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

Yuriy Povstenko,Bożena Woźna-Szcześniak,Renata Kawa,Andrzej Yatsko,Tamara Kyrylych

doi:10.3390/e24010070

Yuriy Povstenko, Bożena Woźna-Szcześniak + Show 3 more

Open Access

https://doi.org/10.3390/e24010070

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Abstract

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 . 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.

Highlights

A real solid, as a rule, contains a large number of different type defects: point defects, dislocations, disclinations, slits, inclusions, holes, and cracks
The Laplace transform with respect to time t will be marked by an asterisk, the Hankel transform of the order zero with respect to the radial coordinate r will be specified by a hat, and the Fourier cosine transform with respect to the coordinate z will be denoted by a tilde (s, ξ, and η are the Laplace, Hankel, and Fourier transform variables, respectively)
We have solved the time-fractional heat conduction equation in an infinite solid with an external circular crack with the interior radius R under constant axisymmetric heat flux acting in a circular domain R < r < ρ

Summary

Introduction

A real solid, as a rule, contains a large number of different type defects: point defects, dislocations, disclinations, slits, inclusions, holes, and cracks. It should be emphasized that in the constitutive equations for the heat flux (3) and (4), the generalized thermal conductivity k has the physical dimension [k] =. Whereas the generalized thermal diffusivity a in the time-fractional heat conduction Equation (7) has the physical dimension m2. Kernel K (t − τ ) in the constitutive Equation (1) is expressed in terms of the Mittag-Leffler functions being the generalization of the exponential function In this case, we arrive at the time-fractional telegraph equations and the associated theories of thermal stresses [60,61]. We expand the previous studies [68,69,70,71] on the case of an external circular crack with the interior radius R in an infinite solid under the prescribed heat flux at its surfaces. 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

Formulation of the Problem

The Temperature Field

The Stress Intensity Factor

Concluding Remarks