On the quasi-incompressible finite element analysis of anisotropic hyperelastic materials

Abstract

Quasi-incompressible behavior is a desired feature in several constitutive models within the finite elasticity of solids, such as rubber-like materials and some fiber-reinforced soft biological tissues. The Q1P0 finite element formulation, derived from the three-field Hu-Washizu variational principle, has hitherto been exploited along with the augmented Lagrangian method to enforce incompressibility. This formulation typically uses the unimodular deformation gradient. However, contributions by Sansour (Eur J Mech A Solids 27: [28][29][30][31][32][33][34][35][36][37][38][39] 2007) and Helfenstein et al. (Int J Solids Struct 47:2056-2061 conspicuously demonstrate an alternative concept for analyzing fiber reinforced solids, namely the use of the (unsplit) deformation gradient for the anisotropic contribution, and these authors elaborate on their proposals with analytical evidence. The present study handles the alternative concept from a purely numerical point of view, and addresses systematic comparisons with respect to the classical treatment of the Q1P0 element and its coalescence with the augmented Lagrangian method by means of representative numerical examples. The results corroborate the new concept, show its numerical efficiency and reveal a direct physical interpretation of the fiber stretches.

An energy‐stable mixed formulation for isogeometric analysis of incompressible hyperelastodynamics

“…In this example, we examine the performance of the proposed formulation for an incompressible anisotropic hyperelastic material, which has been designed to describe arterial tissue layers with distributed collagen fibers . We note that this material model used in the compressible regime may lead to nonphysical deformations, and remedies for this issue have been proposed recently . The geometry set‐up and the material model are summarized in Table .…”

“…n en 61440 120 120 5760 90000 n r en 8 1 1 2 5 n eq 1785024 5091 7584 6396 75144 material model used in the compressible regime may lead to nonphysical deformations, 52,53 and remedies for this issue have been proposed recently. 54 The geometry set-up and the material model are summarized in Table 1. The groundmatrix is modeled as an isotropic Neo-Hookean material, with c 1 being the shear modulus.…”

Modeling of Damage Evolution in a Patient-Specific Stenosed Artery upon Stent Deployment

“…Where vol  is the volumetric, iso  the isochoric, isotropic and f  the fiber contribution. Recent investigations demonstrate that using deviatoric invariants for fiber-reinforced materials may lead to some non-physical results when assuming some degree of compressibility for tissue (Nolan et al, 2014;Gültekin, Dal and Holzapfel, 2018). To avoid these non-physical results, henceforth, we use the right Cauchy-Green tensor C (instead of C ) in the last term of eq.…”

Numerical analyses of the interrelation between extracellular smooth muscle orientation and intracellular filament overlap in the human abdominal aorta

“…In order to enforce incompressibility and to avoid non-physical effects, as reported in [22], the Augmented Lagrangian method is used. [19] The medial layer of the arterial wall contains smooth muscle cells so that we additively decompose the related isochoric part Ψ of the energy function into an active part, say Ψ a , taking care of the smooth muscle contraction, and into a passive contribution, say Ψ p , representing the structural and mechanical properties of the passive components (ground matrix and collagen fibers), i.e. Ψ = Ψ a + Ψ p .…”

“…Stress and elasticity tensors Since stress and elasticity tensors have already been provided within the continuum mechanical framework section, we focus here on the specific expressions for the (active) isochoric stress and elasticity tensors, more precisely the fictitious contributions to the second Piola-Kirchhoff stress tensor S and the material elasticity tensor ℂ, see Equations 18 and 19. Hence, with the use of Equations 24 and 25 we obtain from (18) and (19)…”