On the generalized flexothermoelasticity of a microlayer

The present work attempts to develop the Lord–Shulman (LS) generalized thermoelasticity for flexoelectric materials. To do this, the energy equation and entropy inequality were developed so that all energy components were included as well. The relation between heat flux and temperature was considered as LS assumption and in the same manner as LS theory, and the final forms of constitutive relations and heat conduction equation were extracted. Then by putting the extracted constitutive relations into the dynamic equations of flexoelasticity, the final dynamic governing generalized equations of thermo-flexoelasticity based on LS thermoelasticity model were derived. The generalized thermo-flexoelasticity equations were extracted in the present work for the first time and can be applied to general flexoelectric materials. The derived thermo-flexoelasticity equations showed new couplings between polarization and temperature and also between heat conduction equation and polarization, in such a way that by putting the LS relaxation time equal to zero, the developed generalized thermo-flexoelasticity model was reduced to classical equations of flexoelasticity with thermal effects. As a case study, a one-dimensional flexoelectric layer was considered and the generalized governing thermo-flexoelasticity equations were derived. Finally, a consistent finite element approach was employed to solve the governing equations. The results under both temperature and traction shock loading were also presented. The numerical results showed that all thermal, elastic, and electrical parameters propagated as waves with finite speed.