On the equivalent shear modulus of composite metamaterials

Here, we put the equivalent shear modulus of metamaterials under scrutiny. A micromorphic model for the equivalent shear modulus of metamaterials is developed. This model represents a metamaterial as a multiscale material that exhibits a microstructural strain (micro-strain) field, which is independent of the macroscopic strain (macro-strain) field of the material. The mechanisms by which the shear modulus of a metamaterial would change and depend on the material size are discussed. It is revealed that the shear modulus of a metamaterial is size and microstructure topology-dependent as long as the micro-strain field is significant and different from the macro-strain field. The conditions at which the shear modulus would be zero, negative, or positive are also defined. A unique material parameter (beta) that depends on the microstructure topology is introduced to define the conditions of the material stability. It is revealed that a metamaterial with a negative shear modulus is stable as long as beta(2) < 0. The proposed micromorphic model is also employed to determine the equivalent shear modulus of composite metamaterials of the form of coated inclusions embedded in a matrix material. It is demonstrated that a composite metamaterial with a giant shear modulus can be produced by employing either one phase of a negative modulus or an interfacial layer between the inclusion and the matrix. In addition, the implementation of an interface between the inclusion and the matrix would give a composite metamaterial with a negative shear modulus.