Nonlinear viscous behavior of the tendon's fascicles from the homogenization of viscoelastic collagen fibers

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The level of the fascicle including collagen fibers, membranes and interstitial fluid is quite representative of the structure of tendons. We presently investigate the effect of the rheology of the components of the tendon fascicles in a multiscale analysis starting from the level of individual collagen fibers organized into bundles, and then into fascicles at the next scale. A configuration of a collagen bundle is conceived in terms of a representative unit cell including a collagen fiber surrounded by a viscous membrane and a physiological fluid. The mixing of solid and fluid components gives rise to an equivalent stress-strain response in tensor format highlighting a long term memory, in addition to instantaneous viscous effects. The kernel function of the hereditary response is determined thanks to the theory of homogenization, relying on the solution of the localization problem over the selected unit cell. Homogenization is the principal factor responsible for both the relaxation phenomena and the nonlinearity due to recruitment of fibrils observed at the fascicle level, although none of the components present at the lower scales is endowed with these properties. The nature of the matrix surrounding the collagen fiber – described as either viscous solid or a biological fluid - is shown to strongly influence the transverse response, but it has a weaker influence on the tensile response of fascicles. The initial waviness and progressive recruitment of the collagen fibers under the effect of the local strain has been integrated in the expression of the elastic and viscous stresses at the scale of the collagen fiber bundle, allowing simulating the nonlinear response of a fascicle. The computed response is able to reproduce the measured physiological response of a real tendon to a uniaxial tensile test for a proper choice of the parameters of the recruitment statistical function.