Non-classical dynamical equations of thick plateswith complete thermomechanical coupling

Abstract

The dynamical governing equation of structures is the basis for solution of dynamics and control of distributed parameter systems. Until now, the refined dynamic equation of plates including the complete thermomechanical coupling has not been seen in the literature on structural dynamics. With the development of the modern aerospace technology, the refined dynamic equation of plates including the thermomechanical coupling effect is an urgent problem to be solved. But the traditional modeling method described in classical theory of plates and shells cannot be used to obtain the dynamic equation of thick plates under heating conditions because of the limitation of classical theory. In this paper, the coupled dynamical problem of plates with complete thermomechanical coupling is investigated based on the three-dimensional thermoelasticity and modern mathematics. The spectral composition method for modeling the structural dynamics is developed by combining the Vieta theorem of algebra with the classical method of operator spectra. In the time domain the refined dynamical equations of plates, which involve the bending and stretching vibrations under coupled thermomechanical conditions, are constructed by using the spectral decomposition of operators and proper gauge conditions to eliminate the non-uniqueness of unknown potential function without using the classical assumption in the theory of plates and shells. The dispersion relations from the refined equations of thermoelastic plates are presented graphically to test and verify the refined theory of thermoelastic plates. The space and time evolution of wave motion in heated plates and its dynamical stability are analyzed and discussed. We can see that the refined governing equations of thermoelastic plates are accurate, which would be used to solve the dynamics and control of thick plates with complete thermomechanical coupling, for instance, to investigate the coupling mechanism, coupled modes and dynamical response. And the modeling for the coupled dynamics of plates can be used in aerospace engineering for thermal protection system design.