Elastic bending analysis of bilayered beams containing a gradient layer by an alternative two-variable method

Aifantis’s strain gradient elasticity theories and Zhang’s two-variable method are used to study elastic bending problems of bilayered micro-cantilever beams, containing a gradient layer, subjected to a transverse concentrated load. The differential element method is used to obtain differential governing equations. The variational method is employed to overcome the difficulty in deriving nonlocal natural boundary conditions, which could not be automatically fulfilled in gradient theories, not like that in classical theories. Then the differential governing equations subjected to the related boundary conditions are solved analytically to obtain the deformation field, which could be degenerated to that in classical elasticity theories. The gradient parameters of epoxy polymeric resin and copper single crystals in the present model are provided by fitting Lam’s and Demir’s experiments. The influences of length and layer thickness on normalized deflection and effective rigidity are discussed in a representative case of a Cu/epoxy polymeric resin beam. Results show that size effect makes the effective rigidity vary more prominently with shorter beam length or larger layer thickness. For given materials, although size effect exists, classical elasticity theories are still valid in some particular combination of three geometric parameters: beam length, upper and lower layer thicknesses.