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Power-law hereditariness of hierarchical fractal bones
Luca Deseri1, Mario Di Paola, Massimiliano Zingales
1Center for Nonlinear Analysis and Department of Mathematical Sciences, Carnegie Mellon University, 4811 Frew Street, Pittsburgh, PA 15213-3890, U.S.A.; Dipartimento di Ingegneria Civile, Ambientale e Meccanica, Universitá degli Studi di Trento, Via Mesiano 77, 38123 Trento, Italy.
Abstract:
In this paper, the authors introduce a hierarchic fractal model to describe bone hereditariness. Indeed, experimental data of stress relaxation or creep functions obtained by compressive/tensile tests have been proved to be fit by power law with real exponent 0 ⩽ β ⩽1. The rheological behavior of the material has therefore been obtained, using the Boltzmann-Volterra superposition principle, in terms of real order integrals and derivatives (fractional-order calculus). It is shown that the power laws describing creep/relaxation of bone tissue may be obtained by introducing a fractal description of bone cross-section, and the Hausdorff dimension of the fractal geometry is then related to the exponent of the power law.
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