Dr. Fatih Ertuğrul Öz
Department of Mechanical Engineering
Faculty of Engineering
Prof. Dr. Nuri Ersoy
Department of Mechanical Engineering
Faculty of Engineering
Design and Mathematics
Faculty of Engineering University of West of England
This paper presents the capabilities of Finite Elements Micromechanical Analysis (FEMA) method for simulating mechanical
behaviour of composite materials. A thermosetting prepreg composite system with unidirectional and cross-ply desginations are
considered in this paper. 3-Dimensional Representative Volume Elements (RVEs) are built according to the fibre volume fraction
of composite materials with a commercial finite elements programme. These models enable to calculate the elastic properties of
the composite materials with great accuracy. In addition to this, stress distributions within the fibres and the matrix can be
investigated and give idea about where and how the damage can develop.
Finite Elements Method (FEM) is a very popular method to predict the mechanical behaviour of materials. It is highly valuable
since it can reduce experimental cost by providing an idea about the mechanical response of the regarding material. Finite
Elements Micromechanical Analysis (FEMA) method has been a very important numerical method to predict the mechanical
behaviour of materials in micro-level for the last two decades. There has been numerous studies performed by using this method.
It can be applied easily by using any commercial finite elements programme. Even though polymeric composite materials are
mostly prefferred, different composites can be analysed with this method. It can provide important idea about the mechanical
response of the regarding materials under different load modes in micro-level.
Simple geometrical models are used in FEMA where the fibres are packed into these geometries according to the fibre volume
fraction of the regarding composite. Generally the fibres are packed into the models with respect to two geometries; square
packed and hexagonally packed arrays as shown in Figure 1. Two different materials should be defined in the graphical user
interface of the commercial FE programme for the constituents and their properties must be known. One should be careful while
entering the material properties. Carbon or glass fibres should be defined as orthotropic or transversely isotropic, whereas matrix
materials, polymeric resins, should be defined as isotropic.
Figure 1. Square and hexagonally packed RVEs.
In this paper, hexagonal and square packed RVEs are taken into consideration to investigate the mechanical behaviour of a carbon
fibre reinforced thermosetting composite system with FEMA method. A commercial FE programme, ABAQUS®  is used for the
analyses. First the FEMA procedure in ABAQUS is described briefly. Then elastic properties of the composite are calculated with
RVEs. Finally the results are compared with available data in literature. It can be seen that the FEMA can provide good agreements
with the experimental values and this proves the efficiency of this method by using simple models and low computational time in
terms of predicting mechanical response of composite materials.
2.1. Representative Volume Elements and Boundary Conditions
Meshed RVEs can be seen in Figure 2. These RVEs can be reduced to simpler geometries, shown with black lines in Figure 2. In
order to provide periodicity and symmetry, unified or periodic boundary conditions [2, 3] should be applied to the model. Basically,
the requirements for the periodic boundary conditions are given in Equations 1-3. Application of these boundary conditions is very
easy in ABAQUS graphical user interface by using symmetric boundary conditions and the “Equation Type” constraints. Definitions
of boundary conditions in ABAQUS for different load conditions are given in Table 1 and Table 2 capital letters. (X,Y,Z) on the left
side of the tables designate the normal direction of planes whereas small letters on top, (x,y,z) show the directions of the nodes.
The material in consideration is a thermosetting prepreg composite system which is a carbon fibre reinforced epoxy matrix, having
57.4% fibre volume fraction. This material system is chosen since the open literature can provide comparison.
Figure 2. RVEs used in this study.
uz (TOP)-uz (BOTTOM)=0 (1)
uy (RIGHT)-uy (LEFT)=0 (2)
ux (FRONT)-ux (BACK)=0 (3)
Table 1. Boundary conditions for normal loading
Table 2. Boundary conditions for shear loading
2.2. Calculation of Elastic Properties
Composite materials have five mechanical elastic constants. They are listed in Table 3. Even though there are six independent
parameters given in Table 3, transverse shear modulus, G23, can be obtained by using the other parameters as shown in Equation
4. A unit strain is applied in regarding direction to provide deformation and the average of the resulting stresses in the model is
equal to that elastic modulus. Three steps are followed to calculate the regarding elastic moduli:i. “Equation type” constraint sums
the stresses in each surface node to a single node in terms of reaction force. The regarding reaction force is noted first, ii. The
reaction force in that single node is divided to regarding surface area. iii. Then, Hooke’s law is applied to calculate that elastic
property. Since the applied strain is unit, equal to 1, the calculated stress in (ii) is the elastic modulus in regarding direction.
Table 3. Elastic constants of composite materials
3.Results and Discussion
The resulting deformations caused by the application of unit strain under each load modes are shown in Figure 3. Only hexagonal
models are presented for unidirectional model in Figure 2. Because, even though the square model has advantage in terms of
simpler geometry, that model cannot provide to obtain the transverse isotropy relation in Equation 4. Because of this, the resulting
stress distributions under transverse shear loading is not accurate.
It can be seen that the resulting stress distributions are different under each unit strains in Figure 3. The average stress values
shown in the legends of each load conditions correspond to the regarding elastic modulus. Their calculated elastic properties and
their comparison with the literature is shown in Table 4. Results show that the FEMA method can provide good correlations with
the experimental results. Thus, it is a simple and efficient numerical method for predictions of mechanical behaviour of materials
as shown in Figure 3 and Table 4.
Although, only the calculation of elastic moduli values with FEMA is included in this paper. The capabilities of this method is not
limited to this only. The process-induced residual stresses, and the effect of residual stresses to the strength of the material can
be investigated with FEMA. The detailed studies regarding the application of FEMA for prediction of process-induced residual
stresses, and to progressive damage analysis of composites under the subsequent loading can be found in literature [4–7] or the
readers who have interest to this subject can get in contact with the authors.
Figure 3. Resulting deformations and stress distributions under different load modes
Table 4. Comparison of elastic moduli values with test results
This paper summarizes the capabilities of Finite Elements Micromechanical Analysis (FEMA) method which can predict the
mechanical behaviour of composite materials in micro level. A thermosetting prepreg composite system with unidirectional and
cross-ply desginations are investigated with square and hexagonally packed RVEs. It is seen that the FEMA method provides
good predictions for elastic moduli values. They are validated by comparing with test results. FEMA results can also present
stress distributions within the fibres and the matrix under different load modes that can give idea about where and how damage
can develop. This method can be adopted for investigation of proces-induced residual stresses and its effect during progressive
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Composites Under Various Types of Loading, Yüksek Lisans Tezi, Boğaziçi University, 2012.
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the 20th International Conference on Composite Materials, Copenhagen – Denmark, 2015.
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Doktora Tezi, Boğaziçi University, 2018.
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