This work is motivated by the current widespread interest in modelling the mechanical response of arterial tissue. A widely used approach within the context of anisotropic nonlinear elasticity is to use an orthotropic incompressible hyperelasticity model which, in general, involves a strain-energy density that depends on seven independent invariants. The complexity of such an approach in its full generality is daunting and so a number of simplifications have been introduced in the literature to facilitate analytical tractability. An extremely popular model of this type is where the strain energy involves only three invariants. While such models and their generalisations have been remarkably successful in capturing the main features of the mechanical response of arterial tissue, it is generally acknowledged that such simplified models must also have some drawbacks. In particular, it is intuitively clear that the correlation of such models with experiment will suffer limitations due to the restricted number of invariants considered. Our purpose here is to use the linearised theory for infinitesimal deformations to provide some guidelines for the development of a more robust nonlinear theory. The linearised theory for incompressible orthotropic materials is developed and involves six independent elastic constants. The general stress-strain law is inverted to provide an expression for the fibre stretch in terms of the stress. We examine the linearised response for simple tension in two mutually perpendicular directions corresponding to the axial and circumferential directions in the artery, obtaining an explicit expression for the fibre stretch in terms of the applied tension, fibre angle and linear elastic constants. The focus is then on determining the range of fibre orientation angles that ensure that the fibres are in tension in these simple tension tests. It is shown that the fibre stretch is positive for both simple tension tests if and only if the fibre angle is restricted to lie between two special angles called generalised magic angles. For the special case where the strain-energy function for the nonlinear model depends only on the three invariants $I_1,I_4,I_6$, it is shown that the corresponding linearised model, called the standard linear model (SLM), depends on three elastic constants and the fibre stretch is positive only in the small range of fibre angles between the classic magic angles $35.26°$ and $54.74°$. However, when the two additional invariants $I_5,I_7$ are included in the nonlinear strain energy so that the corresponding linear model involves four elastic constants, it is shown that the domain of fibre angle for which the stretch is positive is much larger and that the fibre stretch is monotonic with respect to the fibre angle in this range.

这项工作受到当前对动脉组织机械反应建模的广泛兴趣的推动。在各向异性非线性弹性范围内，一种广泛使用的方法是使用正交各向异性不可压缩的超弹性模型，该模型通常涉及依赖于七个独立不变性的应变能密度。这种方法的全面性令人望而生畏，因此在文献中已经引入了许多简化方法，以简化分析的可处理性。这种类型的一种非常流行的模型是应变能仅涉及三个不变量。尽管此类模型及其概括在捕获动脉组织机械反应的主要特征方面非常成功，通常公认的是，这种简化的模型还必须具有一些缺点。特别地，从直觉上很清楚，由于所考虑的不变数的数量有限，此类模型与实验的相关性将受到限制。我们的目的是将线性化理论用于无穷小变形，从而为开发更可靠的非线性理论提供一些指导。发展了不可压缩正交各向异性材料的线性化理论，它涉及六个独立的弹性常数。颠倒了一般的应力-应变定律，以根据应力提供纤维拉伸的表达式。我们检查了在两个相互垂直的方向上对应于动脉轴向和圆周方向的简单张力的线性化响应，在施加的张力，纤维角度和线性弹性常数方面获得纤维拉伸的明确表达式。然后，重点是确定纤维定向角的范围，以确保在这些简单的拉伸测试中纤维处于拉伸状态。结果表明，当且仅当纤维角度被限制在两个被称为广义幻角的特殊角度之间时，纤维拉伸对于两个简单的拉伸测试都是正值。对于特殊情况，其中非线性模型的应变能函数仅取决于三个不变量 结果表明，当且仅当纤维角度被限制在两个被称为广义幻角的特殊角度之间时，纤维拉伸对于两个简单的拉伸测试都是正值。对于特殊情况，其中非线性模型的应变能函数仅取决于三个不变量 结果表明，当且仅当纤维角度被限制在两个被称为广义幻角的特殊角度之间时，纤维拉伸对于两个简单的拉伸测试都是正值。对于特殊情况，其中非线性模型的应变能函数仅取决于三个不变量${\mathrm{一世}}_{\mathrm{1个}}，{\mathrm{一世}}_4，{\mathrm{一世}}_6$，表明相应的线性化模型（称为标准线性模型（SLM））取决于三个弹性常数，并且纤维拉伸仅在经典魔术角之间的纤维角很小的范围内为正 $35.26°$ 和 $54.74°$。但是，当另外两个不变式${\mathrm{一世}}_5，{\mathrm{一世}}_7$ 包含在非线性应变能中，因此相应的线性模型包含四个弹性常数，这表明拉伸为正的纤维角度的范围要大得多，并且纤维拉伸相对于纤维角度是单调的。这个范围。