Resonant modes of two hard inclusions within a soft elastic material and their stress estimate

In this paper, we are concerned with subwavelength resonant modes of two hard inclusions embedding in soft elastic materials to realize negative materials in elasticity. All the $12$ subwavelength resonant frequencies are derived explicitly for general convex resonators. In addition, the resonant modes are categorized into dipolar, quadrupolar, and hybrid groups, facilitating the effective realization of negative mass density, negative shear modulus and double-negative properties (both mass density and shear modulus) in elastic metamaterials. Moreover, we analyze the stress distribution between two hard inclusions when they are closely touching. We also precisely derive the sharp blow-up rates of the gradient estimates of the resonant modes. Our findings show that certain resonant modes have bounded stress estimates when the curvature of the hard inclusions is appropriately designed. These results provide valuable insights into the stability of the design and fabrication of elastic metamaterials. Lastly, we express the scattering wave fields explicitly in terms of resonant modes, offering a clear understanding of their impact on the overall scattering behavior.