Asymptotic Expansion Homogenization Analysis Using Two-Phase Representative Volume Element for Non-periodic Composite Materials
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Abstract
Asymptotic expansion homogenization (AEH) method is a well-known approach based on the assumption of the periodicity of microstructures to obtain the homogenized material properties of composite materials. The main advantage of this method is that it can be used as a multiscale simulation tool. A new AEH method is developed in this study to estimate the homogenized elastic properties of non-periodic composite materials using two-phase representative volume elements (RVEs) composed of the inner phase of a non-periodic composite material and the outer phase of a homogenized material. The AEH method is repeatedly applied to the two-phase RVEs to update the homogenized elastic properties of non-periodic composite materials.
Keywords
Asymptotic expansion homogenization Representative volume element Homogenized elastic propertiesIntroduction
Composites are constituted by a matrix or main material and a reinforcement material which generally has higher specific properties than the matrix. Composites have multiple scale natures with statistically homogeneity above a certain scale, and the shape and distribution of the microstructure influences the macroscopic behavior of the composite materials. Since numerical analysis of composite materials using fine meshes for all the heterogeneities of the microstructure can lead to a huge amount of memory and CPU time, it is necessary to develop an efficient approach to parametrize the material properties of composites with a smaller computational cost.
To overcome this problem, asymptotic expansion homogenization (AEH) method [1] has been proposed as an efficient tool for analyzing composite materials. The methods based on AEH formulation can estimate overall material properties from the mechanical behavior of selected microscale representative volume elements (RVEs). The main advantage of AEH formulation is that the asymptotic series approximation to the homogenization problem not only significantly reduces the degrees of freedom but also allows the characterization of the microstructural stress and strain fields by a process called localization. Mathematical formulations and computational methods for AEH are described in detail by [2, 3, 4]. AEH has been implemented to a wide range of different composite material configurations [5, 6] due to its accuracy and efficiency to estimate the homogenized properties. The AEH method has been also used to modern engineering applications such as nanotechnologies [7, 8], smart composite modelling [9, 10], and the modelling of thin network structures [11].
In all the above mentioned researches, RVEs with specific periodic boundary conditions are chosen based on the need of satisfying the periodicity in the microstructure, which is the fundamental requirement of the conventional AEH method. However, the internal structure of commonly used composite materials is non-periodic. Windowing approaches have been proposed to extract RVE models with periodic arrangement that approximate composite materials with non-periodic structures by satisfying energy equivalence between micro- and macro-scales [12]. In order to apply periodicity constraints to non-periodic geometries, real structure geometries need to be modified to become periodic by perturbating the microfields in the vicinity of RVE boundaries. Hence, it is desirable to develop a new AEH technique using non-periodic RVEs to better estimate the homogenized properties of non-periodic composite materials.
In this work, an AEH method using two-phase RVEs is presented to estimate the homogenized elastic properties of non-periodic composite materials. The two-phase RVEs are composed by an inner phase of the representative unit cell of the composite materials and an outer phase of a homogenized material. The homogenized elastic properties of the non-periodic composite are obtained by updating the elastic properties of the outer phase through iterative AEH analyses. AEH formulation is implemented in ABAQUS user subroutine (UEL), and several examples are analyzed to investigate the effects of the sizes of the inner phase and the outer phase of two-phase RVEs.
The paper is organized as follows. Firstly, “Asymptotic expansion homogenization (AEH) method” section summarizes the AEH mathematical and finite element formulations. “Practical implementation of AEH formulation in ABAQUS” section explains the practical implementation of AEH method in ABAQUS. In “Two-phase representative volume element” section, the AEH method using two-phase RVEs is described through numerical models. Finally, “Verification of homogenized elastic properties of two-phase RVE models” section estimates the homogenized elastic moduli of two-phase RVEs taken randomly from a periodically structured composite to verify the present method. The conclusion is presented in “conclusion” section.
Asymptotic Expansion Homogenization (AEH) Method
AEH Formulation
These equations allow to compute the stress and strain values for a heterogeneous sample for any given point of the macroscale model.
AEH Finite Element Formulation
Practical Implementation of AEH Formulation in ABAQUS
In this study, the AEH formulation is implemented in a user subroutine (UEL) of ABAQUS for practical purposes. Yuan and Fish [13] and Barroqueiro et al. [14] also developed a user subroutine of ABAQUS for the AEH formulation. In order to verify the user subroutine developed in this study, the model previously used by Barroqueiro et al. [14] is analyzed to obtain the homogenized elastic properties of a composite with a periodic array of circular fibers.
Elastic properties of the matrix and the fiber of a composite
Elastic properties | Values |
---|---|
Matrix Young’s modulus E_{m} | 68.3 GPa |
Matrix Poisson’s ratio v_{m} | 0.3 |
Fiber Young’s modulus E_{f} | 379.3 GPa |
Fiber Poisson’s ratio v_{f} | 0.1 |
Homogenized elastic moduli of a composite with regularly distributed circular fibers
Homogenized elastic properties | E_{xx} (GPa) | ν _{ xy} | G_{xy} (GPa) |
---|---|---|---|
The present study | 133.87 | 0.2088 | 45.641 |
Upper bound Hashin–Shtrikman | 167.13 | – | 57.14 |
Lower bound Hashin–Shtrikman | 125.14 | – | – |
Barroqueiro et al. [13] | 133.2 | 0.209 | 45.3 |
Two-Phase Representative Volume Element
In this study, we propose a new AEH approach to estimate the homogenized elastic properties of non-periodic composite materials. A two-phase RVE model with periodic boundary conditions is used by introducing the outer phase of a homogenized material surrounding the inner phase representing a portion of non-periodic composite materials. Since the homogenized elastic properties of non-periodic composite materials are not unknown, it is required to perform AEH analyses in an iterative way. Firstly, the homogenized elastic properties of the RVE without the outer phase are obtained by AEH method, and then they are applied to the outer phase of the two-phase RVE model.
Influence of the Thickness of the Outer Phase
Influence of the Size of the Inner Phase
Homogenized elastic properties obtained by AEH analyses using two-phase RVEs of different window sizes
Models | Two-phase RVEs | |
---|---|---|
E_{xx} (GPa) | E_{yy} (GPa) | |
A | 92.336 | 90.334 |
B | 91.158 | 91.659 |
C | 90.909 | 91.075 |
D | 94.518 | 94.607 |
Fiber volume fractions of non-periodic RVE models
Models | Fiber volume fraction (%) |
---|---|
A | 23.55 |
B | 18.31 |
C | 23.66 |
D | 26.89 |
Verification of Homogenized Elastic Properties of Two-Phase RVE Models
The homogenized elastic properties of composite materials depend on the choice of RVE which characterizes the distribution of fibers. Although the macroscopic behavior of the composite with equally distributed fibers is isotropic, the elastic behavior of the two-phase RVEs is anisotropic due to the non-periodic inner phases of the two-phase RVEs. However, as shown in Fig. 11, the averaged values of the homogenized elastic properties obtained by using the two-phase RVE models get closer as the iteration of AEH analyses increases to the target value obtained by using the periodic unit cell. As a result, the two-phase modelling technique to the AEH method of randomly reinforced unit cells can allow us to obtain improved solutions for non-periodic composite materials by taking the averaged values of the homogenized elastic properties obtained by using two-phase RVEs.
Conclusions
In this study, we have developed a two-phase RVE approach to accommodate the periodicity of boundary conditions in the conventional AEH method for non-periodic composite materials. The two-phase RVEs are composed of the inner phase of a non-periodic composite material and the outer phase of a homogenized material. The AEH method is iteratively applied to the two-phase RVEs, being necessary to update the homogenized elastic properties in each one of the iterations. Numerical experiments show that the two-phase RVEs improve the accuracy of the homogenized elastic properties of non-periodic composite materials compared to the single-phase RVEs. The effects of the thickness of the outer phase and the window size for RVEs are investigated through numerical examples. Since the reinforcement volume fraction plays an important role in the evaluation of the homogenized elastic properties of non-periodic composite materials, the inner phase of the two-phase RVE should be chosen to have the average fiber volume fraction of the composites. In addition, we show that the averaged values of the homogeneous elastic properties obtained by using the two-phase RVEs can better represent the overall elastic properties of the non-periodic composite materials. As a result, the use of two-phase RVEs can be an efficient and effective way to estimate the homogenized elastic properties of non-periodic composite materials.
Notes
Acknowledgements
This research was supported by the EDISON Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (No. 2014M3C1A6038854).
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