Section 3.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.动脉组织本构关系中的切换准则可能会引入应力不连续性，致使纤维呈现双力学行为——同时承受拉应力与压应力——具体表现取决于所采用的切换准则。此外，由此得到的条件本构关系无法区分单向复合材料中的纵向剪切与横向剪切，这与学界预期相悖。因此，圆柱环的周向与伸缩剪切行为与其各向同性对应物相似。为解决上述问题，我们基于格林-拉格朗日应变（Green–Lagrange strains）与赛斯-希尔应变（Seth-Hill strains），提出两类消失匹配广义不变量（vanishing-matched-generalized invariants）。对于单向复合材料而言，该不变量可有效抵消纯压缩状态下纤维的贡献，从而无需再使用切换准则。此外，针对旋转对称纤维分布，所提出的本构关系可在整个分散度范围（即0 ≤ κ ≤ 1/2）内产生正确的力学响应。当同时考虑所有变形工况时，所得结构本构关系的应力表达式可线性化为各向同性胡克定律，这是各类本构关系均期望具备的优良特性。与现有模型相比，我们提出的两类模型——(i) 基于简单消失匹配不变量的本构关系，以及(ii) 多项式形式本构关系——在来自13组健康与病变人类动脉组织的平面双轴数据集中，展现出显著更优的描述能力。