Cansız F Barış Can, Dal Hüsnü, Kaliske Michael
a Institute for Structural Analysis, Technische Universität Dresden , Dresden , Germany.
Comput Methods Biomech Biomed Engin. 2015 Aug;18(11):1160-1172. doi: 10.1080/10255842.2014.881475. Epub 2014 Feb 17.
This contribution presents a novel constitutive model in order to simulate an orthotropic rate-dependent behaviour of the passive myocardium at finite strains. The motivation for the consideration of orthotropic viscous effects in a constitutive level lies in the disagreement between theoretical predictions and experimentally observed results. In view of experimental observations, the material is deemed as nearly incompressible, hyperelastic, orthotropic and viscous. The viscoelastic response is formulated by means of a rheological model consisting of a spring coupled with a Maxwell element in parallel. In this context, the isochoric free energy function is decomposed into elastic equilibrium and viscous non-equilibrium parts. The baseline elastic response is modelled by the orthotropic model of Holzapfel and Ogden [Holzapfel GA, Ogden RW. 2009. Constitutive modelling of passive myocardium: a structurally based framework for material characterization. Philos Trans Roy Soc A Math Phys Eng Sci. 367:3445-3475]. The essential aspect of the proposed model is the account of distinct relaxation mechanisms for each orientation direction. To this end, the non-equilibrium response of the free energy function is constructed in the logarithmic strain space and additively decomposed into three anisotropic parts, denoting fibre, sheet and normal directions each accompanied by a distinct dissipation potential governing the evolution of viscous strains associated with each orientation direction. The evolution equations governing the viscous flow have an energy-activated nonlinear form. The energy storage in the Maxwell branches has a quadratic form leading to a linear stress-strain response in the logarithmic strain space. On the numerical side, the algorithmic aspects suitable for the implicit finite element method are discussed in a Lagrangian setting. The model shows excellent agreement compared to experimental data obtained from the literature. Furthermore, the finite element simulations of a heart cycle carried out with the proposed model show significant deviations in the strain field relative to the elastic solution.
本文提出了一种新颖的本构模型，用于模拟有限应变下被动心肌的正交各向异性率相关行为。在本构层面考虑正交各向异性粘性效应的动机在于理论预测与实验观测结果之间的不一致。鉴于实验观测，该材料被视为近乎不可压缩、超弹性、正交各向异性且粘性的。粘弹性响应通过由一个弹簧与一个麦克斯韦元件并联组成的流变模型来表述。在此背景下，等容自由能函数被分解为弹性平衡部分和粘性非平衡部分。基线弹性响应由霍尔扎佩尔和奥格登的正交各向异性模型[霍尔扎佩尔GA，奥格登RW。2009年。被动心肌的本构建模：基于结构的材料表征框架。《皇家学会哲学学报A：数学、物理和工程科学》。367:3445 - 3475]进行建模。所提出模型的关键方面是考虑每个取向方向的不同松弛机制。为此，自由能函数的非平衡响应在对数应变空间中构建，并相加分解为三个各向异性部分，分别表示纤维、片层和法线方向，每个方向都伴有一个独特的耗散势，用于控制与每个取向方向相关的粘性应变的演化。控制粘性流动的演化方程具有能量激活的非线性形式。麦克斯韦分支中的能量存储具有二次形式，导致对数应变空间中的线性应力 - 应变响应。在数值方面，在拉格朗日设置下讨论了适用于隐式有限元方法的算法方面。与从文献中获得的实验数据相比，该模型显示出极好的一致性。此外，用所提出的模型进行的心脏周期有限元模拟表明，相对于弹性解，应变场存在显著偏差。
