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Modelling of anisotropic growth in biological tissues A new approach and computational aspects

机译：生物组织中各向异性生长的建模新方法和计算方面

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摘要

In this contribution, we develop a theoretical and computational framework for anisotropic growth phenomena. As a key idea of the proposed phenomenological approach, a fibre or rather structural tensor is introduced, which allows the description of transversely isotropic material behaviour. Based on this additional argument, anisotropic growth is modelled via appropriate evolution equations for the fibre while volumetric remodelling is realised by an evolution of the referential density. Both the strength of the fibre as well as the density follow Wolff-type laws. We however elaborate on two different approaches for the evolution of the fibre direction, namely an alignment with respect to strain or with respect to stress. One of the main benefits of the developed framework is therefore the opportunity to address the evolutions of the fibre strength and the fibre direction separately. It is then straightforward to set up appropriate integration algorithms such that the developed framework fits nicely into common, finite element schemes. Finally, several numerical examples underline the applicability of the proposed formulation.
机译：在此贡献中，我们为各向异性生长现象建立了理论和计算框架。作为所提出的现象学方法的关键思想，引入了纤维或结构张量，这允许描述横向各向同性的材料行为。基于该附加论点，通过适当的纤维演化方程对各向异性的增长进行建模，而体积重建则通过参考密度的演化来实现。纤维的强度和密度都遵循Wolff型定律。然而，我们阐述了两种不同的纤维方向演变方法，即相对于应变或相对于应力的对准。因此，开发框架的主要优点之一是有机会分别解决纤维强度和纤维方向的变化。然后可以很容易地设置适当的集成算法，以使开发的框架很好地适合常见的有限元方案。最后，几个数值示例强调了所提出配方的适用性。

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