Journal of Engineering Mechanics ASCE, 120 (1994), 1521-1542.


Milan Jirásek and Zdenek P. Bazant
Northwestern University
Evanston, Illinois 60208, U.S.A.


The conventional nonlocal model, often used as a localization limiter for continuum-based constitutive laws with strain-softening, has been based on an isotropic averaging function. It has recently been shown that this type of nonlocal averaging leads to a model which cannot satisfactorily reproduce experimental results for very different test geometries without modifying the value of the characteristic length depending on geometry. A micromechanically based enrichment of the nonlocal operator by a term taking into account the directional dependence of crack interactions can be expected to improve the performance of the nonlocal model. The aim of this paper is to examine this new model in the context of a simple localization problem reducible to a one-dimensional description. Strain localization in an infinite layer under plain stress is studied using both the old and the new nonlocal formulations. The importance of a renormalization of the averaging function in the proximity of a boundary is demonstrated and the differences between the localization sensitivity of the old and new model are pointed out. In addition to the detection of bifurcations from an initially uniform state, the stable branch of the load-displacement diagram is followed using an incremental procedure.

Summary and Conclusions

The conventional nonlocal model with an isotropic averaging function without renormalization cannot capture strain localization at the boundaries. Localized strain profiles are invariant with respect to a shift and not affected by the proximity of the boundary.

With a renormalized averaging function, the conventional nonlocal model leads to uniform strain increments in the hardening regime and in the softening regime with a very small post-peak slope. The strain increments localize into a band at one boundary if the post-peak slope of the local constitutive law exceeds a certain minimum value, which depends on the size of the layer. Large post-peak slopes of the local constitutive law result into a snapback.

The new nonlocal model, which contains an integral describing the effect of orientation-dependent crack interactions leads to nonuniform strain profiles as soon as the local constitutive law deviates from linearity. The global load-displacement diagram can start softening even before the peak in the local constitutive law is reached. Similarly, the solution can bifurcate already in the (locally) hardening regime.

The present method of analysis has been used to trace the entire loading process and study the evolution of the localized strain profiles. Several local constitutive laws leading to reasonable shapes of the load-displacement diagram have been presented.

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EPFL / July 1, 1996 /