Revisiting bending theories of elastic gradient beams

In this paper, we discuss a problem of Bernoulli-Euler beam models development in the frame of the strain gradient and distortion gradient elasticity theories. Contradiction between some known size-dependent gradient beam models and analytical and numerical three-dimensional solutions found for the beam bending problems is shown. In particular, it is shown that in semi-inverse analytical solutions for a beam pure bending problem, the inverse squared dependence of normalized bending stiffness on the beam thickness could arise only due to wrong definition of the resultant bending moment and improper formulation of the boundary conditions on the top and bottom surfaces of the beam. In the correct semi-inverse solutions, finite values of normalized bending stiffness for ultra-thin gradient beams arise. The obtained results are also confirmed on the basis of 3D FE simulations realized for a pure and cantilever beam bending problems in the frame of simplified strain gradient elasticity theory. As a result, it is shown that the only correct approach for Bernoulli-Euler beam models development in gradient theories should ensures the fulfillment of boundary conditions on the top and bottom surfaces of the beam and corresponds to a model that assumed a uniaxial stress state of the beam. The consistent variational algorithm for the correct gradient beam models derivation is proposed.