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Explicit Solutions in Isotropic Planar Elastostatics

Andreas Granath, Per Åhag, Antti Perälä, Rafał Czyż

TL;DR

Isotropic planar elastostatics with non-vanishing boundary traction is recast as a Neumann problem for the inhomogeneous Cauchy–Riemann equations via a complex-analytic displacement potential $v=u_1-iv_2$, yielding explicit displacement-field formulas in domains conformally equivalent to the unit disk or an annulus. The paper proves solvability theorems for simply connected and annular domains (via biholomorphic maps and the Vaitekhovich–Begehr framework) and provides constructive integral representations for $u$ in terms of the data $(\psi,\gamma)$ and boundary inputs. Three classical geometries—cardioid, eccentric ring, and gear-like domains—are worked out with closed-form displacement expressions and interior von Mises-stress distributions, illustrating the method’s versatility and its cusp-related limitations. The approach offers a unified, explicit-displacement toolkit that avoids intermediate potentials and rational conformal maps, with potential extensions to elastodynamics via separable solutions.

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.

Explicit Solutions in Isotropic Planar Elastostatics

TL;DR

Isotropic planar elastostatics with non-vanishing boundary traction is recast as a Neumann problem for the inhomogeneous Cauchy–Riemann equations via a complex-analytic displacement potential , yielding explicit displacement-field formulas in domains conformally equivalent to the unit disk or an annulus. The paper proves solvability theorems for simply connected and annular domains (via biholomorphic maps and the Vaitekhovich–Begehr framework) and provides constructive integral representations for in terms of the data and boundary inputs. Three classical geometries—cardioid, eccentric ring, and gear-like domains—are worked out with closed-form displacement expressions and interior von Mises-stress distributions, illustrating the method’s versatility and its cusp-related limitations. The approach offers a unified, explicit-displacement toolkit that avoids intermediate potentials and rational conformal maps, with potential extensions to elastodynamics via separable solutions.

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.
Paper Structure (6 sections, 7 theorems, 61 equations, 3 figures)

This paper contains 6 sections, 7 theorems, 61 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. The case of simply connected domains
  4. The case of annular domains
  5. Applying the new method to classic cases
  6. Conclusion

Key Result

Theorem 3.1

Let $\psi\in\mathcal{C}^\alpha(\overline{\mathbb{D}};\mathbb{C})$, $0<\alpha<1$, $\gamma\in \mathcal{C}(\partial\mathbb{D};\mathbb{C})$, $c\in\mathbb{C}$. The Neumann problem for the inhomogeneous Cauchy-Riemann equations in the unit disk: is solvable if and only if for $|z|<1$ The unique solution is then given by

Figures (3)

  • Figure 1: Internal distribution of von Mises stress for different values of $a$
  • Figure 2: Internal distribution of von Mises stress due to a linear load with scaling constant $c=10^{-3}$.
  • Figure 3: Internal distribution of von Mises stress for different values of $a$

Theorems & Definitions (17)

  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Remark
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • proof
  • Theorem 4.1
  • ...and 7 more