  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 problem. Our method caters to
domains that are conformally equivalent to a unit disk or annulus and
emphasizes deriving explicit solutions for the displacement field rather than
the stress tensor, differentiating it from most traditional methods. We explore
solutions for specific traditional cases to demonstrate its efficacy, such as a
cardioid domain, a ring domain with a shifted hole, and a gear-like structure.
This work augments the existing toolbox for researchers and practitioners
addressing isotropic planar elastostatics challenges, presenting a unified and
flexible approach.
