New insights on fractional thermoelasticity from anomalous heat conduction

Anomalous heat transport of low-dimensional nanomaterial, e.g., divergent effective thermal conductivity, has been observed from both atomistic simulations and experimental studies. It is greatly urgent to establish the phenomenological anomalous thermoelastic model, and study the thermoelastic coupling due to strong heat transfer. The aim of this work is to revisit fractional wave-type thermoelastic models from the anomalous heat conductive viewpoint. Firstly, it has been suggested that anomalous heat conduction is due to the second sound, hence wave-type heat conduction is considered: the analogy between wave-type heat conduction and viscoelastic model is given, and thus the connection between Cattaneo-Vernotte and Green-Naghdi heat conductive models is clarified. Secondly, it has been recognized that the divergent thermal conductivity of one-dimensional systems satisfies the fractional order power law, therefore fractional derivative should be incorporated: Fractional order thermoelastic models based on Cattaneo-Vernotte and Green-Naghdi theories are summarized and compared, theoretically. Numerical investigations are conducted by using Laplace transform method, and the plot of thermoelastic responses vs. fractional order parameter shows: for all fractional order range [0, 1], fractional Cattaneo-Vernotte (FCV) I model and fractional Green-Naghdi (FGN) I-III models can predict anomalous thermoelastic responses, i.e., higher temperature and compressive stress than classical thermoelasticity. Furthermore, the history of temperature or stress indicates: FGN II model can predict anomalous responses for all time range. Further systematical studies are expected for Green-Naghdi model and its fractional versions to shed light on anomalous heat conduction and thermoelastic coupling, and to facilitate the applications of nanomaterials due to such anomalous behaviors. (Figure presented.)