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A Meshfree Method based on the Local Partition of Unity for Cohesive Cracks

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A Meshfree Method based on the Local Partition of Unity for Cohesive Cracks
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  Comput Mech (2006)DOI 10.1007/s00466-006-0067-4 ORIGINAL PAPERTimon Rabczuk  ·  Goangseup Zi A meshfree method based on the local partition of unityfor cohesive cracks Received: 8 November 2005 / Accepted: 23 February 2006© Springer-Verlag 2006 Abstract  We will present a meshfree method based on thelocal partition of unity for cohesive cracks. The cracks aredescribed by a jump in the displacement field for particleswhose domain of influence is cut by the crack. Particles withpartially cut domain of influence are enriched with branchfunctions.Crackpropagationisgovernedbythematerialsta-bilitycondition.Duetothesmoothnessandhigherordercon-tinuity,themethodisveryaccuratewhichisdemonstratedforseveralquasistaticanddynamiccrackpropagationexamples. Keywords  Extended element-gree Galerkin method(XEFG)  ·  Cracks  ·  Cohesive models  ·  Dynamic fracture 1 Introduction Different methods have been presented to model cracks infiniteelementandmeshfreemethods.Simpleandrobustmeth-ods are interelement separation models in which cracks aremodelled along element interfaces in the mesh, see Xu andNeedleman[43],CamachoandOrtiz[17],OrtizandPandolfi[33], and Zhou and Molinari [44]. Another simple methodwas proposed by Remmers et al. [37] who introduced crack segments in finite elements. Rabczuk and Belytschko [34]developed a ‘cracking particle’ model in meshfree methodswherediscontinuitiesareintroducedattheparticlepositions.The major advantage of these methods are their robustnessand ease in implementation. However, for certain classes of problems, more accurate methods are needed.The embedded discontinuity model [3, 7, 39] is anothermethod for crack problems. However, its effectiveness in T. Rabczuk ( B )Institute for Numerical Mechanics, Technical University of Munich,Boltzmannstr. 15, 85748 Garching b. Munich, GermanyE-mail: Timon.Rabczuk@lnm.mw.tum.deTel.: +49-89-289915304G. ZiDepartmentofCivilandEnvironmentalEngineering,KoreaUniversity,5-1, Anam-dong, Sungbuk- Ku, Seoul 136-701, KoreaE-mail: g-zi@korea.ac.kr crackdynamicshasstillnotbeenassessedandthesemethodsrequire the crack to propagate one element at a time.A very accurate method for crack problems is the ex-tendedfiniteelementmethod(XFEM)developedbythegroupof Prof. Belytschko [5, 31]. This method is based on the ‘lo-cal’partitionofunity,inwhichthesolutionspaceisenrichedbyaprioriknowledgeaboutthebehaviourofthesolutionnearcracks.Becauseonlythenodesbelongingtotheelementscutby cracks are enriched, the number of additional degrees of freedom for the local enrichment is minimized. The detaileddiscussion about the (local) partition of unity is found in theliterature, e.g. Melenk and Babuska [30]; Chessa et al. [18].This method has been successfully applied to static prob-lems in two and three dimensions, (see e.g. [22, 31, 32, 45,47]) and to dynamic problems ([6, 46]) in two dimensions.Previous meshfree methods for cracking [9, 11, 14, 26,29]intwoandthreedimensionsweretreatedbythesocalledvisibilitycriterionorsomemodificationsofit.Therefore,thesupport is truncated by the line of discontinuity. Other novelapproaches which were able to treat kinked and curve crackswere proposed by [41]. They also enriched the MLS basefunctions  p  around the crack tip and significantly improvedthe convergence behaviour. The major drawback is the needfor an explicit representation of the crack.A meshfree concept of XFEM was proposed by Venturaet al. [41] for linear elastic fracture mechanics in statics. Wewill pursue this idea and extend it to dynamics and cohesivecracks. The advantage of this meshfree method over XFEMis the higher smoothness, non-local interpolation characterand higher order continuity which results in a better stressdistribution around the crack tip which is important for thepropagation of the crack.In XFEM, mainly piecewise linear crack opening is as-sumed (due to the use of low order shape functions) but itis well known that in reality, the crack opening displace-ment is nonlinear, especially near the crack tip. Nonlinearcrack opening relations can be easily incorporated in mesh-free methods. In XFEM, Laborde et al. [27] have shown thata wider support for the crack tip enrichment improves theaccuracy and convergence; more nodes are enriched for the  T. Rabczuk, G. Zi Fig. 1  Crack with partial cut and complete cut domain of influenceparticles cracktipthanintheclassicalXFEM.Meshfreemethodsnatu-rallyhaveawidersupportduetotheirnon-localinterpolationcharacter than FEM. However, this results in difficulties inimposing the Dirichlet boundary conditions at the crack, i.e.the crack has to close at the crack tip. A possible solution isaddressed in the next section.The paper is arranged as follows: in the next section, wewill describe the concept, i.e. the approximation of the jumpinthedisplacement.Themeshfreemethodisbrieflyreviewedafterwards. In Sect. 4, we will give the governing equationsandderivethediscreteequationsinSect.5.Wewillverifyourmodel first for linear elastic fracture mechanics and compareresults to former meshfree methods. Finally, we will solveseveralquasi-staticanddynamiccrackpropagationproblemsand compare the results to experimental data or other data inthe literature. 2 Approximation of the test and trial functions The main idea to capture the crack is to enrich the test andtrial functions with additional unknowns so that the approxi-mation is continuous in the whole domain but discontinuousalong the crack as done in many former methods such asXFEM [31]. Therefore, the test and trial functions are writ-ten in terms of a signed distance function  f   (see Fig. 1): δ u ( X )  =   I  ∈ W  ( X )   I  ( X ) δ u  I  +   I  ∈ W  b ( X )   I  ( X )  H   (  f   I  ( X )) δ a  I  +   I  ∈ W  s ( X )   I  ( X )  K  B K  ( X ) δ b K I   (1) u ( X )  =   I  ∈ W  ( X )   I  ( X )  u  I  +   I  ∈ W  b ( X )   I  ( X )  H   (  f   I  ( X ))  a  I  +   I  ∈ W  s ( X )   I  ( X )  K  B K  ( X )  b K I   (2)where W  ( X )  is the entire domain, W  b ( X )  is the completelycut domain, W  s ( X )  is the partial cut domain, and  H   and  B are the enrichment functions explained later. The first termon the right hand side of Eq. (1) or (2), respectively, is theusual approximation where    I   are the shape functions, and u  I   and  δ u  I   are the parameters. The second and third term isthe enrichment, in which the coefficient  δ a  and  δ b  or  a  and b , respectively, are additional unknowns introduced for thecrack in the variational formulation.  H  (  f   ( X ))  depends on the signed distance function  f   I  ( X ) and is defined as:  H   (  f   I  ( X ))  =  1 if   f   I  ( X ) >  0  H   (  f   I  ( X ))  = − 1 if   f   I  ( X ) <  0 (3)with  f   I  ( X ) =  sign [ n  ·  ( X  I   −  X ) ] min  X  I   −  X  ,  for  X  I   ∈ W  b n  ·  ( X tip  −  X  I  ),  for  X  I   ∈ W  s (4)where  X tip  are the coordinates of the crack tip and  n  is thecrack normal. Only nodes which are located in the domain W  b ( X )  are enriched with the additional unknowns  δ a  and  a .The second term of Eq. (1) or (2) is called the ‘step’ enrich-ment.The third term of Eqs. (1) and (2) is applied around thecrack tip  W  s ( X ) . In linear elastic fracture mechanics,  B  ischosen to be continuous in the whole domain  W  s ( X ) , butdiscontinuous at the crack line: B  =  √  r   sin  θ  2 , √  r   cos  θ  2 , √  r   sin  θ  2 sin  θ, √  r   cos  θ  2 sin  θ    (5)accordingtotheanalyticalsolutionaroundthecracktipwhere r   isthedistanceof  X tothecracktipand θ( X )  =  sin − 1 (  f  / r  ) is the angle between the tangent to the crack line and the seg-ment X − X tip ,seeFig.1.Itiscalledthe‘branch’enrichment.For cohesive cracks, there is no crack tip singularity andthe crack opening displacement, which the cohesive tractiondepends on, may be described by the additional unknown a  only. In XFEM, this procedure is straightforward since itis easy to impose the appropriate boundary conditions, seeFig. 2a, i.e. the crack has to close at the end of the elementedge. This can be accomplished e.g. by not enriching thenodes at the element edge where the crack tip is located asshown in Fig.2a. However, in meshfree methods, this tech-nique cannot be applied analogous to the way in XFEM, seeFig. 2b.Therefore, we keep the branch enrichment for cohesivecracks, but without the crack tip singularity: B  =  r  m sin  θ  2   m  =  1 , 2 , 3 (6)  A meshfree method based on the local partition of unity for cohesive cracks (a) (b) Fig. 2  Crack with enriched nodes in  a  XFEM and  b  meshfree methods While lower order finite elements (that are usually applied)can capture only linear crack opening, meshfree methodshave the advantage to capture more realistic crack openings,as measured in experiments, due to their ease of increasingthe order of continuity. Another advantage is the non-localinterpolation character, i.e. many particles are enriched. Wehavetestedthebranchfunctions,Eq.(6),forcohesivecracks.Furthermore, we shifted the function  H   I  (  f   ( X ))  and  B K  ( X ) bytheirvaluesatthepositionofparticle  I  ,i.e.  H   I  (  f   ( X  I  )) and  B K  ( X  I  ) , respectively: ¯  H  n I   ( X )  =  H  n   f   n ( X )  −  H  n   f   n ( X  I  )   (7) ¯ B mK  ( X )  =  B mK   ( X )  −  B mK   ( X  I  )  (8)which makes the enriched region narrower. To avoid havingheavy notations, we drop  ¯·  in the following sections; unlessmentioned otherwise,  H   and  B  stand for  ¯  H   and  ¯ B  of Eqs.(7)and (8), respectively.We would like to mention that for particles in the blend-ing region, i.e. the particles whose domain of influence is notcutbutinfluencedbythe‘enriched’particles,onlythe‘usual’approximation [first term on the right hand side of Eqs. (1)and (2)] is considered in the approximation of the test andtrial functions.2.1 Tracing the crack pathsThelevelsettechniquesisoftenusedtotracethecrackpaths,Moes et al. [31]; Ventura et al. [41]; Belytschko et al. [12];Rabczuk and Belytschko [35]. We believe it is easier to tracethecrackpathsbypiecewiselinearlines.However,incertaincases,e.g.fornon-linearcrackpaths,theuseoflevelsetscanbe advantageous. The crack is defined by an implicit func-tion  f   whichiszeroalongthecrackpathandhasthevalueof the minimum distance to the crack with plus or minus sign.The choice of the sign is completely arbitrary as long as itis consistent throughout the entire calculation. We will notexplain the crack tracing procedure with level sets in moredetail and refer the interested reader to the literature, e.g.[12, 31, 41]. However, we briefly describe how to treat crack branching and crack intersection which is different from theapproach in [19] and [12] in the sense that we do not useany special branch function in addition to the ‘usual’ enrich-ment. Consider cracks shown in Fig. 3. Let W  1 b  be the set of  II  f  1  (x)=0  f  1  (x)=0  f  2  (x)=0  f  2  (x)=0 a) b) Fig. 3  Support of node I with  a  intersecting discontinuities and  b branching discontinuities nodes whose domain of influence is completely cut by thediscontinuity  f  1 ( X )  =  0 and W  2 b  the corresponding set for  f  2 ( X )  =  0. W  3 b  = W  1 b  W  2 b .Thesameappliesaccordinglyfor nodes whose domain of influence is cut by the crack tipenrichment. We will denote this set of nodes with W  1 s  and W  2 s  . Then the approximation of the displacement may begiven by [19] u ( X )  =   I  ∈ W  ( X )   I  ( X )  u  I  +   I  ∈ W  1 b ( X )   I  ( X )  H   (  f  1 ( X ))  a ( 1 )  I  +   I  ∈ W  2 b ( X )   I  ( X )  H   (  f  2 ( X ))  a ( 2 )  I  +   I  ∈ W  3 b ( X )   I  ( X )  H   (  f  1 ( X ))  H   (  f  2 ( X ))  a ( 3 )  I  +   I  ∈ W  1 s  ( X )   I  ( X )  K  B ( 1 ) K   ( X )  b ( 1 ) K I  +   I  ∈ W  2 s  ( X )   I  ( X )  K  B ( 2 ) K   ( X )  b ( 2 ) K I   (9)Principally, more than two branches can be included at onetimeandabranchedcrackcanbranchagain.AscanbeeasilyseenbyEq.(9),additionalcomplexityisthenintroduced.Wewouldliketomention,thatEq.(9)looksworsethanitissinceonly very few nodes are included in all sets W  . However, acrack branching requires theintroductionof another level setwhich makes the computation cumbersome for many cracks.Zi et al. [47] proposed a computationally more efficientapproach than (9) by modifying the signed distance func-tions so that no cross terms are needed for junction or branchproblems.When two cracks are joining, the crack tip enrichment isremoved. By using the signed distance functions of the pre-existing and approaching crack, the signed distance functionof the approaching crack is modified. Consider Fig. 4. Threedifferent subdomains have to be considered:  (  f  1  <  0 ,  f  2  < 0 ) ,  (  f  1  >  0 ,  f  2  >  0 ) ,  (  f  1  >  0 ,  f  2  <  0 )  as in Fig. 4b or (  f  1  >  0 ,  f  2  <  0 ) ,  (  f  1  >  0 ,  f  2  >  0 ) ,  (  f  1  <  0 ,  f  2  <  0 )  as inFig. 4d. The signed distance function of crack 1 of a point  X  T. Rabczuk, G. Zi (a) (b)(d)(c) Fig. 4  Sign functions for crack junction is then obtained by:  f  1 ( X )  =   f   01  ( X ),  if   f   02  ( X 1 )  f   02  ( X ) >  0  f   02  ( X ),  if   f   02  ( X 1 )  f   02  ( X ) <  0 (10)where the superimposed 0 denotes the sign distance func-tionbeforecrackjunction.Therefore,thefinalapproximationwithout the cross term reads: u ( X )  =   I  ∈ W  ( X )   I  ( X )  u  I  + n c  n = 1   I  ∈ W  b ( X )   I  ( X )  H    f   ( n )  I   ( X )   a ( n )  I  + m t   m = 1   I  ∈ W  s ( X )   I  ( X )  K  B ( m ) K   ( X )  b ( m ) K I   (11)where  n c  and  m t   are the number of cracks that completelyor partially, respectively, cross the domain of influence of the corresponding particle. The test functions are expressedaccording to Eq. (11): δ u ( X )  =   I  ∈ W  ( X )   I  ( X ) δ u  I  + n c  n = 1   I  ∈ W  b ( X )   I  ( X )  H    f   ( n )  I   ( X )  δ  a ( n )  I  + m t   m = 1   I  ∈ W  s ( X )   I  ( X )  K  B ( m ) K   ( X ) δ b ( m ) K I   (12) 3 Meshfree approximation The meshfree approximation can be written as u ( X , t  )  =   I    I  ( X )  u  I  ( t  )  (13)In the EFG-method (see e.g. [8–10]), the shape functions arecalculated as follows:   J   =  p ( X ) T A ( X ) − 1 D ( X  J  )  (14) A ( X )  =   J  p ( X  J  ) p T ( X  J  ) W  ( X  −  X  J  , h )  (15) D ( X  J  )  =  p ( X  J  ) W  ( X  −  X  J  , h )  (16)Hereby, p arethebasepolynomials,  W   isthekernelfunction,and  h  is the size of the domain of influence. To ensure theconservation of angular momentum, the approximation hasto be linear complete. Therefore, the base polynomials arechosen to be  p  =  ( 1 ,  X  , Y  ) .In addition to the fact that the order of continuity canbeincreasedquiteeasily,meshfreemethodshaveadvantagesoverfiniteelementsbecauseoftheirsmoothnessandnonlocalinterpolation character. Better stress distributions around thecrack tip are expected, which must lead to a less-oscillatorycrack propagation.Continuity in meshfree methods is governed by the con-tinuity of the kernel function  W  . We used the cubic  B -Splineas the kernel that is  C  2 . 4 Governing equations ThestrongformofthemomentumequationinatotalLagrang-ian description is given by ̺ 0  ¨ u  = ∇  0  ·  P  +  ̺ 0  b  in   0  \  Ŵ c 0  (17)with boundary conditions: u ( X , t  )  = ¯ u ( X , t  )  on  Ŵ u 0  (18) n 0  ·  P ( X , t  )  = ¯ t 0 ( X , t  )  on  Ŵ t  0  (19) n 0  ·  P −  =  n 0  ·  P +  =  t c 0  on  Ŵ c 0  (20) t c 0  =  t c 0 ( [[ u ]] )  on  Ŵ c 0  (21)  A meshfree method based on the local partition of unity for cohesive cracks where ̺ 0  istheinitialdensity,  ¨ u istheacceleration, P  denotesthe nominal stress tensor,  b  designates the body force,  ¯ u  and ¯ t 0  are the prescribed displacement and traction, respectively, n 0  istheoutwardnormaltothedomainand Ŵ u 0  Ŵ t  0  Ŵ c 0  = Ŵ 0 ,  (Ŵ u 0  Ŵ t  0 )  (Ŵ t  0  Ŵ c 0 )  (Ŵ c 0  Ŵ u 0 )  =  Ø. Moreover,we assume that the stresses  P  at the crack surface  Ŵ c 0  arebounded. Since the stresses are not well defined in the crack,the crack surface  Ŵ c 0  is excluded from the domain   0  whichis considered as an open set. 5 The discrete momentum equation Starting point is the weak form of the momentum equationwhich is given by δ W   =  δ W  int  −  δ W  ext  +  δ W  kin  =  0 (22)where δ W  int  =    0 \ Ŵ c 0 ( ∇ ⊗  δ u ) T :  P  d  0  (23) δ W  ext  =    0 \ Ŵ c 0 ̺ 0  δ u  ·  b  d  0  +   Ŵ t  0 δ u  · ¯ t 0  d Ŵ 0 +   Ŵ c 0 [[ δ u ]] ·  t c 0 d Ŵ 0  (24) δ W  kin  =    0 \ Ŵ c 0 ̺ 0  δ u  · ¨ u  d  0  (25)Substituting the test and trial functions (Eqs. (11) and(12), respectively) into Eqs. (23) to (25), we obtain W  kin  =    0 \ Ŵ c 0 ̺ 0  (  I  ( X ) δ u  I  +   I  ( X )  H    f   ( n )  I   ( X )   δ a ( n )  I  +   I  ( X )  B ( m ) K   δ b ( m ) K I   · (  J  ( X )  ¨ u  J  +   J  ( X )  H    f   ( n )  J   ( X )   ¨ a ( n )  J  +   J  ( X )  B ( m ) K  ¨ b ( m ) K J    d  0  (26) W  int  =    0 \ Ŵ c 0 δ u  I   ∇  0   I  ( X )  ·  P  d  0 +    0 \ Ŵ c 0 δ a ( n )  I   ∇  0   I  ( X )  H    f   ( n )  I   ( X )  +    I  ( X ) ∇  0  H    f   ( n )  I   ( X )  ·  P  d  0 +    0 \ Ŵ c 0  ∇  0   I  ( X )  B ( m ) K  +    I  ( X )  ∇  0 B ( m ) K   δ b ( m ) K I   ·  P  d  0  (27)    0 \ Ŵ c 0 ̺ 0  δ u  ·  b  d  0  =    0 \ Ŵ c 0 ̺ 0  (  I  ( X ) δ u  I  +   I  ( X )  H    f   ( n )  I   ( X )   δ a ( n )  I  +   I  ( X ) B ( m ) K   δ b ( m )  I   ·  b  d  0  (28)   Ŵ t  0 δ u  · ¯ t 0  d Ŵ 0 =   Ŵ t  0    I  ( X )δ u  I   +   I  ( X )  H    f   ( n )  I   ( X )  δ a ( n )  I  +    I  ( X ) B ( m ) K   δ b ( m ) K I   · ¯ t 0  d Ŵ 0  (29)   Ŵ c 0 [[ δ u ]] ·  t c 0 d Ŵ 0 =   Ŵ c 0    I  ( X )δ u  I   +   I  ( X )  H    f   ( n )  I   ( X )  δ a ( n )  I  +    I  ( X )  B ( m ) K   δ b ( m ) K I   ·  t c 0 d Ŵ 0 =   Ŵ c 0    I  ( X )  H    f   ( n )  I   ( X )   δ a ( n )  I  +    I  ( X )  B ( m ) K   δ b ( m ) K I   ·  t c 0 d Ŵ 0  (30)where Eqs. (28) to (30) are for  W  ext . After some algebraicoperationsthefinalformoftheequationofmotionisobtainedby M  I J   · ¨ u  I   =  F ext  I   −  F int  I   (31)with M  I J   =  m uu I J   m ua I J   m ub I J  m au I J   m aa I J   m ab I J  m bu I J   m ba I J   m bb I J    (32) ¨ u  I   =  ¨ u u I  ¨ a  I  ¨ b  IK    (33) F ext  I   =  f  u , ext  I  f  a , ext  I  f  b , ext  IK    (34) F int  I   =  f  u , int  I  f  a , int  I  f  b , int  IK    (35)with m uu I J   =    0 \ Ŵ c 0 ̺ 0    I  ( X )   J  ( X )  d  0 m ua I J   =    0 \ Ŵ c 0 ̺ 0    I  ( X )   J  ( X )  H    f   ( n )  I   ( X )   d  0  , m ua I J   =  m au I J  m ub I J   =    0 \ Ŵ c 0 ̺ 0    I  ( X )   J  ( X )  B ( m ) K   d  0  , m ub I J   =  m bu I J 
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