Keywords: thermopiezomechnics, nonlinear coupling, finite element method, thin-walled structures.

The present paper is devoted to three-dimensional coupled thermo-electro-elasticity theory and its application to two-dimensional smart structures consisting of a master structure with surface bonded piezoelectric layers or patches. To this end a thermodynamically consistent continuum mechanics theory is derived that is based on the conservation of mass, linear and angular momentum, electric charge and energy. The second principle of thermodynamics is used to derive the restrictions for the constitutive equations by means of Coleman-Noll analysis. A functional for the Gibb's free energy is chosen that is quadratic in all independent quantities, except of the term depending solely on the temperature changes. As a result, linear constitutive relations evolve for the second Piola-Kirchhoff stress tensor and the electric displacement vector, and a logarithmic dependency of the entropy on the temperature to account for large temperature changes. The resulting field equations account for the geometrical, electrical and thermal nonlinear effects, the thermo-electro-mechanical coupling effects, as well as the instationary thermal and electric effects. This three-dimensional coupled thermo-electro-elasticity theory is more general than others available in literature, in which usually some linearisation is performed with respect to small deformations, small temperature changes and/or small electric field.

Concerning smart structures, in the vast majority of papers available in the literature coupling of the mechanical, electrical, and eventually thermal field quantities is taken into account in the constitutive equations only. The truly coupled thermo-piezo-elastic analysis, however, does not only account for the temperature and piezoelectric effect on the strains, but is based on the coupling of the mechanical, electrical and thermal quantities in the field equations. So for example it is well known that due to thermo-elastic coupling structures exhibit several non-classical physical effects, like strain rate dependent change of temperature due to tension and compression or damping of vibrations due to heat loss. If additionally piezoelectric effects are considered, the heat conduction equation will depend on the mechanical and electrical field variables as well. Here, our earlier works on geometrically nonlinear theory for piezo-integrated plates and shells are extended by using the thermodynamically consistent continuum mechanics based framework described above and thus includes thermo-piezo-elastic coupling as well as geometrical, electrical and thermal nonlinear effects. Based on weak formulations of the conditions of equilibrium, conservation of electric charge and energy a finite element for plates and shells with integrated piezoelectric layers or patches is developed. The transition from three-dimensional continuum mechanics to two-dimensional theories of plates and shells is performed by using the first-order transverse shear deformation hypothesis, while the transverse electric potential and temperature distribution are assumed to be quadratic and cubic, respectively. The latter is chosen such that various boundary conditions can be accurately imposed, including adiabatic surfaces. The resulting two-dimensional theory is implemented into a finite element exhibiting no kinematical restrictions with respect to the magnitude of strains and rotations. The director is assumed to be inextensible, the rotations are parametrised with the Rodriguez formulation. In order to overcome various types of locking, an assumed natural strain combined with an enhanced assumed strain procedure has been used. The various nonlinear and coupling effects are demonstrated for a piezolaminated steel plate subjected to thermal shock.

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