Search results for " Fractional derivatives"

showing 3 items of 3 documents

A mechanical picture of fractional-order Darcy equation

2015

Abstract In this paper the authors show that fractional-order force-flux relations are obtained considering the flux of a viscous fluid across an elastic porous media. Indeed the one-dimensional fluid mass transport in an unbounded porous media with power-law variation of geometrical and physical properties yields a fractional-order relation among the ingoing flux and the applied pressure to the control section. As a power-law decay of the physical properties from the control section is considered, then the flux is related to a Caputo fractional derivative of the pressure of order 0 ⩽ β ≤ 1 . If, instead, the physical properties of the media show a power-law increase from the control sectio…

Numerical AnalysisAnomalous diffusionApplied MathematicsVolumetric fluxMass flowAnomalous diffusion; Anomalous scaling; Darcy equation; Fractional derivatives; Porous mediaMathematical analysisPorous mediaAnomalous diffusionFluxFractional derivativeViscous liquidDarcy–Weisbach equationFractional calculusModeling and SimulationDarcy equationSettore ICAR/08 - Scienza Delle CostruzioniPorous mediumAnomalous scalingMathematicsCommunications in Nonlinear Science and Numerical Simulation
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Free energy and states of fractional-order hereditariness

2014

AbstractComplex materials, often encountered in recent engineering and material sciences applications, show no complete separations between solid and fluid phases. This aspect is reflected in the continuous relaxation time spectra recorded in cyclic load tests. As a consequence the material free energy cannot be defined in a unique manner yielding a significative lack of knowledge of the maximum recoverable work that can extracted from the material. The non-uniqueness of the free energy function is removed in the paper for power-laws relaxation/creep function by using a recently proposed mechanical analogue to fractional-order hereditariness.

Work (thermodynamics)Materials scienceMaterial stateFractional orderMaterial scienceSpectral lineDissipation rateMaterials Science(all)Modelling and SimulationGeneral Materials ScienceComplex materials; Continuous relaxation; Dissipation rates; Fractional derivatives; Fractional order; Free energy function; Material science; Power law creepFree energyPower-law creep/relaxationComplex materialbusiness.industryMechanical EngineeringApplied MathematicsRelaxation (NMR)Order (ring theory)Free energy functionFractional derivativesStructural engineeringFunction (mathematics)MechanicsFractional derivativeCondensed Matter PhysicsFractional calculusContinuous relaxationCreepMechanics of MaterialsModeling and SimulationPower law creepbusinessSettore ICAR/08 - Scienza Delle CostruzioniEnergy (signal processing)International Journal of Solids and Structures
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Response of nonlinear oscillators with fractional derivative elements under evolutionary stochastic excitations: A Path Integral approach based on La…

2023

In this paper, an approximate analytical technique is developed for determining the non-stationary response amplitude probability density function (PDF) of nonlinear/hysteretic oscillators endowed with fractional element and subjected to evolutionary excitations. This is achieved by a novel formulation of the Path Integral (PI) approach. Specifically, a stochastic averaging/linearization treatment of the original fractional order governing equation of motion yields a first-order stochastic differential equation (SDE) for the oscillator response amplitude. Associated with this first-order SDE is the Chapman–Kolmogorov (CK) equation governing the evolution in time of the non-stationary respon…

Path Integral Laplace’s method of integration Evolutionary excitation Fractional derivativesNuclear Energy and EngineeringMechanical EngineeringAerospace EngineeringOcean EngineeringStatistical and Nonlinear PhysicsSettore ICAR/08 - Scienza Delle CostruzioniCondensed Matter PhysicsCivil and Structural EngineeringProbabilistic Engineering Mechanics
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