Fractional Order Thermoelasticity For Piezoelectric Materials

This work is aimed at establishing a unified fractional thermoelastic model for piezoelectric structures, and shedding light on the influence of different definitions on the transient responses. Theoretically, based upon Cattaneo-type equation, a unified form of fractional heat conduction law is proposed by adopting Caputo Fabrizio fractional derivative, Atangana Baleanu fractional derivative and Tempered Caputo fractional derivative. Then, thermoelastic model of fractional order is formulated for piezoelastic materials by combining the unified heat conduction law and the governing equations of elastic and electric fields. Numerically, the present theoretical model is applied to study the transient responses of piezoelectric medium that is subjected to a thermal shock. The governing equations are analytically derived and numerically solved with the aids of Laplace transform method. The obtained results are graphically illustrated, and the influences of different definitions of fractional derivative and different fractional order are revealed. This work may be helpful for understanding the multi-coupling effect of elastic, thermal and electric fields, and for inspiring further developments of fractional calculus.