Equivalence of Inclusion and Inhomogenity

Let there be a homogeneous ellipsoidal inclusion filling the domain A with the electroelastic properties C embedded into an infinite homogeneous matrix with the electroelastic properties C . Such a composite filling the domain A will be subjected to uniform boundary conditions 

The overall strains and electric field strengths Z of the composite can be either interpreted as a response to the applied stresses and electric flux densities or are the direct implication of the boundary conditions. In absence of the inclusion, they would prevail throughout the domain A. The resulting strains and electric field strengths Z inside the inclusion may be assembled 

the stresses and electric flux densities inside the inclusion can be determined from the corresponding constitutive relation of Eq. (5.3) as 

The constraint matrix T depicts the effect of the constraining matrix on the inclusion and is a function of matrix material properties and ellipsoidal inclusion shape. It represents the piezoelectric analog to Eshelby s tensor in the elastic case, see Dunn and Taya [66]. Expressions for cylindrical inclusions to model fibrous composites are provided by Dunn and Taya [67] (this reference uses a different notation, T is called S). Equating Eqs. (5.12) and (5.13) and making use of Eq. (5.14) to replace the eigenfields Z and Eq. (5.11) to eliminate the perturbation fields Z after some manipulations, leads to 

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To accomplish their specihc task, most of composites require a portion of the inclusion phase that certainly cannot be described as dilute. Therefore, methods based on the equivalence of inclusion and inhomogenity derived above had to be developed to extend the range of applications to practicable volume fractions. 

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