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The steady-state forced vibration of a functionally graded piezoelectric material beam under different loading (electrical, mechanical, and electromechanical) cases and different boundary conditions is studied. Sixteen kinds of situations are considered in the present article. The steady-state responses of the beams under forced mechanical or electric load are obtained by using the state-space-based differential quadrature method. In contrast to other investigations, the present work gets the results by a discrete method using the differential quadrature technique, which can easily get stable results. The results agree very well with other investigations found in some special cases. Finally, some properties of the beams under the corresponding load conditions are discussed.

Blends of linear low density polyethylene (LLDPE) and linear low density polyethylene grafted polyether pentaerythritol mono-maleate (LLDPE-g-PPMM) were prepared by melt mixing. The surface of LLDPE/LLDPE-g-PPMM films with different contents of LLDPE-g-PPMM was characterized through contact angle and FT-IR spectroscopy. The tensile properties and light transmission properties of extruded films, as well as the performance of these films compared with commercial anti-fog films, were determined. The carbonyl polar groups on the surface of LLDPE/ LLDPE-g-PPMM films increased, and the contact angles of water and glycerol decreased when the content of LLDPE-g-PPMM increased. LLDPE/LLDPE-g-PPMM films showed a noticeable reduction in water drop formation as the LLDPE-g-PPMM content was increased. The transmittancy and haze of LLDPE/LLDPE-g-PPMM films were improved when using increased contents of PPMM, which promotes better wetting of the water on the surface.

Most of the previous investigations on the static performances of piezoelectric bimorph cantilevers are based on the elementary theory of piezo-elasticity. In fact, one fundamental problem has not been solved yet from the view of the theory of piezo-elasticity, that is, a piezoelectric bimorph cantilever covered fully with electrodes on the upper and lower surfaces and subjected to a transversely concentrated load at the free end of the beam. The major obstacle to find the solution is the equipotential conditions on both the upper and lower surfaces of the cantilever. In this article, by establishing suitable boundary conditions and introducing an appropriate Airy stress function, the analytical solutions of a piezoelectric bimorph cantilever under three different loading conditions have been obtained. Numerical analysis is also conducted and the results are consistent with the present theoretical predictions. This investigation amends the corresponding works given by other investigations based on the elementary theory. It is found that the amendments to the end deflections of a PZT-4 piezoelectric bimorph cantilever caused by an end shear force, an end moment, or a voltage difference between the upper and lower surfaces are about 10.46%, 12.78%, and 3.52%, respectively. In addition, it is also found that these amendments depend just on the material parameters, have nothing to do with the geometric parameters of the bimorph cantilever. The Airy stress function proposed in this article can also be used to study other different piezoelectric cantilevers including multilayer piezoelectric cantilevers under corresponding loads.

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