Using the method of a priori energy estimates, energy dissipation is proved for the class of hereditary fractional wave equations, obtained through the system of equations consisting of equation of ...motion, strain and fractional order constitutive models, that include the distributed-order constitutive law in which the integration is performed from zero to one generalizing all linear constitutive models of fractional and integer orders, as well as for the thermodynamically consistent fractional Burgers models, where the orders of fractional differentiation are up to the second order. In the case of non-local fractional wave equations, obtained using non-local constitutive models of Hooke- and Eringen-type in addition to the equation of motion and strain, a priori energy estimates yield the energy conservation, with the reinterpreted notion of the potential energy.
This article is part of the theme issue ‘Advanced materials modelling via fractional calculus: challenges and perspectives’.
In this paper, searching for a better chloride ions sub-diffusion system, a multi-term time-fractional derivative diffusion model is proposed for the description of the time-dependent chloride ions ...penetration in reinforced concrete structures exposed to chloride environments. We prove the stability and convergence of the model. We use the modified grid approximation method (MGAM) to estimate the fractional orders and chloride ions diffusion coefficients in the reinforced concrete for the multiterm time fractional diffusion system. And then to verify the efficiency and accuracy of the proposed methods in dealing with the fractional inverse problem, two numerical examples with real data are investigated. Meanwhile, we use two methods of fixed chloride ions diffusion coefficient and variable diffusion coefficient with diffusion depth to simulate chloride ions sub-diffusion system. The result shows that with the new fractional orders and parameters, our multi-term fractional order chloride ions sub-diffusion system is capable of providing numerical results that agree better with the real data than other models. On the other hand, it is also noticed from the numerical solution of the chloride ions sub-diffusion system that setting the variable diffusion coefficient with diffusion depth is more reasonable. And it is also found that chloride ions diffusion coefficients in reinforced concrete should be decreased with diffusion depth which is completely consistent with the theory. In addition, the model can be used to predict the chloride profiles with a time-dependent property.
This article is part of the theme issue ‘Advanced materials modelling via fractional calculus: challenges and perspectives’.
A three-branch viscoelastic model based on fractional derivatives is proposed for the viscoelastic behaviours of solid propellants. The simulation results show a satisfactory agreement with the ...stress relaxation modulus and complex modulus of solid propellants. As a comparison, the static modulus is also characterized by traditional viscoelastic model with integer-order derivatives. Results show that the application of the fractional derivatives to the viscoelastic constitutive model can effectively reduce the number of the required parameters while giving an accurate prediction of viscoelastic behaviours of solid propellants. Moreover, a simple and effective direct search method based on simulated annealing and Powell’s mothed is proposed for the data fitting.
This article is part of the theme issue ‘Advanced materials modelling via fractional calculus: challenges and perspectives’.
In this paper the authors introduce a nonlinear model of fractional-order hereditariness used to capture experimental data obtained on human tendons of the knee. Creep and relaxation data on fibrous ...tissues have been obtained and fitted with logarithmic relations that correspond to power-laws with nonlinear dependence of the coefficients. The use of a proper nonlinear transform allows one to use Boltzmann superposition in the transformed variables yielding a fractional-order model for the nonlinear material hereditariness. The fundamental relations among the nonlinear creep and relaxation functions have been established, and the results from the equivalence relations have been contrasted with measures obtained from the experimental data. Numerical experiments introducing polynomial and harmonic stress and strain histories have been reported to assess the provided equivalence relations.
This article is part of the theme issue ‘Advanced materials modelling via fractional calculus: challenges and perspectives’.
This paper addresses nonlinear viscoelastic behaviour of fractional systems with variable time-dependent fractional order. In this case, the main challenge is that the Boltzmann linear superposition ...principle, i.e. the theoretical basis on which linear viscoelastic fractional operators are formulated, does not apply in standard form because the fractional order is not constant with time. Moving from this consideration, the paper proposes a novel approach where the system response is derived by a consistent application of the Boltzmann principle to an equivalent system, built at every time instant based on the fractional order at that instant and the response at all the previous ones. The approach is readily implementable in numerical form, to calculate either stress or strain responses of any fractional system where fractional order may change with time.
This article is part of the theme issue ‘Advanced materials modelling via fractional calculus: challenges and perspectives’.
The relationship between fractional-order heat conduction models and Boltzmann transport equations (BTEs) lacks a detailed investigation. In this paper, the continuity, constitutive and governing ...equations of heat conduction are derived based on fractional-order phonon BTEs. The underlying microscopic regimes of the generalized Cattaneo equation are thereafter presented. The effective thermal conductivity κeff converges in the subdiffusive regime and diverges in the superdiffusive regime. A connection between the divergence and mean-square displacement 〈|Δx|²〉∼tγ
is established, namely, κeff ~t
γ−1, which coincides with the linear response theory. Entropic concepts, including the entropy density, entropy flux and entropy production rate, are studied likewise. Two non-trivial behaviours are observed, including the fractional-order expression of entropy flux and initial effects on the entropy production rate. In contrast with the continuous time random walk model, the results involve the non-classical continuity equations and entropic concepts.
This article is part of the theme issue ‘Advanced materials modelling via fractional calculus: challenges and perspectives’.
The time-nonlocal generalization of the Fourier law with the ‘long-tail’ power kernel can be interpreted in terms of fractional calculus and leads to the time-fractional heat conduction equation with ...the Caputo derivative. The theory of thermal stresses based on this equation was proposed by the first author (J. Therm. Stresses 28, 83–102, 2005 (doi:10.1080/014957390523741)). In the present paper, the fractional heat conduction equation is solved for an infinite solid with a penny-shaped crack in the case of axial symmetry under the prescribed heat flux loading at its surfaces. The Laplace, Hankel and cos-Fourier integral transforms are used. The solution for temperature is obtained in the form of integral with integrands being the generalized Mittag-Leffler function in two parameters. The associated thermoelasticity problem is solved using the displacement potential and Love’s biharmonic function. To calculate the additional stress field which allows satisfying the boundary conditions at the crack surfaces, the dual integral equation is solved. The thermal stress field is calculated, and the stress intensity factor is presented for different values of the order of the Caputo time-fractional derivative. A graphical representation of numerical results is given.
This article is part of the theme issue ‘Advanced materials modelling via fractional calculus: challenges and perspectives’.
Fractional-order modelling of epoxy resin Machado, J. A. Tenreiro; Lopes, António M.; de Camposinhos, Rui
Philosophical transactions of the Royal Society of London. Series A: Mathematical, physical, and engineering sciences,
05/2020, Volume:
378, Issue:
2172
Journal Article
Peer reviewed
This paper describes epoxy resins by means of electrical impedance spectroscopy (EIS) and the mathematical tool of fractional calculus (FC). Two stages are considered: first, the EIS is used for ...testing the samples and, second, the measured data are approximated using integer and fractional order models. The FC-based modelling describes the epoxy resins using a small number of parameters that reflect their main characteristics. The EIS data gathered for the epoxies samples are compared with those of different adhesives and sealants by means of a hierarchical clustering algorithm that unravels the relationships between the distinct materials.
This article is part of the theme issue ‘Advanced materials modelling via fractional calculus: challenges and perspectives’.
Thermo-poromechanics of fractal media Li, Jun; Ostoja-Starzewski, Martin
Philosophical transactions of the Royal Society of London. Series A: Mathematical, physical, and engineering sciences,
05/2020, Volume:
378, Issue:
2172
Journal Article
Peer reviewed
This article advances continuum-type mechanics of porous media having a generally anisotropic, productlike fractal geometry. Relying on a fractal derivative, the approach leads to global balance laws ...in terms of fractal integrals based on product measures and, then, converting them to integer-order integrals in conventional (Euclidean) space. Proposed is a new line transformation coefficient that is frame invariant, has no bias with respect to the coordinate origin and captures the differences between two fractal media having the same fractal dimension but different density distributions. A continuum localization procedure then allows the development of local balance laws of fractal media: conservation of mass, microinertia, linear momentum, angular momentum and energy, as well as the second law of thermodynamics. The product measure formulation, together with the angular momentum balance, directly leads to a generally asymmetric Cauchy stress and, hence, to a micropolar (rather than classical) mechanics of fractal media. The resulting micropolar model allowing for conservative and/or dissipative effects is applied to diffusion in fractal thermoelastic media. First, a mechanical formulation of Fick’s Law in fractal media is given. Then, a complete system of equations governing displacement, microrotation, temperature and concentration fields is developed. As a special case, an isothermal model is worked out.
This article is part of the theme issue ‘Advanced materials modelling via fractional calculus: challenges and perspectives’.
We investigate, in the distributional setting, the restrictions on the constitutive equation for a fractional Burgers model of viscoelastic fluid that follow from the weak form of the entropy ...inequality under isothermal conditions. The results are generalized, from the Burgers model, to an arbitrary class of linear constitutive equations with fractional derivatives. Our results show that the restrictions obtained here on the coefficients of constitutive equations are weaker when compared with the restrictions obtained by Bagley–Torvik method. We show the precise relation between restrictions derived here and those derived by Bagley–Torvik. We deal with the creep test, for the case when Bagley–Torvik conditions are violated, and new conditions obtained in this work are satisfied. The results show a qualitative difference in the form of creep function.
This article is part of the theme issue ‘Advanced materials modelling via fractional calculus: challenges and perspectives’.
