Accurate prediction of stress in fibers with distributed orientations using generalized high-order structure tensors

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13 Citations (Scopus)

Abstract

The orientation of collagen fibers plays an important role on the mechanics of connective tissues. Connective tissues have fibers with different orientation distributions. The angular integration formulation used to model the mechanics of fibers with distributed orientation is accurate, but computationally expensive for numerical methods such as finite elements. This study presents a formulation based on pre-integrated Generalized High-Order Structure Tensors (GHOST) which greatly improves the accuracy of the predicted stress. Simplifications of the GHOST formulation for transversely-isotropic and planar fiber distributions are also presented. Additionally, the GHOST and the angular integration formulations are compared for different loading conditions, fiber orientation functions, strain energy functions and degrees of fiber non-linearity. It was found that the GHOST formulation predicted the stress of the fibers with an error lower than 10% for uniaxial and biaxial tension. Fiber non-linearity increased the error of the GHOST formulation; however, the error was reduced to negligible values by considering higher order structure tensors. The GHOST formulation produced lower errors when used with an elliptical fiber density function and a binomial strain energy function. In conclusion, the GHOST formulation is able to accurately predict the stress of fibers with distributed orientation without requiring numerous integral calculations. Consequently, the GHOST formulation may reduce the computational effort needed to analyze the mechanics of fibrous tissues with distributed orientations.

Original languageEnglish (US)
Pages (from-to)73-83
Number of pages11
JournalMechanics of Materials
Volume75
DOIs
StatePublished - Jan 1 2014

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Tensors
tensors
formulations
fibers
Fibers
predictions
connective tissue
Mechanics
Tissue
Strain energy
nonlinearity
fiber orientation
collagens
Fiber reinforced materials
simplification
Collagen
Probability density function
Numerical methods
energy

All Science Journal Classification (ASJC) codes

  • Instrumentation
  • Materials Science(all)
  • Mechanics of Materials

Cite this

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title = "Accurate prediction of stress in fibers with distributed orientations using generalized high-order structure tensors",
abstract = "The orientation of collagen fibers plays an important role on the mechanics of connective tissues. Connective tissues have fibers with different orientation distributions. The angular integration formulation used to model the mechanics of fibers with distributed orientation is accurate, but computationally expensive for numerical methods such as finite elements. This study presents a formulation based on pre-integrated Generalized High-Order Structure Tensors (GHOST) which greatly improves the accuracy of the predicted stress. Simplifications of the GHOST formulation for transversely-isotropic and planar fiber distributions are also presented. Additionally, the GHOST and the angular integration formulations are compared for different loading conditions, fiber orientation functions, strain energy functions and degrees of fiber non-linearity. It was found that the GHOST formulation predicted the stress of the fibers with an error lower than 10{\%} for uniaxial and biaxial tension. Fiber non-linearity increased the error of the GHOST formulation; however, the error was reduced to negligible values by considering higher order structure tensors. The GHOST formulation produced lower errors when used with an elliptical fiber density function and a binomial strain energy function. In conclusion, the GHOST formulation is able to accurately predict the stress of fibers with distributed orientation without requiring numerous integral calculations. Consequently, the GHOST formulation may reduce the computational effort needed to analyze the mechanics of fibrous tissues with distributed orientations.",
author = "{Cortes Correales}, {Daniel Humbe} and Elliott, {Dawn M.}",
year = "2014",
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doi = "10.1016/j.mechmat.2014.04.006",
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N2 - The orientation of collagen fibers plays an important role on the mechanics of connective tissues. Connective tissues have fibers with different orientation distributions. The angular integration formulation used to model the mechanics of fibers with distributed orientation is accurate, but computationally expensive for numerical methods such as finite elements. This study presents a formulation based on pre-integrated Generalized High-Order Structure Tensors (GHOST) which greatly improves the accuracy of the predicted stress. Simplifications of the GHOST formulation for transversely-isotropic and planar fiber distributions are also presented. Additionally, the GHOST and the angular integration formulations are compared for different loading conditions, fiber orientation functions, strain energy functions and degrees of fiber non-linearity. It was found that the GHOST formulation predicted the stress of the fibers with an error lower than 10% for uniaxial and biaxial tension. Fiber non-linearity increased the error of the GHOST formulation; however, the error was reduced to negligible values by considering higher order structure tensors. The GHOST formulation produced lower errors when used with an elliptical fiber density function and a binomial strain energy function. In conclusion, the GHOST formulation is able to accurately predict the stress of fibers with distributed orientation without requiring numerous integral calculations. Consequently, the GHOST formulation may reduce the computational effort needed to analyze the mechanics of fibrous tissues with distributed orientations.

AB - The orientation of collagen fibers plays an important role on the mechanics of connective tissues. Connective tissues have fibers with different orientation distributions. The angular integration formulation used to model the mechanics of fibers with distributed orientation is accurate, but computationally expensive for numerical methods such as finite elements. This study presents a formulation based on pre-integrated Generalized High-Order Structure Tensors (GHOST) which greatly improves the accuracy of the predicted stress. Simplifications of the GHOST formulation for transversely-isotropic and planar fiber distributions are also presented. Additionally, the GHOST and the angular integration formulations are compared for different loading conditions, fiber orientation functions, strain energy functions and degrees of fiber non-linearity. It was found that the GHOST formulation predicted the stress of the fibers with an error lower than 10% for uniaxial and biaxial tension. Fiber non-linearity increased the error of the GHOST formulation; however, the error was reduced to negligible values by considering higher order structure tensors. The GHOST formulation produced lower errors when used with an elliptical fiber density function and a binomial strain energy function. In conclusion, the GHOST formulation is able to accurately predict the stress of fibers with distributed orientation without requiring numerous integral calculations. Consequently, the GHOST formulation may reduce the computational effort needed to analyze the mechanics of fibrous tissues with distributed orientations.

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