Mean Fields and Concentration Matrices

Mean stresses and electric flux densities Y as well as mean strains and electric field strengths Z are composed of the corresponding mean fields in the inclusion and matrix phase as indicated by the superscripts i and m. Crossconnecting these fields, the constitutive relations of the homogenized composite, as well as of the individual material phases, may be given as follows  

The unique dependence of strains and electric field strengths as well as stresses and electric flux densities in the individual material phases upon the overall fields of the composite may be formulated with the aid of the concentration 

Starting off with Eq. (5.2b) and subsequent substitution of stresses and electric flux densities Y, y, and y from the constitutive relations of Eq. (5.3), of strains and electric field strengths Z from Eq. (5.2a) and Z from Eq. (5.4), the effective properties may be written as 

the effective properties C of a composite are described in Eqs. (5.6) and (5.7) by the known fiber volrnne fraction i/ and constituent properties C and C , as well as by the concentration matrices S and S of the inclusion phase to be determined by the succeeding theories. 

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