| Resumo : |
The theory presented in this thesis considers not only geometric nonlinearity resulting from large deformations but also accounts for the nonlinearity arising from the interaction between polarization and electric field in a piezoelectric solid. The tensor notation (symbolic and indicial), similar to that employed in the classical continuum mechanics, is used to represent mechanical and electrical quantities in both material and spatial descriptions. Three principles that are fundamental to continuum mechanics are considered: (i) conservation of mass, (ii) balance of linear momentum, and (iii) balance of angular momentum. Since magnetic influence is neglected in piezoelectric solids, concepts associated with electrostatics are of primary concern. The volume element of a polarized dielectric is treated as a single electric dipole, which is used to define the polarization vector in each point of the material, and to derive the electric body force and electric body couple arising from the interaction between polarization and electric field. Electric quantities such as polarization, electric field, and electric displacement are defined as spatial and material quantities in the current and reference configurations, respectively. The principle of balance of energy (first law of thermodynamics) and the Clausius-Duhem inequality (a form of expressing the second law of thermodynamics) are used to derive the electro-thermo-elastic constitutive equations of the material. It is developed a piezoelectric beam theory where the displacement varies according to the Timoshenko assumption, and the electric potential varies quadratically through the thickness of each piezoelectric patch. Small displacement gradients, and consequently small strains, are considered. A mechanical-like beam finite element with superconvergence property is then obtained for linear problems. The element is integrated into a corotational algorithm to investigate plane frames under small strains but large deformations. Numerical examples are then provided to demonstrate the accuracy and effectiveness of the proposed finite element. |