On non-physical response in models for fiber-reinforced hyperelastic materials

Abstract

a b s t r a c tSoft biological tissues are sometimes composed of thin and stiff collagen fibers in a soft matrix leading to a strong anisotropy. Commonly, constitutive models for quasi-incompressible materials, as for soft biological tissues, make use of an additive split of the Helmholz free-energy into a volumetric and a deviatoric part that is applied to the matrix and fiber contribution. This split offers conceptual and numerical advantages. The purpose of this paper is to investigate a non-physical effect that arises thereof. In fact, simulations involving uniaxial stress configurations reveal volume growth at rather small stretches. Numerical methods such as the Augmented Lagrangian method might be used to suppress this behavior. An alternative approach, proposed here, solves this problem on the constitutive level.

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

“…Following an extensive literature overview, a brief theoretical synopsis of anisotropic hyperelasticity was provided. Subsequently, the numerical performance of the classical Q1P0 element with and without the augmented Lagrangian method was examined together with a rather new concept introduced by Sansour [26] and Helfenstein et al [14], who proposed the use of the (unsplit) deformation gradient tensor F for the anisotropic part of the constitutive equations. The results corroborate the new concept namely Q1P0+WAS.…”

“…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.…”

“…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. Alternatively, the multiplicative decomposition of the deformation gradient can be avoided for the anisotropic part, as suggested by Sansour [26] and Helfenstein et al [14], which solves the issue on the constitutive level without using any ad hoc algorithm.…”

Rupture risk in abdominal aortic aneurysms: A realistic assessment of the explicit GPU approach

“…Near-incompressibility is enforced by using a penalty function U = K(J -l) 2 /2, in which the bulk modulus K » /i (Helfenstein et al, 2010) is selected to be at least three orders of magnitude larger than the initial (ground matrix) shear modulus and acts as the penalty parameter. The distributed network of collagen fibers contributes to the mechanical response of the aorta To assess mesh quality, we opt for the Jacobian metric, which is defined as the minimum determinant of the Jacobian matrix, evaluated at each corner and at the center (Stimpson et al, 2007).…”

Physical response of hyperelastic models for composite materials and soft tissues

“…For example, by fixing the shear modulus, Helfenstein et al [12] identified two sets of material constants of the Holzapfel model with and without a volumetric-isochoric split. This split was also known as another source of the non-physical response (Ehlers and Eipper [13]).…”