Finite element approximations for near-incompressible and near-inextensible transversely isotropic bodies

TL;DR

This paper provides a comprehensive theoretical and computational analysis of finite element methods for transversely isotropic elastic bodies, highlighting conditions for well-posedness, error estimates, and strategies to prevent locking in near-incompressible and near-inextensible regimes.

Contribution

It introduces new finite element formulations, including under-integration techniques, to effectively address locking issues in anisotropic elasticity problems.

Findings

Standard formulation is locking-free for moderate anisotropy in near-incompressible limit.

Near-inextensibility causes evident locking behavior.

Under-integration of extensional terms achieves superlinear convergence without locking.

Abstract

This work comprises a detailed theoretical and computational study of the boundary value problem for transversely isotropic linear elastic bodies. General conditions for well-posedness are derived in terms of the material parameters. The discrete form of the displacement problem is formulated for conforming finite element approximations. The error estimate reveals that anisotropy can play a role in minimizing or even eliminating locking behaviour, for moderate values of the ratio of Young's moduli in the fibre and transverse directions. In addition to the standard conforming approximation an alternative formulation, involving under-integration of the volumetric and extensional terms in the weak formulation, is considered. The latter is equivalent to either a mixed or a perturbed Lagrangian formulation, analogously to the well-known situation for the volumetric term. A set of numerical examples confirms the locking-free behaviour in the near-incompressible limit of the standard formulation with moderate anisotropy, with locking behaviour being clearly evident in the case of near-inextensibility. On the other hand, under-integration of the extensional term leads to extensional locking-free behaviour, with convergence at superlinear rates.