Fox H-function representations of temperature and displacement fields in unbounded domains with anomalous decelerating thermal conduction

Recent advances in fractional calculus have highlighted the role of distributed-order operators in modeling anomalous diffusion processes, particularly through bi-fractional diffusion equations of the natural type. In this work, we introduce a distributed-order fractional integral formulation that leads to a generalized bi-fractional Fourier law involving two Riemann–Liouville fractional integrals. The proposed constitutive relation captures a class of anomalous heat conduction characterized by decelerating thermal transport, wherein the effective thermal conductivity is relatively large in the short-time regime and diminishes in the long-time regime. For a quasi-static thermoelastic problem in an unbounded domain, exact analytical solutions for the temperature and displacement fields are derived and expressed in terms of the Fox H-function. Within the quasi-static framework, it is rigorously shown that the appropriate zero initial condition must be imposed on the normal stress rather than on the volumetric strain. A damped cosinusoidal boundary condition at infinity is incorporated and is shown to affect the elastic response due to the infinite propagation speed of mechanical disturbances under the quasi-static assumption. The coupled thermo-mechanical analysis reveals that thermal and mechanical fields exhibit analogous transitional behavior: Decelerating thermal conduction induces a corresponding retardation in the deformation of the medium.