On the Nonexistence of a Feasible Solution in the Context of the Differential Form of Eringen’s Constitutive Model: A Proposed Iterative Model Based on a Residual Nonlocality Formulation

Abstract

The motivation of this paper is to highlight and discuss critically the details of two main aspects related to Eringen’s nonlocal constitutive model. The first aspect is to point out the inconsistency of the integral and differential forms of Eringen’s nonlocal constitutive model. In addition, to point out the ill-posed form and physical infeasibility of the results which may be obtained by using the differential form of the model. The critical analysis focuses on the lack of consistency between the set of the boundary constraints required by the differential form of Eringen’s model and the set of the prescribed boundary conditions of the nonlocal static equilibrium problem. Because of this lack of consistency between the two constitutive forms, it can be concluded that the formulation in context of the differential form is ill-posed and the existence of a feasible solution is questionable and might not be admitted. The second aspect deals with the intractability of the analytical solution of the nonlocal continuum problems based on the integral form of Eringen’s nonlocal constitutive model (IENCM). In the meantime, it mentions the cumbersome of numerical work required by the direct computational methods such as the nonlocal finite element method. The complexity of using the integral form of Eringen’s constitutive model and the lack of existence of a feasible solution by using the differential form of Eringen’s constitutive model lead to the mandatory need for developing an efficient iterative computational approach based on the integral form of Eringen’s constitutive model. In this paper, an iterative computational method, based on the nonlocality residual formulation for nonlocal continuums, capable to investigate different elasticity problems, is proposed. The traditional local continuum solution is taken as an initial solution. To point out the inconsistences between the integral and differential forms of Eringen’s nonlocal constitutive model, and to illustrate the efficiency and capability of the proposed iterative model, four static bending of beam problems with different nature are solved.