Li Beibei, Roper Steven M, Wang Lei, Luo Xiaoyu, Hill N A
School of Mathematics and Statistics, University of Glasgow, Glasgow, UK.
Department of Engineering, Durham University, Durham, UK.
J Math Biol. 2019 Apr;78(5):1277-1298. doi: 10.1007/s00285-018-1309-8. Epub 2018 Nov 19.
We develop a mathematical model for a small axisymmetric tear in a residually stressed and axially pre-stretched cylindrical tube. The residual stress is modelled by an opening angle when the load-free tube is sliced along a generator. This has application to the study of an aortic dissection, in which a tear develops in the wall of the artery. The artery is idealised as a single-layer thick-walled axisymmetric hyperelastic tube with collagen fibres using a Holzapfel-Gasser-Ogden strain-energy function, and the tear is treated as an incremental deformation of this tube. The lumen of the cylinder and the interior of the dissection are subject to the same constant (blood) pressure. The equilibrium equations for the incremental deformation are derived from the strain energy function. We develop numerical methods to study the opening of the tear for a range of material parameters and boundary conditions. We find that decreasing the fibre angle, decreasing the axial pre-stretch and increasing the opening angle all tend to widen the dissection, as does an incremental increase in lumen and dissection pressure.
我们为残余应力作用下且轴向预拉伸的圆柱管中的小轴对称撕裂建立了一个数学模型。当无载管沿一条母线切开时，残余应力由一个开口角度来建模。这适用于主动脉夹层的研究，在主动脉夹层中，动脉壁会出现撕裂。动脉被理想化地视为具有胶原纤维的单层厚壁轴对称超弹性管，采用霍尔扎普费尔 - 加塞尔 - 奥格登应变能函数，并且撕裂被视为该管的增量变形。圆柱体的管腔和夹层内部承受相同的恒定（血液）压力。增量变形的平衡方程由应变能函数推导得出。我们开发了数值方法来研究一系列材料参数和边界条件下撕裂的开口情况。我们发现，减小纤维角度、减小轴向预拉伸以及增大开口角度都倾向于使夹层变宽，管腔和夹层压力的增量增加也会如此。
