Multi thermal waves in a thermo diffusive piezo electric functionally graded rod via refined multi-dual phase-lag model

In the present work, a novel analytical modelis provided for wave dispersion in a piezo-thermoelasticdiffusive functionally graded rod through the multi-phaselag model and thermal activation. The plain strain modelfor thermo piezoelectric functionally graded rod is consid-ered. The complex characteristic equations are obtained byusing normal mode method which satisfies the nonlinearboundary conditions of piezo-thermoelastic functionallygraded rod. The numerical calculations are carried out forcopper material. The results of the variants stress, mechan-ical displacement, temperature and electric distribution,frequency are explored against the geometric parametersand some special parameters graded index, concentrationconstants are shown graphically. The observed results willbe discuss elaborate. The results can be build reasonableattention in piezo-thermoelastic materials and smart mate-rials industry.