Transient Thermoelastic Fields in a Transversely Isotropic Infinite Solid With a Penny-Shaped Crack

1987 ◽  
Vol 54 (4) ◽  
pp. 854-860 ◽  
Author(s):  
N. Noda ◽  
F. Ashida
Keyword(s):  
Heat Conduction ◽  
Heat Conduction Equation ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Transient Heat Conduction ◽  
Thermoelastic Problem ◽  
Conduction Equation ◽  
Thermal Stress Field ◽  
Penny Shaped Crack ◽  
Shaped Crack

The present paper deals with a transient thermoelastic problem for an axisymmetric transversely isotropic infinite solid with a penny-shaped crack. A finite difference formulation based on the time variable alone is proposed to solve a three-dimensional transient heat conduction equation in an orthotropic medium. Using this formulation, the heat conduction equation reduces to a differential equation with respect to the spatial variables. This formulation is applied to attack the transient thermoelastic problem for an axisymmetric transversely isotropic infinite solid containing a penny-shaped crack subjected to heat absorption and heat exchange through the crack surface. Thus, the thermal stress field is analyzed by means of the transversely isotropic potential function method.

Fractional thermoelasticity problem for an infinite solid with a penny-shaped crack under prescribed heat flux across its surfaces

2020 ◽  
Vol 378 (2172) ◽  
pp. 20190289 ◽  
Author(s):  
Y. Povstenko ◽  
T. Kyrylych
Keyword(s):  
Heat Flux ◽  
Heat Conduction ◽  
Stress Field ◽  
Fractional Calculus ◽  
Heat Conduction Equation ◽  
Thermoelasticity Problem ◽  
Conduction Equation ◽  
Penny Shaped Crack ◽  
Shaped Crack ◽  
Fractional Heat Conduction

The time-nonlocal generalization of the Fourier law with the ‘long-tail’ power kernel can be interpreted in terms of fractional calculus and leads to the time-fractional heat conduction equation with the Caputo derivative. The theory of thermal stresses based on this equation was proposed by the first author ( J. Therm. Stresses 28 , 83–102, 2005 ( doi:10.1080/014957390523741 )). In the present paper, the fractional heat conduction equation is solved for an infinite solid with a penny-shaped crack in the case of axial symmetry under the prescribed heat flux loading at its surfaces. The Laplace, Hankel and cos-Fourier integral transforms are used. The solution for temperature is obtained in the form of integral with integrands being the generalized Mittag-Leffler function in two parameters. The associated thermoelasticity problem is solved using the displacement potential and Love’s biharmonic function. To calculate the additional stress field which allows satisfying the boundary conditions at the crack surfaces, the dual integral equation is solved. The thermal stress field is calculated, and the stress intensity factor is presented for different values of the order of the Caputo time-fractional derivative. A graphical representation of numerical results is given. This article is part of the theme issue ‘Advanced materials modelling via fractional calculus: challenges and perspectives’.

Analytical Solution for Three-Dimensional Hyperbolic Heat Conduction Equation with Time-Dependent and Distributed Heat Source

2016 ◽  
Vol 33 (1) ◽  
pp. 65-75 ◽  
Author(s):  
M. R. Talaee ◽  
V. Sarafrazi
Keyword(s):  
Heat Conduction ◽  
Analytical Solution ◽  
Heat Source ◽  
Heat Conduction Equation ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Closed Form Solution ◽  
Time Dependent ◽  
Conduction Equation ◽  
Hyperbolic Heat Conduction

AbstractThis paper is devoted to the analytical solution of three-dimensional hyperbolic heat conduction equation in a finite solid medium with rectangular cross-section under time dependent and non-uniform internal heat source. The closed form solution of both Fourier and non-Fourier profiles are introduced with Eigen function expansion method. The solution is applied for simple simulation of absorption of a continues laser in biological tissue. The results show the depth of laser absorption in tissue and considerable difference between the Fourier and Non-Fourier temperature profiles. In addition the solution can be applied as a verification branch for other numerical solutions.

Sign in / Sign up

Export Citation Format

Share Document