Summary
We consider an inhomogeneity-matrix system from a particular class of compressible hyperelastic materials of harmonic-type undergoing finite plane deformations. We discuss the idea that by adjusting the remote (Piola) stress we can design the shape of the inhomogeneity in such a way that the interior (Piola) stress distribution remains uniform. In fact, using complex variable techniques, we show that a uniform (Piola) stress distribution can be achieved inside the inhomogeneity when the system is subjected to linear remote loading, in which case, the inhomogeneity is necessarily hypotrochoidal with three cusps. We obtain the complete solution of the corresponding inhomogeneity-matrix system in this case. In addition, we consider the more general case when the system is subjected to nonuniform remote loading characterized by stress functions in the form of nth degree polynomials in the complex variable z describing the matrix. In this case, by finding the complete solution of the system, we show that we can again obtain uniform stress inside a hypotrochoidal inhomogeneity with n + 1 cusps. Finally, we illustrate our results with an example.
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Kim, C.I., Schiavone, P. Designing an inhomogeneity with uniform interior stress in finite plane elastostatics. Acta Mech 197, 285–299 (2008). https://doi.org/10.1007/s00707-007-0510-4
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DOI: https://doi.org/10.1007/s00707-007-0510-4