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

E. Ferretti1

DISTART, Scienza delle Costruzioni, Facoltá di Ingegneria, Alma Mater Studiorum, Universitá di Bologna, Viale Risorgimento 2, 40136 (BO), ITALY, e-mail:

Computer Modeling in Engineering & Sciences 2005, 9(1), 19-48.


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 opportunity 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, particularly when the size-effect is involved. In the opinion of the Author, this need lies in the basics themselves of the differential formulation, performing the limit process. Actually, with the reduction of global variables 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 describe more than 0-dimensional (nonlocal) effects. Here it is shown how a length scale is intrinsic in Physics. 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 itself and does not require nonlocal constitutive relationships for modeling nonlocal effects. Obtaining a nonlocal formulation by using local constitutive laws and discrete operators seems to be possible and physically appealing. Numerical results are provided here, showing 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.


Cite This Article

Ferretti, E. (2005). A Local Strictly Nondecreasing Material Law for Modeling Softening and Size-Effect: a Discrete Approach. CMES-Computer Modeling in Engineering & Sciences, 9(1), 19–48.

cc This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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