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We consider the Lame system of linear elasticity with periodicallydistributed inclusions whose elastic parameters have high contrast compared tothe background media. We develop a unified method based on layer potentialtechniques to quantify three convergence results when some parameters of theelastic inclusions are sent to extreme values. More precisely, we study theincompressible inclusions limit where the bulk modulus of the inclusions tendsto infinity, the soft inclusions limit where both the bulk modulus and theshear modulus tend to zero, and the hard inclusions limit where the shearmodulus tends to infinity. Our method yields convergence rates that areindependent of the periodicity of the inclusions array, and are sharper thansome earlier results of this type. A key ingredient of the proof is theestablishment of uniform spectra gaps for the elastic Neumann-Poincare operatorassociated to the collection of periodic inclusions that are independent of theperiodicity.

Uniform convergence for linear elastostatic systems with periodic high contrast inclusions