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Fractional Calculus in Biomechanics: A 3D Viscoelastic Model Using Regularized Fractional Derivative Kernels with Application to the Human Calcaneal Fat Pad

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Abstract

A viscoelastic model of the K-BKZ (Kaye, Technical Report 134, College of Aeronautics, Cranfield 1962; Bernstein et al., Trans Soc Rheol 7: 391–410, 1963) type is developed for isotropic biological tissues and applied to the fat pad of the human heel. To facilitate this pursuit, a class of elastic solids is introduced through a novel strain-energy function whose elements possess strong ellipticity, and therefore lead to stable material models. This elastic potential – via the K-BKZ hypothesis – also produces the tensorial structure of the viscoelastic model. Candidate sets of functions are proposed for the elastic and viscoelastic material functions present in the model, including two functions whose origins lie in the fractional calculus. The Akaike information criterion is used to perform multi-model inference, enabling an objective selection to be made as to the best material function from within a candidate set.

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Authors and Affiliations

  1. Bio Science & Technology Branch, MS 49-3, NASA’s John H. Glenn Research Center at Lewis Field, 21000 Brookpark Road, Cleveland, OH, 44135, USA

    A. D. Freed

  2. Department of Biomedical Engineering, ND-20, The Cleveland Clinic Foundation, 9500 Euclid Avenue, Cleveland, OH, 44195, USA

    A. D. Freed

  3. GNS Gesellschaft für Numerische Simulation mbH, Am Gaußberg 2, 38114, Braunschweig, Germany

    K. Diethelm

  4. Institut Computational Mathematics, Technische Universität Braunschweig, Pockelsstraße 14, 38106, Braunschweig, Germany

    K. Diethelm

Authors
  1. A. D. Freed
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  2. K. Diethelm
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Correspondence to A. D. Freed.

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Dedicated to Prof. Ronald L. Bagley.

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Freed, A.D., Diethelm, K. Fractional Calculus in Biomechanics: A 3D Viscoelastic Model Using Regularized Fractional Derivative Kernels with Application to the Human Calcaneal Fat Pad. Biomech Model Mechanobiol 5, 203–215 (2006). https://doi.org/10.1007/s10237-005-0011-0

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  • DOI: https://doi.org/10.1007/s10237-005-0011-0

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