Statistical characterization of the anisotropic strain energy in soft materials with distributed fibers

Abstract

a b s t r a c tWe discuss analytical and numerical tools for the statistical characterization of the anisotropic strain energy density of soft hyperelastic materials embedded with fibers. We consider spatially distributed orientations of fibers following a tridimensional or a planar architecture. We restrict our analysis to material models dependent on the fourth pseudo-invariant I 4 of the Cauchy-Green tensor, and to exponential forms of the fiber strain energy function aniso . Under different loading conditions, we derive the closed-form expression of the probability density function for I 4 and aniso . In view of bypassing the cumbersome extension-contraction switch, commonly adopted for shutting down the contribution of contracted fibers in models based on generalized structure tensors, for significant loading conditions we identify analytically the support of the fibers in pure extension. For uniaxial loadings, the availability of the probability distribution function and the knowledge of the support of the fibers in extension yield to the analytical expression of average and variance of I 4 and aniso , and to the direct definition of the average second Piola-Kirchhoff stress tensor. For generalized loadings, the dependence of I 4 on the spatial orientation of the fibers can be analyzed through angle plane diagrams. Angle plane diagrams facilitate the assessment of the influence of the pure extension condition on the definition of the stable support of fibers for the statistics related to the anisotropic strain energy density.

Cornea modelling

“…Therefore, the exclusion of compressed fibrils when analysing the cornea should not be taken as an unavoidable necessity, but should be verified using inverse analysis. Moreover, the numerical procedures that must be activated in order to exclude compressed fibrils in a spatial distribution are rather complicated and uncertain, and may spoil the hypothetical advantages of using a purely tensile fibril model [38].…”

“…Although compressed fibrils may have a very reduced stiffness related to local buckling [36,37], the material model used in this study does not exclude the contribution of compressed fibrils. The actual role of compressed fibrils has initiated an interesting discussion concerning a criterion to switch between tension and compression in fiber reinforced material models [38,39]. Although we believe that this criterion would be fundamental in materials made only by fibers, in our model we do not account for it because the particular structure of the cornea, made of collagen fibrils immersed into a matrix of elastin and proteoglycans, is able to provide some confinement to the compressed fibrils ruling out the possibility to observe local buckling.…”

Integrated Heart—Coupling multiscale and multiphysics models for the simulation of the cardiac function

“…This prevents unphysical buckling-type phenomena from creating an ambiguity in the solution [116] (see also more recent contributions supporting the study of fiber stability in terms of statistical description of I 4,f , e.g. [106,67,100]). Decomposition (3.5) for orthotropic fiber-reinforced materials is somewhat controversial as it can lead to unphysical spherical deformations under pure hydrostatic loads [225] or undesired volume growth during uniaxial tension [113].…”

On fibre dispersion modelling of soft biological tissues: a review

“…The subsequent paper of Li et al [18] documents computational aspects of the model, and in particular expressions for the elasticity tensor and the integration boundary that admits only fibres which are extended were provided. The work of Gizzi et al [39] introduced a probability density of I 4 to average I 4 itself over the range of values for I 4 > 1 as a means to exclude compressed fibres in the energy function. They also introduced two dispersion parameters, one as for the standard GST model and another one which is related to the second-order approximation.…”