On bounds of the effective ff ective behavior of particulate composites with imperfect interface

Particulate composites are considered here as multiphase composite in which the interfaces are imperfect. When the interface mechanical properties are those of a linear elastic material, the minimum of potential and complementary energy is used in order to obtain bounds of effective elastic modulus of the composite. Test displacements or stress fields are build and characterized using Green's functions of a comparison homogeneous body, polarization fields and extension of the classical Lippmann-Schwinger equations. Then when spatial distribution of phases are known, in particular for isotropic distribution of phases or patterns, a generalization of Hashin-Shtrikman principle is obtained and lower and upper bounds are proposed.