A Local Strictly Nondecreasing Material Law for Modeling Softening and Size-Effect: a Discrete Approach

In this study nonlocality is discussed with regard to the differential and discrete formulations. Here, nonlocality is found to be a concept attaining not to the description of the material, but to the governing equations. This has made it possible to discuss the op-portunity of introducing nonlocality in the constitutive equations, in order to give respectability to strain-softening damage models. When using the differential formulation, a length scale must be introduced into the material description of a strain-softening modeling, par-ticularly when the size-effect is involved. In the opin-ion of the Author, this need lies in the basics themselves of the differential formulation, performing the limit process. Actually, with the reduction of global vari-ables to point (and instant) variables, we loose metrics. Consequently, metrics must be reintroduced a-posteriori, by means of a length scale, if we want to de-scribe more than 0-dimensional (nonlocal) effects. Here it is shown how a length scale is intrinsic in Phys-ics. Avoiding the limit process, that is, using a discrete formulation, we preserve the length scale of Physics and do not need to recover it. In this sense, it may be asserted that the discrete formulation is nonlocal in it-self and does not require nonlocal constitutive relation-ships for modeling nonlocal effects. Obtaining a nonlo-cal formulation by using local constitutive laws and discrete operators seems to be possible and physically appealing. Numerical results are provided here, show-ing how a formulation using discrete operators and a local constitutive law is able to model softening and size-effect, which is impossible for differential local approaches. The mathematical and physical well-posedness and the existence itself of strain-softening are also discussed.