Finite element techniques for nonlocal elasticity

Integral type nonlocal elasticity has received considerable attention by a number of researchers working on typical problems of fracture mechanics. The crack-tip stress concentration problem, for example, is simplified by this approach where no stress singularities arise and classical stress-based safety criteria can in this way be recovered. The present paper considers a nonlocal elasticity model for linear, homogeneous, isotropic, materials, known as Eringen model, (see e.g. [1-3]) and analyses the behavior of a 1D continuous, made of this material subjected to static loads. Two different finite element techniques, proposed in [3], are analyzed in the framework of small-strain hypotheses; precisely: i) Nonlocal finite element method (NL-FEM) leading to a linear solving global equation system formally equal to that of standard FEM, but characterized by a relevant global nonlocal symmetric stiffness matrix which reflects all the nonlocality features of the problem. This nonlocal stiffness matrix is the result of contributions from all FEs, each of which contributes with a single selfstiffness matrix and a set of as many cross-stiffness matrices as there are other FEs in the mesh. ii) Iterative local finite element method (IL-FEM) which solves the nonlocal problem through a standard FEM technique within an iterative procedure of the type local prediction/nonlocal correction and in which the nonlocality is simulated by an imposed-like correction strain. The latter is updated by means of an apposite (nonlocal) consistency condition. Both the above mentioned methods i) and ii) are theoretically based on specific variational principles formulated in [3]. This paper, which resumes the main findings of an ongoing research carried on by the authors (refer to [4-5] for a more detailed discussion), illustrates the two above cited techniques with respect to a simple 1D nonlocal elastic problem. The results are compared together and with an exact solution achievable for the simple case-study here treated. The last comparison seems to be of particular interest whether because the literature is extremely poor of analytical solutions in the nonlocal elasticity field or because it makes perceptible the potentialities of the FEM-based techniques here illustrated.