Isotoxal star-shaped polygonal voids and rigid inclusions in nonuniform antiplane shear fields. Part I: Formulation and full-field solution

TL;DR

This paper develops full-field analytical solutions for the stress distribution around isotoxal star-shaped voids and inclusions in nonuniform antiplane shear fields, highlighting stress singularities at corners critical for composite design.

Contribution

It introduces a novel analytical approach using complex potentials for star-shaped inclusions in nonuniform shear fields, including special cases like cracks and stiffeners.

Findings

Solutions reveal stress singularities at inclusion corners and crack ends.

Closed-form stress fields are derived for various star-shaped geometries.

Stress blow-up at sharp corners impacts composite strength design.

Abstract

An infinite class of nonuniform antiplane shear fields is considered for a linear elastic isotropic space and (non-intersecting) isotoxal star-shaped polygonal voids and rigid inclusions perturbing these fields are solved. Through the use of the complex potential technique together with the generalized binomial and the multinomial theorems, full-field closed-form solutions are obtained in the conformal plane. The particular (and important) cases of star-shaped cracks and rigid-line inclusions (stiffeners) are also derived. Except for special cases (addressed in Part II), the obtained solutions show singularities at the inclusion corners and at the crack and stiffener ends, where the stress blows-up to infinity, and is therefore detrimental to strength. It is for this reason that the closed-form determination of the stress field near a sharp inclusion or void is crucial for the design of ultra-resistant composites.