School of Engineering and Applied Science, University of Virginia, Charlottesville, VA 22904, USA.
Department of Mechanical Engineering, Dublin City University, Glasnevin, Dublin D09 W6Y4, Ireland; School of Mathematics, Statistics, and Applied Mathematics, National University of Ireland Galway, University Road, Galway, Ireland.
J Mech Behav Biomed Mater. 2018 Dec;88:470-477. doi: 10.1016/j.jmbbm.2018.08.052. Epub 2018 Sep 5.
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,I,I, 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,I 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.
这项工作的动机是目前广泛关注的动脉组织力学响应建模。各向异性非线性弹性范围内的一种广泛使用的方法是使用各向异性不可压缩超弹性模型，该模型通常涉及依赖于七个独立不变量的应变能密度。这种方法在其完全一般性中的复杂性是令人生畏的，因此文献中已经引入了许多简化方法来便于分析可处理性。这种类型的一个非常流行的模型是应变能仅涉及三个不变量。虽然这种模型及其推广在捕捉动脉组织力学响应的主要特征方面取得了显著的成功，但人们普遍认为，这种简化模型也必须有一些缺点。特别是，直观地清楚，由于考虑的不变量数量有限，这种模型与实验的相关性将受到限制。我们在这里的目的是使用线性化的小变形理论为更稳健的非线性理论的发展提供一些指导。不可压缩各向异性材料的线性化理论得到发展，并涉及六个独立的弹性常数。将一般的应力-应变定律反转，以给出纤维拉伸的表达式，其以应力表示。我们研究了对应于动脉的轴向和周向方向的两个相互垂直方向的简单拉伸的线性化响应，得到了纤维拉伸的显式表达式，其以施加的张力、纤维角和线性弹性常数表示。然后重点确定确保纤维在这些简单拉伸测试中处于拉伸状态的纤维取向角度的范围。结果表明，如果纤维角限制在称为广义魔术角的两个特殊角度之间，则两种简单拉伸测试的纤维拉伸都是正的。对于非线性模型的应变能函数仅取决于三个不变量 I、I、I 的特殊情况，结果表明，相应的线性化模型称为标准线性模型（SLM），仅取决于三个弹性常数，并且纤维拉伸仅在纤维角在经典魔术角 35.26°和 54.74°之间的小范围内为正。然而，当将两个附加不变量 I、I 包含在非线性应变能中以使对应的线性模型涉及四个弹性常数时，结果表明纤维角度的范围更大，并且纤维在该范围内拉伸与纤维角度单调相关。
