Fractional-Order Theory of Thermoelasticity. II: Quasi-Static Behavior of Bars

Abstract

This work aims to shed light on the “thermally-anomalous” coupled behavior of slightly deformable bodies, in which the strain is additively decomposed in an elastic contribution and in a thermal part. The macroscopic heat flux turns out to depend upon the time history of the corresponding temperature gradient, and this is the result of a multiscale rheological model developed in Part I of the present study, thereby resembling a long-tail memory behavior governed by a Caputo’s fractional operator. The macroscopic constitutive equation between the heat flux and the time history of the temperature gradient does involve a power law kernel, resulting in the “anomaly” mentioned previously. The interplay between such a thermal flux and elastic and thermal deformability are investigated for a pinned-pinned truss. This allows a focus on the effects of the deviation from Fourier’s law on the thermoelastic coupling. Indeed, the interactions in the presented system are fully coupled because the temperature and displacement field mutually influence one another.