A finite element for soft tissue deformation based on the absolute nodal coordinate formulation


This paper introduces an implementation of the absolute nodal coordinate formulation (ANCF) that can be used to model fibrous soft tissue in cases of three-dimensional elasticity. It is validated against results from existing incompressible material models. The numerical results for large deformations based on this new ANCF element are compared to results from analytical and commercial software solutions, and the relevance of the implementation to the modeling of biological tissues is discussed. Also considered is how these results relate to the classical results seen in Treloar’s rubber experiments. All the models investigated are considered from both elastic and static points of view. For isotropic cases, neo-Hookean and Mooney–Rivlin models are examined. For the anisotropic case, the Gasser–Ogden–Holzapfel model, including a fiber dispersion variation, is considered. The results produced by the subject ANCF models agreed with results obtained from the commercial software. For the isotropic cases, in fact, the numerical solutions based on the ANCF element were more accurate than those produced by ANSYS.

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We would like to thank the Research Foundation of the Lappeenranta University of Technology and the Academy of Finland (Application No. 299033 for funding 519 of Academy Research Fellow) for the generous grants that made this work possible.

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Correspondence to Ajay B. Harish.

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Appendix A: Analytical solution

Appendix A: Analytical solution

The analytical solution which we use in this paper has been outlined here. Firstly, we assume an object without any holes or different types of imperfections. Also, we assume that the material is isotropic in nature. Let us now consider the total volumes of them in the initial configuration, which can be expressed for cylindrical or rectangular bars in the forms

$$\begin{aligned} V_\mathrm{cyl}=\pi LR^2, V_\mathrm{rec}=HWL. \end{aligned}$$

Upon application of load, the largest dimensions of them, i.e., let us assume it to be L, changes by \(\lambda \) times. If we consider the material to be incompressible, then the volume of the object does not change; from here for the cylinder, we have received \(r=\frac{R}{\sqrt{\lambda }}\), and in the rectangular cross-sectional case, we have \(w=\frac{W}{\sqrt{\lambda }}\), \(h=\frac{H}{\sqrt{\lambda }}\), where rhw are dimensions of circular and rectangular cross sections in actual configurations.

The Cauchy stress tensor for the incompressible solids can be given as

$$\begin{aligned} \sigma =-p\mathbf{I }+2\mathbf{F }\left( \frac{\partial {\varPsi }}{\partial \mathbf{C }}\right) \mathbf{F }^T \end{aligned}$$

where \({\varPsi }\) is potential density function, p is a function of hydrostatic stress (which is not determined by the deformation). \(\mathbf{C }\) is the right Cauchy–Green tensor. However, p is not established from deformation; it is possible to receive it from boundary conditions. Our deformation occurs along one of the axes, let’s name this axis z, the components of stress tensors in others, i.e., x and y are equal to zero. From this condition, the form of p is possible to derive and then substitute into \(\sigma _{zz}\). The final expressions for the applied loads from which we can obtain the values of \(\lambda \) and as a result define the total displacements are

$$\begin{aligned} N_\mathrm{cyl}=2\pi \int _{0}^{r} \sigma _{zz} r\mathrm{d}r, N_\mathrm{rec}=\int _{0}^{h} \int _{0}^{w} \sigma _{zz} \mathrm{d}x\mathrm{d}y. \end{aligned}$$

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Obrezkov, L.P., Matikainen, M.K. & Harish, A.B. A finite element for soft tissue deformation based on the absolute nodal coordinate formulation. Acta Mech 231, 1519–1538 (2020). https://doi.org/10.1007/s00707-019-02607-4

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