A non-linear thermo-viscoelastic rheological model based on fractional derivatives for high temperature creep in concrete

Fractional viscoelastic models for power-law materials

“…Further examples include biological materialse.g. tissues, [29][30][31] single cells, 13,29,[32][33][34][35][36] intra and extra cellular components; 10,37,38 gels, 39,40 polymers, 9,41 concrete, 42,43 asphalt, [44][45][46] ice, 9 and food -e.g. cheese, 11,47 dough.…”

“…52,53 This led to the development of a new formalism for the modelling of viscoelastic materials known as fractional viscoelasticity. Fractional viscoelasticity has been applied to complex geological and construction materials such as bitumen (asphalt), 44,45 concrete, 42,43 rock mass, [54][55][56][57][58][59] waxy crude oil, 60,61 as well as polymers and gels, 41,[62][63][64][65] and food. 11 Numerous examples can also be found of fractional viscoelasticity applied to biological materials such as epithelial cells, 66 breast tissue cells, 67,68 lung parenchyma, 69 blood flow, 70,71 as well as red blood cell membranes.…”

A Review on Variable-Order Fractional Differential Equations: Mathematical Foundations, Physical Models, Numerical Methods and Applications

“…In addition, the evidence of the VO nature of the particles dynamics flows was provided by the behavior of an oscillating particle in a fluid [31,32]. Subsequently, Bouras et al [14] developed a novel non-linear thermo-viscoelastic rheological model based on VO time fractional derivative for high temperature creep in concrete, which can be expressed as…”

A full probabilistic solution of the random linear fractional differential equation via the random variable transformation technique

“…In this spirit, our subsequent presentation will be more concise since it follows in broad outline a similar structure as in example 1. Let us consider the random fractional IVP (2), where α=0.7, λ(ω) is a Gaussian random variable with mean 1 and standard deviation 0.1, λ(ω) ∼ N(1;0.1); γ(ω) has a Uniform distribution on interval [2,3], γ(ω) ∼ Un(2,3); and β 0 (ω) has a Gamma distribution with parameters 1 and 2, β 0 ∼ Ga(1;2). Hereinafter, we assume that λ(ω), γ(ω) and β 0 (ω) are independent; this ensures hypothesis H1 is fulfilled.…”

“…Apart from the intrinsec interest of fractional calculus as an extension of the Newton-Leibniz calculus, hence providing smart generalizations of classical results, 1,2 a number of contributions has shown its potentiality to model problems where "memory" plays a key role into the modelling process. Specific examples can be found in different realms, for example, in engineering, where appear problems of viscoelasticity, electromagnetism, etc, whose answers depend upon memory and hereditary properties of materials 3,4 and in Epidemiology, where competition dynamics may reinforce certain genogroups by DNA recombination or mutations, and this would depend on the other genogroups coexisting with them as well as the time this coexistence lasts and their populations. 5 On the one hand, the aforementioned strong impact of fractional differential equations in mathematical modelling applications and, on the other hand, the need of quantifying uncertainty involved in measurements or surveys used to fix the input parameters of fractional differential equations lead to 2 main classes of fractional differential equations with uncertainty, namely, stochastic fractional differential equations (SFDEs) and random fractional differential equations (RFDEs).…”