Uniform convergence for linear elastostatic systems with periodic high contrast inclusions

We consider the Lamé system of linear elasticity with periodically distributed inclusions whose elastic parameters have high contrast compared to the background media. We develop a unified method based on layer potential techniques to quantify three convergence results when some parameters of the elastic inclusions are sent to extreme values. More precisely, we study the incompressible inclusions limit where the bulk modulus of the inclusions tends to infinity, the soft inclusions limit where both the bulk modulus and the shear modulus tend to zero, and the hard inclusions limit where the shear modulus tends to infinity. Our method yields convergence rates that are independent of the periodicity of the inclusions array, and are sharper than some earlier results of this type. A key ingredient of the proof is the establishment of uniform spectra gaps for the elastic Neumann-Poincaré operator associated to the collection of periodic inclusions that are independent of the periodicity.