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Flexoelectricity, defined as the electromechanical coupling between polarization and the strain gradient, becomes interesting at the nanoscale where it largely overcomes piezoelectric coupling. However, nanoscale deformations easily get over linear regime when relatively large displacement are observed. To correctly model such nanoactuators, geometric nonlinearity should be taking into account.

In this work, we propose to model the nonlinear behavior of piezoelectric-flexoelectric nanobeams where von Kaman strain is considered. The model is derived using a general form of the enthalpy density that provides electromechanical couplings to the third order. Using Euler–Bernoulli beam's model and employing the inextensibility conditions, applying the Euler–Lagrange principle and implementing a Galerkin discretization the corresponding reduced-order model is derived. Clamped-clamped boundary conditions are implemented where midplane stretching effect is dominant.

The effects of the nonlinearities on the nanoactuator response is investigated and compared with previously published linear models. The self electric field effect and the gradient nonlocal effect are also investigated. The results show that for the clamped-clamped nanobeam the effective electromechanical coupling is mitigated by the presence of the nonlinearity.