The recently emerged idea of incorporating strain or displacement discontinuities into standard finite element interpolations has triggered the development of powerful techniques that allow efficient modeling of regions with highly localized strains, e.g. of fracture process zones in concrete or shear bands in metals or soils. Following the pioneering work of Ortiz, Leroy, and Needleman (1987), a number of studies on elements with embedded discontinuities have been published during the past decade. It has been 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 report suggests a possible approach to their classification, with special attention to the type of kinematic enhancement and of 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 them gives the optimal element behavior from both static and kinematic points of view.
In addition to the systematic classification and evaluation of existing models, the report presents three novel ideas. First, the constitutive description of a crack as a displacement discontinuity is formulated within the framework of damage theory. Crack closure effects and friction on the crack faces are incorporated. Uniqueness of the response on the element level is analyzed, and criteria limiting the size of the finite element are derived. Second, the embedded crack approach is combined with the more traditional smeared crack approach, which leads to a reduction of secondary cracking due to incorrect separation of nodes by the embedded discontinuity. Third, the smeared part is reformulated as nonlocal. Various criteria for placing the discontinuity are compared, and the optimal technique is identified. Remarkable insensitivity of the resulting model to mesh-induced directional bias is demonstrated.
A number of techniques that enrich 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 have motivated 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. Results reported in the literature show that the nonsymmetric model 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.
The theoretical comparison of the basic frameworks for finite elements with embedded discontinuities has been followed by the implementation of a specific embedded crack model. Constitutive description of a damaging interface inspired by continuum damage mechanics has been proposed, and possible extensions that take into account the stiffness recovery upon crack closure and the effects of friction have been outlined. The displacement discontinuity has been incorporated into a constant strain triangle, based on the SKON formulation. The conditions of uniqueness for such an element have been derived, and an algorithm for the evaluation of internal forces and of the element stiffness matrix has been developed.
Systematic numerical testing has revealed that the standard approach, which introduces the discontinuity right at the onset of cracking, often leads to a misprediction of the discontinuity direction. As the direction has to remain fixed, there is no chance for its adjustment. This inevitably leads to stress locking that must be relaxed by a secondary crack in the same element. Multiple cracking complicates the numerical algorithm and can lead to convergence problems. It has therefore been proposed to use a combined model that represents the early stage of cracking in a smeared manner and introduces a discontinuity only when the crack opens sufficiently wide. If the smeared part is modeled by the rotating crack approach, the crack has a chance to readjust its direction, and there is no need for secondary cracking. The combination of the smeared and embedded descriptions of cracking is appealing from the physical point of view. It is intuitively clear that diffuse damage at early stages of material degradation is adequately described by a model dealing with inelastic strain while highly localized fracture is better modeled by a displacement discontinuity. Examples of simulations of fracture specimens have demonstrated the potential of the developed technique.
Finally, the smeared part of the combined model has been reformulated as nonlocal. It has been demonstrated that, in order to avoid locking, it is essential to enforce continuity of the embedded crack trajectory. Optimal performance in terms of insensitivity to mesh-induced directional bias is obtained if the orientation of the embedded crack is in each element determined from the principal directions of nonlocal (rather than local) strain.
To summarize, the embedded crack approach seems to be a very appealing technique that certainly has a number of important advantages compared to the more traditional approaches. Of course, a large amount of work still remains to be done. For example, a challenging goal would be the extension of the model to three dimensions.
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