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Cheese Specific. Ice Cream Specific. Product Development Centres. Packaging Package Equipment. Filling Machines. By using the notation introduced for a piezoelectric material Jung et al. As shown in Equation 21 , P JiMn contains T , which depends on the position x and therefore does not have mathematical similarity with other physics in general. Due to the mathematical analogy, the Eshelby tensor can be expressed in a similar form.

In the most general case of a medium with arbitrary anisotropy and an ellipsoidal inclusion with three different semi-axes a 1 , a 2 , and a 3 , by simplifying Equation 3 , the elastic Eshelby tensor can be expressed as Mura ,. Similarly, by adding three more degrees of freedom from electric polarization and its coupling with lattice distortion, the pizeoelectric Eshelby tensor is derived as follows Dunn and Taya, ,. Because the piezoelectric coefficients are zero for all centrosymmetric crystals, the piezoelectric Eshelby tensor for an isotropic medium does not exist.

For the conduction or dielectric phenomena described by Equation 11 and physical property tensor K in Table 1 , the Eshelby tensor is obtained from.

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Unlike the Green's function in elasticity, the Green's function in the conduction or dielectric phenomena for a medium with arbitrary symmetry has been derived in a closed form as follows:. The second-order Eshelby tensor can be further simplified for an arbitrary ellipsoidal inclusion as an integral having one variable Giordano and Palla, ; Lee et al. For a thermoelectric material, with the coupling of electrical and heat conduction, the Eshelby tensor can be written as follows Jung et al. We now turn our attention to predict the effective properties of composites based on a mathematically analogous formula.

For example, if we use the Voigt notation for a piezoelectric case, the linear operator matrix which corresponds to the material properties and the Eshelby matrix can be expressed as Equations 29, 30 , respectively:.

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As shown by Equations 29, 30 , the coefficients in the transformed matrix are different for the two set of matrices. In contrast, with the Mandel notation, the two matrices can be expressed as. We then compare the effective property prediction based on the Mori-Tanaka method with a FEA analysis for the simple case involving spherical inhomogeneities embedded in a transversely isotropic medium for piezoelectric and isotropic media for others, as shown in Figure 3.

Figure 3. A Normalized Young's modulus of a particle-reinforced composite. B Effective heat conductivity of the composite. C Normalized effective piezoelectric properties of composites. All properties are normalized with respect to the properties of the matrix. We use the material properties for the piezoelectricity Odegard, and the thermoelectricity Jung et al. The Eshelby tensor for anisotropic medium in various physical problems has been extensively studied in the literature Mura, ; Yu et al.

Although the Eshelby tensor can be obtained numerically, significant efforts have been devoted to derive the explicit expressions either closed-form or analytical expression for the facile application of the homogenization method and to provide better insight into the nature of the tensor. In elasticity, analytic expressions for ellipsoidal shape given in terms of a few integrals are available for spheroidal inclusion in transversely isotropic solids involving five independent elastic constants or any medium with higher symmetry, i. For piezoelectricity, Huang and Kuo suggested the Eshelby tensor expression of spheroidal inclusion in transversely isotropic material with a few integrals.

However, for thermoelectricity, to the best of our knowledge, analytic solutions have not been obtained for an anisotropic medium. In actual composites, the interface between the matrix and an inclusion often has imperfections originating from the manufacturing process or from an inherent lattice mismatch People and Bean, ; Habas et al. An interfacial imperfection in elasticity refers to debonding or slippage, i.

An interfacial imperfection in electrical conduction i. The interfacial imperfection in piezoelectricity considers both displacement jump and the electric potential jump Wang et al. Similarly, the interfacial imperfection in thermoelectricity considers the abrupt discontinuities of both temperature and electric potential Jung et al. Out of a few characteristic methods used to describe an interfacial imperfection in elasticity Qiu and Weng, ; Duan et al. As will be shown later, due to the mathematical similarity, the expressions of the effective stiffness and effective conductivity in the presence of an interfacial imperfection are nearly identical to each other.

It is noteworthy that the interfacial imperfection can be well-treated only for a spherical inclusion, because an ellipsoidal inclusion even with slight deviation from a sphere introduces significantly non-uniform interior field Qu, ; Qu and Cherkaoui, ; Lee et al. In problems involving the elastic deformation, the equality between the normal and tangential interfacial compliances is additionally required to ensure the uniform field inside the inclusion which is critical for the applicability of the Eshelby tensor and the mean field homogenization.

Unfortunately, there exist several studies employing the Mori-Tanaka method to obtain effective physical properties of a composite involving ellipsoidal inhomogeneities in the presence of different normal and tangential compliances Yang et al. However, in addition to the Eshelby tensor, different expressions for X Eff and T should be used because the derivation in Section Single inhomogeneity problem and Mean-field homogenization of the effective stiffness where the problem is solved with the superposition of two sub-problems involving applied and constrained fields, respectively does not account for the additional contribution from the interface imperfection.

We will discuss these issues in detail in the next section.


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We adopt the interface spring model to consider the interfacial damage, as depicted in Figure 4. A displacement jump occurs at the interface due to the spring layer having a vanishing thickness between the infinite matrix and the single inclusion. Figure 4. Schematic of an interface spring model in elasticity. To visualize the interface spring in two directions, we compare undeformed and deformed states.

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The constitutive equations and traction t i equilibrium equation at the interface are expressed as follows,. Formulating the Eshelby inclusion problem by adopting the interfacial condition in Equation 34 , the constrained strain is written as Othmani et al. Equation 35 is shown to reproduce the perfect interface case with zero spring compliance; i. Hence, for this special case, the integral equation can be decomposed as follows,.

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The constrained strain field then can be expressed by a tensor algebraic equation as follows Othmani et al. Therefore, it is proven that the modified Eshelby The tensor S M can be written as follows:. The numerical validation against a finite element analysis on a triclinic single crystal material NaAlSi 3 O 8 with 21 independent elastic constants and 36 independent Eshelby tensor components is presented in Figure 5. This leads to the simplified expression of the modified T tensor T M in the Mori-Tanaka method when the matrix modulus is L 0 and the inhomogeneity modulus is L 1 :.

Figure 5. Thirty-six independent components of a modified Eshelby tensor for a wide range of interfacial damage. We derive the effective modulus L Eff of the composite as follows Lee et al. To confirm the validity of Equation 42 , we compare the theoretical prediction with the effective Young's modulus of a composite for a wide range of interface damage as obtained by a FEA See Figure 6. The equation can also be formulated in terms of the effective inclusion, and we concisely summarize the formulation in Appendix B.

A detailed discussion of this issue can be found in work by Dogri's group Othmani et al. Figure 6. The Poisson's ratio of the two phases is 0. We now turn our attention to other physical phenomena where interfacial imperfections play an important role, specifically the conduction problem. The interfacial resistance augments an additional surface integration term in the eigen-intensity problem which corresponds to the eigenstrain problem in elasticity , as follows Quang et al. Figure 7.

A Schematic of the Kapitza resistance at the interface, and B temperature jump across the interface. It has been found that the heat flux within a spherical inclusion is uniform in the presence of interfacial thermal resistance. In an earlier work of ours, we proved that the modified Eshelby tensor for a matrix with arbitrary anisotropy can be written as follows Lee et al. Here, I is the second-order identity tensor and S is the Eshelby tensor for the thermal conduction problem. We can also obtain the modified localization tensor subjected to the constant flux boundary condition q 0 as.

Figure 8. A Normalized heat flux within an inhomogeneity with respect to the interfacial thermal resistance in a single inhomogeneity problem. Figure 9. The radius of the particle used in A—C is 1 mm. For a thermoelectric composite, we refer to our work on micromechanics homogenization considering both the interfacial thermal and electrical resistance Jung et al. Our homogenization study provides new insight on the effect of interfacial thermal and electrical resistance on the thermoelectric properties. In piezoelectricity, the modified Eshelby tensor has been derived by violating the Fubini's theorem by changing integral order Wang et al.

Hence, the effective piezoelectric property has not been obtained correctly in the presence of interfacial imperfections considering both displacement jump and electric potential jump. With the adaptation of Mandel notation and the careful mathematical derivation, the expression for the effective piezoelectric property is expected to be similar to that of the effective thermoelectric property Jung et al.

We close the section by providing numerically obtained interior field inside ellipsoidal reinforcement Figure 10 , which clearly demonstrate the limited applicability of the homogenization method in the presence of interfacial imperfection. Figure In this paper, we introduce a series of recent studies which attempted to obtain the effective physical properties of composites by taking into account the interfacial damage and anisotropy of the matrix, providing a universal formulation for different physical phenomena based on their mathematical analogies.

First, with the specific example of elasticity, we introduced the concept of the Eshelby tensor and discussed a how single-inclusion problem can be related to a single inhomogeneity problem, which ultimately is applied to mean-field homogenization to predict the effective properties of composites. Second, based on a mathematical analogy, we show that mean-field homogenization equations for different physical phenomena can be represented by linear equations involving symmetric matrices with different dimensions by adapting Mandel's notation.

Third, we extend our discussion by taking into account two common issues in realistic applications: imperfections in a matrix-inhomogeneity interface and anisotropy of the matrix.


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  • Although we have discussed the homogenization method while covering a wide range of issues, there remain a few issues requiring further developments. We categorize the advanced issues of homogenization theory into the six problems of i various physical phenomena, ii anisotropy of the matrix, iii matrix-filler interfacial imperfections, iv nonlinear responses, v time-dependent responses, and vi high volume fractions of reinforcements.

    Although issues i - iii have been investigated extensively, we nonetheless cannot take into account important non-spherical reinforcement materials including ellipsoidal types such as carbon nanotubes and graphene in the presence of interfacial imperfections due to the non-uniform interior field. There are a few studies of composites showing highly nonlinear such as hyperelastic and time-dependent such as viscoelastic responses Friebel et al. Studies to overcome aforementioned challenges are underway in our research group with the formation of an international collaborative research network.

    SR designed the structure of the review paper. SR and SL wrote the manuscript. SR and SL. SR, SL, and JJ contributed to the summary of theoretical derivations, prepared figures, and discussed the mathematical analogy among different physical problems. JL contributed the theoretical derivations. YK contributed the effective inclusion method section. The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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