Geodesic
Caputo Derivatives in Viscoelasticity: A Non-Linear Finite-Deformation Theory for Tissue
Fractional calculus and applied analysis, Tome 10 (2007) no. 3, pp. 219-248

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The popular elastic law of Fung that describes the non-linear stress- strain behavior of soft biological tissues is extended into a viscoelastic material model that incorporates fractional derivatives in the sense of Caputo. This one-dimensional material model is then transformed into a three-dimensional constitutive model that is suitable for general analysis. The model is derived in a configuration that differs from the current, or spatial, configuration by a rigid-body rotation; it being the polar configuration. Mappings for the fractional-order operators of integration and differentiation between the polar and spatial configurations are presented as a theorem. These mappings are used in the construction of the proposed viscoelastic model.
Keywords: Hyper-Elasticity, Hypo-Elasticity, Viscoelasticity, Soft Biological Tissue, Three-Dimensional Material Model, Caputo Derivative, Polar Configuration, Fractional Polar Derivative, Fractional Polar Integral, 26A33, 74B20, 74D10, 74L15 
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     author = {Freed, Alan and Diethelm, Kai},
     title = {Caputo {Derivatives} in {Viscoelasticity:} {A} {Non-Linear} {Finite-Deformation} {Theory} for {Tissue}},
     journal = {Fractional calculus and applied analysis},
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     url = {http://geodesic.mathdoc.fr/item/FCAA_2007_10_3_a0/}
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Freed, Alan; Diethelm, Kai. Caputo Derivatives in Viscoelasticity: A Non-Linear Finite-Deformation Theory for Tissue. Fractional calculus and applied analysis, Tome 10 (2007) no. 3, pp. 219-248. http://geodesic.mathdoc.fr/item/FCAA_2007_10_3_a0/