Hybrid finite element formulation for electrostrictive materials: static and buckling analysis of beam

A nonlinear electromechanical coupled static finite element formulation for electrostrictive materials is proposed. This formulation includes the quadratic dependence of strain with polarization, valid at a constant temperature and excludes hysteresis. The present formulation uses linear finite element analysis for stress and strain evaluation along with the numerical solution of the nonlinear constitutive equation using Newton - Raphson technique only within each electrostrictive elements and hence this formulation is named as hybrid finite element formulation. Polarization is an explicit independent variable in this formulation and the nonlinear equations at each electrostrictive elements are solved for this variable. Since the nonlinear constitutive equation is a function of polarization and tends to infinity for certain values of polarization, the Newton - Raphson technique is specially modified in order to guarantee the convergence of the solution. A simple technique for obtaining the initial guess of the solution for Newton - Raphson technique is also proposed which gives faster convergence of the solution. Since the polarization is an explicit independent variable in the present formulation, the assumption, made in most of the finite element formulations [15, 16, 17, 18], that polarization is approximately equal to electric displacement has been relaxed. The developed static finite element formulation has been extended to solve buckling problems of electrostrictive beams. Analytical solutions have been developed for static and buckling analysis of electrostrictive beams. The developed finite element formulation results are compared with that of the analytical solution results and it has been found that the results are in very good agreement. The proposed finite element formulation is computationally very efficient than any other available nonlinear finite element formulation for electrostrictive materials. The proposed finite element formulation proves its very high computational efficiency especially in case of buckling analyses as well as in case of electrostrictive patches embedded in large structures.