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Analysis of General Groundwater Flow Equation with Fractal Derivative
Published in Abdon Atangana, Mathematical Analysis of Groundwater Flow Models, 2022
Mashudu Mathobo, Abdon Atangana
In applied mathematics and mathematical analysis, the fractal derivative is a nonstandard type of derivative in which the variable, such as t, has been scaled according to tα. (Yang, 2012) argued that the fractal derivative can be categorized as a special local fractional derivative. This type of derivative can be applied to solve problems associated with a discontinuous media and equations with fractal derivatives can easily be solved. Fractal derivatives have also been employed as an alternative modeling approach to the classical Fick’s second law, where it is used to derive the linear anomalous transport-diffusion equation underlying the diffusion process (Chen et al., 2010).
Prediction of triaxial drained creep behaviors of interactive marine-terrestrial deposit soils by fractal derivative
Published in European Journal of Environmental and Civil Engineering, 2022
Gao Luchao, Dai Guoliang, Wan Zhihui, Zhu Mingxing, Zhu Wenbo
Fractal derivative theory, which involves the use of local operators without convolution integrals, has been applied in a number of studies to study the creep behaviors of materials (Chen, 2006; Chen et al., 2010a; Chen et al., 2010b; Sun et al., 2013; Liang et al., 2016; Cai et al., 2018). For instance, Cai et al. (2016) proposed fractal Maxwell and Kelvin models to analyze the creep behaviors of geotechnical materials. Yao et al. (2019) proposed a new unsteady fractal dashpot based on the fractal derivative and a novel unsteady fractal derivative creep model to study the creep behaviors of soft interlayers. Su et al. (2017) proposed fractal Maxwell and Bingham models to characterize the rheological behaviors of materials such as polymethyl methacrylate, polypropylene copolymer glass, and waxy potato starch. However, variations in sedimentary environments have made it difficult to examine the creep behaviors of IMT deposits via fractal derivative theory.
A hybrid phenomenological model for thermo-mechano-electrical creep of 1–3 piezocomposites
Published in Philosophical Magazine, 2019
The polymer matrix is viscoelastic in nature. It is passive, soft and ductile, compared to the active, hard and brittle piezoceramic fibres. Thus, a time-dependent deformation is inevitable in the epoxy matrix when acted upon by creep loads. Its passive nature is prone to creep. Hence, 1–3 piezocomposites as a whole undergoes creep evolution primarily because of passivity of epoxy matrix. From the experimental results, it is observed that the evolution is quite significant up till 600 s and the creep variables do not saturate soon, say within 100 s or so. This suggests that the material has some fluid-like behaviour in terms of electro-mechanical response. To model this, the concept of fractal geometry and scaling transformation is introduced. The concept of fractal geometry and scaling transformation gives rise to fractal calculus [35]. A fractal derivative model is developed in this context, wherein the dependent variable has an exact differential with unit order and the independent variable has a positive non-integer order between 0 and 1. This is obtained from the scaling transformations derived from theories of fractal calculus [36]. Fractal derivative viscoelastic models have been developed earlier to predict the evolution of creep variables. This subtle notion is exercised in this work. The local derivative analogy is exercised to capture the continuous evolution of creep variables. Thus the evolution equations are modified and written as