An Accelerated Newmark Scheme for Integrating the Equation of Motion of Nonlinear Systems Comprising Restoring Elements Governed by Fractional Derivatives

Visco-elastic behavior through fractional calculus: An easier method for best fitting experimental results

“…The remarkable result in Eq. (13) for the constitutive law of a visco-elastic material directly descends in assuming that the kernel in the convolution integral is of power law type (Schiessel and Blumen, 1993;Bagley and Torvik, 1979, 1983a,b, 1986Evangelatos and Spanos, 2011;Schmidt and Gaul, 2002;Di Paola et al, 2011). Then we may affirm that integral equation with kernel of exponential type leads to ordinary differential equation, while when the kernel is of power law type leads to fractional differential equation.…”

Bending test for capturing the vivid behavior of giant reeds, returned through a proper fractional visco-elastic model

“…In all other cases, results in terms of the Riemmann-Liouville or Caputo's fractional derivative are quite different to each another, and fractional differential equations involving Riemann-Liouville fractional derivative show some inconsistencies in terms of initial conditions (Samko et al, 1993;Podlubny, 1999;Hilfer, 2000;Evangelatos and Spanos, 2011). Contrary, such a problem disappears when working in terms of Caputo's fractional derivative.…”

“…(i) solving the fractional differential equations of the continuous Euler-Bernoulli beam (since fractional operators are involved and readers are referred to Caputo and Mainardi (1971), Gonsovski and Rossikhin (1973), Stiassnie (1979), Bagley and Torvik (1983), Bagley and Torvik (1986), Samko et al (1993), Podlubny (1999), Hilfer (2000), Schmidt and Gaul (2002), Mainardi and Gorenflo (2007), Mainardi (2010), Evangelatos and Spanos (2011));…”

Bending test for capturing the fractional visco-elastic parameters: Theoretical and experimental investigation on giant reeds

“…For these reasons in the second part of the last century a lot of researches have been carried out enforcing the knowledge of fractional hereditary materials (Caputo and Mainardi [12], Gonsovski and Rossikhin [13], Stiassnie [14], Bagley and Torvik [15][16], Schmidt and Gaul [17], Mainardi and Gorenflo [18], Mainardi [19], Evangelatos and Spanos [20]). Another fractional model to capture the relaxation function is the fractional Kelvin-Voigt model depicted in Fractional Kelvin-Voigt Model whose constitutive law is given by…”