Deficiencies in numerical models of anisotropic nonlinearly elastic materials

Abstract

Incompressible nonlinearly hyperelastic materials are rarely simulated in Finite Element numerical experiments as being perfectly incompressible because of the numerical difficulties associated with globally satisfying this constraint. Most commercial Finite Element packages therefore assume that the material is slightly compressible. It is then further assumed that the corresponding strain-energy function can be decomposed additively into volumetric and deviatoric parts. We show that this decomposition is not physically realistic, especially for anisotropic materials, which are of particular interest for simulating the mechanical response of biological soft tissue. The most striking illustration of the shortcoming is that with this decomposition, an anisotropic cube under hydrostatic tension deforms into another cube instead of a hexahedron with non-parallel faces. Furthermore, commercial numerical codes require the specification of a 'compressibility parameter' (or 'penalty factor'), which arises naturally from the flawed additive decomposition of the strainenergy function. This parameter is often linked to a 'bulk modulus', although this notion makes no sense for anisotropic solids; we show that it is essentially an arbitrary parameter and that infinitesimal changes to it result in significant changes in the predicted stress response. This is illustrated with numerical simulations for biaxial tension experiments of arteries, where the magnitude of the stress response is found to change by several orders of magnitude when infinitesimal changes in 'Poisson's ratio' close to the perfect incompressibility limit of 1/2 are made.

Finite element implementation of a new model of slight compressibility for transversely isotropic materials

“…However, following Sansour (2008), the recent work of Ní Annaidh et al (2013), Gilchrist et al (2014) has shown that this decomposition is invalid for compressible anisotropic materials. This is best illustrated by simulating the hydrostatic tension of a transversely isotropic cube, which unrealistically deforms into another larger cube instead of a rectangular cuboid because of the additive split of the SEF (Ní Annaidh et al 2013;Gilchrist et al 2014). This deficiency has also been shown for a transversely isotropic sphere deforming into another sphere (Vergori et al 2013).…”

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.…”

A homeostatic-driven turnover remodelling constitutive model for healing in soft tissues

“…For supra-physiological loadings, the injury produced could potentially introduce changes in the water content of the tissue, resulting in a compressible material. In some cases, the adequacy of the quasi-incompressibility hypothesis in soft tissues subjected to physiological loading has also been debated [93,94]. The possibility of cavitational damage arising in soft tissues has also been put forth [95].…”