A confocally multicoated elliptical inclusion under antiplane shear: Some new results

The problem of a confocally multicoated elliptical inclusion in an unbounded matrix subjected to an antiplane shear is studied. Making use of the complex potentials and conformal mapping techniques, we show that the multiple coatings can be analyzed through a recurrence procedure in the transformed domain, while remaining explicit in detail and transparent overall. Particularly, the effect of the multiple confocal coatings is mathematically represented by a (2 × 2) array alone, resulting from a serial multiplication of matrices of the same order. Further we prove the following proposition. If the displacement prescribed at the remote boundary of the matrix is a polynomial of degree j in the position coordinates x_i, the stresses, at the innermost core are polynomials of degree j-1, j-3,..., in x_i. This result is universally true provided that all elliptical interfaces are confocal, while no regard is paid to the number of coatings, their constituent properties and area fractions. Explicit expressions for the stresses at the innermost core are obtained in simple, closed forms.