An External Circular Crack in an Infinite Solid Under Axisymmetric Heat Flux Loading in the Framework of Fractional Thermoelasticity

In a real solid there are different types of defects. During sudden cooling, near cracks, there can appear high thermal stresses. In this paper, the time-fractional heat conduction equation is studied in an infinite space with an external circular crack with the interior radius R in the case of axial symmetry. The surfaces of a crack are exposed to the constant heat flux loading in a circular ring R<r<ρ. The stress intensity factor is calculated as a function of the order of time-derivative, time, and the size of a circular ring and is presented graphically.

Analysis of the Properties of Fractional Heat Conduction in Porous Electrodes of Lithium-Ion Batteries

Research on the heat transfer characteristics of lithium-ion batteries is of great significance to the thermal management system of electric vehicles. The electrodes of lithium-ion batteries are composed of porous materials, and thus the heat conduction of the battery is not a standard form of diffusion. The traditional heat conduction model is not suitable for lithium-ion batteries. In this paper, a fractional heat conduction model is used to study the heat transfer properties of lithium-ion batteries. Firstly, the heat conduction model of the battery is established based on the fractional calculus theory. Then, the temperature characteristic test was carried out to collect the temperature of the battery in various operating environments. Finally, the temperature calculated by the fractional heat conduction model was compared with the measured temperature. The results show that the accuracy of fractional heat conduction model is higher than that of traditional heat conduction model. The fractional heat conduction model can well simulate the transient temperature field of the battery. The fractional heat conduction model can be used to monitor the temperature of the battery, so as to ensure the safety and stability of the battery performance.

Non-differentiable solutions for non-linear local fractional heat conduction equation

Fractional calculus has many advantages. Under consideration of this paper is a (2+1)-dimensional non-linear local fractional heat conduction equation with arbitrary degree non-linearity. Backlund transformation of a reduced form of the local fractional heat conduction equation is constructed by Painleve analysis. Based on the Backlund transformation, some exact non-differentiable solutions of the local fractional heat conduction equation are obtained. To gain more insights of the obtained solutions, two solutions are constrained to a Cantor set and then two spatio-temporal fractal structures with profiles of these two solutions are shown. This paper further reveals by local fractional heat conduction equation that fractional calculus plays important role in dealing with non-differentiable problems.