### Abstract

Despite distinct mechanical functions, biological soft tissues have a common microstructure in which a ground matrix is reinforced by a collagen fibril network. The microstructural properties of the collagen network contribute to continuum mechanical tissue properties that are strongly anisotropic with tensile-compressive asymmetry. In this study, a novel approach based on a continuous distribution of collagen fibril volume fractions is developed to model fibril reinforced soft tissues as nonlinearly elastic and anisotropic material. Compared with other approaches that use a normalized number of fibrils for the definition of the distribution function, this representation is based on a distribution parameter (i.e. volume fraction) that is commonly measured experimentally while also incorporating pre-stress of the collagen fibril network in a tissue natural configuration. After motivating the form of the collagen strain energy function, examples are provided for two volume fraction distribution functions. Consequently, collagen second-Piola Kirchhoff stress and elasticity tensors are derived, first in general form and then specifically for a model that may be used for immature bovine articular cartilage. It is shown that the proposed strain energy is a convex function of the deformation gradient tensor and, thus, is suitable for the formation of a polyconvex tissue strain energy function.

Original language | English |
---|---|

Pages (from-to) | 706-715 |

Number of pages | 10 |

Journal | Mathematics and Mechanics of Solids |

Volume | 16 |

Issue number | 7 |

DOIs | |

Publication status | Published - Sep 2011 |

### Fingerprint

### Keywords

- articular cartilage
- collagen fibril network
- distribution function
- polyconvexity
- volume fraction

### ASJC Scopus subject areas

- Materials Science(all)
- Mathematics(all)
- Mechanics of Materials

### Cite this

*Mathematics and Mechanics of Solids*,

*16*(7), 706-715. https://doi.org/10.1177/1081286510387866

**Modeling the collagen fibril network of biological tissues as a nonlinearly elastic material using a continuous volume fraction distribution function.** / Shirazi, Reza; Vena, Pasquale; Sah, Robert L.; Klisch, Stephen M.

Research output: Contribution to journal › Article

*Mathematics and Mechanics of Solids*, vol. 16, no. 7, pp. 706-715. https://doi.org/10.1177/1081286510387866

}

TY - JOUR

T1 - Modeling the collagen fibril network of biological tissues as a nonlinearly elastic material using a continuous volume fraction distribution function

AU - Shirazi, Reza

AU - Vena, Pasquale

AU - Sah, Robert L.

AU - Klisch, Stephen M.

PY - 2011/9

Y1 - 2011/9

N2 - Despite distinct mechanical functions, biological soft tissues have a common microstructure in which a ground matrix is reinforced by a collagen fibril network. The microstructural properties of the collagen network contribute to continuum mechanical tissue properties that are strongly anisotropic with tensile-compressive asymmetry. In this study, a novel approach based on a continuous distribution of collagen fibril volume fractions is developed to model fibril reinforced soft tissues as nonlinearly elastic and anisotropic material. Compared with other approaches that use a normalized number of fibrils for the definition of the distribution function, this representation is based on a distribution parameter (i.e. volume fraction) that is commonly measured experimentally while also incorporating pre-stress of the collagen fibril network in a tissue natural configuration. After motivating the form of the collagen strain energy function, examples are provided for two volume fraction distribution functions. Consequently, collagen second-Piola Kirchhoff stress and elasticity tensors are derived, first in general form and then specifically for a model that may be used for immature bovine articular cartilage. It is shown that the proposed strain energy is a convex function of the deformation gradient tensor and, thus, is suitable for the formation of a polyconvex tissue strain energy function.

AB - Despite distinct mechanical functions, biological soft tissues have a common microstructure in which a ground matrix is reinforced by a collagen fibril network. The microstructural properties of the collagen network contribute to continuum mechanical tissue properties that are strongly anisotropic with tensile-compressive asymmetry. In this study, a novel approach based on a continuous distribution of collagen fibril volume fractions is developed to model fibril reinforced soft tissues as nonlinearly elastic and anisotropic material. Compared with other approaches that use a normalized number of fibrils for the definition of the distribution function, this representation is based on a distribution parameter (i.e. volume fraction) that is commonly measured experimentally while also incorporating pre-stress of the collagen fibril network in a tissue natural configuration. After motivating the form of the collagen strain energy function, examples are provided for two volume fraction distribution functions. Consequently, collagen second-Piola Kirchhoff stress and elasticity tensors are derived, first in general form and then specifically for a model that may be used for immature bovine articular cartilage. It is shown that the proposed strain energy is a convex function of the deformation gradient tensor and, thus, is suitable for the formation of a polyconvex tissue strain energy function.

KW - articular cartilage

KW - collagen fibril network

KW - distribution function

KW - polyconvexity

KW - volume fraction

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UR - http://www.scopus.com/inward/citedby.url?scp=80052842365&partnerID=8YFLogxK

U2 - 10.1177/1081286510387866

DO - 10.1177/1081286510387866

M3 - Article

AN - SCOPUS:80052842365

VL - 16

SP - 706

EP - 715

JO - Mathematics and Mechanics of Solids

JF - Mathematics and Mechanics of Solids

SN - 1081-2865

IS - 7

ER -