for inhomogeneity in an infinite medium. The author will consider the twophase MoriTanaka model for predicting the elastic moduli of FRP composites for future work.
Conflict of Interest
The authors declare that there is no conflict of interest regarding the publication of this paper.
Acknowledgments
This computational facility support from the Department of Mechanical and Nuclear Engineering at the University of New Mexico is gratefully acknowledged.
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