One main feature of metametarials is the occurrence of a negative dynamic mass density that is produced when an inner local resonance is present. The inner resonance can be obtained in composite materials containing composite inclusions. For suitable ratios of the physical properties of the constituting materials, the composite inclusions act as spring-mass systems. The scaling of physical properties leading to such an inner resonance and the associated effective dynamic properties of materials containing composite inclusions are recalled briefly. The resonance frequencies and dynamic mass densities are obtained in a closed form for materials containing cylindrical composite fibers or spherical composite inclusions, after solving the related boundary value elasticity problems.