On Eulerian constitutive equations for modeling growth and residual stresses in arteries.

K Y Volokh
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Abstract

Recently Volokh and Lev (2005) argued that residual stresses could appear in growing arteries because of the arterial anisotropy. This conclusion emerged from a continuum mechanics theory of growth of soft biological tissues proposed by the authors. This theory included Lagrangian constitutive equations, which were formulated directly with respect to the reference configuration. Alternatively, it is possible to formulate Eulerian constitutive equations with respect to the current configuration and to 'pull them back' to the reference configuration. Such possibility is examined in the present work. The Eulerian formulation of the constitutive equations is used for a study of arterial growth. It is shown, particularly, that bending resultants are developed in the ring cross-section of the artery. These resultants may cause the ring opening or closing after cutting the artery in vitro as it is observed in experiments. It is remarkable that the results of the present study, based on the Eulerian constitutive equations, are very similar to the results of Volokh and Lev (2005), based on the Lagrangian constitutive equations. This strengthens the authors' argument that anisotropy is a possible reason for accumulation of residual stresses in arteries. This argument appears to be invariant with respect to the mathematical description.

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欧拉本构方程在动脉生长和残余应力模拟中的应用。
最近Volokh和Lev(2005)认为,由于动脉的各向异性,残余应力可能出现在生长的动脉中。这一结论来源于作者提出的生物软组织生长的连续介质力学理论。这个理论包括拉格朗日本构方程,它是直接根据参考位形表述的。另外,也有可能将欧拉本构方程与当前构型相结合,并将其“拉回”到参考构型。本文对这种可能性进行了探讨。本构方程的欧拉公式用于动脉生长的研究。特别是,在动脉的环形横截面上出现了弯曲产物。这些结果可能导致环在体外切割动脉后打开或关闭,正如实验中观察到的那样。值得注意的是,本研究基于欧拉本构方程的结果与Volokh和Lev(2005)基于拉格朗日本构方程的结果非常相似。这加强了作者的论点,即各向异性是动脉中残余应力积累的可能原因。就数学描述而言,这个论证似乎是不变的。
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