On Nonlocality and Locality: Differential and Discrete Formulations

One of the main research fields in past years concerns the modeling of heterogeneous materials. For these materials, the use of the classical local continuum concept does not seem to be adequate. The classical local continuum concept leads to constitutive models falling within the category of simple nonpolar materials. For these materials, the stress at a given point uniquely depends on the current values, and possibly also the previous history, of deformation and temperature at that point only. Beginning with Krumhansl, Rogula, Eringen, Kunin, and Kr\uf6ner, the idea was promulgated that heterogeneous materials should properly be modeled by some type of nonlocal continuum. Nonlocal continua are continua in which the stress at a certain point is not a function of the strain at the same point, but a function of the strain distribution over a certain representative volume of the material centered at that point. The aim of the present paper is to show that nonlocal constitutive relations between stress and strain tensors are not strictly needed to construct a material model. They are required only if a differential formulation is used for modeling nonlocality, since differential operators are local. The physical well-posedness of nonlocality is discussed with regard to the differential and discrete formulations. Nonlocality was found to be a concept not attaining to the description of the material, but of the phenomenon. This made it possible to discuss the opportunity of using nonlocality in order to give respectability to strain-softening damage models. The mathematical and physical well-posedness and the existence of strain-softening are also discussed. When using the differential formulation, a length scale must be introduced into the material description of a strain-softening modeling. This need has been here justified on the basis of the geometrical information which has been lost in performing the limit process. It was shown how, avoiding the limit process, a length scale is intrinsically taken into account into a discrete formulation. Thus, the discrete formulation turns out to be more appealing than the differential formulation with nonlocal approach, from the physical point of view