On the consistency of the nonlocal strain gradient elasticity

The nonlocal strain gradient elasticity theory is being widely used to address structural problems at the micro- and nano-scale, in which size effects cannot be disregarded. The application of this approach to bounded solids shows the necessity to fulfil boundary conditions, derived from an energy variational principle, to achieve equilibrium, as well as constitutive boundary conditions inherent to the formulation of the constitutive equation through convolution integrals. In this paper we uncover that, in general, is not possible to accomplish simultaneously the boundary conditions, which are all mandatory in the framework of the nonlocal strain gradient elasticity, and therefore, the problems formulated through this theory have no solution. The model is specifically applied to the case of static axial and bending behaviour of Bernoulli-Euler beams. The corresponding governing equation in terms of displacements results in a fourth-order ODE with six boundary conditions for the axial case, and in a sixth-order ODE with eight boundary conditions for the bending case. Therefore, the problems become overconstrained. Three study cases will be presented to reveal that all the boundary conditions cannot be simultaneously satisfied. Although the ill-posedness has been pointed out for an elastostatic 1D problem, this characteristic holds for other structural problems. The conclusion is that the nonlocal strain gradient theory is not consistent when applied to finite structures and leads to problems with no solution in a general case.