Nonlinear flexure mechanics of mixture unified gradient nanobeams

The mixture unified gradient theory of elasticity is invoked for the nanoscopic study of the nonlinear flexure mechanics of nanobeams. A mixed variational framework is conceived based on ad hoc functional space of kinetic test fields, wherein a complete set of governing equations, classical and non-standard boundary conditions, and the constitutive relations are integrated into a solitary functional. The size-effect phenomenon associated with the nonlinear flexure of nanobeams are efficiently realized as the stress gradient theory, the strain gradient theory, and the classical elasticity theory are consistently unified within the framework of the mixture unified gradient elasticity. A viable numerical approach is introduced based on the conceived mixed variational framework while the autonomous series solution of the kinematic and kinetic field variables is implemented. As the kinetic field variables are directly determined rather than utilizing post-computations, the established consistent variational theory provides an advantageous means for efficacious approximate approaches. The nonlinear flexural response of the mixture unified gradient beams with kinematics constraints of practical interest in nano-mechanics is detected and thoroughly compared with the linear counterparts in terms of the characteristic parameters. The detected nonlinear flexural characteristics of nano-sized beams can pave the way ahead in the nonlinear mechanics of nano-structures. •Consistent variational framework for the mixture unified gradient elasticity theory.•Unification of stress gradient model, strain gradient theory and classical elasticity.•Rigorous nanoscopic analysis of the nonlinear flexure mechanics of nanobeams.•Introduction of an effective numerical approach based on mixed variational functional.•Applying independent series solutions of the kinematic and kinetic field variables.