Fractional heat conduction equation and associated thermal stress

A quasi-static uncoupled theory of thermoelasticity based on the heat conduction equation with a time-fractional derivative of order alpha is proposed. Because the heat conduction equation in the case 1 less than or equal to alpha less than or equal to 2 interpolates the parabolic equation (alpha = 1) and the wave equation (alpha = 2), the proposed theory interpolates a classical thermoelasticity and a thermoelasticity without energy dissipation introduced by Green and Araghdi. The Caputo fractional derivative is used. The stresses corresponding to the fundamental solutions of a Cauchy problem for the fractional heat conduction equation are found in one-dimensional and two-dimensional cases.