Abstract
Generalised Structure Tensors (GSTs) are used to formulate constitutive models for anisotropic fibre-reinforced materials in which fibres are dispersed. The GST approach has been applied so far to models based on invariants I4 and I5 (I6 and I7). These anisotropic invariants capture the effect of deformation on each fibre family in isolation, unlike the invariant I8, which couples two fibre families. We extend the GST approach to models based on the invariant I8. We consider two different formulations and for each model derive expressions for stress and elasticity tensors in both the general case and for axisymmetric distributions. We apply the proposed formulation to the hyperelastic Holzapfel–Ogden model for myocardium and obtain a modified model, in which fibre dispersion is consistently accounted for in every term of the strain-energy function. We demonstrate that when accounting for fibre dispersion in the coupling term, the effect on the predicted material response can be significant and may also reduce material symmetry.
Highlights
Many soft biological tissues can be regarded as elastic solid composites, consisting of an incompressible and isotropic matrix, which is reinforced by one or several families of fibres
One can properly incorporate fibre dispersion data into material models that include invariant I8 and, in principle, into any hyperelastic constitutive model, since Generalised Structure Tensors (GSTs)-based expressions are available for every anisotropic invariant in the set I1, ... , I9, which forms a functional basis for an arbitrary strain-energy function [21,22]
Even though the relevance of fibre exclusion to invariant I8 remains to be examined from the physical standpoint and considering material stability [24], all existing methods for fibre exclusion can be straightforwardly applied to the proposed formulations, since the fourth-order GST Ĥ and the structurelike tensor Hare defined in terms of the second-order GSTs
Summary
Many soft biological tissues can be regarded as elastic solid composites, consisting of an incompressible and isotropic matrix, which is reinforced by one or several families of fibres. Local variability of microscopic organisation is found in many tissues and can be captured at the continuum level using orientation density functions (ODFs), which express the probability of observing certain orientations within a representative volume element. This statistical datum is acquired by histological examinations using modern imaging techniques: see, e.g., [8].
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