Induced fields in isolated elliptical inhomogeneities due to imposed polynomial fields at infinity

ABSTRACT

The Eshelby inhomogeneity problem plays a crucial role in the micromechanical analysis of the effective mechanical behaviour of inhomogeneous media since it provides a mechanism to predict interior fields associated with ellipsoidal inhomogeneities. In the context of linear elasticity, Eshelby showed that given an isolated elliptical (two dimensions) or ellipsoidal (three dimensions) inhomogeneity embedded in a homogeneous material of infinite extent, then for any uniform strain or traction imposed in the far field, the induced strain inside the inhomogeneity is also uniform. In the case of non-uniform far-field conditions, Eshelby showed that if the loading is a polynomial of order n, the associated interior field is characterized by a polynomial of the same order. This is often called ‘Eshelby's polynomial conservation theorem’. Since then, the problem has been studied by many, but in most cases for the uniform loading scenario, i.e. when strains or tractions in the far field are uniform. However, in many applications, e.g. permittivity, conductivity, elasticity, etc., the case of non-uniform conditions is also of interest and furthermore, methods to deal with non-elliptical and non-ellipsoidal inhomogeneities are required. In this work, for prescribed non-uniform polynomial far-field conditions, we introduce a method to approximate interior fields for isolated inhomogeneities of elliptical shape. This subproblem is relevant for approximating effective properties of numerous composites since constituent inhomogeneities are often of this form, or limiting forms, e.g. layered and fibre reinforced composites. We verify that the obtained results agree with the polynomial conservation property and with results determined using conformal mappings or the classical circle inclusion theorem. We close with a discussion of how the method can be straightforwardly extended to the case of non-elliptical inhomogeneities.

KEYWORDS:

• Eigenstrain
• elliptical inhomogeneity
• inclusion
• conformal mapping
• Eshelby's tensor
• circle theorem
• polynomial conservation property

AMS CLASSIFICATIONS:

• 30E10
• 42A10
• 65D99
• 35Q74

Acknowledgments

Parnell is grateful to the Engineering and Physical Sciences Research Council for funding his fellowship (EP/L018039/1).

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Calvo has been partially supported through Ministerio de Economía y Competitividad [MTM2014-53309-P] of Spain and from the Consejería de Economía, Innovación, Ciencia y Empleo, Junta de Andalucía through Research Group Grants [FQM-309].