Journal of Engineering Mechanics ASCE, 128 (2002), 1119-1149.


Zdenek P. Bazant
Northwestern University, Evanston, IL
Milan Jirásek
Swiss Federal Institute of Technology
LSC - ENAC, EPFL, 1015 Lausanne, Switzerland


Modeling of the evolution of distributed damage such as microcracking, void formation and softening frictional slip necessitates strain-softening constitutive models. The nonlocal continuum concept has emerged as an effective means for regularizing the boundary value problems with strain softening, capturing the size effects and avoiding spurious localization that gives rise to pathological mesh sensitivity in numerical computations.  A great variety of nonlocal models have appeared during the last two decades.  This paper reviews the progress in the nonlocal models of integral type, and discusses their physical justifications, advantages and numerical applications.

Concluding Thoughts and Future Path

While two decades ago strain-softening damage models were regarded as controversial, the nonlocal concept has by now rendered them respectable and their use realistic.  Nonlocality is now generally accepted as the proper approach for regularizing the boundary problems of continuum damage mechanics, for capturing the size effect, and for avoiding spurious localization giving rise to pathological mesh sensitivity.

The last two decades of research gave birth to a wide variety of nonlocal models, with differences that are not quite justified by diversity in the types of materials and practical application. One may now expect a period of crystalization in which many artificially complex or oversimplified models will fade, being recognized as superfluous, and only a few will gain a permanent pedestal in the pantheon of knowledge.

Speculations though could be made in this regard, they are better left to future scrutiny.  The strength of each model's pedestal will be judged by physics of the microstucture, and the permanence of that pedestal will be decided by passage of each model through the sieve of practical applications. Doubtless much further research lies ahead, and polemics will enliven the path into the future.

The complete paper can be downloaded in an electronic form: PostScript, PDF

EPFL / 16 October 2002 /