Explicit Solutions in Isotropic Planar Elastostatics

TL;DR

This paper introduces a new method for explicitly solving displacement fields in isotropic planar elastostatics, especially for complex geometries and non-zero traction conditions, inspired by complex analysis techniques.

Contribution

It presents a novel approach based on the Neumann problem for inhomogeneous Cauchy-Riemann equations, applicable to conformally equivalent domains, and provides explicit solutions for classical geometries.

Findings

Explicit solutions for displacement fields in complex geometries

Effective handling of non-zero traction boundary conditions

Demonstrated applicability on classical domain examples

Abstract

Addressing the intricate challenges in plane elasticity, especially with non-vanishing traction and complex geometries, requires innovative methods. This paper offers a novel approach, drawing inspiration from the Neumann problem for the inhomogeneous Cauchy-Riemann equations. Our method applies to domains conformally equivalent to a unit disk or an annulus, focusing on deriving explicit solutions for the displacement field rather than the stress tensor, which distinguishes it from most traditional approaches. We explore solutions for specific classical cases to demonstrate its efficacy, such as a cardioid domain, a ring domain with a shifted hole, and a gear-like structure. This work enhances the toolkit for researchers and practitioners tackling isotropic planar elastostatic challenges with a unified and flexible approach.