Uniformity of stresses inside a parabolic inhomogeneity in finite plane elastostatics

Using complex variable techniques, we establish the uniformity of the internal Piola stresses within a parabolic inhomogeneity embedded in a matrix from a particular class of compressible hyperelastic materials of harmonic type subjected to uniform remote Piola stresses. In addition, two Piola stress components inside the inhomogeneity simply coincide with their remotely prescribed counterparts in the matrix, while the two other internal Piola stress components can be determined via the solution of a quadratic equation, and are found to be independent of the single geometric parameter characterizing the inhomogeneity–matrix system. We derive a complete solution to the corresponding boundary value problem describing the mechanical behavior of the composite.