A new formulation of slight compressibility for arterial tissue and its Finite Element implementation

In order to avoid the numerical difficulties in locally enforcing the incompressibility constraint using the displacement formulation of the Finite Element Method, slight compressibility is typically assumed when simulating the mechanical response of arterial tissue. The current standard method of accounting for slight compressibility of hyperelastic soft tissue assumes an additive decomposition of the strain-energy function into a volumetric and a deviatoric part. This has been shown, however, to be inconsistent with the linear theory and results in cubes retaining their cuboid shape under hydrostatic tension and compression, which seems at variance with the reinforcement of arterial tissue with two families of collagen fibres. A remedy for these defects is proposed here, a solution which generalises the current standard model of slight compressibility to include two additional terms, one of which is quadratic in the invariants and the other quadratic in . Experimental data are used to motivate typical values for the associated material constants of these additional terms. Some simulations are performed to allow contrasts and comparisons to be made between the current standard model of slight compressibility and its generalisation proposed here.

为规避采用有限元法（Finite Element Method）位移列式局部施加不可压缩性约束时遭遇的数值求解困难，在模拟动脉组织的力学响应过程中，通常会假设材料具备轻微可压缩性。当前用于描述超弹性软组织轻微可压缩性的标准方法，是将应变能函数通过加法分解为体积分量与偏量分量。然而已有研究证实，该方法与线性理论并不自洽，且会导致立方体在静水拉伸与压缩条件下仍保持长方体外形，这与动脉组织受两类胶原纤维增强的实际力学特性相悖。针对上述缺陷，本文提出一种修正方案：将现有轻微可压缩性标准模型进行推广，新增两项附加项——其中一项关于不变量呈二次形式，另一项则关于某一未明确变量呈二次形式。本文利用实验数据确定了新增附加项对应的材料常数的典型取值。此外，本文开展了若干仿真实验，用于对比现有轻微可压缩性标准模型与本文提出的推广模型的差异。