Exact Mechanical Hierarchy of Non-Linear Fractional-Order Hereditariness

Abstract

Non-local time evolution of material stress/strain is often referred to as material hereditariness. In this paper, the widely used non-linear approach to single integral time non-local mechanics named quasi-linear approach is proposed in the context of fractional differential calculus.The non-linear model of the springpot is defined in terms of a single integral with separable kernel endowed with a non-linear transform of the state variable that allows for the use of Boltzmann superposition. The model represents a self-similar hierarchy that allows for a time-invariance as the result of the application of the conservation laws at any resolution scale. It is shown that the non-linear springpot possess an equivalent mechanical hierarchy in terms of a functionally-graded elastic column resting on viscous dashpots with power-law decay of the material properties. Some numerical applications are reported to show the capabilities of the proposed model.

Fractional‐order poromechanics for a fully saturated biological tissue: Biomechanics of meniscus

“…The presence of biological tissues with a marked presence of hydrated collagen, as in meniscus tissue, makes, however, largely ineffective the elastic models of the material since a marked‐time dependent behavior of the representative volume element is observed also in absence of fluid‐filling material pores 13,16,27–34 . As a consequence a mathematical formulation that can be used to represent this mechanical behavior is the linear theory of material hereditariness that is nowadays described by the so‐called Fractional‐Order hereditariness (FOH).…”

“…The presence of biological tissues with a marked presence of hydrated collagen, as in meniscus tissue, makes, however, largely ineffective the elastic models of the material since a marked-time dependent behavior of the representative volume element is observed also in absence of fluid-filling material pores. 13,16,[27][28][29][30][31][32][33][34] As a consequence a mathematical formulation that can be used to represent this mechanical behavior is the linear theory of material hereditariness that is nowadays described by the so-called Fractional-Order hereditariness (FOH). In this setting the formalism of fractional-calculus, lately used in several biomechanical contexts 18,21,25,26 allows to replace the well-known constitutive equations of classical linear elasticity with their-fractional-order counterparts, involving, as additional parameters the derivation order β 0, 1 ½ .…”

Two-scale, non-local diffusion in homogenised heterogeneous media

“…On the other hand, a one-dimensional diffusion problem in a bounded homogeneous medium is studied in [76], wherein Darcy's equation is generalised with a fractional integral in space. Furthermore, in the context of hierarchical materials, such as bones and ligaments, a generalised viscoelastic approach has been proposed to describe their rheological properties by using fractional derivatives and integrals [3,30], while numerical methods have been developed for the case of hereditary-ageing materials in [16]. Additionally, we notice that in [81] the analytical and numerical solution of a generalised heat conduction equation was studied by considering a fractional time derivative instead of the first-order partial time derivative of the temperature.…”