A finite element formulation for highly anisotropic incompressible elastic solids

Abstract

A numerical approach is developed for the solution of problems of materials with extremely strong directions. Small deformations of a transversely isotropic linear elastic solid, reinforced by a single family of inextensible fibres, are considered. The kinematic constraint equations of incompressibility and inextensibility in the fibre direction lead to the appearance of an arbitrary hydrostatic pressure and an arbitrary tension stress in the constitutive equations. A Galerkin approach is used to discretize the virtual work and weak form of the constraint equations. Independent interpolation of the displacement, pressure and tension fields leads to a mixed system of equations, with characteristic zero-diagonal terms. The assumption of plane stress conditions in the plane of the fibres results in a simplified displacement-tension formulation, analogous to the primitive-variable formulation of Stokes flow. A mixed penalty approximation is then employed to solve for displacement and tension stress fields. Computations are carried out using a biquadratic displacement element with discontinuous bilinear tension stress interpolation. The formulation is used to solve a number of simple beam problems and the results compared to closed-form solutions.

Pathologies associated with the numerical analysis of hyper‐anisotropic materials

“…Based on the above findings, the use of an anisotropic irreducible formulation is limited to somewhat unconstrained meshes and/or to unreasonably low values of p l , . These conclusions are not unlike those associated with irreducible elements used in the analysis of nearly incom- 5 3.20 x 104 pressible media. As such, the usefulness of the present irreducible formulation is limited.…”

“…Equation (5) suggests that the magnitude and sign of the eigenvalue with the shape of its associated eigenvector can provide a significant amount of information concerning the behaviour of a specific element or patch of elements. For the present study, the following ranges of eigenvalues are of particular importance:…”

“…(1) Zero eigenoalues. From equation (5) it is evident zero or near zero eigenvalues decide the degree of singularity of the system. They correspond to a load shape that invokes infinite response.…”

“…In the present study, element stability was studied through an eigenvalue analysis. (5) The fibre orientation also influences the number of 'spurious' modes associated with an element. If the fibre orientation corresponds to a co-ordinate direction, the element behaviour may, under limiting conditions, be effected.…”

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

“…Mixed finite element approximations leading to stable formulations have been developed and implemented for near‐incompressibility and near‐inextensibility in the work of Zdunek et al, whereas transverse anisotropy has been studied, also using various mixed finite element formulations, in the works of Zdunek and Rachowicz . A mixed approach allied with a Lagrange multiplier or perturbed Lagrange multiplier formulation of the inextensibility constraint forms the basis of a computational study in the work of Wriggers et al In earlier work, a corresponding numerical study, using a mixed three‐field approach for the linear problem, has been carried out in the work of Brádaigh and Pipes . The recent contribution presents a locking‐free implementation for nonlinear transversely isotropic elasticity, the inextensibility constraint in the fibre direction being accommodated using a perturbed Lagrangian formulation.…”