Numerical modeling of strong discontinuities Milan Jirásek Laboratory of Structural and Continuum Mechanics Swiss Federal Institute of Technology (EPFL) CH-1015 Lausanne, Switzerland
[email protected] ABSTRACT. The
paper presents an overview of modern approximation techniques for explicit resolution of strong discontinuities (displacement jumps) that can run arbitrarily across a finite element mesh. Two types of such formulations are covered: (i) elements with embedded discontinuities, which introduce internal degrees of freedom that are eliminated on the element level, and (ii) extended finite elements with additional global degrees of freedom, based on the partition-of-unity concept. RÉSUMÉ. Cet article présente une vue d’ensemble des techniques d’approximation modernes pour la résolution explicite des fortes discontinuités (sauts dans le champ des déplacements) qui peuvent traverser de manière quelconque un maillage d’éléments finis. Deux extensions de la méthode des éléments finis y sont décrites : (i) celle qui ajoute des degrés de liberté internes, éliminés au niveau de l’élément, et (ii) celle baptisée X-FEM ("eXtended Finite Element Method"), qui utilise des degrés de liberté globaux supplémentaires, et dont le concept est basé sur la partition de l’unité. KEYWORDS:
discontinuities, cohesive crack, extended finite elements, partition of unity
discontinuités, fissure avec zone de cohésion, extension de la méthode des éléments finis, partition de l’unité
MOTS-CLÉS :
Revue française de génie civil. Volume 6 - n 6/2002, pages 1133 à 1146
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1. Introduction The preceding paper [JIR 02] gave an overview of three main classes of models that provide an objective description of strain localization, and then discussed in more detail one of those classes, namely formulations regularized by spatial integrals or by gradient terms. The present paper is focused on models that treat highly localized strains as strong discontinuities (jumps in the displacement field). Standard finite element approximations cannot properly capture the discontinuous character of the displacement field corresponding to localized fracture. In the context of smeared-crack models, this deficiency can lead to a spurious stress transfer across a widely open crack [JIR 98]. Discrete-crack models with special interfaces ˇ between conventional elements [SAO 81, CER 94] do not suffer by this pathology, but they require frequent remeshing in order to allow for crack propagation in the correct direction. The recently emerged idea of incorporating strain or displacement discontinuities into standard finite element interpolations 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. The discontinuities can have an arbitrary orientation, which makes it much easier to capture a propagating crack or softening band without remeshing. This class of methods, collectively called elements with embedded discontinuities, was inspired by the pioneering work of Ortiz et al. [ORT 87] and Belytschko et al. [BEL 88]. The early works used weak (strain) discontinuities, but the idea was later extended to strong (displacement) discontinuities [DVO 90, KLI 91, OLO 94, SIM 94]. A systematic classification and critical evaluation of embedded discontinuity models within a unified framework was presented in [JIR 00a], with the conclusion that there exist three main groups of such formulations, called statically optimal symmetric (SOS), kinematically optimal symmetric (KOS), and statically and kinematically optimal nonsymmetric (SKON). The SOS formulation works with a natural stress continuity condition, but it does not properly reflect the kinematics of a completely open crack. On the other hand, the KOS formulation describes the kinematic aspects satisfactorily, but it leads to an awkward relationship between the stress in the bulk of the element and the tractions across the discontinuity line. Optimal performance is achieved with the nonsymmetric SKON formulation, which uses a very natural stress continuity condition and reasonably represents complete separation at late stages of the fracturing process. This is the formulation to be described next. In section 6, we will present a different approach to the modeling of discontinuities, based on the concept of partition of unity [MEL 96] and refered to as the extended finite element method [MOË 99, SUK 00, DAU 00, MOË 02].
2. Triangular element with embedded displacement discontinuity The optimal combination of static and kinematic equations for elements with embedded discontinuities first appeared in [DVO 90], even though their exact nature is
Modeling of strong discontinuities
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y
(a)
(b) 1
- +
1 3
s 2
n 3 α
2
en es
x
(c)
(d) 1
(e) σx
1 3 3
2
2
τxy σy
ts tn
sen+ces cen-se s
Figure 1. Constant-strain triangle with an embedded displacement discontinuity not easy to understand from that paper. A very similar quadrilateral element based on simple and instructive physical considerations was constructed in [KLI 91]. The same technique was then applied to a constant-strain triangle [OLO 94]. A general version of the SKON formulation for an arbitrary type of parent element was outlined in [SIM 94] and fully described in [OLI 96]. Consider a triangular element crossed by a discontinuity (Fig. 1a). The displacement field can be decomposed into a continuous part and a discontinuous part due to the opening and sliding of a crack (Fig. 1b). The same decomposition applies to the nodal displacements of a finite element. Instead of smearing the displacement jump over the area of the element and replacing it by an equivalent inelastic strain, as is done by standard smeared crack models (Fig. 1c), the discontinuity can be represented by additional degrees of freedom, and , corresponding respectively to the normal (opening) and tangential (sliding) component of the displacement jump and collected . The contribution of the displacement jump is then in a column matrix subtracted from the nodal displacement vector, , and only the part of the nodal displacements produced by the continuous deformation serves as input for the evaluation of strains in the bulk material, (Fig. 1d). This leads to kinematic equations in the form [1] !
#"
!"
%$ is the column matrix of engineering strain components, where is the standard strain-displacement matrix, and is a matrix reflecting the effect of the displacement jump on the nodal displacements.
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In general, the displacement jump could be approximated by a suitable function, for example a polynomial one. This approximation need not be continuous on interelement boundaries. For triangular elements with a linear displacement interpolation, the strains and stresses in the bulk are constant in each element, and so it is natural to approximate the displacement jump also by a piecewise constant function. This is why we describe the jump in each element by only two parameters, and . These additional degrees of freedom have an internal character and can be eliminated on the element level, which yields the global equilibrium equations written exclusively in terms of the standard unknowns—nodal displacements. From Fig. 1d it is clear that if the discontinuity line separates node 3 from nodes 1 and 2 (in local numbering), the crack-effect matrix is given by '( * ( *
( * ( *
.0/ / / / / /
-,
1
( * ( * *
&
* +
) ,
+
[2]
where + 32547698 , ,:36=?8 , and 8 is the angle between the normal to the crack (discontinuity line) and the global @ -axis; see Fig. 1a. Strains in the bulk material generate certain stresses, AB are here computed from the equations of linear elasticity, AFGHI
CD! C "
