The elastic problem for fractal media: basic theory and finite element formulation

Heat transfer through fractal-like porous media: A tessellated continuum approach

“…(20) and (22) is achieved with w r ¼ w s and F À1 Á v r ¼ v s . Similarly, the right-hand sides of Eqs.…”

“…The k indicating the kth generation of pre-fractal has been dropped for convenience. Equation (20) can be related to Eq. (21) via the hole-fill map; geometrical considerations give rise to Nanson's differential relationships: …”

“…Transport approaches involving fractals have been considered by Tarasov [13,14] and more recently by Ostoja-Starzewski [15,16]. Their approach (like others [17][18][19][20]) is founded on the application of fractional derivatives. These theories are untested and physically unrealisable in many respects since fractional derivatives do not readily arise from the governing physics.…”

Wave Propagation in Heterogeneous Media with Local and Nonlocal Material Behavior

“…Use of fractional calculus on a fractal set in solid mechanics is first presented in [9] where the Kolwankar-Gangal derivative [28] is suggested as a fractional derivative and the virtual work principle is proposed for a finite element solution. The elastic potential for fractal media is proposed in [10] and scaling properties of kinematic and equilibrium equations are investigated with examples given on the Cantor set. Balance laws for thermoelastic fractal media are presented in [39] based on the local fractional derivative proposed in [50].…”

Nonlinear dynamics for local fractional Burgers’ equation arising in fractal flow

“…Recently, the local fractional calculus was successfully applied to non-differentiable problems arising in the areas of solid mechanics [33], heat transfer and wave propagation [34], diffusion [35], hydrodynamics [36], vehicular traffic flow [37] and other topics [38][39][40][41][42] (see also references therein).…”