On the pure traction problem of linear elasticity: a regularized formulation and its robust approximation

TL;DR

This paper introduces a regularized finite element approach for the pure traction problem in linear elasticity, ensuring unique, convergent solutions without extra boundary conditions, and effectively handling loading incompatibilities.

Contribution

A novel regularized formulation and finite element method for the pure traction problem that guarantees uniqueness, convergence, and robustness against loading incompatibilities.

Findings

The regularized approach yields unique solutions that converge to minimal norm solutions.

The method effectively handles non-equilibrated loadings in discretized problems.

Numerical examples demonstrate the approach's robustness and accuracy.

Abstract

The pure traction problem of elasticity appears frequently in engineering applications, and its complexity stems from the fact that its solution is unique only up to (infinitesimal) rigid body motions. When finite elements are employed to approximate this problem, one solution is typically singled out by applying carefully selected boundary conditions on the discrete model or by imposing global constraints on the deformation. However, neither of these strategies is both simple and computationally efficient. In this work, we propose a new approach to solving the pure traction problem that overcomes existing limitations. Our method builds on a regularized form of the problem whose solution is shown to be unique, converges to the original solution of minimal norm, and can be approximated with finite elements in a straightforward way, without additional degrees of freedom. Additionally, we analyze the situation in which the approximation of the solution domain renders the loading of the discretized problem non-equilibrated, making the problem ill-posed. In this case, we propose a regularized predictor--corrector finite element formulation that handles the incompatibilities of the loading, providing a solution that converges to that of the original Neumann problem as the mesh size and the regularizing parameter tend to zero. Numerical examples illustrate the effectiveness of the proposed approach for representative problems in mechanics where pure traction boundary conditions appear.