Bounds on the overall properties of composites with debonded frictionless interfaces

In the present work the homogenization problem of periodic composites with nonlinear hyperelastic constituents and debonded frictionless interfaces is formulated as a minimum problem of the free- or complementary-energy of the unit cell of the composite over the convex set of the admissible strain or stress fields, respectively. Because of the unilateral contact condition at the debonded interfaces, the overall constitutive behavior of the composite turns out to be nonlinear even when the constituents are linearly hyperelastic. In this particular case, the homogenized free- and complementary-energy density functionals of the composite are positively homogeneous of degree two, i.e. from a mechanical point of view, the overall constitutive behavior of the composite is nonlinear conewise multimodular. Approximations of the proposed variational principles, suitable for numerical computations, are built up by adopting the finite-element method. In this way, rigorous upper bounds on the homogenized free- and complementary-energy density functionals are obtained. Then, lower bounds are derived by using the conjugacy relation between the homogenized energy density functionals.