A geometrically nonlinear beam model based on the second strain gradient theory

The geometrically nonlinear governing differential equation of motion and corresponding boundary conditions of small-scale Euler–Bernoulli beams are achieved using the second strain gradient theory. This theory is a non-classical continuum theory capable of capturing the size effects. The appearance of many higher-order material constants in the formulation can certify that it appropriately assesses the behavior of extremely small-scale structures. A hinged–hinged beam is chosen as an example to lay out the nonlinear size-dependent static bending and free vibration behaviors of the derived formulation. The results of the new model are compared with the previously obtained results based on the strain gradient theory and the classical theory.