A method to transform a nonlocal model into a gradient one within elasticity and plasticity

• We give contributions to claerify relationship between gradient and nonlocal models.
• A consistent derivation of gradient models from nonlocal one is proposed.
• A light on gradient boundary conditions and nonlocal boundary layer is given.
• A generalized Principle of Virtual Power is given for Nonlocal (integral models).
• An equivalence between nonlocal and complementary gradient is given.

A method based on the principle of the virtual power (PVP) is presented, by which a mechanical problem of nonlocal elasticity, or plasticity, is transformed into one of gradient nature. Different Taylor series expansion techniques are applied to the driving local strain fields of the nonlocal problem, either full spatial expansion within the bulk volume, or uni-directional expansion along the normal to the thin boundary layer. This, at the limit when the boundary layer thickness tends to zero, makes the PVP of the nonlocal model transform itself into one featuring a counterpart gradient model. Also, for a class of “associated” nonlocal and gradient elasticity models (i.e. the kernel function of the former is the Green function of the latter), the mutual relationship is discussed and the existence of some related complementary formats is pointed out together with their computational relevance.