On fibre dispersion in anisotropic soft biological tissues using fourth-order structural tensors

Abstract

Developments in microscopic imaging techniques for soft biological tissues have greatly improved understanding of their underlying microstructures, notably the three-dimensional orientation and dispersion of fibrous constituents. In structural constitutive modelling, assuming that the collagenous and muscle fibres in soft tissues are perfectly aligned along a prescribed orientation can limit the ability to accurately predict the experimentally detected stress–strain response of biological samples. The objective of this paper is to extend the representation of hyperelasticity for anisotropic bodies using fourth-order structural tensors (FOSTs) to incorporate the effects of fibre dispersion according to the generalised structural tensor (GST) approach. Dispersion within a family of reinforcing collagen fibres is a common phenomenon in soft tissues which alters their overall mechanical behaviour. The FOST-based representation of the strain energy density function (SEF) separates the constitutive properties of the continuum from the nonlinear descriptions of geometric deformations; with each of these being described by different fourth-order tensors. The constant-valued material and structural properties of the continuum reside within fourth-order material tensors referred to as the anisotropic Lamé tensors. Nonlinearity in the model is then solely applied to the components of the displacement gradient tensor in fourth-order strain measures. The framework provided leads to a unique set of fourteen generalised FOSTs for two-direction preferred hyperelasticity including fibre families which are dispersed according to normalised distributions. In the absence of dispersion, the model collapses to the conventional FOST-based model. Models for the hyperelastic response of arterial wall and passive ventricular myocardium tissues are postulated and show good agreement with experimental data using minimal ground-state material constants.