A non-local two-dimensional foundation model

Classical foundation models such as the Pasternak and the Reissner models have been recently reformulated within the framework of non-local mechanics, by using the gradient theory of elasticity. To contribute to the research effort in this field, this paper presents a two-dimensional foundation model built by using a mechanically based non-local elasticity theory, recently proposed by the authors. The foundation is thought of as an ensemble of soil column elements resting on an elastic base. It is assumed that each column element is acted upon by a local Winkler-like reaction force exerted by the elastic base, by contact shear forces and volume forces due, respectively, to adjacent and non-adjacent column elements. As in the Pasternak model, the contact shear forces involve the second-order derivative of the column element displacement. The volume forces are non-local forces assumed to depend (1) on the relative displacement between the interacting column elements through power-law distance-decaying attenuation functions and (2) on the product between the volumes of the interacting column elements. As a result, the equilibrium equations are fractional differential equations, for which a numerical solution can be readily found based on the finite difference method. Solutions are built for different foundation shapes and loading conditions.