Area-Preserving Configuration of Holes in an Elastic Solid Under Plane Deformation: Implications for Mechanical Insensitivity of Liquid Inclusions

Abstract

Abstract. This article studies an inverse problem in plane elasticity concerning the preservation of the cross-sectional area of a hole under remote loading. The problem is motivated primarily by the need to stabilize internal fluid environments in flexible structures and microfluidic systems, for which deformation-induced area change may lead to an internal pressure increment and consequently alter the mechanical response of the surrounding structure. Restricting attention to plane deformation, we consider a cylindrical fluid inclusion of arbitrary shape and seek combinations of far-field loading and inclusion geometry such that the cross-sectional area of the inclusion remains unchanged. Under this condition, the pressure increment inside a compressible liquid inclusion vanishes, and the problem can be equivalently treated as that of an area-invariant, traction-free hole in an infinite elastic plane. By exploiting the complex variable method for plane elasticity, we derive for an arbitrary constant far-field loading the explicit condition imposed on the configuration of the hole for the corresponding area-invariance requirement. It is shown that in terms of combinations of symmetric hole shapes and pure shear loadings, the area invariance is achieved only when the symmetry axis of the hole is aligned with one of the principal directions of the shear loading, except for the cases of special symmetric hole shapes (e.g., circular, hypocycloidal, regularly polygonal) in which the area invariance can be always ensured for arbitrary orientations under pure shear loading. For biaxial tensile/compressive loadings, several representative numerical examples are presented to illustrate the evolution of the desired hole shapes relative to the ratio of the biaxial loading.