Computer Methods in Applied Mechanics and Engineering, 188 (2000), 307-330
COMPARATIVE STUDY ON FINITE ELEMENTS WITH EMBEDDED CRACKS
Swiss Federal Institute of Technology
The recently emerged concept
of strain or displacement discontinuities embedded into standard
finite elements is a powerful technique that allows efficient
modeling of zones with highly localized
strains, such as the fracture process zone in concrete or the shear bands
in metals or soils.
Following the seminal paper by Ortiz, Leroy, and Needleman,
a number of studies
on elements with embedded discontinuities have been published during
the past decade.
It was demonstrated that local enrichments of the
displacement and/or strain interpolation can improve the performance
of finite element models in problems with strain localization.
of approaches proposed in the literature calls for a comparative study
that would present the diverse techniques in a unified framework,
point out their common features and differences, and find their limits
There are many aspects in which individual formulations differ, such
as the type of discontinuity (weak/strong), variational principle used
for the derivation of basic equations, constitutive law, etc.
The present paper suggests a possible approach to
with special attention to the type of kinematic enhancement and of
the stress continuity condition. The differences between individual
formulations are elucidated by applying them to the simplest finite
element---the constant strain triangle. The sources of stress locking
(spurious stress transfer) reported by some authors are analyzed.
It is shown that there exist three major classes of models with
but only one of the classes gives an optimal element behavior from
both static and kinematic point of view.
A number of techniques enriching the standard finite element interpolation
by additional terms corresponding to a displacement or strain discontinuity
have been presented within a unified framework and critically evaluated.
It has been shown that there exist three major classes of these models,
called here statically optimal symmetric (SOS),
kinematically optimal symmetric (KOS), and statically and
kinematically optimal nonsymmetric (SKON).
The SOS formulation cannot properly reflect the
kinematics of a completely open crack but it gives a natural stress
continuity condition, while the KOS formulation describes the
kinematic aspects satisfactorily but leads to an awkward relationship
between the stress in the bulk of the element and the tractions across
the discontinuity line.
These findings motivate the development
of the nonsymmetric SKON formulation, which combines the strong points
of each of the symmetric formulations. It is not variationally consistent
but leads to an improved numerical performance.
This formulation deals with a very natural stress
continuity condition and is capable of properly representing complete
separation at late stages of the fracturing process, without any locking
effects (spurious stress transfer). The price to pay is the loss of symmetry
of the tangential stiffness matrix. Also, some doubts can be raised
regarding thermodynamic consistency of a model based on the SKON
formulation. Nevertheless, results
reported in the literature show that this formulation can be used
with success in numerical simulations of localized cracking and shear
It is also worth noting
that the SKON formulation does not require any specification of the ``length''
of the localization band. This is an important advantage because such
length is in general not an objective quantity and its value depends
on the (partially ambiguous) rule for the positioning of the discontinuity
inside the element.
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25 September 2000 /