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

An equivalence between generalized Maxwell model and fractional Zener model

“…Through this process, various fractional derivative models can be obtained, such as fractional Maxwell model, fractional Kelvin-Vogit model, fractional Zener model (Mainardi 2010) and more complex model as shown in Arikoglu (2014). These models have been widely adopted to describe the relaxation and creep behaviors of elastomers (Di Paola et al 2011) and natural materials (Cataldo et al 2015), dynamic behavior of biological tissue (Kohandel et al 2005) and other solids (Rossikhin and Shitikova 2010), and visco-elastic Euler-Bernoulli beam (Di Paola et al 2013). Fan et al (2015) and Yu et al (2015) have developed numerical algorithm to obtain the model parameters for fractional derivative models.…”

A numerical integration approach for fractional‐order viscoelastic analysis of hereditary‐aging structures

“…With respect to the derivation of the relevant relaxation function, it can be obtained by introducing the fractional-order creep function (9) in Equation (4), that is, J(t, t 0 ) = J F (t, t 0 ). However, the multiple integrals now involved in Equation (4) are not so trivial to be solved, especially for large numbers assigned to the variable k. A possible way to reduce the complexity of such fractional-order integrals is given by the introduction of the Grünwald-Letnikov approximation.…”

“…Moreover, the clamped beam is characterized by a unitary cross-section area and length, A = 1m 2 and L = 1m, respectively; and we suppose to apply a compressive load of 1000 kN, after 1000 days from concrete casting. One of the main advantage of considering the problem depicted in Figure 4 is that it corresponds to the definition of the creep function; therefore, the strain undergone by the clamped beam due to the load applied at t 0 is well known and can be exactly evaluated through Equation (9). The value of the creep function to be mostly used as reference solution by the following accuracy analyses is J F (10 000, 1000) = 6.57 * 10 −5 ; hence, we set t 0 = 1000 days and t f = 10 000 days.…”

“…The application of fractional calculus to linear viscoelasticity covers many decades and finds its bases with Nutting and Gemant . All studies on this topic, such as, proved the success of using fractional differential operators in the description of creep and relaxation phenomena exhibited by hereditary materials, like polymers, rubbers and biological tissues. Fractional‐order calculus was also adopted to capture the time‐dependent behavior of non‐aging concrete beams .…”

Stability and resonance conditions of second-order fractional systems

“…The transfer function (6) can be used in mechanical engineering to model behavior of viscoelastic systems, where models with fractional damping part based on springs and dash-pots are widely used, for example by Gaul et al (1991), Schiessel et al (1995), Lewandowski and Chorazyczewski (2010), Moreau et al (2010), Meral et al (2010), Castaldo (2013), Grzesikiewicz et al (2013), Dai et al (2015), Zopf (2015), Zerpa et al (2015), and Cataldo (2015). Hence, constitutive equations with fractional derivatives permit noncausal responses provoked by transient vibrations of a damping model to be avoided (Gaul et al, 1991).…”