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Anisotropy of a discrete fiber icosahedron model for fibrous tissues exhibited for large deformations

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Abstract

Early work on modeling the mechanical response of fibrous tissues suggested a structural model based on integration of angular distributions of fiber bundles over the surface of a unit sphere. This paper considers a discrete fiber model, based on a regular icosahedron with six fiber bundles defined by the twelve uniformly distributed vertices on a unit sphere. Like the structural model, the icosahedron model introduces a weighted sum of the strain energies of the six fiber bundles with parameters that characterize the density and strength of each fiber bundle as well as the undulation of the fibers in the bundle. It is shown that even when all fiber directions are identical and weighted evenly, this discrete model exhibits anisotropic response for large deformations. The reason for this anisotropic response is that the uncoupled strain energies of the fiber bundles do not allow for coupling of the strains in the fiber directions that are needed to form the principal invariants of strain. Anisotropic strain invariants based on structural tensors defined by a regular icosahedron model are discussed that can characterize isotropy and material anisotropy.

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Acknowledgments

The author would like to acknowledge comments of Y Lanir which were used to improve this paper.

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  1. Faculty of Mechanical Engineering, Technion-Israel Institute of Technology, 32000, Haifa, Israel

    M. B. Rubin

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Appendix.: Geometry of the discrete icosahedron fiber model

Appendix.: Geometry of the discrete icosahedron fiber model

1.1 A.1 Geometry of a regular icosahedron

The six unit vectors NI parallel to the directions of opposing vertices of a regular icosahedron are defined by the

$$ \begin{array}{@{}rcl@{}} \mathbf{N}_{1} &=& \mathbf{e}_{1}, \quad \mathbf{N}_{2} = \frac{1}{\sqrt{5}} \mathbf{e}_{1} + \frac{2}{\sqrt{5}} \mathbf{e}_{2}, \\ \mathbf{N}_{3} &=& \frac{1}{\sqrt{5}} \mathbf{e}_{1} + \frac{1}{2} \left( 1 - \frac{1}{\sqrt{5}} \right) \mathbf{e}_{2} + \sqrt{\frac{1}{2} \left( 1 + \frac{1}{\sqrt{5}} \right)} \mathbf{e}_{3}, \\ \mathbf{N}_{4} &=& \frac{1}{\sqrt{5}} \mathbf{e}_{1} - \frac{1}{2} \left( 1 + \frac{1}{\sqrt{5}} \right) \mathbf{e}_{2} + \sqrt{\frac{1}{2} \left( 1 - \frac{1}{\sqrt{5}} \right)} \mathbf{e}_{3}, \\ \mathbf{N}_{5} &=& \frac{1}{\sqrt{5}} \mathbf{e}_{1} - \frac{1}{2} \left( 1 + \frac{1}{\sqrt{5}} \right) \mathbf{e}_{2} - \sqrt{\frac{1}{2} \left( 1 - \frac{1}{\sqrt{5}} \right)} \mathbf{e}_{3}, \\ \mathbf{N}_{6} &=& \frac{1}{\sqrt{5}} \mathbf{e}_{1} + \frac{1}{2} \left( 1 - \frac{1}{\sqrt{5}} \right) \mathbf{e}_{2} - \sqrt{\frac{1}{2} \left( 1 + \frac{1}{\sqrt{5}} \right)} \mathbf{e}_{3}. \end{array} $$
(A.1)

Also, the symmetric matrix βIJ in (6) is given by

$$ \beta^{IJ} = \frac{1}{8} \left( \begin{array}{rrrrrr} 9 & -1 & -1 & -1 & -1 & -1 \\ -1 & 9 & -1 & -1 & -1 & -1 \\ -1 & -1 & 9 & -1 & -1 & -1 \\ -1 & -1 & -1 & 9 & -1 & -1 \\ -1 & -1 & -1 & -1 & 9 & -1 \\ -1 & -1 & -1 & -1 &-1 & 9 \end{array}\right). $$
(A.2)

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Rubin, M.B. Anisotropy of a discrete fiber icosahedron model for fibrous tissues exhibited for large deformations. Mech Soft Mater 4, 2 (2022). https://doi.org/10.1007/s42558-022-00040-7

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