Size-dependent nonlinear analysis of three-dimensional cantilever beams with strain gradient effects using mixed finite element method

In this study, the geometric nonlinear behavior of three-dimensional size-dependent beams is investigated using the mixed finite element method. Nonlinear von-Karman terms are considered in the strain-- displacement relation to capture the geometric nonlinear deformations at higher load magnitudes. The torsion-- bending deformation of size-dependent beams is studied using the simplified form of rotation gradient theory. The governing equations and related boundary conditions were derived using the variational principle. The Newton-Raphson iteration procedure for solving nonlinear governing equations is coded in Python using a stable 11-node tetrahedral C0 element via the robust open-source finite element FEniCS platform. The accuracy and convergency of this formulation are examined via existing results. The effects of Poisson's ratio, thickness, and dimensionless length scale parameters on the displacement and rotation fields were explored. By varying the location of the applied load through the width of the beam, the effect of combined bending-torsion of the cantilever size-dependent beam is studied.