Journal of Engineering Mechanics ASCE, 120 (1994), 1521-1542.
LOCALIZATION ANALYSIS OF NONLOCAL MODEL
Milan Jirásek and Zdenek P. Bazant
BASED ON CRACK INTERACTIONS
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
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)
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.
Please send me an email
if you wish to receive the complete paper.
July 1, 1996 /