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

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 minimising 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. KEYWORDS elasticity, finite element methods, solids Int J Numer Methods Eng. 2019;117:693-712.wileyonlinelibrary.com/journal/nme

Discontinuous Galerkin approximations for near-incompressible and near-inextensible transversely isotropic bodies

“…We recall the same behaviour as stated in [23], that is, extensional locking for P 1 CG except for the angles 0, where the material is very stiff, and π/2, where the extensional term tends to 0.…”

“…We recall the same behaviour as stated in [23], that is, extensional locking for P 1 CG except for the angles 0,…”

“…The compliance coefficients S ij are given in [23], as is the analytical solution. We measure the vertical tip displacement at corner C with meshing 2 × 80 × 16 elements.…”

“…Note that a(•, •) is symmetric and corresponds to a positive definite operator, as shown in [23], meaning that the weak problem is well-posed. Regarding regularity of the solution, for Ω bounded and convex, for example, and f and t defined as in the line after (14), there exists a unique solution u ∈ [H 2 (Ω)] d (see for example [20,7]).…”

“…In recent work [23], the authors have studied the well-posedness of boundary value problems involving transversely isotropic elastic materials. They have also investigated theoretically and computationally the use of conforming finite element approximations, paying attention to both near-incompressibility and near-inextensibility.…”