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# Numerical modeling of str ong discontinuities Milan Jirsek Labor atory of Structur al and Continuum Mec hanics Swiss eder al Institute of ec hnolo gy EPFL CH Lausanne Switzerland Milan PDF document - DocSlides

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Numerical modeling of str ong discontinuities Milan Jirásek Labor atory of Structur al and Continuum Mec hanics Swiss eder al Institute of ec hnolo gy (EPFL) CH-1015 Lausanne Switzerland Milan.Jir asek@epﬂ.c ABSTRA CT The paper pr esents an vervie of modern appr oximation tec hniques for xplicit es- olution of str ong discontinuities (displacement jumps) that can run arbitr arily acr oss ﬁnite element mesh. wo types of suc formulations ar co ver ed: (i) elements with embedded dis- continuities, whic intr oduce internal de gr ees of fr eedom that ar eliminated on the element le vel, and (ii) xtended ﬁnite elements with additional global de gr ees of fr eedom, based on the partition-of-unity concept. RÉSUMÉ. Cet article présente une vue d’ensemble des tec hniques d’appr oximation modernes pour la résolution xplicite des fortes discontinuités (sauts dans le hamp des déplacements) qui peuvent tr aver ser de manièr quelconque un mailla d’éléments ﬁnis. Deux xtensions de la méthode des éléments ﬁnis sont décrites (i) celle qui ajoute des de grés de liberté internes, éliminés au niveau de l’élément, et (ii) celle baptisée X-FEM ("eXtended inite Element Me- thod"), qui utilise des de grés de liberté globaux supplémentair es, et dont le concept est basé sur la partition de l’unité. KEYW ORDS: discontinuities, cohesive cr ac k, xtended ﬁnite elements, partition of unity MO TS-CLÉS discontinuités, ﬁssur avec zone de cohésion, xtension de la méthode des éléments ﬁnis, partition de l’unité Re vue française de génie ci vil. olume 6/2002, pages 1133 1146

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1134 Re vue française de génie ci vil. olume 6/2002 1. Intr oduction The preceding paper [JIR 02 ga an ervie of three main classes of models that pro vide an objecti description of strain localization, and then discussed in more detail one of those classes, namely formulations re gularized by spatial inte grals 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 con- te xt of smeared-crack models, this deﬁcienc can lead to spurious stress transfer across widely open crack [JIR 98 ]. Discrete-crack models with special interf aces between con entional elements [SA 81 CER 94 do not suf fer by this pathology ut the require frequent remeshing in order to allo for crack propagation in the correct direction. The recently emer ged idea of incorporating strain or displacement discontinuities into standard ﬁnite element interpolations triggered the de elopment of po werful techniques that allo ef ﬁcient modeling of re gions with highly localized strains, e.g. of fracture process zones in concrete or shear bands in metals or soils. The discontinuities can ha an arbitrary orientation, which mak es it much easier to capture propagating crack or softening band without remeshing. This class of meth- ods, collecti ely called elements with embedded discontinuities as inspired by the pioneering ork of Ortiz et al. [OR 87 and Belytschk et al. [BEL 88 ]. The early orks used weak (strain) discontinuities, ut the idea as later xtended to strong (displacement) discontinuities [D 90 KLI 91 OLO 94 SIM 94 ]. systematic classiﬁcation and critical aluation of embedded discontinuity mod- els within uniﬁed frame ork as presented in [JIR 00a ], with the conclusion that there xist three main groups of such formulations, called statically optimal symmet- ric (SOS), kinematically optimal symmetric (K OS), and statically and kinematically optimal nonsymmetric (SK ON). The SOS formulation orks with natural stress con- tinuity condition, ut it does not properly reﬂect the kinematics of completely open crack. On the other hand, the OS formulation describes the kinematic aspects sat- isf actorily ut it leads to an wkw ard relationship between the stress in the ulk of the element and the tractions across the discontinuity line. Optimal performance is achie ed with the nonsymmetric SK ON formulation, which uses ery natural stress continuity condition and reasonably represents complete separation at late stages of the fracturing process. This is the formulation to be described ne xt. In section 6, we will present dif ferent approach to the modeling of discontinuities, based on the concept of partition of unity [MEL 96 and refered to as the xtended ﬁnite element method [MOË 99 SUK 00 00 MOË 02 ]. 2. riangular element with embedded displacement discontinuity The optimal combination of static and kinematic equations for elements with em- bedded discontinuities ﬁrst appeared in [D 90 ], en though their xact nature is

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Modeling of strong discontinuities 1135 (a) (b) (c) (d) ce -se (e) se +ce xy Figur 1. Constant-str ain triangle with an embedded displacement discontinuity not easy to understand from that paper ery similar quadrilateral element based on simple and instructi physical considerations as constructed in [KLI 91 ]. The same technique as then applied to constant-strain triangle [OLO 94 ]. general ersion of the SK ON formulation for an arbitrary type of parent element as outlined in [SIM 94 and fully described in [OLI 96 ]. Consider triangular element crossed by discontinuity (Fig. 1a). The displace- ment ﬁeld can be decomposed into continuous part and discontinuous part due to the opening and sliding of crack (Fig. 1b). The same decomposition applies to the nodal displacements of ﬁnite element. Instead of smearing the displacement jump er the area of the element and replacing it by an equi alent inelastic strain, as is done by standard smeared crack models (Fig. 1c), the discontinuity can be represented by additional de grees of freedom, and corresponding respecti ely to the normal (opening) and tangential (sliding) component of the displacement jump and collected in column matrix The contrib ution of the displacement jump is then subtracted from the nodal displacement ector and only the part of the nodal dis- placements produced by the continuous deformation serv es as input for the aluation of strains in the ulk material, (Fig. 1d). This leads to kinematic equations in the form [1] where ! #" %$ !" is the column matrix of engineering strain components, is the standard strain-displacement matrix, and is matrix reﬂecting the ef fect of the displacement jump on the nodal displacements.

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1136 Re vue française de génie ci vil. olume 6/2002 In general, the displacement jump could be approximated by suitable function, for xample polynomial one. This approximation need not be continuous on inter element boundaries. or triangular elements with linear displacement interpolation, the strains and stresses in the ulk are constant in each element, and so it is natural to approximate the displacement jump also by piece wise constant function. This is why we describe the jump in each element by only tw parameters, and These additional de grees of freedom ha an internal character and can be eliminated on the element le el, which yields the global equilibrium equations written xclusi ely in terms of the standard unkno wns—nodal displacements. From Fig. 1d it is clear that if the discontinuity line separates node from nodes and (in local numbering), the crack-ef fect matrix is gi en by & -, .0/ [2] where 32547698 ,:36<;>=?8 and is the angle between the normal to the crack (discontinuity line) and the global -axis; see Fig. 1a. Strains in the ulk material generate certain stresses, AB CD! C" !" which are here computed from the equations of linear elasticity AFGHI [3] where is the elastic stif fness matrix (for plane stress or plane strain). Note that, in general, the constituti la for the ulk material could be nonlinear The tractions transmitted by the crack, are link ed to the separation ector (displacement jump) by another constituti la that describes the gradual de elopment of stress-free crack. One speciﬁc form of this la will be presented in Section 3. The stresses in the ulk and the tractions across the crack must satisfy certain con- ditions that xpress internal equilibrium and serv as static equations corresponding to the internal de grees of freedom, The most natural requirement is that the traction ector be equal to the stress tensor contracted with the crack normal, similar to static boundary conditions. This internal equilibrium (traction continuity) condition can be deri ed from equilibrium of an elementary triangle with one side on the discontinuity line; see Fig. 1e. In the engineering notation it reads AFLJ [4] where +5M O, [5]

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Modeling of strong discontinuities 1137 is stress rotation matrix. or linear triangles with constant displacement jump, both and are constant in each element, and so condition [4] can be satisﬁed xactly In general it ould ha to be enforced in weak sense. Finally the nodal forces are aluated from the standard relation PRQ S TVUXWY ZZ[H5 [6] where Z\H is the area of the element. or the constant-strain triangle, the kinematic relations and the traction continuity condition follo quite naturally from simple physical considerations. De elopment of more complicated elements with embedded discontinuities is often done within the frame ork of enhanced assumed strain (EAS) methods, and such elements are sometimes refered to as EAS elements (which can be some what confusing). 3. raction-separation law in damage ormat The basic equations presented in the preceding section must be completed by la that links the traction transmitted by the discontinuity to the displacement jump. One possible type of such la as proposed in [JIR 01a in the form J]&^ G [7] where is dimensionless scalar compliance parameter olving from zero to inﬁnity and G_a`cb # %ed [8] is stif fness matrix corresponding to reference intermediate stage of the de gradation process. Before crack initiation, the alue of is zero. or simplicity it is assumed here that crack initiation is controlled by the Rankine criterion of maximum principal stress. This means that the discontinuity line is inserted perpendicular to the direction of maximum principal stress, and the shear traction at the instant of crack initiation is zero. The olution of is described by the loading-unloading conditions in the Karush- uhn-T uck er form, $hg iej $Xi [9] The loading function characterizing the elastic domain is deﬁned as <$ Rklnm D<o [10] where is scalar measure of the separation ector called the equi alent sepa- ration (analogous to the equi alent strain in continuum damage mechanics), and

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1138 Re vue française de génie ci vil. olume 6/2002 is suitable function describing the dependence of the compliance parameter on the equi alent separation during monotonic loading. In [JIR 01a it as proposed to set Bp Gq # [11] and lnm r s nm [12] where is scalar function describing the traction-separation curv for Mode-I crack- ing. Alternati ely the traction-separation la could be formulated within the frame- ork of plasticity; see e.g. [OLO 94 ]. This is especially useful for the description of cohesi zones in metals or shear bands in soils. Due to space limitations, the details cannot be presented here. 4. Ev aluation of inter nal or ces and tangent stiffness In an incremental-iterati analysis of structure discretized by ﬁnite elements with embedded discontinuities, the nodal displacements are computed iterati ely from the global equilibrium equations, and the main tasks on the le el of one ﬁnite element are to aluate the internal forces and the tangent stif fness matrix for gi en increment of nodal displacements. Substituting [3] and [1] into the traction continuity condition [4], we obtain use- ful xpression for the traction ector in terms of the kinematic ariables, Ju GvHIwDrxyw [13] where we ha denoted xz [14] Expression [13] for the traction ector substituted into the traction-separation la [7] yields the equation x{}|~^ Gq< x [15] that links the nodal displacements separation ector and compliance parameter If is kno wn, the separation ector can be solv ed from [15], taking into account that the compliance parameter may change during the step according to [9]–[10]. Adopting the usual numerical scheme, equation [15] is ﬁrst solv ed under the assump- tion of constant damage (unloading), i.e., with ept equal to its alue at the end of the pre vious step. If the computed separation ector satisﬁes the condition %$ then the solution is admissible, otherwise it is necessary to solv [15] with replaced by lnm < The conditions under which this problem has unique solution were studied in [JIR 00b ], where it as sho wn that uniqueness may be lost not only for

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Modeling of strong discontinuities 1139 elements that are too lar ge ut also for badly shaped elements. Once the separation ector has been determined, the strain, stress and internal forces are easily aluated by substituting into equations [1], [3] and [6]. The tangent stif fness matrix can be deri ed by combining the rate forms of the basic equations and is gi en by the formula [JIR 01a H H5‚ Gƒ| x{„…J I†Y‡x [16] where Hˆ‚Z\H5 GvH5 is the elastic element stif fness matrix and Š‰ l-‹#‰Œ is the gradient of the loading function with respect to the separation ector Recalling the deﬁnition [14] of matrix xpression [16] can be re written in the equi alent form Z[H Ž GqH GHIhŠ^ G‘| GvHIh„…J I†Y‡ GqH“’” [17] This formula has the same structure as the standard xpression •Z G for the tangent element stif fness of con entional constant-strain triangle with nonlinear material. It is interesting to note that the “equi alent tangent material stif fness Ga–GqH GH5y• G—| GqH5h˜…J n†X‡ GH [18] depends not only on the constituti parameters (elastic ulk stif fness GH reference cohesi stif fness compliance parameter ), ut also on the matrices and that reﬂect the orientation of the discontinuity and on matrix that reﬂects the size and shape of the ﬁnite element. The dependence on orientation (crack direction) means that crack-induced anisotrop is tak en into account, and the dependence on element size is typical feature of smeared models based on the fracture ener gy concept; see Section 3.2 in [JIR 02 ]. Ho we er the present model contains additional information on the kinematics of the ailure process, because matrix is af fected by the position of the discontinuity with respect to the element and matrix depends on the element shape. or con entional element, the element stif fness is symmetric if and only if the material stif fness is symmetric. or the present element with an embedded dis- continuity this happens only if (i) is scalar multiple of and (ii) h is scalar multiple of The ﬁrst condition is related to the traction-separation la and is equi v- alent to symmetry of the tangent stif fness that links the rates of the separation ector and of the traction ector or the equi alent separation deﬁned by formula [11], the gradient ector ™‰Œl-‹#‰ hš›9l-‹#›œm 5‰rm ‹#‰ 9[™lž GOD‹9 # is indeed colin- ear with the traction ector JŸ^ GOD‹ The second condition is related xclusi ely to geometrical properties. It is satisﬁed only if the discontinuity line is parallel to one of the element sides [JIR 00a ]. Therefore, en if the material stif fness is symmetric, symmetry of the structural stif fness is disturbed by the kinematic and static equations. This is why the present SK ON formulation is called nonsymmetric. There xist tw types of symmetric formulations (K OS and SOS), in which only one group of equa- tions (either kinematic or static) is postulated and the dual group is deri ed from the principle of virtual ork. Ho we er the equations obtained in this ay are not “natu- ral and the resulting formulations ha se ere dra wbacks; see [JIR 00a for detailed analysis.

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1140 Re vue française de génie ci vil. olume 6/2002 5. racing of the discontinuity path The elements described in the preceding sections can easily accommodate pre- xisting discontinuities, e.g., rock joints or delaminating interf aces, ut the principal interest lies in modeling of olving discontinuities such as propagating cracks. In this latter case, the position of the discontinuity is not kno wn in adv ance and the simula- tion starts with all elements in their “vir gin state. discontinuity se gment is inserted in an element only when certain initiation criterion is satisﬁed. The simplest ap- proach is to formulate this criterion in terms of the stress in the ulk material, e.g., as the Rankine criterion of maximum principal stress, and to place the discontinuity se gment in the element center perpendicular to the maximum principal stress direc- tion. Ho we er if this is done in each element independently of the others, numerical dif ﬁculties often appear The are caused by the ﬁnite size of the incremental steps and by the limited accurac of the constant-strain triangular elements. (a) (b) (c) (d) Figur 2. Evolution of the fr actur pr ocess zone in the centr al part of notc hed beam under thr ee-point bending

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Modeling of strong discontinuities 1141 One possible remedy as proposed in [JIR 01b ]. The embedded discontinuity model as combined with smeared (continuum-based) description of inelastic pro- cesses. It as ar gued that early stages of the fracture process in quasibrittle materi- als are adequately described by distrib uted damage while the macroscopic crack that forms at later stages is naturally treated as displacement discontinuity The best re- sults are obtained with nonlocal formulation of the continuum part of the combined model. strong discontinuity is inserted in an element when the damage parameter attains critical le el. This is of course an ad hoc criterion that cannot be deri ed in rigorous ay ut it seems to ork reasonably well. The traction-separation la go erning the discontinuity must be adjusted so as to ensure that the erall ener gy dissipation remains correct. The progressi ailure of notched beam under three- point bending is illustrated in Fig. 2. The gre color marks the re gion in which the damage parameter eeps gro wing (acti part of the process zone). The material in the ak of the propagating crack with decreasing cohesi tractions is unloading, and so in this re gion the damage parameter does not gro an more. (a) (b) Figur 3. Embedded cr ac tr ajectory for model with enfor ced continuity of the cr ac path, with dir ection determined (a) fr om the local str ain, (b) fr om the nonlocal str ain

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1142 Re vue française de génie ci vil. olume 6/2002 mak sure that the propagating discontinuity cleanly separates the ﬁnite ele- ment model into tw parts, it is good to enforce continuity of the crack path (note that the interpolation of the crack opening is still only piece wise continuous, with jumps at element boundaries). When ne discontinuity se gment is inserted in an element that meets the initiation criterion, it is check ed whether one of its neighboring ele- ments is already crossed by discontinuity and, if it is the case, the ne wly inserted discontinuity is placed such that it passes through the intersection of the neighboring discontinuity se gment with the edge shared by both elements. If the orientation of the discontinuity line is determined from the local orientation of principal strain ax es in each element separately the resulting crack trajectory is tortuous (Fig. 3a). Real cracks in quasibrittle materials are not perfectly smooth, ut their roughness is con- troled by the material microstructure while the present tortuosity is purely numerical, dependent on the structure of the ﬁnite element mesh. An ef ﬁcient remedy is to de- termine the direction of the discontinuity from the principal directions of the nonlocal strain. This approach gi es correct crack trajectory en on highly biased meshes, such as that in the bottom part of Fig. 3b 6. Resolution of discontinuities by extended ﬁnite elements Elements with embedded discontinuities pro vide better kinematic description of discontinuous displacement ﬁelds than pure continuum models that smear the dis- placement jumps uniformly er the entire element, ut the still ha certain lim- itations. Their main disadv antage is that the strain approximations in the tw parts of the element separated by discontinuity are not independent. or instance, using constant-strain triangle, the strains in these tw parts are approximated by the same constant tensor Of course, one could use higher -order formulation with spatially ariable strain approximation, ut there al ays remains certain constraint that does not allo modeling of the beha vior of tw separated material bodies in full generality ne class of methods that ercome this dra wback emer ged ery recently en though similar idea could be traced back to the so-called manifold method, de el- oped in the conte xt of discontinuous deformation analysis [SHI 92 ]. Conceptually this ne approach to modeling of discontinuities can be considered as particular case of the partition-of-unity method [MEL 96 ]. The general idea of that method is that the approximation space spanned by standard basis (e.g., by the standard ﬁnite element shape functions) is enriched by products of the standard basis functions with special functions selected by the user and constructed, e.g., from the analytical solution of the problem under some simplifying assumptions. This permits the incorporation of priori kno wledge about the character of the problem and its solutions. The enriched displacement approximation is written in the form ”@ŒF¡œ¢£¥¤ Q>§ ‡¨ ©rˆª «¬ 5®s¯9°#± ©r% ² [19]

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Modeling of strong discontinuities 1143 where s´“µ is the number of nodes of the ﬁnite element model; ¶R¸· ¹¹¹ s´“µ are the standard shape functions; ¶o¸· ¹¹¹ º´ µ are the standard displacement DOFs; »¼¸· ¹¹¹%½ are the enrichment functions; QR¿ ¹¹¹À½y is the set of inte gers that indicate which enrichment functions are acti ated at node and are the additional DOFs associated with node and enrichment function The trick is that the global enrichment functions are multiplied by the nodal shape functions The products inherit from good approximation properties and from limited support. Consequently the enrichment has local character and the resulting stif fness matrix is sparse (with proper renumbering it remains banded). The standard approximation is usually enriched only locally in certain re gion of interest, e.g., in localization zone, and the ne wly added de grees of freedom can be associated with nodes of the xisting mesh, without the need for changing the topology Owing to the partition-of-unity property of the standard shape functions (their sum is equal to one at an point ), the enrichment functions can be reproduced xactly This general idea as adapted for linear elastic fracture mechanics, with the en- richment constructed using the singular near -tip asymptotic ﬁelds and simple Hea vi- side functions [MOË 99 ]. The method as later called the eXtended Finite Element Method (X-FEM). It can ef ﬁciently handle three-dimensional cracks [SUK 00 and en branching and intersecting cracks [D 00 ]. big adv antage of this technique is that the displacement interpolation is conforming, with no incompatibilities between elements, and that the strains on both sides of stress-free crack are fully decoupled. The partition-of-unity concept is also applicable to cohesi crack models. ells and Sluys [WEL 01 enriched the interpolation functions by products of the Hea viside function with standard ﬁnite element shape functions that correspond to the nodes of those elements that are intersected by the crack. Moës and Belytschk [MOË 02 added special non-singular enrichments around the crack tip, moti ated by asymptotic analysis of the strain ﬁeld at the tip of cohesi crack. These ele gant formulations seem to ercome the dif ﬁculties associated with the piece wise constant interpola- tion of the displacement jump used by embedded discontinuity models, and the en restore the symmetry of the stif fness matrix. Their implementation is, ho we er some- what more dif ﬁcult, because it is necessary to add ne global de grees of freedom during the simulation and to reﬁne the inte gration scheme in the enriched area around the crack. The impro ed resolution of discontinuities by the xtended ﬁnite element method is illustrated by the schematic tests in Fig. 4. square piece of material di vided into tw parts by ertical stress-free crack (top ro w) is ﬁrst subjected to relati motion of the tw parts (middle ro w), and then the right part is compressed in the direction parallel to the crack (bottom ro w). Fig. 4a depicts the actual physical process. Fig. 4b sho ws the approximation obtained with standard bilinear ﬁnite element. The relati motion of the tw parts is transformed into normal and shear strain, and the forces imposed on the right part of the body inﬂuence the deformation of the left part. An element with an embedded discontinuity (Fig. 4c) can cleanly reproduce the rigid-body separation ut forces parallel to the crack are still transmitted from the

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1144 Re vue française de génie ci vil. olume 6/2002 (a) (b) (c) (d) (e) Figur 4. Illustr ation of separ ation tests: (a) eal body split into two indepen- dent parts, (b) standar ﬁnite element, (c) element with embedded discontinuity (d-e) xtended ﬁnite element right part to the left one. This is because the formulation allo ws displacement jump ut the strain in the ulk material is still interpolated in continuous manner The approximation obtained with the xtended ﬁnite element method can be thought of as using tw independent erlayed elements (Fig. 4d). The edges of these elements are plotted by dotted and dashed lines, resp. Solid circles mark standard nodes while empty circles mark enriched nodes. The displacement interpolation constructed with the “dotted element is alid to the left of the crack, and that constructed with the “dashed element is alid to the right of the crack (Fig. 4e). In this ay both the separation and the deformation of one part can be reproduced xactly The fore going xample considered only one single element. Fig. 5a sho ws an as- sembly of xtended triangular ﬁnite elements modeling partially crack ed body All the elements that are crossed by the crack can be thought of as doubled. Each of the “child elements pro vides an approximation alid only on one side of the crack and is connected to the standard nodes on this side and to special enrichment nodes (mark ed by empty circles) on the other side. The enrichment nodes are introduced at the same initial locations as the standard nodes ut their displacements are completely indepen- dent of the standard ones (Fig. 5b). The nodes connected by the edge at which the crack tip is located are not enriched, to mak sure that the displacement interpolation along this edge is continuous. As is clear from Fig. 5c, the displacement jump is in- terpolated in continuous, piece wise linear manner The deformation on one side of the crack is fully independent of the deformation on the other side.

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Modeling of strong discontinuities 1145 (a) (b) (c) Figur 5. Extended ﬁnite element method: (a) ﬁnite element mesh with added de gr ees of fr eedom ar ound cr ac k, (b) displacement appr oximation using doubled elements and nodes, (c) esulting discontinuous displacement appr oximation 7. Concluding emarks This short paper could pro vide only an elementary introduction into the wide and comple subject of numerical approximations with uilt-in discontinuities. This ﬁeld of research has been olving ery ast in recent years and man interesting appli- cations are currently being de eloped. or the sak of simplicity we considered the strong discontinuity as crack, ut it could be as well slip line or an interf ace be- tween tw dif ferent materials. The main idea of the xtended ﬁnite element method can be adapted for analysis of weak discontinuities and propagating fronts that appear e.g., in phase transformation, solidiﬁcation, dessication or leaching problems. 8. Refer ences [BEL 88] “A ﬁnite element with embedded localization zones”, Computer Methods in Applied Mec hanics and Engineering ol. 70, 1988, p. 59–89. CER 94] “Discrete crack modeling in concrete structures”, PhD thesis, Uni- ersity of Colorado, Boulder Colorado, 1994. [D 00] “Arbitrary branched and intersecting cracks with the xtended ﬁnite element method”, International ournal for Numerical Methods in Engineering ol. 48, 2000, p. 1741–1760. [D 90] “Finite elements with displacement interpolated embedded localization lines insensiti to mesh size and distortions”, Com- puter Methods in Applied Mec hanics and Engineering ol. 90, 1990, p. 829–844. [JIR 98] “Analysis of rotating crack model”, ournal of Engineering Mec hanics, ASCE ol. 124, 1998, p. 842–851. [JIR 00a] “Comparati study on ﬁnite elements with embedded cracks”, Com- puter Methods in Applied Mec hanics and Engineering ol. 188, 2000, p. 307–330.

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1146 Re vue française de génie ci vil. olume 6/2002 [JIR 00b] “Conditions of uniqueness for ﬁnite elements with embedded cracks”, Pr oceedings of the Sixth International Confer ence on Computational Plasticity Barcelona, 2000, CD-R OM. [JIR 01a] “Embedded crack model: I. Basic formulation”, International ournal for Numerical Methods in Engineering ol. 50, 2001, p. 1269 1290. [JIR 01b] “Embedded crack model: II. Combination with smeared cracks”, International ournal for Numerical Methods in Engineering ol. 50, 2001, p. 1291–1305. [JIR 02] “Objecti modeling of strain localization”, Re vue fr ançaise de génie civil 2002, in this issue. [KLI 91] “Finite element with inner softening band”, ournal of Engineering Mec hanics, ASCE ol. 117, 1991, p. 575–587. [MEL 96] “The partition of unity ﬁnite element method: Ba- sic theory and applications”, Computer Methods in Applied Mec hanics and Engineering ol. 39, 1996, p. 289–314. [MOË 99] “A ﬁnite element method for crack gro wth without remeshing”, International ournal for Numerical Methods in Engineering ol. 46, 1999, p. 131–150. [MOË 02] “Extended ﬁnite element method for cohesi crack gro wth”, Engineering actur Mec hanics ol. 69, 2002, p. 813–833. [OLI 96] “Modelling strong discontinuities in solid mechanics via strain softening constituti equations. art 1: Fundamentals. art 2: Numerical Simulation”, International ournal for Numerical Methods in Engineering ol. 39, 1996, p. 3575–3624. [OLO 94] “Inner softening bands: ne approach to localization in ﬁnite elements”, Eds., Compu- tational Modelling of Concr ete Structur es Pineridge Press, 1994, p. 373–382. [OR 87] RT “A ﬁnite element method for localized ailure analysis”, Computer Methods in Applied Mec hanics and Engineering ol. 61, 1987, p. 189–214. [SA 81] “Interacti ﬁnite element analysis of reinforced concrete: frac- ture mechanics approach”, PhD thesis, Cornell Uni ersity Ithaca, Ne ork, 1981. [SHI 92] “Modeling rock joints and blocks by manifold method”, Roc Mec hanics, Pr oceedings of the 33r .S. Symposium Santa Fe, Ne Me xico, 1992, p. 639–648. [SIM 94] “A ne approach to the analysis and simulation of strain softening in solids”, Eds., ac- tur and Dama in Quasibrittle Structur es London, 1994, FN Spon, p. 25–39. [SUK 00] “Extended ﬁnite element method for three-dimensional crack modeling”, International ournal for Numerical Meth- ods in Engineering ol. 48, 2000, p. 1549–1570. [WEL 01] “A ne method for modelling cohesi cracks using ﬁnite elements”, International ournal for Numerical Methods in Engineering ol. 50, 2001, p. 2667–2682.

Jir asekep64258c ABSTRA CT The paper pr esents an vervie of modern appr oximation tec hniques for xplicit es olution of str ong discontinuities displacement jumps that can run arbitr arily acr oss 64257nite element mesh wo types of suc formulations a ID: 23791

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Numerical modeling of str ong discontinuities Milan Jirásek Labor atory of Structur al and Continuum Mec hanics Swiss eder al Institute of ec hnolo gy (EPFL) CH-1015 Lausanne Switzerland Milan.Jir asek@epﬂ.c ABSTRA CT The paper pr esents an vervie of modern appr oximation tec hniques for xplicit es- olution of str ong discontinuities (displacement jumps) that can run arbitr arily acr oss ﬁnite element mesh. wo types of suc formulations ar co ver ed: (i) elements with embedded dis- continuities, whic intr oduce internal de gr ees of fr eedom that ar eliminated on the element le vel, and (ii) xtended ﬁnite elements with additional global de gr ees of fr eedom, based on the partition-of-unity concept. RÉSUMÉ. Cet article présente une vue d’ensemble des tec hniques d’appr oximation modernes pour la résolution xplicite des fortes discontinuités (sauts dans le hamp des déplacements) qui peuvent tr aver ser de manièr quelconque un mailla d’éléments ﬁnis. Deux xtensions de la méthode des éléments ﬁnis sont décrites (i) celle qui ajoute des de grés de liberté internes, éliminés au niveau de l’élément, et (ii) celle baptisée X-FEM ("eXtended inite Element Me- thod"), qui utilise des de grés de liberté globaux supplémentair es, et dont le concept est basé sur la partition de l’unité. KEYW ORDS: discontinuities, cohesive cr ac k, xtended ﬁnite elements, partition of unity MO TS-CLÉS discontinuités, ﬁssur avec zone de cohésion, xtension de la méthode des éléments ﬁnis, partition de l’unité Re vue française de génie ci vil. olume 6/2002, pages 1133 1146

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1134 Re vue française de génie ci vil. olume 6/2002 1. Intr oduction The preceding paper [JIR 02 ga an ervie of three main classes of models that pro vide an objecti description of strain localization, and then discussed in more detail one of those classes, namely formulations re gularized by spatial inte grals 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 con- te xt of smeared-crack models, this deﬁcienc can lead to spurious stress transfer across widely open crack [JIR 98 ]. Discrete-crack models with special interf aces between con entional elements [SA 81 CER 94 do not suf fer by this pathology ut the require frequent remeshing in order to allo for crack propagation in the correct direction. The recently emer ged idea of incorporating strain or displacement discontinuities into standard ﬁnite element interpolations triggered the de elopment of po werful techniques that allo ef ﬁcient modeling of re gions with highly localized strains, e.g. of fracture process zones in concrete or shear bands in metals or soils. The discontinuities can ha an arbitrary orientation, which mak es it much easier to capture propagating crack or softening band without remeshing. This class of meth- ods, collecti ely called elements with embedded discontinuities as inspired by the pioneering ork of Ortiz et al. [OR 87 and Belytschk et al. [BEL 88 ]. The early orks used weak (strain) discontinuities, ut the idea as later xtended to strong (displacement) discontinuities [D 90 KLI 91 OLO 94 SIM 94 ]. systematic classiﬁcation and critical aluation of embedded discontinuity mod- els within uniﬁed frame ork as presented in [JIR 00a ], with the conclusion that there xist three main groups of such formulations, called statically optimal symmet- ric (SOS), kinematically optimal symmetric (K OS), and statically and kinematically optimal nonsymmetric (SK ON). The SOS formulation orks with natural stress con- tinuity condition, ut it does not properly reﬂect the kinematics of completely open crack. On the other hand, the OS formulation describes the kinematic aspects sat- isf actorily ut it leads to an wkw ard relationship between the stress in the ulk of the element and the tractions across the discontinuity line. Optimal performance is achie ed with the nonsymmetric SK ON formulation, which uses ery natural stress continuity condition and reasonably represents complete separation at late stages of the fracturing process. This is the formulation to be described ne xt. In section 6, we will present dif ferent approach to the modeling of discontinuities, based on the concept of partition of unity [MEL 96 and refered to as the xtended ﬁnite element method [MOË 99 SUK 00 00 MOË 02 ]. 2. riangular element with embedded displacement discontinuity The optimal combination of static and kinematic equations for elements with em- bedded discontinuities ﬁrst appeared in [D 90 ], en though their xact nature is

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Modeling of strong discontinuities 1135 (a) (b) (c) (d) ce -se (e) se +ce xy Figur 1. Constant-str ain triangle with an embedded displacement discontinuity not easy to understand from that paper ery similar quadrilateral element based on simple and instructi physical considerations as constructed in [KLI 91 ]. The same technique as then applied to constant-strain triangle [OLO 94 ]. general ersion of the SK ON formulation for an arbitrary type of parent element as outlined in [SIM 94 and fully described in [OLI 96 ]. Consider triangular element crossed by discontinuity (Fig. 1a). The displace- ment ﬁeld can be decomposed into continuous part and discontinuous part due to the opening and sliding of crack (Fig. 1b). The same decomposition applies to the nodal displacements of ﬁnite element. Instead of smearing the displacement jump er the area of the element and replacing it by an equi alent inelastic strain, as is done by standard smeared crack models (Fig. 1c), the discontinuity can be represented by additional de grees of freedom, and corresponding respecti ely to the normal (opening) and tangential (sliding) component of the displacement jump and collected in column matrix The contrib ution of the displacement jump is then subtracted from the nodal displacement ector and only the part of the nodal dis- placements produced by the continuous deformation serv es as input for the aluation of strains in the ulk material, (Fig. 1d). This leads to kinematic equations in the form [1] where ! #" %$ !" is the column matrix of engineering strain components, is the standard strain-displacement matrix, and is matrix reﬂecting the ef fect of the displacement jump on the nodal displacements.

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1136 Re vue française de génie ci vil. olume 6/2002 In general, the displacement jump could be approximated by suitable function, for xample polynomial one. This approximation need not be continuous on inter element boundaries. or triangular elements with linear displacement interpolation, the strains and stresses in the ulk are constant in each element, and so it is natural to approximate the displacement jump also by piece wise constant function. This is why we describe the jump in each element by only tw parameters, and These additional de grees of freedom ha an internal character and can be eliminated on the element le el, which yields the global equilibrium equations written xclusi ely in terms of the standard unkno wns—nodal displacements. From Fig. 1d it is clear that if the discontinuity line separates node from nodes and (in local numbering), the crack-ef fect matrix is gi en by & -, .0/ [2] where 32547698 ,:36<;>=?8 and is the angle between the normal to the crack (discontinuity line) and the global -axis; see Fig. 1a. Strains in the ulk material generate certain stresses, AB CD! C" !" which are here computed from the equations of linear elasticity AFGHI [3] where is the elastic stif fness matrix (for plane stress or plane strain). Note that, in general, the constituti la for the ulk material could be nonlinear The tractions transmitted by the crack, are link ed to the separation ector (displacement jump) by another constituti la that describes the gradual de elopment of stress-free crack. One speciﬁc form of this la will be presented in Section 3. The stresses in the ulk and the tractions across the crack must satisfy certain con- ditions that xpress internal equilibrium and serv as static equations corresponding to the internal de grees of freedom, The most natural requirement is that the traction ector be equal to the stress tensor contracted with the crack normal, similar to static boundary conditions. This internal equilibrium (traction continuity) condition can be deri ed from equilibrium of an elementary triangle with one side on the discontinuity line; see Fig. 1e. In the engineering notation it reads AFLJ [4] where +5M O, [5]

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Modeling of strong discontinuities 1137 is stress rotation matrix. or linear triangles with constant displacement jump, both and are constant in each element, and so condition [4] can be satisﬁed xactly In general it ould ha to be enforced in weak sense. Finally the nodal forces are aluated from the standard relation PRQ S TVUXWY ZZ[H5 [6] where Z\H is the area of the element. or the constant-strain triangle, the kinematic relations and the traction continuity condition follo quite naturally from simple physical considerations. De elopment of more complicated elements with embedded discontinuities is often done within the frame ork of enhanced assumed strain (EAS) methods, and such elements are sometimes refered to as EAS elements (which can be some what confusing). 3. raction-separation law in damage ormat The basic equations presented in the preceding section must be completed by la that links the traction transmitted by the discontinuity to the displacement jump. One possible type of such la as proposed in [JIR 01a in the form J]&^ G [7] where is dimensionless scalar compliance parameter olving from zero to inﬁnity and G_a`cb # %ed [8] is stif fness matrix corresponding to reference intermediate stage of the de gradation process. Before crack initiation, the alue of is zero. or simplicity it is assumed here that crack initiation is controlled by the Rankine criterion of maximum principal stress. This means that the discontinuity line is inserted perpendicular to the direction of maximum principal stress, and the shear traction at the instant of crack initiation is zero. The olution of is described by the loading-unloading conditions in the Karush- uhn-T uck er form, $hg iej $Xi [9] The loading function characterizing the elastic domain is deﬁned as <$ Rklnm D<o [10] where is scalar measure of the separation ector called the equi alent sepa- ration (analogous to the equi alent strain in continuum damage mechanics), and

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1138 Re vue française de génie ci vil. olume 6/2002 is suitable function describing the dependence of the compliance parameter on the equi alent separation during monotonic loading. In [JIR 01a it as proposed to set Bp Gq # [11] and lnm r s nm [12] where is scalar function describing the traction-separation curv for Mode-I crack- ing. Alternati ely the traction-separation la could be formulated within the frame- ork of plasticity; see e.g. [OLO 94 ]. This is especially useful for the description of cohesi zones in metals or shear bands in soils. Due to space limitations, the details cannot be presented here. 4. Ev aluation of inter nal or ces and tangent stiffness In an incremental-iterati analysis of structure discretized by ﬁnite elements with embedded discontinuities, the nodal displacements are computed iterati ely from the global equilibrium equations, and the main tasks on the le el of one ﬁnite element are to aluate the internal forces and the tangent stif fness matrix for gi en increment of nodal displacements. Substituting [3] and [1] into the traction continuity condition [4], we obtain use- ful xpression for the traction ector in terms of the kinematic ariables, Ju GvHIwDrxyw [13] where we ha denoted xz [14] Expression [13] for the traction ector substituted into the traction-separation la [7] yields the equation x{}|~^ Gq< x [15] that links the nodal displacements separation ector and compliance parameter If is kno wn, the separation ector can be solv ed from [15], taking into account that the compliance parameter may change during the step according to [9]–[10]. Adopting the usual numerical scheme, equation [15] is ﬁrst solv ed under the assump- tion of constant damage (unloading), i.e., with ept equal to its alue at the end of the pre vious step. If the computed separation ector satisﬁes the condition %$ then the solution is admissible, otherwise it is necessary to solv [15] with replaced by lnm < The conditions under which this problem has unique solution were studied in [JIR 00b ], where it as sho wn that uniqueness may be lost not only for

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Modeling of strong discontinuities 1139 elements that are too lar ge ut also for badly shaped elements. Once the separation ector has been determined, the strain, stress and internal forces are easily aluated by substituting into equations [1], [3] and [6]. The tangent stif fness matrix can be deri ed by combining the rate forms of the basic equations and is gi en by the formula [JIR 01a H H5‚ Gƒ| x{„…J I†Y‡x [16] where Hˆ‚Z\H5 GvH5 is the elastic element stif fness matrix and Š‰ l-‹#‰Œ is the gradient of the loading function with respect to the separation ector Recalling the deﬁnition [14] of matrix xpression [16] can be re written in the equi alent form Z[H Ž GqH GHIhŠ^ G‘| GvHIh„…J I†Y‡ GqH“’” [17] This formula has the same structure as the standard xpression •Z G for the tangent element stif fness of con entional constant-strain triangle with nonlinear material. It is interesting to note that the “equi alent tangent material stif fness Ga–GqH GH5y• G—| GqH5h˜…J n†X‡ GH [18] depends not only on the constituti parameters (elastic ulk stif fness GH reference cohesi stif fness compliance parameter ), ut also on the matrices and that reﬂect the orientation of the discontinuity and on matrix that reﬂects the size and shape of the ﬁnite element. The dependence on orientation (crack direction) means that crack-induced anisotrop is tak en into account, and the dependence on element size is typical feature of smeared models based on the fracture ener gy concept; see Section 3.2 in [JIR 02 ]. Ho we er the present model contains additional information on the kinematics of the ailure process, because matrix is af fected by the position of the discontinuity with respect to the element and matrix depends on the element shape. or con entional element, the element stif fness is symmetric if and only if the material stif fness is symmetric. or the present element with an embedded dis- continuity this happens only if (i) is scalar multiple of and (ii) h is scalar multiple of The ﬁrst condition is related to the traction-separation la and is equi v- alent to symmetry of the tangent stif fness that links the rates of the separation ector and of the traction ector or the equi alent separation deﬁned by formula [11], the gradient ector ™‰Œl-‹#‰ hš›9l-‹#›œm 5‰rm ‹#‰ 9[™lž GOD‹9 # is indeed colin- ear with the traction ector JŸ^ GOD‹ The second condition is related xclusi ely to geometrical properties. It is satisﬁed only if the discontinuity line is parallel to one of the element sides [JIR 00a ]. Therefore, en if the material stif fness is symmetric, symmetry of the structural stif fness is disturbed by the kinematic and static equations. This is why the present SK ON formulation is called nonsymmetric. There xist tw types of symmetric formulations (K OS and SOS), in which only one group of equa- tions (either kinematic or static) is postulated and the dual group is deri ed from the principle of virtual ork. Ho we er the equations obtained in this ay are not “natu- ral and the resulting formulations ha se ere dra wbacks; see [JIR 00a for detailed analysis.

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1140 Re vue française de génie ci vil. olume 6/2002 5. racing of the discontinuity path The elements described in the preceding sections can easily accommodate pre- xisting discontinuities, e.g., rock joints or delaminating interf aces, ut the principal interest lies in modeling of olving discontinuities such as propagating cracks. In this latter case, the position of the discontinuity is not kno wn in adv ance and the simula- tion starts with all elements in their “vir gin state. discontinuity se gment is inserted in an element only when certain initiation criterion is satisﬁed. The simplest ap- proach is to formulate this criterion in terms of the stress in the ulk material, e.g., as the Rankine criterion of maximum principal stress, and to place the discontinuity se gment in the element center perpendicular to the maximum principal stress direc- tion. Ho we er if this is done in each element independently of the others, numerical dif ﬁculties often appear The are caused by the ﬁnite size of the incremental steps and by the limited accurac of the constant-strain triangular elements. (a) (b) (c) (d) Figur 2. Evolution of the fr actur pr ocess zone in the centr al part of notc hed beam under thr ee-point bending

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Modeling of strong discontinuities 1141 One possible remedy as proposed in [JIR 01b ]. The embedded discontinuity model as combined with smeared (continuum-based) description of inelastic pro- cesses. It as ar gued that early stages of the fracture process in quasibrittle materi- als are adequately described by distrib uted damage while the macroscopic crack that forms at later stages is naturally treated as displacement discontinuity The best re- sults are obtained with nonlocal formulation of the continuum part of the combined model. strong discontinuity is inserted in an element when the damage parameter attains critical le el. This is of course an ad hoc criterion that cannot be deri ed in rigorous ay ut it seems to ork reasonably well. The traction-separation la go erning the discontinuity must be adjusted so as to ensure that the erall ener gy dissipation remains correct. The progressi ailure of notched beam under three- point bending is illustrated in Fig. 2. The gre color marks the re gion in which the damage parameter eeps gro wing (acti part of the process zone). The material in the ak of the propagating crack with decreasing cohesi tractions is unloading, and so in this re gion the damage parameter does not gro an more. (a) (b) Figur 3. Embedded cr ac tr ajectory for model with enfor ced continuity of the cr ac path, with dir ection determined (a) fr om the local str ain, (b) fr om the nonlocal str ain

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1142 Re vue française de génie ci vil. olume 6/2002 mak sure that the propagating discontinuity cleanly separates the ﬁnite ele- ment model into tw parts, it is good to enforce continuity of the crack path (note that the interpolation of the crack opening is still only piece wise continuous, with jumps at element boundaries). When ne discontinuity se gment is inserted in an element that meets the initiation criterion, it is check ed whether one of its neighboring ele- ments is already crossed by discontinuity and, if it is the case, the ne wly inserted discontinuity is placed such that it passes through the intersection of the neighboring discontinuity se gment with the edge shared by both elements. If the orientation of the discontinuity line is determined from the local orientation of principal strain ax es in each element separately the resulting crack trajectory is tortuous (Fig. 3a). Real cracks in quasibrittle materials are not perfectly smooth, ut their roughness is con- troled by the material microstructure while the present tortuosity is purely numerical, dependent on the structure of the ﬁnite element mesh. An ef ﬁcient remedy is to de- termine the direction of the discontinuity from the principal directions of the nonlocal strain. This approach gi es correct crack trajectory en on highly biased meshes, such as that in the bottom part of Fig. 3b 6. Resolution of discontinuities by extended ﬁnite elements Elements with embedded discontinuities pro vide better kinematic description of discontinuous displacement ﬁelds than pure continuum models that smear the dis- placement jumps uniformly er the entire element, ut the still ha certain lim- itations. Their main disadv antage is that the strain approximations in the tw parts of the element separated by discontinuity are not independent. or instance, using constant-strain triangle, the strains in these tw parts are approximated by the same constant tensor Of course, one could use higher -order formulation with spatially ariable strain approximation, ut there al ays remains certain constraint that does not allo modeling of the beha vior of tw separated material bodies in full generality ne class of methods that ercome this dra wback emer ged ery recently en though similar idea could be traced back to the so-called manifold method, de el- oped in the conte xt of discontinuous deformation analysis [SHI 92 ]. Conceptually this ne approach to modeling of discontinuities can be considered as particular case of the partition-of-unity method [MEL 96 ]. The general idea of that method is that the approximation space spanned by standard basis (e.g., by the standard ﬁnite element shape functions) is enriched by products of the standard basis functions with special functions selected by the user and constructed, e.g., from the analytical solution of the problem under some simplifying assumptions. This permits the incorporation of priori kno wledge about the character of the problem and its solutions. The enriched displacement approximation is written in the form ”@ŒF¡œ¢£¥¤ Q>§ ‡¨ ©rˆª «¬ 5®s¯9°#± ©r% ² [19]

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Modeling of strong discontinuities 1143 where s´“µ is the number of nodes of the ﬁnite element model; ¶R¸· ¹¹¹ s´“µ are the standard shape functions; ¶o¸· ¹¹¹ º´ µ are the standard displacement DOFs; »¼¸· ¹¹¹%½ are the enrichment functions; QR¿ ¹¹¹À½y is the set of inte gers that indicate which enrichment functions are acti ated at node and are the additional DOFs associated with node and enrichment function The trick is that the global enrichment functions are multiplied by the nodal shape functions The products inherit from good approximation properties and from limited support. Consequently the enrichment has local character and the resulting stif fness matrix is sparse (with proper renumbering it remains banded). The standard approximation is usually enriched only locally in certain re gion of interest, e.g., in localization zone, and the ne wly added de grees of freedom can be associated with nodes of the xisting mesh, without the need for changing the topology Owing to the partition-of-unity property of the standard shape functions (their sum is equal to one at an point ), the enrichment functions can be reproduced xactly This general idea as adapted for linear elastic fracture mechanics, with the en- richment constructed using the singular near -tip asymptotic ﬁelds and simple Hea vi- side functions [MOË 99 ]. The method as later called the eXtended Finite Element Method (X-FEM). It can ef ﬁciently handle three-dimensional cracks [SUK 00 and en branching and intersecting cracks [D 00 ]. big adv antage of this technique is that the displacement interpolation is conforming, with no incompatibilities between elements, and that the strains on both sides of stress-free crack are fully decoupled. The partition-of-unity concept is also applicable to cohesi crack models. ells and Sluys [WEL 01 enriched the interpolation functions by products of the Hea viside function with standard ﬁnite element shape functions that correspond to the nodes of those elements that are intersected by the crack. Moës and Belytschk [MOË 02 added special non-singular enrichments around the crack tip, moti ated by asymptotic analysis of the strain ﬁeld at the tip of cohesi crack. These ele gant formulations seem to ercome the dif ﬁculties associated with the piece wise constant interpola- tion of the displacement jump used by embedded discontinuity models, and the en restore the symmetry of the stif fness matrix. Their implementation is, ho we er some- what more dif ﬁcult, because it is necessary to add ne global de grees of freedom during the simulation and to reﬁne the inte gration scheme in the enriched area around the crack. The impro ed resolution of discontinuities by the xtended ﬁnite element method is illustrated by the schematic tests in Fig. 4. square piece of material di vided into tw parts by ertical stress-free crack (top ro w) is ﬁrst subjected to relati motion of the tw parts (middle ro w), and then the right part is compressed in the direction parallel to the crack (bottom ro w). Fig. 4a depicts the actual physical process. Fig. 4b sho ws the approximation obtained with standard bilinear ﬁnite element. The relati motion of the tw parts is transformed into normal and shear strain, and the forces imposed on the right part of the body inﬂuence the deformation of the left part. An element with an embedded discontinuity (Fig. 4c) can cleanly reproduce the rigid-body separation ut forces parallel to the crack are still transmitted from the

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1144 Re vue française de génie ci vil. olume 6/2002 (a) (b) (c) (d) (e) Figur 4. Illustr ation of separ ation tests: (a) eal body split into two indepen- dent parts, (b) standar ﬁnite element, (c) element with embedded discontinuity (d-e) xtended ﬁnite element right part to the left one. This is because the formulation allo ws displacement jump ut the strain in the ulk material is still interpolated in continuous manner The approximation obtained with the xtended ﬁnite element method can be thought of as using tw independent erlayed elements (Fig. 4d). The edges of these elements are plotted by dotted and dashed lines, resp. Solid circles mark standard nodes while empty circles mark enriched nodes. The displacement interpolation constructed with the “dotted element is alid to the left of the crack, and that constructed with the “dashed element is alid to the right of the crack (Fig. 4e). In this ay both the separation and the deformation of one part can be reproduced xactly The fore going xample considered only one single element. Fig. 5a sho ws an as- sembly of xtended triangular ﬁnite elements modeling partially crack ed body All the elements that are crossed by the crack can be thought of as doubled. Each of the “child elements pro vides an approximation alid only on one side of the crack and is connected to the standard nodes on this side and to special enrichment nodes (mark ed by empty circles) on the other side. The enrichment nodes are introduced at the same initial locations as the standard nodes ut their displacements are completely indepen- dent of the standard ones (Fig. 5b). The nodes connected by the edge at which the crack tip is located are not enriched, to mak sure that the displacement interpolation along this edge is continuous. As is clear from Fig. 5c, the displacement jump is in- terpolated in continuous, piece wise linear manner The deformation on one side of the crack is fully independent of the deformation on the other side.

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Modeling of strong discontinuities 1145 (a) (b) (c) Figur 5. Extended ﬁnite element method: (a) ﬁnite element mesh with added de gr ees of fr eedom ar ound cr ac k, (b) displacement appr oximation using doubled elements and nodes, (c) esulting discontinuous displacement appr oximation 7. Concluding emarks This short paper could pro vide only an elementary introduction into the wide and comple subject of numerical approximations with uilt-in discontinuities. This ﬁeld of research has been olving ery ast in recent years and man interesting appli- cations are currently being de eloped. or the sak of simplicity we considered the strong discontinuity as crack, ut it could be as well slip line or an interf ace be- tween tw dif ferent materials. The main idea of the xtended ﬁnite element method can be adapted for analysis of weak discontinuities and propagating fronts that appear e.g., in phase transformation, solidiﬁcation, dessication or leaching problems. 8. Refer ences [BEL 88] “A ﬁnite element with embedded localization zones”, Computer Methods in Applied Mec hanics and Engineering ol. 70, 1988, p. 59–89. CER 94] “Discrete crack modeling in concrete structures”, PhD thesis, Uni- ersity of Colorado, Boulder Colorado, 1994. [D 00] “Arbitrary branched and intersecting cracks with the xtended ﬁnite element method”, International ournal for Numerical Methods in Engineering ol. 48, 2000, p. 1741–1760. [D 90] “Finite elements with displacement interpolated embedded localization lines insensiti to mesh size and distortions”, Com- puter Methods in Applied Mec hanics and Engineering ol. 90, 1990, p. 829–844. [JIR 98] “Analysis of rotating crack model”, ournal of Engineering Mec hanics, ASCE ol. 124, 1998, p. 842–851. [JIR 00a] “Comparati study on ﬁnite elements with embedded cracks”, Com- puter Methods in Applied Mec hanics and Engineering ol. 188, 2000, p. 307–330.

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1146 Re vue française de génie ci vil. olume 6/2002 [JIR 00b] “Conditions of uniqueness for ﬁnite elements with embedded cracks”, Pr oceedings of the Sixth International Confer ence on Computational Plasticity Barcelona, 2000, CD-R OM. [JIR 01a] “Embedded crack model: I. Basic formulation”, International ournal for Numerical Methods in Engineering ol. 50, 2001, p. 1269 1290. [JIR 01b] “Embedded crack model: II. Combination with smeared cracks”, International ournal for Numerical Methods in Engineering ol. 50, 2001, p. 1291–1305. [JIR 02] “Objecti modeling of strain localization”, Re vue fr ançaise de génie civil 2002, in this issue. [KLI 91] “Finite element with inner softening band”, ournal of Engineering Mec hanics, ASCE ol. 117, 1991, p. 575–587. [MEL 96] “The partition of unity ﬁnite element method: Ba- sic theory and applications”, Computer Methods in Applied Mec hanics and Engineering ol. 39, 1996, p. 289–314. [MOË 99] “A ﬁnite element method for crack gro wth without remeshing”, International ournal for Numerical Methods in Engineering ol. 46, 1999, p. 131–150. [MOË 02] “Extended ﬁnite element method for cohesi crack gro wth”, Engineering actur Mec hanics ol. 69, 2002, p. 813–833. [OLI 96] “Modelling strong discontinuities in solid mechanics via strain softening constituti equations. art 1: Fundamentals. art 2: Numerical Simulation”, International ournal for Numerical Methods in Engineering ol. 39, 1996, p. 3575–3624. [OLO 94] “Inner softening bands: ne approach to localization in ﬁnite elements”, Eds., Compu- tational Modelling of Concr ete Structur es Pineridge Press, 1994, p. 373–382. [OR 87] RT “A ﬁnite element method for localized ailure analysis”, Computer Methods in Applied Mec hanics and Engineering ol. 61, 1987, p. 189–214. [SA 81] “Interacti ﬁnite element analysis of reinforced concrete: frac- ture mechanics approach”, PhD thesis, Cornell Uni ersity Ithaca, Ne ork, 1981. [SHI 92] “Modeling rock joints and blocks by manifold method”, Roc Mec hanics, Pr oceedings of the 33r .S. Symposium Santa Fe, Ne Me xico, 1992, p. 639–648. [SIM 94] “A ne approach to the analysis and simulation of strain softening in solids”, Eds., ac- tur and Dama in Quasibrittle Structur es London, 1994, FN Spon, p. 25–39. [SUK 00] “Extended ﬁnite element method for three-dimensional crack modeling”, International ournal for Numerical Meth- ods in Engineering ol. 48, 2000, p. 1549–1570. [WEL 01] “A ne method for modelling cohesi cracks using ﬁnite elements”, International ournal for Numerical Methods in Engineering ol. 50, 2001, p. 2667–2682.

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