Градиентная модель изгиба неоднородной пьезоэлектрической балки

Abstract

The problem of bending a horizontally inhomogeneous piezoelectric cantilever beam is studied taking into account scale effects. The lower and upper surfaces of the beam are electroplated. Three types of loading are considered: load uniformly distributed along the length; transverse force at the other end of the beam; supply of electric potential to the upper electrode. The beam bending is modeled based on the Euler-Bernoulli hypotheses and the quadratic distribution of the electric potential. Based on the application of the variational principle of gradient electroelasticity, a system of differential equations of bending and electrostatics is obtained, as well as an extended range of boundary conditions. To find the bending moment and deflection of the middle line of a homogeneous beam, exact analytical expressions are obtained. In the case of an inhomogeneous beam at large values of the scale parameter, the solution is based on the shooting method. On specific examples, the calculations of moments and deflection are carried out, both in the case of a homogeneous and inhomogeneous beam. The influence of the mechanical and electrostatic gradient parameters, the electromechanical coupling coefficient, and the inhomogeneity parameter on the distribution of deflections has been clarified.