Asymptotic analysis for Cauchy stresses in a semi-infinite space with anomalous thermal conduction

Fractional derivative is a promising mathematical tool which has been found to successfully simulate a variety of heat/mass transfer phenomena. In this work, an asymptotic analysis for a thermoelastic problem in the semi-infinite space (known as Danilovskaya problem ) is presented for temperature and stresses when the fractional Fourier law with Riemann-Liouville fractional derivative replaces the classical version. Exact formulas, in both the short-time and long-time domains, are derived in terms of the Fox H -function. All closed-form expressions are validated by comparing them with numerical results. The anomalous diffusion limit is a special case of the derived asymptotic solutions whenever the thermoelastic coupling vanishes ( ε → 0 ε → 0 ). Whilst the temperature solution shows two different behaviors in the short-time and the long-time domains in the presence of thermoelastic coupling, this transitional behavior disappears when ( ε → 0 ε → 0 ), where both the short-time and long-time solutions coincide with the one presented by the ancestors (Schneider and Wyss, 1989). The anomalous thermal conduction relationship adapted from the Compte and Metzler suggestion for anomalous diffusion coefficient, (Compte and Metzler, 1997), κ α=κ τ 1-α κ α = κ τ 1 - α , where κ κ is the classical Fourier thermal conductivity, τ τ is a characteristic time constant, and 0<α <1 0 < α < 1 , is shown to present different anomalous thermal conductivities (low thermal conduction κ α<κ κ α < κ or high “divergent” thermal conduction κ α>κ κ α > κ , α \ <\ 1 α < 1 ) depending on certain choices for the constants τ τ and α α .