Reduced micromorphic model in orthogonal curvilinear coordinates and its application to a metamaterial hemisphere

The reduced micromorphic model (RMM) is used to study the effect of the applied force on a hemisphere made of phononic crystals that belongs to the metamaterials group. The strain tensor, the micro-strain tensor and the coupling between them are the kinematic relations used to measure the deformation and micro-deformation of the representative volume element of these materials. The free energy function, the constitutive relations, the field equations, and the boundary conditions are presented firstly in the Cartesian coordinate. Then, the orthogonal curvilinear coordinates are introduced as a general coordinate to describe the physical quantities included in the RMM. The spherical coordinates are deduced as a special case from the curvilinear coordinates to study the deformation and micro-deformation for the hemisphere. The kinematic relations and the governing equations of the model are considered to changing with the radius of the hemisphere only. The analytical solutions of the field equations are also obtained by using the Frobenius series satisfying the given boundary conditions and consequently the value of the physical constants of the problem is determined. Numerical applications for the obtained results are introduced with discussion. The results showed that the displacement has a greater effect rather than the micro-strain, when it is measured relative to the classical physical quantities while the micro-strain has a greater effect rather than the displacement, when it is measured relative to the nanoscale physical quantities.