Modelling of transversely isotropic nonlinear incompressible soft tissues using spectral invariants

Abstract

In isotropic elasticity, numerous strain energy functions with different types of invariants are developed to serve certain purposes. This wealth of functions has partly contributed to the knowledge of the mechanical behaviour of isotropic elastic solids. In general, soft tissues are not isotropic but can be modelled as transversely isotropic solid. The knowledge of the mechanical behaviour of transversely isotropic elastic solids is not as profound as isotropic solid. Hence, the need to develop accurate strain energy functions to understand the mechanical behaviour of transversely isotropic soft tissues. In isotropic elasticity, phenomenological strain energy functions with principal stretches have certain attractive features from both the mathematical and physical viewpoints. These forms of strain energy have been widely and successfully used in prediction of elastic deformations. This research is an extension from classical invariants of isotropic models to characterize transversely isotropic soft tissues with spectral invariants. In order to obtain a specific form of the strain energy function from an experiment, it is convenient to have explicit and analytic expressions for the derivatives of the strain energy function with respect to its invariants. Three of the invariants are the principal extension ratios and the other two are the cosines of the angles between the principal directions of the right stretch tensor and the material preferred direction. These direct physical interpretations of the invariants shows that the model has an experimental advantage where a triaxial test can vary a single invariant while keeping the remaining invariants fixed. The symmetrical and orthogonal properties developed here are similar to that possessed by a strain energy function of an isotropic elastic solid written in terms of principal stretches. A specific constitutive model was applied to biological soft tissues and the model compares well with existing experimental data.