Micromechanics and homogenization

This part of the theory manual

  • discusses the general concepts of micromechanics and homogenization of heterogeneous media,
  • presents the main micromechanics and homogenization techniques utilized in SMART+.

1. General micromechanics concepts

All the materials present internal stuctures that vary and are often very complex: this is called the materials microstructure. In the microstructure local variations of the matter appear: these are the heterogeneities. The heterogeneities induce physical fields (strain, temperature …) highly non-homogeneous in the matter. For example, a metal is composed by crystals, called grains, which have different oriantations each other: this is the notion of a polycrystal. Certain material properties depend on the interatomic distance (like for instance the elastic stiffness tensor \(L_{ijkl}\)) and, in the general case, this distance varies in the microstructure.
There are many ways to observe the microstructure of materials, but the latest methods allow to observe the microstructure in three dimensions. One of these ways is to combine EBSD analysis with a FIB (Focused Ion Beam) to perform texture analysis layer by layer. Another possibility is to analyze the material using a microtomograph. Many materials can be analyzed with this method (polycrystalline materials, composite materials with random / continuous reinforcement, architectured materials, porous, etc).

Physical origin of heterogeneities

In polycrystalline media, it has been observed that the components of the the elastic stiffness tensor \(L_{ijkl}\) depend on the interatomic distance \(d\). For instance, a material with a cubic structure has a linear constitutive law of the form (in Voigt notation):
\[\boldsymbol{L}=\left(\begin{matrix}
L_{1111} & L_{1122} & L_{1122} & 0 & 0 & 0 \\
L_{1122} & L_{1111} & L_{1122} & 0 & 0 & 0 \\
L_{1122} & L_{1122} & L_{1111} & 0 & 0 & 0 \\
0 & 0 & 0 & L_{1212} & 0 & 0 \\
0 & 0 & 0 & 0 & L_{1212} & 0 \\
0 & 0 & 0 & 0 & 0 & L_{1212}
\end{matrix}\right).\]
This relation has been established in the local coordinate of the crystal, which means that the axis 1 is in the [1 0 0] direction. Proper rotation of the elastic stiffness tensor is required if the coordinate systems of the crystal and the laboratory frame (where the boundary conditions are imposed) are not the same. The constitutive law expressed in the laboratory coordinate system will vary from one grain to another, as a function of the grain orientation. Thus, the grain orientation is already one source of heterogeneity in the microstructure.
There are of course additional sources of heterogeneities. In steel, for example, a perlitic phase can be formed in the grain boundaries. This phase consists of ferrite lamellae (phase –) and cementite lamellae (Fe\(_3\)C phase). In this case, even if the crystals are oriented in the same way, the difference in composition induces a variation of the elastic constants between ferrite and perlite lamellae. Moreover, with regard to steel, there are also phenomena like the formation of martensitic phase, the residual austenite, etc. How is it possible to obtain the overall behavior of the material in these cases?

1.1 Definition of homogenization

The problem of homogezization can be stated as:

Considering a given heterogeneous microstructure, how the overall properties of an equivalent homogeneous medium can be obtained?

The definition of the equivalent medium can not be done without specifying the characteristic sizes of the heterogeneities with respect to the studied structure. In a metallic material for example, several characteristic scales can be considered:

  • The scale of the structure,
  • the scale of the crystalline structure,
  • the scale of the intragranular structure (precipitates, lamellae perlite / martensite etc),
  • the scale of the crystallographic lattice.

It becomes evident that the appropriate choice of the characteristic scale for the heterogeneities is not obvious. As one considers finer scales, the behavior of the material is more accuaretely defined. But this also means that the representative volume of the medium becomes smaller. Moreover, below a certain scale continuum mechanics are unable to describe the material behavior, since the interatomic interactions must be considered discretely. These methods are not included in SMART+. What is important in the modeling of heterogeneous materials is the choice of the most suitable scale.
The classical homogenization approaches are valid when the characteristic size of the studied microstructure is orders of magnitude smaller than the characteristic size of the structure. Thus, a ty.pical volume that is large enough to be representative of the microstructure may be considered as a material point. This notion leads to the definition of a Representative Volume Element (RVE), which plays a very important role in choosing an appropriate modeling strategy. The scope of homogenization is to replace the real heterogeneous material of the RVE with a fictitious homogeneous material that is able to provide the same stress and strain fields in the structure.

In homogenization theory a composite is described through the introduction of two suitable scales. The first scale, the microscopic or representative volume element (RVE), represents the microstructure considering different material constituents and their geometry. The second scale, the macroscopic, considers the overall body as an imaginary homogeneous medium. At the macroscale, the continuum body occupies the space \(\bar{S}\) with volume \(\bar{V}\) and bounded by a surface \(\partial\bar{S}\) with normal unit vector \(\bar{n}_i\). Each macroscopic point is assigned with a position vector \(\bar{x}_i\) in \(\bar{S}\). On the other hand, the representative volume element (RVE) occupies the space \(S\) with volume \(V\) and is bounded by the surface \(\partial S\) with normal vector \(n_i\). Each microscopic point is assigned with a position vector \(x_i\) in \(S\). In the sequel, a bar above a symbol denotes a macroscopic quantity or variable. The standard methodology is to connect macroscopic quantities through volume averaging of their microscopic counterparts over the RVE (Hill 1967), i.e.
\[\bar{\sigma}_{ij}=\frac{1}{V}\int_{S}\sigma_{ij}\textrm{d}V,~~~\bar{\varepsilon}_{ij}=\frac{1}{V}\int_{S}\varepsilon_{ij}\textrm{d}V.\]

1.2 Average theorems

Consider a typical RVE. For the following theorems, the classical assumptions of the small strain theory are considered and the following definitions are stated (Nguyen, 1988):
1. Kinematically admissible strain \(\varepsilon_{ij}\) is called every symmetric second order tensor that is related with a displacement vector \(u_i\) through the relation
\[\varepsilon_{ij}=\frac{1}{2}\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right).\]
2. Statically admissible stress \(\sigma_{ij}\) is called every symmetric second order tensor that satisfies the equilibrium equation (ignoring body forces)
\[\frac{\partial\sigma_{ij}}{\partial x_j}=0.\]

1.2.1 Average stress theorem

Assume that at each point of the boundary surface \(\partial S\) the traction vector \(t^0_i=\sigma^0_{ij}n_j\) with \(\sigma^0_{ij}\) constant is applied. Then, the volume average of the stresses inside the RVE is equal to the boundary stress tensor, i.e.
\[\frac{1}{V}\int_{S}\sigma_{ij}\textrm{d}V=\sigma^0_{ij}.\]

1.2.2 Average strain theorem

Assume that at each point of the boundary surface \(\partial S\) the displacement vector \(u^0_i=\varepsilon^0_{ij}x_j\) with \(\varepsilon^0_{ij}\) constant is applied. Then, the volume average of the strains inside the RVE is equal to the boundary strain tensor, i.e.
\[\frac{1}{V}\int_{S}\varepsilon_{ij}\textrm{d}V=\varepsilon^0_{ij}.\]

1.2.3 Periodicity conditions average theorem

Assume that at each point of a periodic RVE \(S\) the displacement vector \(u^0_i\) is given by the relation \(u^0_i=\varepsilon^0_{ij}x_j+z_i\) with \(\varepsilon^0_{ij}\) constant and \(z_i\) a periodic vector. Then, the volume average of the strains inside the RVE is equal to the strain tensor \(\varepsilon^0_{ij}\), i.e.
\[\frac{1}{V}\int_{S}\varepsilon_{ij}\textrm{d}V=\varepsilon^0_{ij}.\]

These theorems can be proven using the Green-Ostrogradsky (divergence) theorem. For the detailed proofs the interested reader is refered to Nemat-Nasser and Hori (1999) and Suquet (1987).

1.3 Hill’s lemma and Hill-Mandel theorem

The Hill’s lemma connects the mechanical energies of the two scales (macro and micro) and is expressed in the following way:
Let \(\varepsilon_{ij}\) be a kinematically admissible strain and \(\sigma_{ij}\) be a statically admissible stress. Then it holds
\[\frac{1}{V}\int_{S}{\sigma_{ij}\varepsilon_{ij}}\textrm{d}V-
\bar{\sigma}_{ij}\bar{\varepsilon}_{ij}=
\frac{1}{V}\int_{\partial S}{\left(u_i-\bar{\varepsilon}_{ij}x_j\right)
\left(\sigma_{iq}n_q-\bar{\sigma}_{iq}n_q\right)}\textrm{d}A.\]
The proof of this lemma can be found in Qu and Cherkaoui (2006). A direct consequence of the last expression is the Hill-Mandel theorem, stated as:

Let \(\varepsilon_{ij}\) be a kinematically admissible strain and \(\sigma_{ij}\) be a statically admissible stress. Then the three types of boundary conditions:

  1. \(u_i=\bar{\varepsilon}_{ij}x_j\) on \(\partial S\),
  2. \(\sigma_{iq}n_q=\bar{\sigma}_{iq}n_q\) on \(\partial S\),
  3. \(u_i=\bar{\varepsilon}_{ij}x_j+z_i\), with \(z_i\) periodic and \(\sigma_{iq}n_q\) anti-periodic,

satisfy the micro-macro energy equivalence
\[\frac{1}{V}\int_{S}{\sigma_{ij}\varepsilon_{ij}}\textrm{d}V=\bar{\sigma}_{ij}\bar{\varepsilon}_{ij}.\]

1.4 Concentration tensors

In the case of linearly elastic heterogeneous materials, one can express the relation between the micro and macroquantities (stress and strain) through appropriate concentration tensors,
\[\varepsilon_{ij}(\boldsymbol{x})=A_{ijkl}(\boldsymbol{x})\bar{\varepsilon}_{kl},~~~\sigma_{ij}(\boldsymbol{x})=B_{ijkl}(\boldsymbol{x})\bar{\sigma}_{kl},\]
where \(A_{ijkl}\) and \(B_{ijkl}\) are the strain and stress concentration tensors respectively. From these definitions it can easily be shown that
\[\frac{1}{V}\int_{S}A_{ijkl}\textrm{d}V=\mathcal{I}_{ijkl},~~~\frac{1}{V}\int_{S}B_{ijkl}\textrm{d}V=\mathcal{I}_{ijkl},\]
where \(\mathcal{I}_{ijkl}\) denotes the fourth order symmetric identity tensor. This formulation is suitable for a disordered heterogeneous material, knowing that the fields are heterogeneous from one point to another in the medium. When the medium is constructed by \(N+1\) separate phases, it is possible to define the average quantities per phase,
\[{\sigma}^{r}_{ij}=\frac{1}{V_r}\int_{S_r}\sigma_{ij}\textrm{d}V,~~~{\varepsilon}^{r}_{ij}=\frac{1}{V_r}\int_{S_r}\varepsilon_{ij}\textrm{d}V,\]
where \(S_r\) and \(V_r\) denote the space and the volume of the \(r_{\textrm{th}}\) phase. Moreover, strain and stress concentration tensors per phase can be defined which are independent of the position vector,
\[{\varepsilon}^{r}_{ij}=A^{r}_{ijkl}\bar{\varepsilon}_{kl},~~~{\sigma}^{r}_{ij}=B^{r}_{ijkl}\bar{\sigma}_{kl}.\]
From these definitions it can easily be shown that
\[\sum_{r=0}^{N}c^rA^r_{ijkl}=\mathcal{I}_{ijkl},~~~\sum_{r=0}^{N}c^rB^r_{ijkl}=\mathcal{I}_{ijkl},~~~c^r=\frac{V_r}{V}.\]
In the case of periodic media, separate phases can also be defined. However, the periodic homogenization theory considers that the deformation and stress fields are heterogeneous everywhere, and that the average fields per phase are not representative. Thus, in periodic media the concentration tensors are written as in the case of disordered heterogeneous materials,
\[\varepsilon_{ij}(\boldsymbol{x})=A_{ijkl}(\boldsymbol{x})\bar{\varepsilon}_{kl},~~~\sigma_{ij}(\boldsymbol{x})=B_{ijkl}(\boldsymbol{x})\bar{\sigma}_{kl}.\]
The stresses and strains in the RVE are connected through the relations,
\[\varepsilon_{ij}(\boldsymbol{x})=M_{ijkl}(\boldsymbol{x}){\sigma}_{kl}(\boldsymbol{x}),~~~\sigma_{ij}(\boldsymbol{x})=L_{ijkl}(\boldsymbol{x}){\varepsilon}_{kl}(\boldsymbol{x}),\]
where the elastic stiffness tensor, \(L_{ijkl}\), and the elastic compliance tensor, \(M_{ijkl}\), depend on the position in a disordered heterogeneous material. In a similar manner, the macroscopic elastic stiffness tensor, \(\bar{L}_{ijkl}\), and the macroscopic elastic compliance tensor, \(\bar{M}_{ijkl}\), of a composite can be defined by the relations
\[\bar{\sigma}_{ij}=\bar{L}_{ijkl}\bar{\varepsilon}_{kl},~~~\bar{\varepsilon}_{ij}=\bar{M}_{ijkl}\bar{\sigma}_{kl}.\]
With the help of the concentration tensors, these macroscopic tensors can be easily shown that are given by the expressions
\[\bar{L}_{ijkl}=\sum_{r=0}^{N}c^rL^r_{ijmn}A^r_{mnkl},~~~\bar{M}_{ijkl}=\sum_{r=0}^{N}c^rM^r_{ijmn}B^r_{mnkl},\]
for a composite consisting of \(N+1\) separate phases and
\[\bar{L}_{ijkl}=\frac{1}{V}\int_{S}L_{ijmn}A_{mnkl}\textrm{d}V,
~~~\bar{M}_{ijkl}=\frac{1}{V}\int_{S}M_{ijmn}B_{mnkl}\textrm{d}V,\]
for a disordered heterogeneous material or a composite with periodic structure.

1.5 Eshelby problem

Here the Eshelby inclusion problem and the Eshelby inhomogeneity problem are discussed. Further details can be found in Mura (1987), Qu and Cherkaoui (2006).

First, the difference between inclusion and inhomogeneity needs to be clarified:

  • An inclusion \(\Omega\) of a body \(S\) is a region where uniform eigenstrains \({\varepsilon}^*_{ij}\) appear. These eigenstrains are considered generally inelastic and they disturb the total strains \({\varepsilon}_{ij}\) in the body.
  • An inhomogeneity \(\Omega\) of a body \(S\) is a region where the material properties are different with regard to the rest of the body.

1.5.1 Eshelby tensor

The Eshelby inclusion problem considers an inclusion \(\Omega\) in an infinite elastic body with elastic stiffness tensor \({L}^0_{ijkl}\). From a practical point of view, this is equivalent with the case of a very small inclusion \(\Omega\) inside a finite elastic body \(S\). At the boundary of the body, the tractions and the displacement field are assumed zero. No body forces on this body are assumed. In mathematical formalism, the Eshelby problem solves the set of equations
\[\begin{array}{l}
\displaystyle{\frac{\partial\sigma_{ij}}{\partial x_j}=0} ~~~~\textrm{in } S, \\
{\sigma}_{ij}={L}^0_{ijkl}({\varepsilon}_{kl}-{\varepsilon}^*_{kl}) ~~~~\textrm{in } S, \\
{\sigma}_{ij}{n}_{j}=0 ~~~~\textrm{on } \partial S,\\
{\varepsilon}^*_{ij}=\left\{\begin{array}{ll}
\textrm{constant}\neq{0} & \textrm{in } \Omega\,,\\
{0}& \textrm{in } S-\Omega\,.\end{array}\right.
\end{array}\]
Of course the total strains have different distribution inside and outside the inclusion. Eshelby (1957) has proven that inside the inclusion the total strains are uniform and they are given by the expression
\[{\varepsilon}_{ij}=\textrm{constant}=\mathcal{S}_{ijkl}{\varepsilon}^*_{kl} ~~~~\textrm{in }\Omega,\]
where \(\mathcal{S}_{ijkl}\) is the fourth order Eshelby tensor that depends on the shape of the inclusion and the material properties \({L}^0_{ijkl}\). For special inclusion geometries and isotropic materials this tensor has analytical form (Mura, 1987). For anisotropic materials the Eshelby tensor can be computed numerically (Gavazzi and Lagoudas, 1990). Considering an ellipsoidal inclusion with main axes \(a_1\), \(a_2\) and \(a_3\), the Eshelby tensor is given by the formula
\[\mathcal{S}_{ijkl}=
\displaystyle{\frac{1}{8\pi}{\mathcal{L}}^0_{mnkl}
\int_{-1}^{1}\int_{0}^{2\pi}\left(G_{imjn}+G_{jmin}\right)\textrm{d}\omega \textrm{d}\zeta_{3}},\]
with
\[\begin{array}{c}\displaystyle{G_{ijkl}=\hat{\zeta}_k\hat{\zeta}_l{\mathcal{Z}}_{ij}^{-1}, ~~~\hat{\zeta}_1=\sqrt{1-\zeta_3}~\frac{\cos\omega}{a_1}, ~~~
\hat{\zeta}_2=\sqrt{1-\zeta_3}~\frac{\sin\omega}{a_2},} \\
\displaystyle{\hat{\zeta}_3=\frac{\zeta_3}{a_3}, ~~~
{\mathcal{Z}}_{ik}=\mathcal{L}^{0}_{ijkl}\hat{\zeta}_j\hat{\zeta}_l.}\end{array}\]

1.5.2 Inhomogeneity problem

The inhomogeneity problem considers a small inhomogeneity \(\Omega\) with elastic stiffness tensor \({L}^1_{ijkl}\) in a finite elastic body \(S\) with elastic stiffness tensor \({L}^0_{ijkl}\). The body is subjected to uniform surface traction \({\sigma}^0_{ij}{n}_j\). For this type of problem the equivalent inclusion method is utilized. In mathematical formalism, the following set of equations needs to be solved:
\[\begin{array}{l}
\displaystyle{\frac{\partial\sigma_{ij}}{\partial x_j}=0} ~~~~\textrm{in } S,\\
{\sigma}_{ij}=\left\{\begin{array}{ll}
{L}^0_{ijkl}{\varepsilon}_{kl} & \textrm{in } S-\Omega,\\
{L}^1_{ijkl}{\varepsilon}_{kl} & \textrm{in } \Omega,\end{array}\right.\\
{\sigma}_{ij}{n}_j={\sigma}^0_{ij}{n}_j ~~~~\textrm{in } \partial S.\end{array}\]
Eshelby (1957) expresses this problem as the sum of two simpler problems:

  1. A boundary value problem with constant surface tractions \({\sigma}^0_{ij}{n}_j\), where the inhomogeneity has the same properties with the rest of the material. In mathematical formalism, this is expressed as
    \[\begin{array}{l}
    \displaystyle{\frac{\partial\sigma^0_{ij}}{\partial x_j}=0} ~~~~\textrm{in } S,\\
    {\sigma}^0_{ij}={L}^0_{ijkl}{\varepsilon}^0_{kl} ~~~~\textrm{in } S,\\
    \textrm{tractions}~~~{\sigma}^0_{ij}{n}_j ~~~~\textrm{in } \partial S.\end{array}\]
    The solution of this problem is trivial and is expressed as uniform stress \({\sigma}^0_{ij}\) and uniform strain \({\varepsilon}^0_{ij}\) inside the entire body \(S\).
  2. A boundary value problem with zero surface tractions, where the inhomogeneity has been substituted by an equivalent inclusion problem. Thus, it is assumed that the region \(\Omega\) has the same properties with the rest of the body and the disturbance due to the inhomogeneity is recovered by an imaginary eigenstrain. In mathematical formalism, this is expressed as
    \[\begin{array}{l}
    \displaystyle{\frac{\partial\tilde{\sigma}_{ij}}{\partial x_j}=0} ~~~~\textrm{in } S,\\
    \tilde{\sigma}_{ij}={L}^0_{ijkl}(\tilde{\varepsilon}_{kl}-{\varepsilon}^*_{kl}) ~~~~\textrm{in } S,\\
    \tilde{\sigma}_{ij}{n}_j={0} ~~~~\textrm{in } \partial S,\\
    {\varepsilon}^*_{ij}=\left\{\begin{array}{ll}
    \textrm{constant}\neq{0} & \textrm{in } \Omega\,,\\
    {0}& \textrm{in } S-\Omega\,.\end{array}\right.\end{array}\]
    Acording to the Eshelby inclusion problem, inside the inclusion holds
    \[\tilde{\varepsilon}_{ij}=\textrm{constant}=\mathcal{S}_{ijkl}{\varepsilon}^*_{kl} ~~~~\textrm{in }\Omega.\]

All the problems are elastic, thus the principle of superposition, holds. This means that the solution of the initial problem is the sum of the solutions of the two simpler problems. Focusing only in the region inside the inclusion, it holds
\[{\sigma}_{ij}={\sigma}^0_{ij}+\tilde{\sigma}_{ij} ~~~~\textrm{and }
{\varepsilon}_{ij}={\varepsilon}^0_{ij}+\tilde{\varepsilon}_{ij} ~~~~\textrm{in } \Omega.\]
After proper calculations, it can be proven that (Qu and Cherkaoui, 2006)
\[{\varepsilon}_{ij}=\textrm{constant}={T}_{ijkl}{\varepsilon}^0_{kl}~~~~\textrm{in } \Omega,\]
where \({T}_{ijkl}\) is called strain interaction tensor and is given by
\[{T}_{ijkl}=\left(\mathcal{I}_{ijkl}+\mathcal{S}_{ijpq}({L}^0_{pqmn})^{-1}({L}^1_{mnkl}-{L}^0_{mnkl})\right)^{-1},\]
and \(\mathcal{I}_{ijkl}\) denotes the fourth order symmetric identity tensor.

2. Mori Tanaka

The presentation of the Mori-Tanaka approach in this section is based on the discussion of Qu and Cherkaoui (2006).

Consider a composite material, composed by a matrix material (index 0) and \(N\) inhomogeneities of ellipsoidal shape. The matrix is occupying the space \(S_0\) and has volume \(V_0\), while the \(r_{\textrm{th}}\) inhomogeneity occupies the space \(S_r\) and has volume \(V_r\). Obviously
\[S=S_0\bigcup S_1\bigcup…\bigcup S_r\bigcup…\bigcup S_N ~~~\textrm{and}~~~ V=V_0+V_1+…+V_r+…+V_N.\]
For a typical inhomogeneity with elastic stiffness tensor \(L^r_{ijkl}\) (\(r=1,2,…,N\)), the effects of the other inhomogeneities are communicated to it through the strain and stress fields in its surrounding matrix material with elastic stiffness tensor \(L^0_{ijkl}\). Although the strain and stress fields vary from one location to another inside the matrix, the average strains, \({\varepsilon}^{0}_{ij}=\displaystyle{\frac{1}{V_0}\int_{S_0}\varepsilon_{ij}\textrm{d}V}\) , and stresses, \({\sigma}^{0}_{ij}=\displaystyle{\frac{1}{V_0}\int_{S_0}\sigma_{ij}\textrm{d}V}\), in the matrix represent good approximations of the actual fields of the matrix surrounding each inhomogeneity, when a large number of inhomogeneities exist and are randomly distributed in the matrix. Also, it would be reasonable to assume that the absence of only one inhomogeneity will not affect the overall elastic behavior of the composite. In other words, when the \(r_{\textrm{th}}\) inhomogeneity is removed and replaced by the matrix material, the averages \({\varepsilon}^{0}_{ij}\) and \({\sigma}^{0}_{ij}\) will not change. Therefore, as far as the \(r_{\textrm{th}}\) inhomogeneity is concerned, it can be viewed as an ellipsoidal inhomogeneity with stiffness tensor \(L^r_{ijkl}\), placed within a uniform matrix of stiffness tensor \(L^0_{ijkl}\), which had been subjected to the uniform strain \({\varepsilon}^{0}_{ij}\). Under this assumption and after proper calculations, the strain concentration tensor of the \(r_{\textrm{th}}\) inhomogeneity is given by
\[A^r_{ijkl}=T^r_{ijmn}\left(c^0\mathcal{I}_{mnkl}+\sum_{q=0}^N c^qT^q_{mnkl}\right)^{-1},\]
where
\[{T}^r_{ijkl}=\left(\mathcal{I}_{ijkl}+\mathcal{S}^r_{ijpq}({L}^0_{pqmn})^{-1}({L}^r_{mnkl}-{L}^0_{mnkl})\right)^{-1},~~~
c^r=\frac{V_r}{V},\]
and \(\mathcal{S}_r\) is the Eshelby tensor of the \(r_{\textrm{th}}\) inhomogeneity embedded in the matrix material. For the matrix it holds \({T}^0_{ijkl}=\mathcal{I}_{ijkl}\).
The connection between the strain interaction tensor \({T}^r_{ijkl}\) and the stress interaction tensor \({H}^r_{ijkl}\) is given by the relation
\[{H}^r_{ijkl}=\mathcal{L}^r_{ijpq}{T}^r_{pqmn}({L}^0_{mnkl})^{-1}.\]
For the matrix it holds \({H}^0_{ijkl}=\mathcal{I}_{ijkl}\). The stress concentration tensor of the \(r_{\textrm{th}}\) inhomogeneity can be expressed as
\[B^r_{ijkl}=H^r_{ijmn}\left(c^0\mathcal{I}_{mnkl}+\sum_{q=0}^N c^qH^q_{mnkl}\right)^{-1}.\]

3. Self consistent

The presentation of the self consistent approach in this section is based on the discussion of Qu and Cherkaoui (2006).

Consider a composite material, composed by a matrix material (index 0) and \(N\) inhomogeneities of ellipsoidal shape. The matrix is occupying the space \(S_0\) and has volume \(V_0\), while the \(r_{\textrm{th}}\) inhomogeneity occupies the space \(S_r\) and has volume \(V_r\). Obviously
\[S=S_0\bigcup S_1\bigcup…\bigcup S_r\bigcup…\bigcup S_N ~~~\textrm{and}~~~ V=V_0+V_1+…+V_r+…+V_N.\]
In the self consistent method it is assumed that the macroscopic stiffness tensor \(\bar{L}_{ijkl}\) is already known. If there are numerous inhomogeneities in the composite, the macroscopic properties are not affected by the absence of one inhomogeneity. When the composite is subjected to either displacement or traction boundary conditions, one may envision that the effects of the applied loads and the interaction between the inhomogeneities can be accounted for by assuming that the \(r_{\textrm{th}}\) inhomogeneity is placed within a homogeneous matrix of elastic stiffness \(\bar{L}_{ijkl}\) that had been subjected to the strain tensor \(\bar{\varepsilon}_{ij}\). Under this assumption and after proper calculations, the strain concentration tensor of the \(r_{\textrm{th}}\) inhomogeneity is given by
\[A^r_{ijkl}=\left(\mathcal{I}_{ijkl}+\bar{\mathcal{S}}^r_{ijpq}(\bar{L}_{pqmn})^{-1}({L}^r_{mnkl}-\bar{L}_{mnkl})\right)^{-1},~~~
c^r=\frac{V_r}{V},\]
where \(\bar{\mathcal{S}}_r\) is the Eshelby tensor of the \(r_{\textrm{th}}\) inhomogeneity embedded in the homogenized material. The connection between the strain concentration tensor \({A}^r_{ijkl}\) and the stress concentration tensor \({B}^r_{ijkl}\) is given by the relation
\[B^r_{ijkl}=L^r_{ijpq}A^r_{pqmn}(\bar{L}_{mnkl})^{-1}.\]
The self consistent approach yields implicit equations for the macroscopic properties, thus iterative computational scheme is required.

4. Periodic homogenization

Consider the mechanical constitutive law
\[\sigma_{ij}=L_{ijkl}\varepsilon_{kl},\]
where \(\sigma_{ij}\) is statically admissible stress, \(\varepsilon_{kl}\) is kinematically admissible strain and \(L_{ijkl}\) is periodic in the RVE. In the periodic homogenization the displacements in the microscopic level are considered to be expressed as
\[u_i=\bar{\varepsilon}_{ij}x_j+z_i,\]
with \(z_i\) being a periodic vector. By introducing the space of the microscopic periodic test functions
\[{\mathcal{V}}=\left\{\eta_i: ~~\eta_i\,,\frac{\partial \eta_i}{\partial x_j}\in\mathcal{L}^2(S)\,, ~~\eta_i ~\textrm{periodic on}~ \partial S\right\},\]
the weak form of the microscale equilibrium equation can be written as
\[\begin{aligned}
\frac{1}{V}\int_S\frac{\partial \eta_i}{\partial x_j}\sigma_{ij}\textrm{d}V=0,
&& \forall\eta_i\in{\mathcal{V}}.\end{aligned}\]
The last expression can be written as
\[\frac{1}{V}\int_S\frac{\partial \eta_i}{\partial x_j}\sigma_{ij}\textrm{d}V=
\frac{1}{V}\int_S\frac{\partial \eta_i}{\partial x_j}L_{ijkl}\varepsilon_{kl}\textrm{d}V=
\frac{1}{V}\int_S\frac{\partial \eta_i}{\partial x_j}L_{ijkl}\left(\varepsilon_{kl}+\frac{\partial z_i}{\partial x_j}\right)\textrm{d}V=0.\]
Assuming known macroscopic strain, the solution of the above equation is written as
\[z_m=\bar{\varepsilon}_{kl}U_{klm},\]
where the third order tensor \(U_{klm}\) is given by
\[\frac{1}{V}\int_S\frac{\partial \eta_i}{\partial x_j}\left(L_{ijkl}+L_{ijmn}\frac{\partial U_{klm}}{\partial x_n}\right)\textrm{d}V=0,\]
or, in local form,
\[\frac{\partial}{\partial x_j}\left(L_{ijkl}+L_{ijmn}\frac{\partial U_{klm}}{\partial x_n}\right)=0.\]
The last equation represents the microscale problem. The macroscopic stress is thus given by
\[\begin{array}{rl}\bar{\sigma}_{ij}=&\displaystyle{\frac{1}{V}\int_S{\sigma_{ij}}\textrm{d}V=\frac{1}{V}\int_S{L_{ijmn}\varepsilon_{mn}}\textrm{d}V
=\frac{1}{V}\int_S{L_{ijmn}\left(\bar{\varepsilon}_{mn}+\frac{\partial z_m}{\partial x_n}\right)}\textrm{d}V} \\
=&\displaystyle{\frac{1}{V}\int_S{\left(L_{ijkl}+L_{ijmn}\frac{\partial U_{klm}}{\partial x_n}\right)}\textrm{d}V~\bar{\varepsilon}_{kl}}.\end{array}\]
The last expression indicates that we can identify the macroscopic elasticity tensor \(\bar{L}_{ijkl}\) as
\[\bar{L}_{ijkl}=\frac{1}{V}\int_S{L_{ijmn}\left(\mathcal{I}_{mnkl}+\frac{\partial U_{klm}}{\partial x_n}\right)}\textrm{d}V.\]
Finally, the strain concentration tensor is given by
\[{A}_{ijkl}=\mathcal{I}_{ijkl}+\frac{1}{2}\left(\frac{\partial U_{kli}}{\partial x_j}+\frac{\partial U_{klj}}{\partial x_i}\right).\]

4.1 Composite laminates

Consider a composite laminate, consisting of \(N\) different layers that are repeated in a periodic fashion along the \(x_1\) axis. In the RVE of the composite, each layer has volume fraction \(c^{(k)}=\displaystyle{\frac{V_k}{V}}\) and elastic stiffness tensor \({L}_{ijkl}^{(k)}\) (\(k=1,2,…,N\)). It obviously holds \(\displaystyle \sum_{r=1}^Nc^{(r)}=1\). For the computational details of this subsection the interested reader is referred to Chatzigeorgiou et al (2015).

In the RVE of the composite laminate the periodic part of the displacement, \({z}_i\), is uniform along the \(x_2\) and \(x_3\) axes and presents non-uniformity only on the \(x_1\) axis. Introducing for each layer \(k\) the 3\(\times\)3 matrices
\[\begin{aligned}
\boldsymbol{L}^{(k)}_{nn}=&\left(\begin{array}{ccc}
L^{(k)}_{1111} & L^{(k)}_{1121} & L^{(k)}_{1131} \\
L^{(k)}_{2111} & L^{(k)}_{2121} & L^{(k)}_{2131} \\
L^{(k)}_{3111} & L^{(k)}_{3121} & L^{(k)}_{3131} \end{array}\right),
&\boldsymbol{L}^{(k)}_{tn}=&\left(\begin{array}{ccc}
L^{(k)}_{2211} & L^{(k)}_{2221} & L^{(k)}_{2231} \\
L^{(k)}_{3311} & L^{(k)}_{3321} & L^{(k)}_{3331} \\
L^{(k)}_{2311} & L^{(k)}_{2321} & L^{(k)}_{2331} \end{array}\right),
\\ \\
\boldsymbol{L}^{(k)}_{nt}=&\left(\begin{array}{ccc}
L^{(k)}_{1122} & L^{(k)}_{1133} & L^{(k)}_{1123} \\
L^{(k)}_{2122} & L^{(k)}_{2133} & L^{(k)}_{2123} \\
L^{(k)}_{3122} & L^{(k)}_{3133} & L^{(k)}_{3123} \end{array}\right),
&\boldsymbol{L}^{(k)}_{tt}=&\left(\begin{array}{ccc}
L^{(k)}_{2222} & L^{(k)}_{2233} & L^{(k)}_{2223} \\
L^{(k)}_{3322} & L^{(k)}_{3333} & L^{(k)}_{3323} \\
L^{(k)}_{2322} & L^{(k)}_{2333} & L^{(k)}_{2323} \end{array}\right),
\\ \\
\boldsymbol{U}_n^{(k)}=&\left(\begin{array}{ccc}
U_{111}^{(k)} & U_{211}^{(k)} & U_{311}^{(k)} \\
U_{112}^{(k)} & U_{212}^{(k)} & U_{312}^{(k)} \\
U_{113}^{(k)} & U_{213}^{(k)} & U_{313}^{(k)} \end{array}\right),
&\boldsymbol{U}_t^{(k)}=&\left(\begin{array}{ccc}
U_{221}^{(k)} & U_{331}^{(k)} & U_{231}^{(k)} \\
U_{222}^{(k)} & U_{332}^{(k)} & U_{232}^{(k)} \\
U_{223}^{(k)} & U_{333}^{(k)} & U_{233}^{(k)} \end{array}\right),
\end{aligned}\]
the microscale problem in the local form can be written as a system of ordinary differential equations,
\[\begin{aligned}\frac{\textrm{d}}{\textrm{d}x_1}\left(\boldsymbol{L}_{nn}^{(k)}\frac{\textrm{d}\boldsymbol{U}_n^{(k)}}{\textrm{d}x_1}+
\boldsymbol{L}_{nn}^{(k)}\right)=\boldsymbol{0}\,, &&
\frac{\textrm{d}}{\textrm{d}x_1}\left(\boldsymbol{L}_{nn}^{(k)}\frac{\textrm{d}\boldsymbol{U}_t^{(k)}}{\textrm{d}x_1}+
\boldsymbol{L}_{nt}^{(k)}\right)=\boldsymbol{0}.\end{aligned}\]
The solution of the above linear system is
\[\frac{\textrm{d}\boldsymbol{U}_n^{(k)}}{\textrm{d}x_1}=\left(\boldsymbol{L}_{nn}^{(k)}\right)^{-1}
\left(\boldsymbol{m}_{n}-\boldsymbol{L}_{nn}^{(k)}\right), ~~~~
\frac{\textrm{d}\boldsymbol{U}_t^{(k)}}{\textrm{d}x_1}=\left(\boldsymbol{L}_{nn}^{(k)}\right)^{-1}
\left(\boldsymbol{m}_{t}-\boldsymbol{L}_{nt}^{(k)}\right),\]
where
\[\boldsymbol{m}_{n}=\left(\sum_{r=1}^{N}c^{(r)}\left(\boldsymbol{L}_{nn}^{(r)}\right)^{-1}\right)^{-1},
~~~~\boldsymbol{m}_{t}=\boldsymbol{m}_{n}\left(\sum_{r=1}^{N}c^{(r)}\left(\boldsymbol{L}_{nn}^{(r)}\right)^{-1}\boldsymbol{L}_{nt}^{(r)}\right).\]
Finally, the macroscopic elastic stiffness tensor is expressed as
\[\begin{array}{rll}
\bar{\boldsymbol{L}}_{nn}=&
\left(\begin{array}{ccc}
\bar{L}_{1111} & \bar{L}_{1121} & \bar{L}_{1131} \\
\bar{L}_{2111} & \bar{L}_{2121} & \bar{L}_{2131} \\
\bar{L}_{3111} & \bar{L}_{3121} & \bar{L}_{3131} \end{array}\right)=&
\displaystyle{\left(\sum_{r=1}^{N}c^{(r)}\left(\boldsymbol{L}_{nn}^{(r)}\right)^{-1}\right)^{-1}}, \\ \\
\bar{\boldsymbol{L}}_{nt}=&
\left(\begin{array}{ccc}
\bar{L}_{1122} & \bar{L}_{1133} & \bar{L}_{1123} \\
\bar{L}_{2122} & \bar{L}_{2133} & \bar{L}_{2123} \\
\bar{L}_{3122} & \bar{L}_{3133} & \bar{L}_{3123} \end{array}\right)=&
\displaystyle{\bar{\boldsymbol{L}}_{nn}\left(\sum_{r=1}^{N}c^{(r)}\left(\boldsymbol{L}_{nn}^{(r)}\right)^{-1}\boldsymbol{L}_{nt}^{(r)}\right)}, \\ \\
\bar{\boldsymbol{L}}_{tn}=&\left(\begin{array}{ccc}
\bar{L}_{2211} & \bar{L}_{2221} & \bar{L}_{2231} \\
\bar{L}_{3311} & \bar{L}_{3321} & \bar{L}_{3331} \\
\bar{L}_{2311} & \bar{L}_{2321} & \bar{L}_{2331} \end{array}\right)=&
\left(\bar{\boldsymbol{L}}_{nt}\right)^T, \\ \\
\bar{\boldsymbol{L}}_{tt}=&\left(\begin{array}{ccc}
\bar{L}_{2222} & \bar{L}_{2233} & \bar{L}_{2223} \\
\bar{L}_{3322} & \bar{L}_{3333} & \bar{L}_{3323} \\
\bar{L}_{2322} & \bar{L}_{2333} & \bar{L}_{2323} \end{array}\right)=&
\displaystyle{\sum_{r=1}^{N}c^{(r)}\left(\boldsymbol{L}^{(r)}_{tn}\left(\boldsymbol{L}_{nn}^{(r)}\right)^{-1}
\left(\bar{\boldsymbol{L}}^{}_{nt}-\boldsymbol{L}^{(r)}_{nt}\right)+\boldsymbol{L}^{(r)}_{tt}\right)}.\end{array}\]

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