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Asymptotic Expansion Homogenization Analysis Using Two-Phase Representative Volume Element for Non-periodic Composite Materials

  • Diego Ariza Colera
  • Hyun-Gyu KimEmail author
Original Research
  • 31 Downloads

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 properties 

Introduction

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

Consider a composite structure with spatially periodic fibers. The homogenized material properties of the composite can be determined by using a representative unit-cell or a representative volume element (RVE) model in AEH method. The global model X is described by coordinates xi and the RVE or the microscale model Y is described by coordinates yi (see Fig. 1).
Fig. 1

Schematic of asymptotic expansion homogenization method

The parameter \( \in \) relates both microscale and macroscale characteristic dimensions. As the difference between the two scales is noticeable big, the value for \( \in \) must be rather small \( \left( { \in \ll 1} \right) \)
$$ {\mathbf{y}} = \frac{\varvec{x}}{ \in }. $$
(1)
Due to the periodicity in the microscale model Y, the relation between the microscale and macroscale models for the elasticity tensor D is
$$ D_{ijkl}^{\varepsilon } \left( \varvec{x} \right) = D_{ijkl} \left( {\frac{\varvec{x}}{ \in }} \right), $$
(2)
\( D_{ijkl} \) represents the elastic properties of Y. Once \( D_{ijkl} \) is computed, the elasticity tensor for the macroscale \( D_{ijkl}^{\varepsilon } \left( \varvec{x} \right) \) can be determined in coordinates xi.
The linear-elasticity problem is governed by the following equations:
$$ \frac{{\partial \sigma_{ij}^{ \in } }}{{\partial x_{j}^{ \in } }} + f_{i} = 0 \quad {\text{in}}\;\varOmega , $$
(3)
$$ \varepsilon_{ij}^{ \in } = \frac{1}{2}\left( {\frac{{\partial u_{i}^{ \in } }}{{\partial x_{j}^{ \in } }} + \frac{{\partial u_{j}^{ \in } }}{{\partial x_{i}^{ \in } }}} \right), $$
(4)
$$ \sigma_{ij}^{ \in } = D_{ijkl}^{ \in } \varepsilon_{kl}^{ \in } , $$
(5)
$$ u_{i}^{ \in } = \bar{u}_{i} \quad in\quad \varGamma_{u} , $$
(6)
$$ \sigma_{ij}^{ \in } n_{j} = \bar{t}_{i} \quad in\quad \varGamma_{t} , $$
(7)
where \( \sigma_{ij} \) are the components of the Cauchy stress tensor, and \( \varepsilon_{ij} \) are the components of the strain tensor in the macroscale coordinates xi. The repeated indices in the above equations imply the summation convention over all three coordinates.
The displacement field for a two-scale material can be approximated by asymptotic series expansion in ϵ as
$$ u_{i}^{ \in } \left( \varvec{x} \right) = u_{i}^{\left( 0 \right)} \left( {\varvec{x},\varvec{y}} \right) + \in u_{i}^{\left( 1 \right)} \left( {\varvec{x},\varvec{y}} \right) + \in^{2} u_{i}^{\left( 2 \right)} \left( {\varvec{x}, \varvec{y}} \right) + \cdots , $$
(8)
where \( u_{i}^{\left( r \right)} \left( {\varvec{x}, \varvec{y}} \right) \) are periodic functions of order \( r \) of the displacement field. As two different coordinate systems, xi and yi, are considered in the derivation of the equations, the following chain rule expansion can be considered as
$$ \frac{\partial }{{\partial x_{i}^{ \in } }} = \frac{\partial }{{\partial x_{i} }} + \frac{1}{ \in }\frac{\partial }{{\partial y_{i} }}. $$
(9)
Then, applying the chain rule stated in Eq. (9) on Eq. (8), we can obtain strain and stress components:
$$ \varepsilon_{ij}^{ \in } = \in^{ - 1} \varepsilon_{ij}^{\left( 0 \right)} + \in^{0} \varepsilon_{ij}^{\left( 1 \right)} + \in^{1} \varepsilon_{ij}^{\left( 2 \right)} + \cdots , $$
(10)
$$ \sigma_{ij}^{ \in } = \in^{ - 1} \sigma_{ij}^{\left( 0 \right)} + \in^{0} \sigma_{ij}^{\left( 1 \right)} + \in^{1} \sigma_{ij}^{\left( 2 \right)} + \cdots , $$
(11)
where
$$ \varepsilon_{ij}^{\left( 0 \right)} = \frac{1}{2}\left( {\frac{{\partial u_{i}^{\left( 0 \right)} }}{{\partial y_{j} }} + \frac{{\partial u_{j}^{\left( 0 \right)} }}{{\partial y_{i} }}} \right). $$
(12)
Substituting Eq. (8) into the strain–displacement relationship in Eq. (4) and applying the result into the stress–strain relation in Eq. (5), we have an expression dependent on the coefficient ϵ:
$$ \in^{ - 2} \frac{{\partial \sigma_{ij}^{\left( 0 \right)} }}{{\partial y_{j} }} + \in^{ - 1} \left( {\frac{{\partial \sigma_{ij}^{\left( 0 \right)} }}{{\partial x_{j} }} + \frac{{\partial \sigma_{ij}^{\left( 1 \right)} }}{{\partial y_{j} }}} \right) + \in^{0} \left( {\frac{{\partial \sigma_{ij}^{\left( 1 \right)} }}{{\partial x_{j} }} + \frac{{\partial \sigma_{ij}^{\left( 2 \right)} }}{{\partial y_{j} }} + f_{i} } \right) + \cdots = 0. $$
(13)
Since Eq. (13) is valid for any given ϵ\( \to 0 \), any coefficient of the powers of ϵ is zero. Thus, we obtain the following set of differential equations:
$$ \frac{{\partial \sigma_{ij}^{\left( 0 \right)} }}{{\partial y_{j} }} = 0, $$
(14)
$$ \frac{{\partial \sigma_{ij}^{\left( 0 \right)} }}{{\partial x_{j} }} + \frac{{\partial \sigma_{ij}^{\left( 1 \right)} }}{{\partial y_{j} }} = 0, $$
(15)
$$ \frac{{\partial \sigma_{ij}^{\left( 1 \right)} }}{{\partial x_{j} }} + \frac{{\partial \sigma_{ij}^{\left( 2 \right)} }}{{\partial y_{j} }} + f_{i} = 0. $$
(16)
Considering the Dirichlet and Neumann boundary conditions in Eqs. (6) and (7), the boundary conditions become
$$ \in^{0} u_{i}^{\left( 0 \right)} + \in^{1} u_{i}^{\left( 1 \right)} + \in^{2} u_{i}^{\left( 2 \right)} + \cdots = \bar{u}_{i} \quad in\quad \varGamma_{u} . $$
(17)
$$ \left( { \in^{ - 1} \sigma_{ij}^{\left( 0 \right)} + \in^{0} \sigma_{ij}^{\left( 1 \right)} + \in^{1} \sigma_{ij}^{\left( 2 \right)} + \cdots } \right)n_{j} = \bar{t}_{i} \quad in\quad \varGamma_{t} . $$
(18)
Since Eq. (15) defines the relation between both microscale and macro scale stresses, the microscale perturbation displacement \( u_{i}^{\left( 1 \right)} \) can be expressed as
$$ u_{i}^{\left( 1 \right)} \left( {\varvec{x}, \varvec{y}} \right) = - \chi_{i}^{kl} \left( \varvec{y} \right) \frac{{\partial u_{k}^{\left( 0 \right)} }}{{\partial x_{l} }}\left( \varvec{x} \right) + \tilde{u}_{i}^{\left( 1 \right)} \left( \varvec{x} \right), $$
(19)
where \( \tilde{u}_{i}^{\left( 1 \right)} \left( \varvec{x} \right) \) are the integration constants in yi, usually considered as zero, and \( \chi_{i}^{kl} \) are the Y-periodic components of the characteristic displacement tensor χ. The characteristic displacement tensor χ is the solution of the variational problem:
$$ \mathop \int \limits_{Y}^{{}} D_{ijkl} \frac{{\partial \chi_{k}^{mn} }}{{\partial y_{l} }}\frac{{\partial v_{i} }}{{\partial y_{j} }}dy = \mathop \int \limits_{Y}^{{}} D_{ijmn} \frac{{\partial v_{i} }}{{\partial y_{j} }}dy\quad \forall v_{i} \in V_{Y} , $$
(20)
where \( V_{Y} \) is the set of Y-periodic continuous and regular functions with zero average value in Y.
Combining Eqs. (19) into (16) and considering the Y periodicity of \( u_{i}^{\left( 2 \right)} \) in y, the term \( \frac{{\partial \sigma_{ij}^{\left( 2 \right)} }}{{\partial y_{j} }} \) can only be equal to zero:
$$ \frac{\partial }{{\partial x_{j} }}D_{ijmn}^{h} \frac{{\partial u_{m}^{\left( 0 \right)} }}{{\partial x_{n} }} + f_{i} = 0. $$
(21)
Then, the homogenized values of an Y-periodicity given function ф can be defined as
$$ \Phi_{Y} = \frac{1}{\left| Y \right|}\mathop \int \limits_{Y}^{{}} \Phi \left( {x,y} \right)dY. $$
(22)
Considering the appropriate boundary conditions for the elastic problem defined by the macroscale equation and applying the average operator, the homogenized elastic properties become
$$ D_{ijmn}^{h} = \frac{1}{\left| Y \right|}\mathop \int \limits_{Y}^{{}} D_{ijkl} \left[ {\delta_{km} \delta_{\ln } - \frac{{\partial \chi_{k}^{mn} }}{{\partial y_{l} }}} \right]dY. $$
(23)
Once the macroscale problem has been solved by using the homogenized elastic properties, stress and strain fields at the microscale level can be obtained by a localization process. This process can be seen as the inverse of homogenization. The localization process for stress and strain fields are respectively defined as
$$ \sigma_{ij}^{\left( 1 \right)} \left( {\varvec{x},\varvec{y}} \right) = D_{ijkl} \left( \varvec{y} \right)\left( {\delta_{km} \delta_{ln} - \frac{{\partial \chi_{k}^{mn} }}{{\partial y_{l} }}} \right)\frac{{\partial u_{m}^{\left( 0 \right)} }}{{\partial x_{n} }}, $$
(24)
$$ \varepsilon_{ij}^{\left( 1 \right)} \left( {\varvec{x},\varvec{y}} \right) = \frac{1}{2}\left( {\delta_{ik} \delta_{jl} + \delta_{il} \delta_{jk} } \right)\left( {\delta_{km} \delta_{ln} - \frac{{\partial \chi_{k}^{mn} }}{{\partial y_{l} }}} \right)\frac{{\partial u_{m}^{\left( 0 \right)} }}{{\partial x_{n} }}. $$
(25)

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

The finite element approximation of the microscale equation in Eq. (20) is defined as
$$ \mathop \int \limits_{{Y^{e} }}^{{}} \varvec{B}^{\varvec{T}} \varvec{DB}dY \chi = \mathop \int \limits_{{Y^{e} }}^{{}} \varvec{B}^{\varvec{T}} \varvec{D}dY, $$
(26)
where \( Y^{e} \) is a single representative volume element, D is the elasticity matrix, and B is the strain-nodal displacement matrix defined below:
$$ \varvec{\varepsilon}= \varvec{Bu}\text{,} $$
(27)
$$ \varvec{\sigma}= \varvec{DBu}\text{.} $$
(28)
In the case of a hexahedral RVE in \( y_{1} \in \left[ {0, y_{1}^{0} } \right], y_{2} \in \left[ {0, y_{2}^{0} } \right] and y_{3} \in \left[ {0, y_{3}^{0} } \right] \), the periodic boundary conditions are applied on the edges as
$$ \chi_{i}^{jk} \left( {0,y_{2} ,y_{3} } \right) = \chi_{i}^{jk} \left( {y_{1}^{0} ,y_{2} ,y_{3} } \right), $$
(29)
$$ \chi_{i}^{jk} \left( {y_{1} ,0,y_{3} } \right) = \chi_{i}^{jk} \left( {y_{1} ,y_{2}^{0} ,y_{3} } \right), $$
(30)
$$ \chi_{i}^{jk} \left( {y_{1} ,y_{2} ,0} \right) = \chi_{i}^{jk} \left( {y_{1} ,y_{2} ,y_{3}^{0} } \right). $$
(31)
The homogenized elasticity tensor in Eq. (23) can be obtained by
$$ \varvec{D}^{H} = \mathop \sum \limits_{e = 1}^{ne} \frac{{Y^{e} }}{Y}\varvec{D}^{e} \left( {\varvec{I} - \varvec{B}^{e}\varvec{\chi}^{k} } \right), $$
(32)
where Y is the total volume of the unit cell, and \( Y^{e} , \)\( \varvec{D}^{\varvec{e}} , \)\( \varvec{B}^{\varvec{e}} \) and \( \varvec{\chi}^{\varvec{k}} \) denote element volume, elemental elasticity tensor, elemental integration matrix and elemental elasticity matrix, respectively.
When it comes to the macroscale analysis, the homogenized elastic properties of the RVE, Exx, Eyy and νxy can be obtained straightforward by the following relation:
$$ \left[ {D^{H} } \right] = \left[ {\begin{array}{*{20}l} {\frac{{E_{xx} }}{{1 - \nu_{xy}^{2} }}} \hfill & {\frac{{_{xy} E_{yy} }}{{1 - \nu_{xy}^{2} }}} \hfill & 0 \hfill \\ {\frac{{_{xy} E_{xx} }}{{1 - \nu_{xy}^{2} }}} \hfill & {\frac{{E_{yy} }}{{1 - \nu_{xy}^{2} }}} \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & {G_{xy} } \hfill \\ \end{array} } \right]. $$
(33)
In the same manner, microscale strain and stress fields can be obtained by
$$ \varvec{\varepsilon}^{\left( 1 \right)} \left( {x,y} \right) = \left( {\varvec{I} - \varvec{B}^{e}\varvec{\chi}^{e} } \right)\varvec{\varepsilon}^{\left( 0 \right)} \left( {x,y} \right), $$
(34)
$$ \varvec{\sigma}^{\left( 1 \right)} \left( {x,y} \right) = \varvec{D}^{e} \left( {\varvec{I} - \varvec{B}^{e}\varvec{\chi}^{e} } \right)\varvec{\varepsilon}^{\left( 0 \right)} \left( {x,y} \right). $$
(35)

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.

The matrix material of the structure is boron, reinforced with an aluminum fiber being 47% of the total volume of the RVE. The mechanical properties of both materials are given in Table 1. Figure 2a shows the RVE mesh, and the periodic boundary conditions stated in Eqs. (29), (30) and (31), are applied on the RVE mesh with equal number of nodes on each of the sides, which ties the node of one side with its opposite in the other side.
Table 1

Elastic properties of the matrix and the fiber of a composite

Elastic properties

Values

Matrix Young’s modulus Em

68.3 GPa

Matrix Poisson’s ratio vm

0.3

Fiber Young’s modulus Ef

379.3 GPa

Fiber Poisson’s ratio vf

0.1

Fig. 2

RVE with periodic boundary conditions: a RVE mesh, and b characteristic deformation of RVE

The characteristic deformation of the RVE obtained by ABAQUS using the UEL developed in this study is shown in Fig. 2b. The homogenized elastic properties of the composite are compared in Table 2. It can be clearly appreciated that the results obtained in this study are within the upper and lower homogenized bounds defined for transversely isotropic composites with random arrangement of fibers by Hashin–Shtrikman [15]:
$$ E^{{H^{ + } }} = \frac{{9K^{{H^{ + } }} G^{{H^{ + } }} }}{{3K^{{H^{ + } }} + G^{{H^{ + } }} }}, $$
(36)
$$ E^{{H^{ - } }} = \frac{{9K^{{H^{ - } }} G^{{H^{ - } }} }}{{3K^{{H^{ - } }} + G^{{H^{ - } }} }}, $$
(37)
with
$$ G^{{H^{ + } }} = G_{f} + \frac{{F_{v} - 1}}{{\frac{1}{{G_{f} - G_{m} }} - \frac{{F_{v} \left( {2G_{f} + K_{f} } \right)}}{{2G_{f} \left( {G_{f} + K_{f} } \right)}}}}, $$
(38)
$$ G^{{H^{ - } }} = G_{m} + \frac{{F_{v} }}{{\frac{1}{{G_{f} - G_{m} }} - \frac{{\left( {2G_{m} + K_{m} } \right)\left( {F_{v} - 1} \right)}}{{2G_{m} \left( {G_{m} + K_{m} } \right)}}}}, $$
(39)
$$ K^{{H^{ + } }} = K_{f} + \frac{{1 - F_{v} }}{{\frac{1}{{K_{m} - K_{f} }} + \frac{{F_{v} }}{{\left( {K_{f} + \mu_{f} } \right)}}}}, $$
(40)
$$ K^{{H^{ - } }} = K_{m} + \frac{{F_{v} }}{{\frac{1}{{K_{f} - K_{m} }} + \frac{{1 - F_{v} }}{{\left( {K_{m} + \mu_{m} } \right)}}}}, $$
(41)
where \( K_{i} = \frac{{E_{i} }}{{3\left( {1 - 2\nu_{i} } \right)}} \), \( \mu_{i} = \frac{{E_{i} }}{{2\left( {1 + \nu_{i} } \right)}} \), the super index m and f stand for matrix and fiber, respectively and \( F_{v} \) is the volume reinforcement fraction. In addition, the homogenized elastic moduli are almost identical to the results obtained previously by Barroqueiro et al. [13]. Note that the RVE mesh in this study is different from that in the Ref. [14].
Table 2

Homogenized elastic moduli of a composite with regularly distributed circular fibers

Homogenized elastic properties

Exx (GPa)

ν xy

Gxy (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.

The RVE without the outer phase is referred as iteration 0, as shown in Fig. 3a. Figure 3b shows a two-phase RVE composed of the inner phase of a non-periodic composite material and the outer phase of a homogenized material. Several iterations of AEH analyses need to be carried out until a convergence solution is obtained. In each of the iterations, it is necessary to update the elastic properties of the outer phase to the homogenized values obtained in the previous iteration.
Fig. 3

AEH method using two-phase RVE model: a single-phase RVE in the 0th iteration, and b a two-phase RVE composed of the inner phase of a non-periodic composite material and the outer phase of a homogenized material

Influence of the Thickness of the Outer Phase

An important design parameter that needs to be considered in the two-phase RVEs is the thickness of the outer phase. In this part, the influence of the thickness of the outer phase is investigated for a composite with randomly distributed fibers. A boron matrix with randomly reinforced aluminium inclusions taking 20.79% of the total volume of the composite is considered as shown in Fig. 4. The material properties of boron and aluminium given in Table 1 are used for the AEH analyses.
Fig. 4

A composite with randomly distributed fibers

For the parametrization of the thickness of the outer phase, two-phase RVEs of the smallest unit cell \( 1.0 \times 1.0 \) in Fig. 4 with four different thicknesses of the outer phase, 0.1, 0.2, 0.3 and 0.4, as shown in Fig. 5, are considered in this study. Characteristic deformations and stress distributions of the two-phase RVEs are presented in Fig. 6. AEH analyses are performed repeatedly for these four two-phase RVEs to obtain converged values of homogenized elastic properties.
Fig. 5

Two-phase RVE models with different thicknesses of the outer phase

Fig. 6

Characteristic deformations and stress distributions of two-phase RVE models with different thicknesses of the outer phase

As plotted in Fig. 7, the homogenized elastic properties approach converged values as the thickness of the outer phase increases. Since the inner phase of the unit cell in Fig. 5 is not periodic, the homogenized elastic properties \( E_{xx} \) and \( E_{yy} \) of the two-phase RVEs are different. For the two-phase RVEs considered in this study, the thickness of the outer phase of the two-phase RVEs should be larger than around 30–40% of the length of the RVE side to obtain converged values. In other words, a small thickness of the outer phase of two-phase RVEs can lead to an error in the estimation of the homogeneous elastic properties of non-periodic composite materials.
Fig. 7

Homogenized elastic properties of two-phase RVEs as a function of the thickness of the outer phase

Influence of the Size of the Inner Phase

In this part, we investigate the effect of the size of the inner phase in the determination of the homogenized elastic properties of a composite with non-periodic structures. For this purpose, the four different window sizes showed in Fig. 8 from the boron-aluminium composite in Fig. 4 are considered in this study. The thicknesses of the outer phase of the two-phase RVEs are taken by 30% of the length of the RVE side.
Fig. 8

Two-phase RVEs with different sizes of the inner phase

Since the fiber reinforcement is non-periodic, homogenized elastic properties depend on the choice of window domain for RVEs. Table 3 lists the results obtained by AEH analyses using single-phase and two-phase RVEs. As the size of the RVE increases, the homogenized elastic properties in x and y directions get closer to each other, because the RVEs contain more fibers that are placed randomly in a matrix. Note that a composite with randomly distributed fibers shows a transversely isotropic behavior in a macroscale. When the size of the RVE is small, uneven distribution of fibers leads to orthotropic behavior.
Table 3

Homogenized elastic properties obtained by AEH analyses using two-phase RVEs of different window sizes

Models

Two-phase RVEs

Exx (GPa)

Eyy (GPa)

A

92.336

90.334

B

91.158

91.659

C

90.909

91.075

D

94.518

94.607

Since the four two-phase RVEs in Fig. 8 have different fiber volume fractions (see Table 4), the homogenized elastic properties obtained by AEH analyses using the two-phase RVEs can be different. Ideally, the size of RVE should be chosen to contain all information necessary for describing the behavior of a composite. Fiber volume fraction can be statically representative of the microgeometry of a composite [16]. Fiber volume fraction can be an important factor to determine homogenized elastic moduli of a composite with randomly distributed fibers. Hence, it is important to take the fiber volume fraction of the two-phase RVE as the fiber volume fractions of the entire region of a composite with randomly distributed fibers.
Table 4

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

A composite with periodically distributed fibers is analyzed to verify the present AEH method using two-phase RVEs. The periodic unit cell is taken as the RVE model to obtain the reference homogenized elastic properties of the composite in Fig. 9. In order to investigate the performance of AEH method using two-phase RVEs in Fig. 9b, we take three different windows for RVEs as shown in Fig. 9a. The thickness of the outer phase is taken by 30% of the length of the RVE size as proposed in the previous part.
Fig. 9

a A composite with periodically distributed fibers, and b two-phase RVE models taken from the composite with periodically distributed fibers

All the inner phases of the two-phase RVEs in Fig. 9 have almost similar fiber volume fraction of 12.52% which is the fiber volume fraction of the periodic unit cell. The finite element meshes and characteristic stresses of the two-phase RVEs are shown in Fig. 10. The homogenized elastic properties obtained by using the two-phase RVEs are plotted in Fig. 11. The expected elastic modulus and Poisson’s ratio obtained by using the periodic unit cell RVE shown in Fig. 10 are 79.177 GPa and 0.2874, respectively. The first iteration of the present method uses single-phase RVE models without outer phases. As the iteration of AEH analyses increases, the homogenized elastic properties approach the converged values, as shown in Fig. 11.
Fig. 10

Finite element meshes and characteristic stresses of two-phase RVEs for a composite with equally distributed fibers

Fig. 11

Homogenized elastic properties obtained by using two-phase RVE models: a model A, b model B, and c model C

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).

References

  1. 1.
    J.L. Bensoussan, G. Lions, Papanicolaou, asymptotic analysis for periodic structures, Studies in mathematics and its applications, vol. 5 (Elsevier, Amsterdam, 1978)Google Scholar
  2. 2.
    P.W. Chung, K.K. Tamma, R.R. Namburu, Asymptotic expansion homogenization for heterogeneous media: computational issues and applications. Compos. Part. A 32, 1291–1301 (2001)CrossRefGoogle Scholar
  3. 3.
    J. Pinho-da-Cruz, J.A. Oliveira, F. Teixeira-Dias, Asymptotic homogenisation in linear elasticity. Part I: mathematical formulation and finite element modelling. Comput. Mater. Sci. 45, 1073–1080 (2009)CrossRefGoogle Scholar
  4. 4.
    J. Pinho-da-Cruz, J.A. Oliveira, F. Teixeira-Dias, Asymptotic homogenisation in linear elasticity. Part II: finite element procedures and multiscale applications. Comput. Mater. Sci. 45, 1081–1096 (2009)CrossRefGoogle Scholar
  5. 5.
    X. Zhuang, Q. Wang, H. Zhu, A 3D computational homogenization model for porous material and parameters identification. Comput. Mater. Sci. 96, 536–548 (2014)CrossRefGoogle Scholar
  6. 6.
    J. Barroqueiro, J. Dias-de-Oliveira, A. Pinho-da-Cruz, Andrade-Campos, Multiscale analysis of heat treatments in steels: theory and practice. Fin. Element. Anal. Des. 114, 39–56 (2016)CrossRefGoogle Scholar
  7. 7.
    A.L. Kalamkarov, A.V. Georgiades, S.K. Rokkam, V.P. Veedu, M.N. Ghasemi-Nejhad, Analytical and numerical technique to predict carbon nanotube properties. Int. J. Solids Struct. 43, 6832–6854 (2006)CrossRefzbMATHGoogle Scholar
  8. 8.
    Y.S. Song, J.R. Youn, Modeling of effective elastic properties for polymer based carbon nanotube composites. Polymer 47, 1741–1748 (2006)CrossRefGoogle Scholar
  9. 9.
    A.L. Kalamkarov, A.V. Georgiades, Asymptotic homogenization models for smart composite plates with rapidly varying thickness: part I – theory. Int. J. Multiscale Comput. Eng. 2, 133–148 (2004)CrossRefGoogle Scholar
  10. 10.
    A.V. Georgiades, A.L. Kalamkarov, Asymptotic homogenization models for smart composite plates with rapidly varying thickness: part II – applications. Int. J. Multiscale Comput. Eng. 2, 149–172 (2004)CrossRefGoogle Scholar
  11. 11.
    K.S. Challagulla, A.V. Georgiades, A.L. Kalamkarov, Asymptotic homogenization modeling of thin composite network structures. Compos. Struct. 79, 432–444 (2007)CrossRefGoogle Scholar
  12. 12.
    H.J. Böhm, A short introduction to basic aspects of continuum micromechanics (CDL-FMD-Report, TU Wien, Vienna, 1998)Google Scholar
  13. 13.
    Z. Yuan, J. Fish, Toward realization of computational homogenization in practice. Int. J. Numer. Methods Eng. 73, 361–380 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    J. Barroqueiro, J. Dias-de-Oliveira, A. Pinho-da-Cruz, Andrade-Campos, Practical implementation of asymptotic expansion homogenisation in thermoelasticity using a commercial simulation software. Compos. Struct. 141, 117–131 (2016)CrossRefGoogle Scholar
  15. 15.
    Z. Hashin, S. Shtrikman, A variational approach to the theory of the elastic behaviour of polycrystals. J. Mech. Phys. Solids 10, 343–352 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    C. Pelissou, J. Baccou, Y. Monerie, F. Perales, Determination of the size of the representative volume element for random quasi-brittle composites. Int. J. Solids Struct. 46, 2842–2855 (2009)CrossRefzbMATHGoogle Scholar

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© Korean Multi-Scale Mechanics (KMSM) 2019

Authors and Affiliations

  1. 1.Department of Mechanical and Automotive EngineeringSeoul National University of Science and TechnologySeoulSouth Korea

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