Estimations for Nonsymmetric Effective Coefficients

During the past several years, several papers have been devoted to the study of composite periodic materials. In a simpler manner, the thermic conductivity of a composite material is considered here. If the periodicity directions do not coincide with the coordinate axes, the elastic matrix of components is not symmetric. In this case, estimations are obtained for the effective coefficients that do not contain the inverse of an elastic and effective matrix. Also, nonsymmetry may be related to the position of the fiber (inclusion), to which it is nonsymmetric with respect to the symmetry axes of the periodicity cell (generally a parallelepipedon). In this case, a configuration is given for which the effective coefficients a12= a21may be negative or positive, even if the initial corresponding coefficients are equal to zero. This fact agrees with the numerical results given at the beginning of §] 2.