On volumetric locking‐free behaviour of p‐version finite elements under finite deformations

Abstract

SUMMARYWe demonstrate the locking-free properties of the displacement formulation of p-finite elements when applied to nearly incompressible Neo-Hookean material under finite deformations. For an axisymmetric model problem we provide semi-analytical solutions for a nearly incompressible Neo-Hookean material exploited to investigate the robustness of p-FEM with respect to volumetric locking. An analytical solution for the incompressible case is also derived to demonstrate the convergence of the compressible numerical solution towards the incompressible case when the compression modulus is increased.

A parallel spectral element method for dynamic three-dimensional nonlinear elasticity problems

“…Because the interior modes of an element are coupled only with the boundary modes (i.e. vertex/edge/face modes) of the same element, in the implementation we arrange the vector U such that the vertex modes are followed by the edge modes, face modes, and the interior modes, and a Schur complement is performed on the linear system (19) to condense out all the interior modes. This results in a smaller linear system of equations about the boundary modes only, which is then solved with the conjugate gradient solver.…”

“…Eqs. (19) and (31), with the preconditioned conjugate gradient solver. This solver is invoked once every time step for linear elastodynamic problems, and once every Newton-Raphson iteration (within a time step) for geometrically nonlinear elastodynamic problems.…”

“…In linear elasticity the high-order methods possess many advantages over ''classical FEMs", such as considerably higher convergence rates, the flexibility of using large aspect ratios of elements without significant deterioration in accuracy, locking-free behavior with respect to thickness for plate and shell-like structures, and with respect to the Poisson ratio for nearly incompressible materials. The efficiency and advantages of p-FEMs were also extended in recent years to nonlinear problems such as elasto-plasticity [20,15], flow-structure interaction [30,31], and finite-deformation problems with follower loads [26,14,35,19], demonstrating that p-FEM's advantages carry over to nonlinear solid mechanics problems.…”

p-FEMs in biomechanics: Bones and arteries

“…These methods have been extended to non-linear problems, first for plasticity [2,3], and thereafter to isotropic hyperelasticity [4] and to nearly-incompressible hyperelasticity [5].…”

“…The p-FEM based on the displacement formulation has been shown to be efficient in the framework of finite-deformations for isotropic hyperelastic materials [4,14] and that it is locking free for nearly-incompressible hyperelastic materials [5,15] thus it is expected to be especially attractive for modeling arteries.…”