Geometries of inhomogeneities with minimum field concentration

This paper is devoted to the study of geometries of inhomogeneities with minimum strain or stress concentration. The solutions are achieved by the indirect method of first deriving lower bounds and then constructing the geometries to attain the lower bounds. In particular, we show that a new class of geometries, namely, E-inclusions and periodic E-inclusions, are the optimal geometries with minimum field concentrations. We also obtain the explicit relation between the shape matrix of E-inclusion and remote applied strain which will be convenient for engineering applications of these new geometries.