Computer Methods in Applied Mechanics and Engineering, 188 (2000), 307-330


Milan Jirásek
Swiss Federal Institute of Technology
LSC -DGC, EPFL, 1015 Lausanne, Switzerland


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. The multitude 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 of applicability. 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 their classification, 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 embedded discontinuities 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 banding. 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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EPFL / 25 September 2000 /