A robust anisotropic hyperelastic formulation for the modelling of soft tissue

Abstract

The Holzapfel-Gasser-Ogden (HGO) model for anisotropic hyperelastic behaviour of collagen fibre reinforced materials was initially developed to describe the elastic properties of arterial tissue, but is now used extensively for modelling a variety of soft biological tissues. Such materials can be regarded as incompressible, and when the incompressibility condition is adopted the strain energy Ψ of the HGO model is a function of one isotropic and two anisotropic deformation invariants. A compressible form (HGO-C model) is widely used in finite element simulations whereby the isotropic part of Ψ is decoupled into volumetric and isochoric parts and the anisotropic part of Ψ is expressed in terms of isochoric invariants. Here, by using three simple deformations (pure dilatation, pure shear and uniaxial stretch), we demonstrate that the compressible HGO-C formulation does not correctly model compressible anisotropic material behaviour, because the anisotropic component of the model is insensitive to volumetric deformation due to the use of isochoric anisotropic invariants. In order to correctly model compressible anisotropic behaviour we present a modified anisotropic (MA) model, whereby the full anisotropic invariants are used, so that a volumetric anisotropic contribution is represented. The MA model correctly predicts an anisotropic response to hydrostatic tensile loading, whereby a sphere deforms into an ellipsoid. It also computes the correct anisotropic stress state for pure shear and uniaxial deformation. To look at more practical appli- cations, we developed a finite element user-defined material subroutine for the simulation of stent deployment in a slightly compressible artery. Significantly higher stress triaxiality and arterial compliance are computed when the full anisotropic invariants are used (MA model) instead of the isochoric form (HGO-C model).Keywords: Anisotropic, Hyperelastic, Incompressibility, Finite element, Artery, Stent Nomenclature I -identity tensor Ψ -Helmholtz free-energy (strain-energy) function Ψ vol -volumetric contribution to the free energy Ψ aniso -anisotropic contribution to the free energy Ψ iso -isotropic contribution to the isochoric free energy Ψ aniso -anisotropic contribution to the isochoric free energy σ -Cauchy stress σ -deviatoric Cauchy stress q -von Mises equivalent stress σ hyd -hydrostatic (pressure) stress F -deformation gradient J -determinant of the deformation gradient; local ratio of volume change C -right Cauchy-Green tensor I 1 -first invariant of C I 4,6 -anisotropic invariants describing the deformation of reinforcing fibres F -isochoric portion of the deformation gradient C -isochoric portion of the right Cauchy-Green deformation tensor I 1 -first invariant of C I 4,6 -isochoric anisotropic invariants a 0i , i = 4, 6 -unit vector aligned with a reinforcing fibre in the reference configuration a i , i = 4, 6 -updated (deformed) fibre direction (= Fa 0i ) κ 0 -isotropic bulk modulus µ 0 -isotropic shear modulus k i , i = 1, 2 -anisotropic material co...

Effect of Wall Flexibility on the Deformation during Flow in a Stenosed Coronary Artery

“…We obtain the syntax for Neo-Hookean model from [18]; further we confirmed that our UMAT code is correct by comparing with the formulation in [19] for the same Holzapfel (MA) model.…”

Crimping and deployment of metallic and polymeric stents -- finite element modelling

“…To consider compressible deformation, the HGO-C model was suggested, with the anisotropic part expressed by isochoric invariants (insensitive to volumetric deformation). However, Nolan et al [26] found that the HGO-C model was unable to simulate compressible anisotropic behavior correctly, because it used isochoric anisotropic invariants which is insensitive to volumetric deformation. Consequently, they formulated a modified anisotropic (MA) model by using the full anisotropic invariants which accounted for a volumetric anisotropic contribution.…”

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

“…Several studies, e.g., Helfenstein et al [14], Annaidh et al [1] and Nolan et al [22], have reported the erroneous analysis results of fiber-reinforced anisotropic material models for soft biological tissues (Weiss et al [37], Holzapfel et al [16] and Rubin and Bodner [25]) when they are mistakenly used in the compressible domain; e.g., a sphere reinforced with one family of fibers would be deformed into a sphere with a smaller size upon hydrostatic pressure instead of taking on an ellipsoidal shape. One remedy for (ii) is to implement the computationally (rather) expensive augmented Lagrangian method to bring the analysis towards the incompressibility limit, see Glowinski and Le Tallec [8,9] and Simo and Taylor [31] among others.…”