Energy dissipation for hereditary and energy conservation for non-local fractional wave equations

Abstract

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’.

Fractional Burgers wave equation on a finite domain

“…On the other hand, the profiles in the case of Model VII, being also damped oscillatory, resemble to the sequence of two excitation processes followed by two relaxation processes, since profiles repeatedly change their convexity from concave to convex, as clearly visible from Figure 7b. Again, good agreement between curves obtained analytically through (59), using parameters from Table 1, and by ab initio numerical Laplace transform inversion is observed.…”

“…The particular interest is the behavior of displacement and stress, obtained as a response to the boundary conditions assumed as the Heaviside step function, since the stress for prescribed displacement of rod's free end correspond to the relaxation modulus, while displacement for prescribed stress acting on rod's free end correspond to the creep compliance, that are studied in [47] for the thermodynamically consistent Burgers models. Writing the constitutive equation of viscoelastic body in terms of relaxation modulus found application in proving the dissipativity properties of the hereditary fractional wave equations using a priori energy estimates in [59]. Note, the relaxation modulus represents the time-evolution of stress, obtained from the constitutive equation for strain prescribed as the step function, while the creep compliance represents the time-evolution of strain, obtained from the constitutive equation for stress prescribed as the step function.…”

Advanced materials modelling via fractional calculus: challenges and perspectives

“…Numerical results on 2D structure prove that such operators are intrinsically capable of capturing the complex linear-to-nonlinear dynamic transitions resulting from the translation of dislocations as well as the creation and annihilation of bonds between particles. Finally, the contribution to the theme issue of Zorica & Oparnica [105] focuses on timefractional-wave equations modelling hereditary viscoelastic behaviour and space-fractional-wave equations associated with existing non-local elasticity models. For a number of fractional-wave equations, the authors provide rigorous mathematical evidence of energy dissipation and energy conservation.…”

Theoretical Analysis of Fractional Viscoelastic Flow in Circular Pipes: General Solutions

“…The comprehensive overview summarizing state-of-the-art practical applications of FC has been recently published by The Royal Society Publishing. The sixteen-paper issue entitled "Advanced materials modeling via fractional calculus: challenges and perspectives" [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27] covers applications of constant-order (CO) and variable-order (VO) fractional differential operators to several fundamental phenomena. These include anomalous diffusion, [13,16] heat conduction [14,27], fractional viscoelasticity of fluids [19], and materials [12,18,22].…”

“…The sixteen-paper issue entitled "Advanced materials modeling via fractional calculus: challenges and perspectives" [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27] covers applications of constant-order (CO) and variable-order (VO) fractional differential operators to several fundamental phenomena. These include anomalous diffusion, [13,16] heat conduction [14,27], fractional viscoelasticity of fluids [19], and materials [12,18,22]. The approach to model viscoelastic properties of materials with VO FC operators is undoubtedly among the most promising ones, as it allows for the consideration of fractional order dynamics with respect to time, space, and material variables [22].…”