Section 2.zip from Switchless constitutive relation for arterial tissues: eliminating all discontinuities in mechanical response

The switching criterion in the constitutive relations for arterial tissues could introduce stress discontinuities and cause fibres to exhibit dual behaviour—simultaneously experiencing tension and compression—depending on the switching criteria. Furthermore, the resulting conditional constitutive relations do not distinguish between longitudinal and transverse shear in unidirectional composites, contrary to expectations. Consequently, the azimuthal and telescopic shear behaviours of a cylindrical annulus resemble that of their isotropic counterpart. To work though these concerns, we propose two classes of vanishing-matched-generalized invariants, based on the Green–Lagrange and Seth-Hill strains. For unidirectional composites, this invariant effectively nullifies fibre contributions in pure compression, thereby eliminating the need for switching criteria. Furthermore, for rotationally symmetric fibre distributions, the proposed relations produce the correct mechanical response for the entire range of dispersion, i.e. 0 ≤ κ ≤ 1/2. The resulting structural constitutive relation for the stress linearizes to the isotropic Hooke’s Law when considering all deformations simultaneously—a desirable feature of any constitutive relation. Compared to the existing models, both proposed classes—(i) the simple vanishing-matched invariants-based relations and (ii) polynomial forms—demonstrate significantly improved descriptive capability across 13 sets of planar biaxial data from various healthy and diseased human arterial tissues.