Variational principles and bounds for elastic inhomogeneous materials with coherent imperfect interfaces

Some composites and nanomaterials can be appropriately considered as multiphase materials in which the interfaces are imperfect and described by the interface stress model. The minimum potential and complementary energy principles of linear elasticity are extended to such inhomogeneous materials and to bracketing their effective elastic properties. By constructing simple trial strain and stress fields, the explicit first-order upper and lower bounds are derived for the effective elastic moduli of multiphase materials consisting of spherical or cylindrical inhomogeneities embedded in a matrix. Numerical results are provided to illustrate the dependence of the upper and lower bounds on the interface-to-volume ratio.