A Robust Cohesive Zone Model for Cyclic Loading

Transcription

1 14 h Inernaional LS-DYNA Users Conference Session: Simulaion A Robus Cohesive Zone Model for Cyclic Loading Ala Tabiei and Wenlong Zhang Deparmen of Mechanical Engineering Universiy of Cincinnai, Cincinnai, OH Absrac Cohesive elemen approach is a promising way o simulae crack propagaion. Commonly used cohesive laws include bilinear law, rapezoidal law and exponenial law. However, research has found ha when exponenial cohesive law is unloading or reloading, he racion curve canno remain coninuous when he mixed mode raio changes. (Kreging 005) [1] In his paper, he disconinuiy behavior of exponenial law is discussed and a remedy is given o handle ha. Insead of using a consan unloading slope, we use an unloading and reloading slope ha changes wih mixed mode raio, like he damage parameer used in bilinear model. Improved Xu-Needleman s exponenial cohesive law will be used as an example o show his improvemen. Is applicaion in cyclic loading for faigue failure is presened. Keywords: Cohesive zone model; Improved Xu-Needleman cohesive law; Bilinear cohesive law; Unloading behavior; Reloading behavior; Coninuiy; Cyclic loading Inroducion The concep of cohesive zone, firsly conceived by Dugdale (1960) [] and Barenbla (196) [3], is used o handle he process zone ahead of crack ip ha linear elasic fracure mechanics (LEFM) doesn work well when he process zone is no sufficienly small compared o he srucural size. I reas he process zone as a cohesive zone in which here exiss racion beween wo virual surfaces ahead of he crack ip and i degrades as he separaion beween he surfaces increases, and when he racion decrease o zero, he virual surfaces are considered o form real crack surfaces. The area under he racion separaion curve is he energy consumed o open he surfaces, hus conneced wih physical model. This mehod is sraighforward and powerful. I can be used o handle fracure problems whose geomery has or doesn have iniial cracks (or blun crack), and he laer one is hard o do in LEFM. In 1976, Hillerborg [4] firs applied cohesive zone model in finie elemen mehod o simulaion crack iniiaion and growh in concree beam. In 1994 Xu and Needleman [5] firs insered cohesive inerface elemens o every elemen inerface o allow arbirary dynamic crack propagaion and produced good resuls. These cohesive inerface elemens have zero hickness and are insered prior o he beginning of simulaion, and i s called inrinsic approach. The success of Xu and Needleman brough an encouraging insigh ino he fracure simulaion ha crack propagaion can be simulaed in a much easier way where no complicaed failure crieria and opologies bu only he informaion of cohesive zone model is needed. Similar o Xu and Needleman, Oriz (1999) [6] used zero hickness cohesive elemen in a way ha insead of having cohesive inerface elemens a he beginning of simulaion, hey are insered progressively o locaions where cerain crierion is me, like sress reaching maerial srengh and his is referred o as exrinsic approach. There are oher ways o imbed cohesive zone models ino fracure simulaions, like using Pariion of Uniy Mehod (PUM) in Exended Finie Elemen Mehod [7], or smeared crack approach [8]. June 1-14,

2 Session: Simulaion 14 h Inernaional LS-DYNA Users Conference Large aenion has been pu on cohesive racion separaion law (TSL) since i is he only informaion conrols he cohesive zone model behavior. Various TSLs are proposed by differen people, he bigges differen beween hese TSLs is he shape. The mos common shapes are bilinear (Camanho 003) [9], exponenial (Xu 1993)[5], and rapezoidal (Tvergaard V, Huchinson 199) [10]. Alhough he shape of TSL are differen, hey all have common feaures ha he area under he racion-separaion curve represens he criical energy release rae, and he maximum racion represens he maximum ensile or shear srengh in he maerial. Differen sudies have been carried ou on he influence of cohesive law shape on he simulaion resul. K. Volokh 004 [11] used differen cohesive laws on a block peel es and showed he shape of cohesive zone model is imporan. Giulio Alfano 005 [1] conduced a comprehensive sudy and concluded ha for a ypical double canilever beam es he soluion is pracically independen from he shape of cohesive law, whereas up o 15% difference in he maximum load is recorded in a rigid compac specimen. Among hese various cohesive models we are especially ineresed in he exponenial cohesive model since we observed some disconinuiy in he racion curve as i unloads or reloads in a differen mixed mode raio, and i will be elaboraed in deail in secion. A cohesive law is called irreversible if i considers damage accumulaion and is unloading pah don follow he same loading pah. When irreversible cohesive law is used, an unloading and reloading pah needs o be defined o deermine he unloading and reloading behavior. In his paper, bilinear and exponenial cohesive law s racion separaion law will be briefly reviewed and hen heir unloading and reloading behavior will be examined. A disconinuous behavior during he reloading process of exponenial law is observed (Figure 1) when i s reloading a a differen mixed mode raio. The mixed mode raio we used in his paper is he raio beween angenial separaion rae and normal separaion rae: r = n Two condiions in combinaion caused his disconinuiy behavior: (1) Unloading and reloading behavior is described using a consan slope, and his is wha commonly used in lieraure [13-15]. () The mixed mode raio r changes during he unloading or reloading process. Figure 1: Disconinuiy in Improved Xu-Needleman s model when reload a differen mixed mode raio using consan unloading/reloading slope Bilinear cohesive law and is unloading and reloading behavior 1- June 1-14, 016

3 14 h Inernaional LS-DYNA Users Conference Session: Simulaion We will sar by looking a a bilinear cohesive law by Camanho (003) [9]. I is formulaed based on damage mechanics and B-K (1997) mixed mode crierion [16] and was implemened in LS-DYNA s maerial library as MAT_138 [17]. In bilinear cohesive law, i considers he mixed mode separaion (Equaion 1) as an inpu and uses i o calculae he damage parameer (Equaion ), which is incorporaed in he expression of racion separaion relaionship: δ = δ 1 + δ + δ 3 (1) Where δ 1 and δ corresponds o he in plane shear separaion; δ 3 is he normal separaion and is aken as 0 when i s negaive. x is he Macaulay bracke: x if x > 0 x = { 0 if x 0 () The damage parameer is: D = min ( δf δ max δ 0 δ max δ F, 1) (3) δ0 Where δ 0 (Equaion 4) is he mixed mode separaion corresponding o maximum racion, i corresponds o he poin when damage iniiaes. δ F (Equaion 5) is he mixed mode failure separaion, exceeding which he cohesive elemen will be considered failed and deleed. δ max records he hisory of mixed mode separaion δ and akes he larges posiive separaion in hisory. In his way he damage parameer canno decrease and his is how he irreversible behavior is conrolled. The damage parameer ranges from 0 o 1 and when i reaches 1 he cohesive elemen is considered oally damaged and no able o carry any more load. δ 0 = δ 0 I δ β II (4) (δ 0 II ) + (βδ 0 I ) δ F = { G IIC S (1 + β ) δ 0 [( EN α ) + ( ETβ α 1 α ) ] G IC G IIC δ 3 > 0 δ 3 0 (5) In hese equaions, δ I = δ 3 and δ II = δ 1 + δ ; β = δ II δ I, α is parameer deermined by user. Afer all he damage parameer is define, he racion can be expressed as: Normal seperaion: T = { E (1 D) δ 3 0 < δ 3 < δ F E Scale Facor δ 3 δ 3 < 0 (6) Tangenial separaion: S = G (1 D) δ 1 (or δ ) (7) Where E and G are elasic and shear modulus in cohesive law, and hey are usually aken as a much higher value han he elasic modulus of bulk maerial o minimize he exra compliance. June 1-14,

4 Session: Simulaion 14 h Inernaional LS-DYNA Users Conference The way he damage parameer is formulaed in bilinear model makes i incorporae he mixed mode behavior hus no maer how he mixed mode raio changes, he bilinear cohesive law can always remain coninuous. As an example, a 3D and D plo is given in Figure, in he firs cycle of loading, he mixed mode raio is r = 1; in he reloading cycle, mixed mode raio changes o r = Figure : Coninuous behavior of bilinear cohesive law during unloading and reloading process when mixed mode raio changes Exponenial cohesive law and is disconinuiy during unloading and reloading process The exponenial cohesive law proposed by Xu and Needleman (1994) [5]was formed poenial based. For exponenial cohesive law, however, he damage parameer usually used in lieraure is no closely relaed o mixed mode separaion. In his paper an improved Xu-Needleman s cohesive law is adoped as an example. I is proposed by (Bosch 006) [18] o address he energy inconsisency in he mixed mode decohesion process. The energy consisency is esed by separaing he cohesive elemen in normal direcion firs for a cerain disance n before failure, hen separae in he angenial direcion ill failure. The normal and angenial energy versus normal displacemen in he firs sep is ploed in Figure 3 (a). And we can see in he improved exponenial cohesive model he oal energy is consan no maer how n changes. This is no he case in he bilinear model (Figure 3 (b)). 1-4 June 1-14, 016

5 14 h Inernaional LS-DYNA Users Conference Session: Simulaion Figure 3: (a) Energy consisency of improved exponenial cohesive model; (b) Energy inconsisency of bilinear cohesive model The modified Xu-Needleman s cohesive law has he following form: φ( n, ) = φ n [1 (1 + n ) exp ( n ) exp ( )] (8) δ n δ n T n = φ = φ n ( n ) exp ( n ) exp ( n δ n δ n δ n δ ) (9) T = φ = φ ( ) (1 + n ) exp ( n ) exp ( δ δ δ n δ n δ ) (10) Where δ n = φ n /(et n ), and δ = φ /(T n e/). Park (011) [44] saed ha poenial based models proposed under he condiion of monoonic separaion pah needs independen unloading and reloading relaions. The unloading and reloading behavior of he improved Xu-Needleman cohesive law is usually did by linear inerpolaion, like in paper of Kullari 014 [15]. When cohesive zone sars o unload, a raio beween he racion and separaion a ha poin is calculaed and used as he unloading and reloading slope. The damage parameer in normal and angenial direcion is given as: d n = 1 exp ( n,max δ n d = 1 exp (,max δ δ ) exp (,max δ ) (11) ) exp ( n,max ) (1 + n,max ) (1) δ n δ n And racion during he unloading and reloading process is calculaed by using: { T n = (1 d n )K n n T = (1 d )K (13) Where K n = G IC δ n = e T n /G IC and K = G IIC δ = et /G IIC are he iniial slope of he cohesive law in normal and angenial direcion. This works fine when mixed mode raio doesn change during he process, as shown in Figure 4. June 1-14,

6 Session: Simulaion 14 h Inernaional LS-DYNA Users Conference Figure 4: Linear inerpolaion for unloading and reloading behavior of improved exponenial law when he mixed mode raio doesn change However, when mixed mode raio changes, for example, from r = 1 in he firs load cycle o r = 0.5 in he second load cycle, he racion curve canno remain coninuous, like in Figure 5. This is because unlike he damage parameer in bilinear law, which is consanly updaing wih mixed mode raio hrough δ max, δ 0 and δ F, he inerpolaion based slope for exponenial law can only consider he mixed mode sae when i sar o unload. Figure 5: Disconinuiy during unloading and reloading process of improved exponenial law using consan slope when he mixed mode raio changes The consequence of his disconinuiy is no ye discussed in lieraure o he bes of auhor s knowledge, bu such disconinuiy is for sure undesirable in numerical simulaion and could possibly cause some numerical insabiliies. For crack propagaion problems, where unloading and reloading is likely o happen as energy keeps being released a he crack ip; For faigue simulaion where millions of cycles of reloading happen, i is possible ha he mixed mode raio changes during his process. So i is imporan o guaranee a coninuous and robus unloading and reloading behavior of he cohesive law. Remedies o fix he disconinuiy in unloading and reloading process of exponenial cohesive law when mixed mode raio changes Kreging in 005 [1] has alk abou his disconinuiy and ried o solve i using exrapolaion o updae he consan slope in he new load cycle. His remedy is based on an assumpion ha mixed mode raio doesn change wihin a load cycle. When he cohesive elemen sars o unload, he mixed mode separaion cycle1 is recorded and he corresponding angenial separaion and normal separaion n are used o calculae he unloading slope. When a new loading cycle sars, he mixed mode raio r for ha cycle is capured and exrapolaion is did o find a new pair of angenial and normal separaion, n ha will reach he mixed mode separaion cycle1 of he firs cycle. And his new pair of separaions is used o calculae he new reloading slope. I works well when mixed mode raio is consan wihin one loading cycle (Figure 6). 1-6 June 1-14, 016

7 14 h Inernaional LS-DYNA Users Conference Session: Simulaion Figure 6: Reloading behavior using Rene s exrapolaion mehod when mixed mode raio is consan wihin a cycle However, when he mixed mode raio changes during reloading period, for example i firs reloads following r = 1 hen changes o r = before i reaches maximum separaion max of las cycle, he curve can no longer be coninuous using Rene s exrapolaion mehod because he unloading and reloading slope using his mehod is consan wihin one loading cycle hus canno capure ha change (Figure 7). Figure 7: Disconinuiy in exponenial cohesive law using exrapolaion mehod when mixed mode raio changes wihin a load cycle To fix ha a new coninuous damage parameer ha considers he mixed mode raio change is proposed in his paper. The idea is o release he angenial separaion in damage facor for normal racion, and release he normal separaion in damage facor for angenial racion (Equaion 14, 15). Where n,max and,max are defined as: d n = exp ( n,max δ n ) exp ( δ ) (14) d = exp (,max δ ) exp ( n ) (1 + n ) (15) δ n δ n n,max = max (16) June 1-14,

8 Session: Simulaion 14 h Inernaional LS-DYNA Users Conference,max = max n (17) In his way, he mixed mode ineracion is considered for boh he unloading, reloading slope and he poin on max separaion envelop of las cycle. Then he racion will be calculaed using he same equaion: { T n = d n K n n T = d K when m m,max (18) When m = n + reaches he separaion envelope again, he racion separaion relaionship will follow he improved exponenial cohesive law again and m,max will increase correspondingly. Using he new damage parameer helps remain he coninuiy of cohesive law no maer wha loading and unloading pah is like, as shown in Figure 8, in which he mixed mode raio changed during he reloading period. Noe bilinear cohesive model also has he capabiliy o remain coninuous even when mixed mode raio changes wihin one load cycle. Figure 8: Coninuiy in exponenial cohesive law using proposed unloading and reloading mehod when mixed mode raio changes wihin a load cycle Implemenaion of he cohesive zone model o inerface faigue crack growh CZM has been used on faigue predicion by differen auhors. ([19][0]) Roe 003 [13]adoped Xu-Needleman s exponenial cohesive model for cyclic loading faigue analysis. His model incorporaed ypical damage evoluion laws: (1) damage only sars o accumulae if a deformaion is greaer han a criical magniude; () The incremen of damage is relaed o he incremen of deformaion a he curren load level; (3) There exis an endurance limi he sress below which can proceed cyclically wihou producing failure. Based on hese hree requiremens he gave he evoluion equaion: D c = [ T C δ σ f ] H( δ 0 ) and D c 0 (19) max Where H is Heaviside funcion. The presen damage evoluion law is formulaed using effecive cohesive zone quaniies. The resulan separaion,, and is incremen are defined as: 1-8 June 1-14, 016

9 14 h Inernaional LS-DYNA Users Conference Session: Simulaion The resul racion, T is defined as: = n +, = (0) T = T + S (1) A new parameer called cohesive zone endurance limi σ f is incorporaed via C f, he raio of σ f and he iniial undamaged cohesive normal srengh: C f = σ f σ max,0, 0.0 < C f < 1.0 () Afer he damage rae is calculaed, he curren cohesive srengh is hen scale down by σ max = σ max,0 (1 D), τ max = τ max,0 (1 D), (3) During he unloading and reloading process where he curren separaion is smaller han he larges value of las cycle, he racion is calculaed by T n = T n,max + k n ( u n u n,max ) (4) T = T,max + k ( u u,max ) (5) Where k n, k used in Roe s paper are consan values. In his paper ha consan slope is replaced by a parameer ha changes wih mixed mode raio o accoun for mixed mode raio change beween each loading cycles. An example is carried ou o show schemaically how our unloading and reloading mehodology of exponenial cohesive law help he coninuiy of he cohesive law. Make he angenial and normal separaion change cyclically and in such a way ha he mixed mode raio changes beween each cycle (Figure 9). Using consan unload and reload slope, we can see i he disconinuiy in he cohesive law is very obvious (Figure 10 (a)). However, for our proposed unloading and reloading mehodology, he cohesive law remains coninuous very well (Figure 10 (b)).anoher verificaion is done by making he separaion a funcion of sin (), like shown in figure 11. Again consan unloading slope (Figure 1 (a)) shows disconinuiy and our proposed mehod is no (Figure 1 (b)). June 1-14,

10 Session: Simulaion 14 h Inernaional LS-DYNA Users Conference Figure 9: Cyclic separaion hisory in cohesive model Figure 10: (a) Cyclic loading of exponenial cohesive law using consan unloading and reloading slope (b) Cyclic loading of exponenial cohesive law using proposed unloading and reloading mehodology Figure 11: Cyclic separaion hisory in cohesive model 1-10 June 1-14, 016

11 14 h Inernaional LS-DYNA Users Conference Session: Simulaion Figure 1: (a) Cyclic loading of exponenial cohesive law using consan unloading and reloading slope (b) Cyclic loading of exponenial cohesive law using proposed unloading and reloading mehodology Conclusion In his paper a brief review of cohesive zone model is carried ou, and he unloading and reloading behavior of bilinear and exponenial law is analyzed. I s found ha he commonly used consan unloading slope mehod for exponenial law could lead o disconinuiy when he mixed mode raio changes. Thus a new mehodology for unloading and reloading behavior of exponenial law is proposed and verified on faigue cyclic loading using wo differen loading hisories. The resul proves is abiliy o reain coninuiy under arbirary load and unload hisory ha consan unloading slope mehod couldn have. References [1] Kreging R. Cohesive zone models: Towards a robus implemenaion of irreversible behaviour. Philips Applied Technologies 005. [] Dugdale D. Yielding of seel shees conaining slis. J Mech Phys Solids 1960;8: [3] Barenbla GI. The mahemaical heory of equilibrium cracks in brile fracure. Adv Appl Mech 196;7: [4] Hillerborg A, Modéer M, Peersson P. Analysis of crack formaion and crack growh in concree by means of fracure mechanics and finie elemens. Cem Concr Res 1976;6: [5] Xu X, Needleman A. Numerical simulaions of fas crack growh in brile solids. J Mech Phys Solids 1994;4: [6] Oriz M, Pandolfi A. Finie-deformaion irreversible cohesive elemens for hree-dimensional crack-propagaion analysis. In J Numer Mehods Eng 1999;44: [7] Wells G, Sluys L. A new mehod for modelling cohesive cracks using finie elemens. In J Numer Mehods Eng 001;50: [8] Bors Rd, Remmers JJ, Needleman A, Abellan M. Discree vs smeared crack models for concree fracure: bridging he gap. In J Numer Anal Mehods Geomech 004;8: [9] Camanho PP, Davila C, De Moura M. Numerical simulaion of mixed-mode progressive delaminaion in composie maerials. J Composie Maer 003;37: [10] Tvergaard V, Huchinson JW. The influence of plasiciy on mixed mode inerface oughness. J Mech Phys Solids 1993;41: [11] Volokh KY. Comparison beween cohesive zone models. Communicaions in Numerical Mehods in Engineering 004;0: [1] Alfano G. On he influence of he shape of he inerface law on he applicaion of cohesive-zone models. Composies Sci Technol 006;66: [13] Roe K, Siegmund T. An irreversible cohesive zone model for inerface faigue crack growh simulaion. Eng Frac Mech 003;70:09-3. June 1-14,

12 Session: Simulaion 14 h Inernaional LS-DYNA Users Conference [14] Nguyen O, Repeo E, Oriz M, Radovizky R. A cohesive model of faigue crack growh. In J Frac 001;110: [15] Kolluri M, Hoefnagels J, van Dommelen J, Geers M. Irreversible mixed mode inerface delaminaion using a combined damage-plasiciy cohesive zone enabling unloading. In J Frac 014;185: [16] Benzeggagh M, Kenane M. Measuremen of mixed-mode delaminaion fracure oughness of unidirecional glass/epoxy composies wih mixed-mode bending apparaus. Composies Sci Technol 1996;56: [17] LSDYNA keyword user's manual, volume II, maerial models [18] Van den Bosch M, Schreurs P, Geers M. An improved descripion of he exponenial Xu and Needleman cohesive zone law for mixed-mode decohesion. Eng Frac Mech 006;73: [19] Bouvard J, Chaboche J, Feyel F, Gallerneau F. A cohesive zone model for faigue and creep faigue crack growh in single crysal superalloys. In J Faigue 009;31: [0] Turon A, Cosa J, Camanho P, Dávila C. Simulaion of delaminaion in composies under high-cycle faigue. Composies Par A: applied science and manufacuring 007;38: June 1-14, 016