Cohesive zone models towards a robust implementation of irreversible behaviour

Transcription

1 Cohesve zoe models towards a robust mplemetato of rreversble behavour Reé Kregtg rd February 5 MT5. Supervso: dr.r. O. va der Slus (Phlps Appled Techloges) dr.r. R.H.J. Peerlgs prof.dr.r. M.G.D. Geers

2 Cotets Itroducto Cohesve zoe models 4. Itroducto Geeral theory MSC.Marc user subroutes Implemetato ssues. Numercal tegrato Hstory depedecy Ucoupled rreversblty Coupled rreversblty Bechmark test 5 Coclusos ad recommedatos 4 Refereces A Dervato of taget stffesses 8

3 Chapter Itroducto Deformato ad fracture are complex processes, whch exhbt the teractos betwee the dfferet phases the materal ad betwee mcro- ad macrocracks, whch may tate ad propagate smultaeously. Phlps Appled Techloges ad more specfcally the process modellg group (PPM) vestgates falure behavour of ceramc matrx compostes for optmzato purposes. These compostes cosst of a ceramc matrx wth metal clusos. For ths reaso, these compostes ca exhbt a combato of brttle ad ductle behavour. Fracture these heterogeeous materals cossts of dfferet stages: tato of vods ad mcrocracks radom stes throughout the body, the subsequet growth, teracto, clusterg, coalescece. Ths leads to the formato of tal macrocracks, ther growth ad, fally, propagato of oe of the cracks up to complete falure. Ital stresses, preset the composte due to processg, have a large fluece o the resultg behavour of the composte, (va der Slus, 4). Over the last decades several umercal methods have bee developed to smulate falure mechasms materals. The Fte Elemet Method (FEM) provdes a way to predct the falure behavour of materals, order to optmze the mcrostructure of these compostes. For ths purpose, approprate cotuum mechacal falure models have to be used to descrbe mcrostructural falure mechasms. Several falure models are avalable to ths ed. Fracture mechacs, cotuum damage mechacs ad XFEM are possble methods to smulate falure behavour materals. The scale at whch fracture occurs s equal to the scale at whch the mcrostructure s evaluated, therefore fracture wll to be smulated explctly. Also, fracture ca occur at physcal terfaces. A method lke cotuum damage mechacs s therefore t very sutable. Sce the behavour ca be quas-brttle, the use of fracture mechacs s also t recommeded. A dfferet techque to smulate crack tato ad crack growth s the use of so-called cohesve laws mplemeted fte elemets. The bass for cohesve zoe models ca be traced back to the works of Dugdale (96) ad Bareblatt (96). The mplemetato of these cohesve zoe models s rather straghtforward commercally avalable fte elemet packages. Calculatos o crack tato ad crack growth are possble for both ductle ad brttle materals. However, there are stll a umber of ssues that eed to be solved before ths approach ca be used for realstc smulatos.

4 Ths report focusses o a few umercal ssues whch stll affect the behavour of cohesve zoes. The goal s to develop a robust cohesve zoe mplemetato whch cludes hstory depedet behavour. A mplemetato MSC.Marc whch has already bee developed by Marco va de Bosch (4) wll be used as the startg pot. Also, prevous research has show that spurous oscllatos of computed stresses ca occur uder certa crcumstaces whch may lead to udesrable results. The occurrece of these oscllatos wll be dscussed, as well as some remedes. Fally a bechmark test wll be smulated to determe f the resultg cohesve zoe behavour (cludg hstory depedet behavour) s robust. The bechmark test wll also be used to verfy the computed stresses.

5 p us = [MPa] h = [mm] l =.5 [mm] h tch = [mm] ν =. eff < eff = Chapter Cohesve zoe models. Itroducto eff < eff = The vewpot from whch cohesve zoe models orgate regards fracture as a gradual phe- whch separato takes place across a exteded crack tp, or cohesve zoe, me ad s ressted by cohesve tractos (Ortz ad Padolf, 999). Thus cohesve zoe elemets do t represet ay physcal materal, but descrbe the cohesve forces whch occur whe materal elemets (such as gras) are beg pulled apart. Therefore cohesve zoe elemets = t + are placed betwee cotuum (bulk) elemets, as show Fgure.. eff >,old T Bulk elemets Cohesve zoe elemets Bulk elemets Fgure.: Applcato of cohesve zoe elemets alog bulk elemet boudares Whe damage growth occurs these cohesve zoe elemets ope order to smulate crack tato or crack growth. Sce the crack path ca oly follow these elemets, the drecto of crack propagato strogly depeds o the presece (or absece) of cohesve zoe elemets, mplyg the crack path s mesh depedet. However, refg the mesh reduces ths problem. I two dmesos, tractos ca occur the rmal ad the shear drecto. The descrpto of the falure behavour s defed by tracto-separato laws. These relatos descrbe the tractos as a fucto of separatos ad determe the costtutve behavour of cohesve zoe models. There s a great varety tracto separato laws (Chadra et al., ) but they all exhbt the same global behavour. As the cohesve surfaces separate, the tracto frst creases utl a mum s reached, ad subsequetly the tracto decreases to zero, 4

6 eff < eff = whch results complete (local) separato. Ths holds for both the rmal ad the shear drecto. A schematc example of a tracto-separato curve s show Fgure.. eff = t + eff >,old φ tracto T dsplacemet Fgure.: Example of tracto-separato curve There are a umber of factors whch play a mportat role the resultg falure behavour. For stace, the area uder the tracto separato curve correspods wth the eergy eeded for separato (φ). The tal stffess of the cohesve zoe model has a large fluece o the overall elastc deformato ad should be very hgh order to obta realstc results. It s show by Chadra et al. () that the form of the tracto-separato relatos plays a mportat role the macroscopc mechacal respose of the system. The cohesve zoe model whch s used the followg s the expoetal model by Xu ad Needlema (994). Ths model provdes smooth tracto-separato curves ad may therefore be more stable tha dscotuous models, such as the blear model.. Geeral theory The tracto vector T actg at the cohesve surface s derved from a terfacal potetal (Xu ad Needlema, 994) wth rmal ad tagetal compoets T ad T t, respectvely: T = φ() () (.) wth = (, ) The potetal ca be wrtte as: ( φ(, ) = φ +φ exp δ ){[ r+ δ ] [ q r q+ ( r q r ) δ ] ( )} exp t δt (.) 5

7 η = η = + p p Where δ ad δ t represet characterstc separatos, such a way that T (δ ) = σ ad lus = [MPa] T t (δ t / E-modulus = [MPa] ) = τ. σ ad τ represet the mum values of the rmal tracto ad h = [mm] h = [mm] the shear tracto respectvely. Furthermore, q = φ t /φ ad r = /δ, where s the l =.5 [mm] l =.5 [mm] value of whe complete shear separato has take place wthout resultg rmal h tch = [mm] h tch = [mm] teso (T = ). q wll be take equal to oe ad r equal to zero. The resultg equatos for ν =. the rmal ad shear tractos are derved ν by =. combg (.) ad (.) wth exp(...) = e (...) as: T = φ ( exp ){ ( ) exp t δ δ δ δt + q [ ( )][ exp t r δt r ]} eff < (.) δ ( ){ ( ) } ( φ r q T t = δt q + exp ) ( ) exp t r δ δ δt (.4) eff < eff = φ ad φ t are the areas uder the rmal tracto-separato curve ad the shear tracto curve respectvely. They represet the amout of work eeded for complete separato. Ths ca be see for q =, r =, ad assumg that T = T (, = ), T t = T t ( =, ), for whch case the so-called ucoupled tractos are obtaed. Usg T (δ ) = σ ad T t (δ t / eff < ) = τ the followg relatos eff for < φ ad φ t ca the be obtaed: eff = φ = σ exp() δ, φ t = eff = exp()/ τ δ t (.5) eff = t + eff >,old T / σ.5.8 T / δ η = η = + p p eff = The rmalzed tracto curves for ucoupled rmal separato ad shear separato are show Fgure. ad Fgure.4. I these fgures T /σ ad T t /τ represet the eff = t + eff >,old T t / τ..4.6 / δ t Fgure.: Normal tracto curve for the ucoupled modellg Fgure.4: Shear tracto curve for the ucoupled modellg dmesoless rmal ad shear tractos. /δ ad /δ t represet the dmesoless rmal ad shear opegs. The rmal tracto-separato curve shows that startg from a opeg of zero ad creasg the separato also creases the tracto utl a mum value s reached at δ. After that the cohesve force decreases utl the cohesve zoe loger has ay stffess rmal drecto. Whe the cohesve zoe s gve a separato dsplacemet egatve drecto the tracto rapdly becomes more egatve order to 6

8 eff < prevet peetrato. The shear tracto separato curve does t show such a behavour for egatve separatos. Separatos egatve drecto merely lead to shear tractos the eff = egatve drecto, whch are opposte to eff = those for a postve. As was already metoed, fgures. ad.4 llustrate the ucoupled relatos. Next, the couplg effect wll be llustrated, as for ths case, t holds that: T = T (, ) ad T t = T t (, ) (.6) eff < eff < eff = The relatos (.) ad (.4) correspod eff = wth surfaces, whch are show fgures.5 ad.6. Fgure.5 shows that a -zero value of results a lower curve for the T ( ) relato. O the other had, Fgure.6 shows that a -zero value of results a lower curve for the T t ( ) relato. For w, ths potetal ad ts dervatves ca oly be used for two eff = t + eff >,old T /σ 4 eff < T t /δ /δ t /δ /δ t t eff = t + eff >,old T t /τ.5.5 Fgure.5: Normal tracto surface Fgure.6: Shear tracto surface dmesos. Ths ca be exteded to three dmesos by addg a ew tagetal tracto to the set of equatos. Ths ew tracto wll be drected perpedcularly to the other two. However, ths wll t be dscussed here. For formato regardg a three dmesoal mplemetato see Goçalves et al. ().. MSC.Marc user subroutes The cohesve zoe elemet has bee mplemeted as a user elemet the commercal fte elemet package MSC.Marc. The user subroutes have bee wrtte by va de Bosch et al. (4). The cohesve zoe elemet s defed as a four de elemet wth two tegrato pots, lyg o the le A-B. The locato of the tegrato pots depeds o whch umercal tegrato scheme s chose, see Fgure.7. The defto of the elemet des s mportat. The des must be assged couterclockwse drecto, the last de (4) must be opposte to the frst de () ad the thrd de must be opposte to the secod de. 7

9 eff = eff < eff = = t + eff >,old T A p p A B Newto Cotes η = η = η = + B Gauss 4 A θ e e t B e y e x Fgure.7: Cohesve zoe elemet, show wth des ad legth AB ad the locato of tegrato pots the Gauss ad Newto-Cotes schemes The ma subroute s uselem, ths subroute s called by MSC.Marc every tme the elemet stffess matrx ad the teral force colum s eeded for the calculato. For ths purpose uselem exports the curret dal coordates ad the total dal dsplacemets to czbehav. Wth these quattes czbehav calculates the separatos ad. Frst the curret coordates are used to calculate the legth ad the oretato of the ceter le AB, see Fgure.7. Next, the locatos of the tegrato pots are determed wth respect to A-B. After that terpolato of the dal dsplacemets s used to determe ad at the tegrato pots. The elemet stffess matrx s calculated uselem usg K e = l + wth D for the ucoupled case D = N T DNdη (.7) T (.8) ad the teral force colum s calculated usg f e t = l + N T Tdη (.9) 8

10 wth T = [ Tt T ] (.) D s the cosstet taget stffess matrx, l s the legth (wdth) of the elemet ad N s the matrx of shape fuctos. T s the colum of calculated tractos. The tractos are calculated accordg to equatos (.) ad (.4). The followg subroutes are used by MSC.Marc uselem : Ths s the ma subroute, the elemet stffess matrx ad teral forces are calculated here. czbehav : The costtutve behavour of the cohesve zoe elemet s mplemeted ths subroute. It s called by uselem for the actual calculato of the stffess matrx ad the teral force colum. Ths subroute cotas the tracto separato characterstcs ad stffesses as well as dal trasformatos ad a declarato of the tegrato pots. plotv : Ths subroute wrtes the status of the cohesve zoe elemets to the marc postprocessg fle, t ca also be used to wrte dfferet quattes to fle f eeded. elevar : Ths subroute wrtes the opegs ad ad the tractos T ad T t to a fle as well as desred addtoal quattes. 9

11 η = η = η = + p p lus = [MPa] h = [mm] l =.5 [mm] h tch = [mm] ν =. eff < eff = Chapter Implemetato eff < ssues. Numercal tegrato A hgh tal stffess of the cohesve zoe elemets s ecessary whe tryg to obta a physcally realstc model. Whe the tal stffess s too low addtoal deformato wll occur, caused by the decreased overall (bulk) stffess. Therefore the stffess must be suffcetly eff < hgh. However, a hgh stffess ca result eff < so-called spurous oscllatos. Ths meas that eff = the tracto profle exhbts a oscllatory eff = behavour that has physcal meag. Research by Schellekes ad de Borst (99) has show that these oscllatos are caused by the combato of a hgh tracto gradet wth oe elemet ad a Gauss tegrato scheme. Ths ca be llustrated by meas of a test case whch was also devsed by Schellekes ad eff = t + eff >,old L d η = η = η = + p p E-modulus = [MPa] h = [mm] l =.5 [mm] h tch = [mm] cohesve zoe elemets ν =. eff = h eff = t + eff >,old Y Z X Fgure.: Test case set up Fgure.: Test case mesh de Borst (99), see Fgure. ad Fgure.. The beam has a legth L of 5[mm] ad a heght h of 5[mm]. A tal crack s serted at the bottom of the beam wth legth d = [mm]. The cohesve zoes are placed drectly above ths crack. The bulk materal has a Youg s modulus of [MPa] ad a Posso s rato of.. Fve des whch le the mddle drectly above the ceterle are gve a very small dsplacemet dowward, ulke the test case by Schellekes ad de Borst (99) whch a dowward pressure was appled. Ths

12 eff = dsplacemet s kept small sce the rmal opeg of the cohesve zoe elemets eeds to be kept very small ( δ, δ t ) order to vestgate the fluece of the tal stffess. Two stuatos are smulated, usg cohesve zoe elemets wth a low stffess (τ = [MPa], eff < δ = δ t =.[mm]) ad wth a hgh stffess eff < (τ = [MPa], δ = δ t = E-4[mm]). Parameters q ad r are take oe ad zero eff = eff = respectvely. Fgures. ad.4 show the computed rmal tractos the cohesve zoes. Fgure. shows oscllatos the tracto profle, but Fgure.4 does show oscllatos. eff = t + eff >,old heght [mm] T T [MPa] T [MPa] eff = eff = t + eff >,old heght [mm] Fgure.: Tracto curve usg Gauss tegrato ad τ = [MPa] Fgure.4: Tracto curve usg Gauss tegrato ad τ = [MPa] To plot the tracto profles use was made of the tegrato pot values, to avod possble accuraces due to the extrapolato of tegrato pot values to dal pot values. A possble way to remove the spurous oscllatos s by usg a dfferet tegrato scheme stead of Gauss. Schellekes ad de Borst (99) have show that the Newto-Cotes tegrato scheme does t suffer from ths problem. Ths tegrato scheme has a frst order accuracy stead of the secod order accuracy whch s obtaed whe usg the Gauss scheme. However, f the occurrece of spurous oscllatos has a sgfcat (log term) effect, t may be wse to evertheless use the Newto-Cotes scheme stead of the Gauss scheme. The oly parameters that eed to be chaged the subroute are the locatos of the tegrato pots sce the weght factors are the same for two pot Gauss tegrato ad Newto-Cotes tegrato. Fgures.5 ad.6 show the results for the same smulato but usg the Newto-Cotes tegrato scheme. These fgures show that the use of the Newto-Cotes tegrato scheme does t result spurous oscllatos the tracto profle.

13 eff p lus = [MPa] eff = h = [mm] l =.5 t + [mm] hotch eff > =,old [mm] ν =. T eff = < eq(.) heght [mm] T = eff eq(.) =.5 T = eff eq(.) = T T [MPa] T [MPa] eff p E-modulus = [MPa] eff = h = [mm] l =.5 t + [mm] hotch eff > =,old [mm] ν =. T eff = < eq(.) Fgure.5: Tracto curve usg Newto- Fgure.6: Tracto curve usg Newto- Cotes tegrato ad τ = [MPa] Cotes tegrato ad τ = [MPa] eff < eff < eff = eff = Ather way to avod the oscllatos s to use mesh refemet, sce ths heretly lowers the tracto gradet over oe elemet whe applyg the same load. The result s show Fgure.7 usg twce the amout of cohesve zoes. Ths fgure shows that the spurous oscllatos are t preset whe mesh refemet s used. eff = t + eff >,old heght [mm] T [MPa] T [MPa] 6 8 eff = t + eff >,old heght [mm] heght [mm] Fgure.7: Tracto profle usg a fe mesh, Gauss tegrato ad τ = [MPa] Fgure.8: Tracto profle usg a fe mesh, Newto-Cotes tegrato ad τ = [MPa] The cause of the spurous oscllatos seems to le the egemodes whch a elemet has whe usg Gauss tegrato as opposed to Newto-Cotes tegrato (Schellekes ad de Borst, 99). The dfferet egemodes (except for the three rgd body modes ad the zero-eergy mode whch represets exteso of the cohesve zoe alog ts legth) for the Newto-Cotes tegrato scheme are show Fgure.9. These egemodes have bee obtaed by computg the egevalues ad egevectors of the elemet stffess matrx. Ths s

14 eff < eff = eff = t + eff >,old 4. λ =.6. =.6. = = 4.66 λ λ λ eff < eff = eff = t + eff >,old 4. =.777. = λ λ =.6 4. = 4.66 λ λ Fgure.9: Egemodes ad egevalues wth Newto-Cotes tegrato Fgure.: Egemodes ad egevalues wth Gauss tegrato doe for a gve combato of ad, usg Gauss tegrato ad Newto-Cotes tegrato. The problem les the couplg betwee degrees-of-freedom of the dfferet des. Ths s llustrated by Fgure., whch shows the egemodes ad egevalues usg a Gauss tegrato scheme. Whe usg Gauss tegrato a sgle dal degree-of-freedom ca be actvated dfferet egemodes, ths s t the case whe usg the Newto-Cotes tegrato scheme. However t s t completely clear why ths would result spurous oscllatos. A alteratve explaato, whch s also t completely rgorous s that for Gauss tegrato there s a teracto betwee the des at the left ed of the elemet ad those at the rght ed, whereas for the Newto-Cotes tegrato the opegs at both eds ca be vared depedetly.. Hstory depedecy The curret mplemetato does t exhbt hstory depedet behavour or rreversble behavour. Whe uloadg, the same tracto curve s followed as durg loadg. Ths s show Fgure.. Ths mples that to acheve uloadg oe must crease the tracto. Ths s t realstc sce damage s regarded as a rreversble process. It s assumed that order to acheve realstc rreversble behavour, uloadg should occur a lear way to the org. To ths ed, oe or more hstory parameters wll be troduced the model. Ths ca be doe for both the rmal ad the shear drecto or for the both of them a coupled fasho. I the followg, three dfferet forms of hstory depedecy are mplemeted ad tested. Frst a ucoupled hstory depedet model wll be mplemeted ad tested. Followg the same framework, coupled hstory depedet behavour wll be mplemeted. Fally a dfferet approach wll be used to acheve coupled hstory depedet behavour.

15 eff = eff = t + eff >,old T / σ / δ Fgure.: Curret mplemetato, subsequet loadg ad uloadg shows reversblty.. Ucoupled rreversblty To obta hstory depedet behavour a hstory parameter eeds to be troduced. Ths parameter s chose as ad ζ = { ω (τ) τ t} wth =,t (.) ω = { f δ c, = f > (.) ω t = δ c t (.) ζ s defed as the hstory parameter for separatos rmal drecto ad ζ t for separatos tagetal drecto. δ c ad δ c t are crtcal separatos rmal ad shear drecto respectvely. These separatos are chose as the separatos where the correspodg tractos are almost zero. Sce the tracto curves approach zero very slowly for larger separatos, the exact value of these parameters s t very mportat. Damage evoluto s already accouted for the free eergy potetal φ. Sce the tractos are derved from ths potetal, damage evoluto s also accouted for the tracto relatos. Ths damage evoluto ca be see the tracto curves as softeg behavour. There s eed to degrade the tractos wth some sort of hstory depedet damage parameter 4

16 eff < for cotued loadg. It s oly ecessary eff < to determe whether the cohesve zoe s loadg eff = or uloadg ad what drecto ths eff takes = place. Sce the cohesve zoe elemet should experece a dfferet behavour for uloadg tha for loadg, the tracto-separato equatos should be exteded wth a uloadg mode. Ths exteso s made for the rmal ad tagetal resposes separately ad should result the followg behavour. Whe loadg eff = t + eff >,old ω t = t δ t c ω t > ζ t,old ζ t = ζ t,old T t = α t = α t ζ t = ω t α t = Tt = eq(b.4) eff = t + eff >,old ζ = ω T = eq(b.) α = T ω = δ c ω > ζ,old ζ = ζ,old T = α T = α Fgure.: Flowchart for tagetal loadg Fgure.: Flowchart for rmal loadg takes place, the orgal relatos, (.) ad (.4), should be used to calculate the tractos. Whe uloadg takes place rmal or tagetal drecto the last calculato o the curve durg the loadg process for ths drecto determes the uloadg stffess α for rmal drecto ad α t for tagetal drecto. Whe the elemet s subsequetly loaded, these uloadg relatos should the be used utl the prevous mum value s aga reached, ths s the reload path. α = T,, wth =,t (.4) The tracto-separato relatos case of uloadg are hereby smplfed to T = α T t = α t (.5) Ths s llustrated Fgure. for the tagetal drecto ad Fgure. for the rmal drecto. Note aga that the two drectos are treated a completely separate way. The smulatos are dsplacemet cotrolled. The test case cossts of two materal elemets wth oe cohesve zoe elemet betwee. The results are obtaed by respectvely loadg, uloadg ad after that creased loadg, see Fgure.4. The same has bee doe for the shear drecto whch was respectvely loaded, uloaded, loaded opposte drecto, ad uloaded, see Fgure.5. The results are show fgures.6 ad.7. 5

17 t eff = eff < t + eff >,old eff = ν =. T t = eff < eq(.) eff = eff = t + eff >,old. Normal loadg/ uloadg. Normal Tloadg t eff = eff < t + eff >,old eff = Fgure.4: Prescrbed rmal dsplacemets comp of stress α σ T δ, dsplacemet Y ν =. T t = eff < eq(.) eff = eff = t + eff >,old.shear loadg/.shear loadg/ uloadg uloadg Fgure.5: Prescrbed shear dsplacemets comp of stress dsplacemet X x Fgure.6: Ucoupled rreversblty: Normal tracto curve Fgure.7: Ucoupled rreversblty: Shear tracto curve.. Coupled rreversblty Itroducto The prevous mplemetato of hstory depedet behavour was doe a ucoupled way,.e. the evoluto of the hstory parameter ζ dd t have a fluece o the tracto behavour tagetal drecto ad vce versa. Ths mples that a cohesve zoe elemet whch s loaded rmal drecto utl t s damaged, ad subsequetly uloaded, stll has the tal stffess tagetal drecto. Ths s t very realstc for most materals ad we therefore attempt to couple the hstory depedet tracto-separato behavour. To acheve the couplg, the complete, coupled tracto-separato relatos eed to be used for loadg ad the tagetal stffess matrx should also be exteded. The orgal taget stffess matrx oly cotaed the dagoal terms. The ew cosstet taget stffess matrx D for 6

18 loadg also cotas the cross-dervatves: D = T T (.6) The expressos D are gve appedx B e equatos (A.)-(A.4). Oe effectve hstory parameter Based o the work of Ortz ad Padolf (999), rreversble behavour s mplemeted usg a sgle hstory parameter stead of ζ ad ζ t. Ths parameter s defed as = { eff (τ) τ t} (.7) wth eff the effectve opeg dsplacemet eff = t + wth = (, ) (.8) takes place whe eff = ad eff, ad uloadg (or reloadg) whe eff <. Whe loadg occurs, the tractos from equatos (.) ad (.4) ad the correspodg stffesses are used. For uloadg, the followg relatos for T ad T t are used T = T,, + T,, (.9) T t = T t,, + T t,, (.) The uloadg stffesses used equatos (.9) ad (.), T, / j, where,j =,t, are evaluated for the mum values of ad reached durg the prevous loadg. However, whe a cohesve zoe elemet s subjected to mxed mode loadg, the tractos durg reloadg are t calculated correctly. If ths reloadg takes place same other drecto tha the uloadg, however, these stffesses do t correspod wth the udamaged loadg respose ths drecto. As a cosequece, a jump the tractos may occur at the trasto from the uloadg/reloadg relatos to the loadg relatos (.)-(.4), see Fgure.8. Ths does t seem very realstc; t seems more realstc that the rmal tracto follows the dashed le stead. For ths reaso, a dfferet approach s eeded to calculate the tractos durg uloadg ad reloadg. To calculate the tractos durg reloadg correctly, the reloadg stffesses should be take to accout the ew (re-)loadg drecto ad the loadg respose ths drecto. For ths purpose the tractos are terpolated betwee the org ad the tractos at whch the loadg surface eff = s reached the ew loadg drecto. Ths s show Fgure.9. Ths fgure shows the damage surface gve by eff = the dsplacemet space. Ths surface s somewhat smlar to the yeld surface for plastcty a stress space. The surface of ca oly grow, ths occurs whe eff = ad eff >. The dea s w that for a gve (, ) for whch eff < the tractos are learly terpolated betwee the 7

19 eff < eff = eff = t + eff >,old T T.5 x eff = t + eff >,old eff < eff = Fgure.8: Curret mxed mode loadg result Fgure.9: Damage surface dsplacemet space org ad the tractos at whch the damage surface wll be reached for cotued loadg the same drecto ( the space). Ths effectvely meas that the opegs ad are frst scaled by a factor / eff, the tractos assocated wth these scaled opegs are computed, ad these tractos are scaled back by multplyg them by eff /. The resultg tractos ca thus be wrtte as T ul T ul t = ( eff T, ) eff eff = ( eff T t, ) eff eff (.) (.) where T (, ) ad T t (, ) are the tracto-separato relatos used for loadg as gve by (.) ad (.4). Note that ths approach also works properly for uform paths, sce the tersecto wth the loadg surface s automatcally updated for such paths. Ths mplemetato bascally works the same as the ucoupled mplemetato, wth as hstory parameter stead of ζ ( = or t). Fgure. shows the resultg flowchart for ths approach. The taget stffesses assocated wth (.) ad (.), whch are to be used uloadg/reloadg, are gve Appedx B, equatos (A.4)-(A.7). Ths mplemetato has bee tested for mxed mode loadg, for whch the prescrbed dsplacemets are show Fgure.. The elemet s frst loaded ad uloaded both rmal ad shear drecto, wth the rate of rmal dsplacemet equal to the rate of shear dsplacemet. Subsequetly, a mxed mode load s appled wth the rate of shear dsplacemet beg larger tha the rate of rmal dsplacemet. The resultg tractos are show fgures. ad.. The tracto jump whch s see Fgure.8 s abset, the tracto curves appear to have bee mproved. 8

20 T > α t T = α t η = eff < η = eff = η = + p p lus = [MPa] h = [mm] l =.5 [mm] h tch = [mm] ν =. eff < eff = eff < eff = eff = t + eff >,old E-modulus = eff = [MPa] h = t + [mm] eff = t + l eff = >.5,old [mm] h tch = [mm] ν =. eff >,old = Teq(B.6-B.9) eff < T > α t T = α t η = eff < η = eff = η = + p p T 5 5 eff = cremet [ ] Fgure.: Flowchart for coupled mplemetato usg oe hstory parameter mxed mode Fgure.: Prescrbed dsplacemets for loadg T x eff < eff = eff = t + eff >,old dsplacemet [mm] T t 6 x x Fgure.: Normal tractos for mxed mode loadg Fgure.: Shear tractos for mxed mode loadg 9

21 T = α t η = η = η = + p p lus = [MPa] h = [mm] l =.5 [mm] h tch = [mm] ν =. eff < Chapter 4 Bechmark test eff = eff = I order to obta quattatve data o the occurrg stresses durg the delamato process a bechmark test has bee developed by Daves (). Ths bechmark test volves a double catlever beam (DCB) whch s pulled apart mode I. The bechmark problem s show Fgure 4.. The eds of the beam are gve a dsplacemet u/ the y-drecto, whle the other ed of the beam s etrely fxed x ad y drecto. The cohesve zoes are placed o the dotted le ths fgure. Whe loadg the beam eff < ths mode the top ad bottom parts eff < of the beam are frst bet over the [mm] eff = log tal crack. Subsequetly, the eff cohesve = zoes start to ope, whch results a decrease of the stffess. Whe the cohesve zoes are all completely opeed the beam starts to bed aga, ths tme over the total legth of [mm]. The mesh of the fal result s show Fgure 4.. Aalytcal relatos are avalable to verfy the umercal results. eff = t + eff >,old u/ u/ mm lcase mm T = α t η = η = η = + p p E-modulus = [MPa] h = [mm] l =.5 [mm] h tch = [mm] ν =. eff < eff = t + eff >,old depth = mm mm Ic: Tme:.e+ Y Z X Fgure 4.: Bechmark test set up Fgure 4.: Bechmark deformed mesh The aalytcal relatos are obtaed by combg the lear elastc beam theory wth lear elastc fracture mechacs. The results are P = uei a for tal bedg (4.)

22 T = α ζ t T > α t T = α t η = (bg Ic EI) P = η = / a for delamato (4.) η = + wth p p P = vertcal force o beam edge E-modulus = [MPa] b = beam thckess h = [mm] G Ic = l = eergy.5 [mm] release rate for mode I E h = tch elastcty = [mm] modulus ν =. I = momet of erta a = legth of tal crack The two equatos eff < that determe the resultg force-dsplacemet curve are equatos (4.) ad (4.). Equato (4.) s derved from classc beam theory ad relates the (perpedcular) dsplacemet (u) to the bedg force (P ). A smlar relato for the fal state of the complete opeg ca be obtaed by replacg a by the legth of the beam. Equato (4.) eff = relates the eergy release rate to the forces eeded for delamato. Ths eergy release rate correspods wth φ ad s therefore a parameter the costtutve behavour of the t cohesve zoe elemets. Ths relato descrbes the decreasg force whe the cohesve zoe elemets are opeg. The load P as a fucto of the dsplacemet u/ s show Fgure 4.. The dashed les eff < correspod wth the aalytcal solutos as gve above. The umercal soluto follows the eff = aalytcal relatos qute closely. Fgure 4. shows that the beam regas some stffess whe the opeg exceeds 8[mm], ths pot correspods wth the beam bedg purely over [mm] (full legth). The results obtaed by Alfa ad Crsfeld () show a less tha 4 eff = t + eff >,old P [N] u/ [mm] Fgure 4.: Numercal ad aalytcal results for bechmark test smooth curve for the delamato part wth may oscllatos. Ths may be the result of the

23 eff = use of a blear costtutve model. The oscllatos were reduced by makg the cohesve zoe more ductle, ths s acheved by lowerg τ (whle keepg the eergy G Ic the same). The coupled eff rreversble < mplemetato should be able to descrbe the opeg ad closg of the beam. eff However, = durg the tal phase of the opeg, there are areas whch are compressed. Ths ca be see Fgure 4.4, wth the dark grey areas represetg a dowward dsplacemet ad the lght grey areas a upward dsplacemet. These areas cosst of multple eff = t + eff >,old Y Z X Fgure 4.4: Areas of compresso durg the tal opeg cohesve zoe elemets. They are t a umercal artfact, but reflect a wave-lke theoretcal soluto, cf. a beam o a elastc foudato. Because the elemets are compressed before they are beg pulled apart, eff = at some pot ad >. Ths leads to umercal dffcultes, the tractos ad the stffesses cat be calculated for eff =. A extra statemet has bee added to the uloadg mplemetato to make sure that for eff, the tractos are zero ad the stffesses are calculated usg the relatos for loadg. The results of subsequetly opeg ad closg the double catlever beam are show Fgure 4.5. The fgure shows that the uloadg also takes place a lear fasho to the org, whch seems realstc.

24 h = [mm] l =.5 [mm] h tch = [mm] ν =. eff < eff = eff < eff = eff = t + eff >,old P [N] u/ [mm] Fgure 4.5: Bechmark results for subsequet opeg ad closg of the double catlever beam

25 Chapter 5 Coclusos ad recommedatos Coclusos Spurous oscllatos: The vestgato to the spurous oscllatos dd t have the desred depth. It s t yet clear what the exact cause of these oscllatos s, other tha t has to do wth coupled egemodes. However, ths behavour does t eed to be a problem, sce t ca be avoded by usg mesh refemet or a Newto-Cotes tegrato scheme. Irreversblty: The ucoupled ad the frst attempt at coupled mplemetato of rreversblty have ther shortcomgs. The frst mplemetato exhbts couplg betwee the rmal ad the tagetal drecto. Ths mplemetato s therefore t very useful whe coupled loads are appled, such as a mxed mode load. Nether s t very useful whe subsequet loadg s appled dfferet drectos. Ths model has bee modfed to show coupled behavour, by rgorously couplg the hstory parameters to each other. Ths mplemetato s able to descrbe subsequet loadg, but t s too stable to apply a complex load or to apply a load to multple cohesve zoe elemets, such as the bechmark problem. Ths s due to the extesve use of several crtera ad statemets. The fal coupled mplemetato s able to descrbe both subsequet ad mxed mode loadg, because of the terpolato betwee the mum effectve opeg ad the org. Ths results a model whch the uloadg path ca vary, but the tractos ad the stffesses ca stll be calculated. Bechmark test: The model seems to descrbe the aalytcal soluto qute well. The computed force-dsplacemet curve les very close to the aalytcal soluto. Subsequetly opeg ad closg of the beam leads to the desred result, sce uloadg s lear to the org. Recommedatos There are some ssues that stll deserve atteto regardg the fal mplemetato, as oe ca t yet call the model robust. For stace, t s debatable whether damage should grow 4

26 for compresso as t s assumed here. To avod ths, the effectve opeg ca be made depedet of whe the cohesve zoe s compressed. Furthermore, a three dmesoal mplemetato of rreversble behavour would be useful. Ths wll allow more realstc smulatos tha s possble at the momet. Regardg a three dmesoal mplemetato, t s eve more mportat that the model s able to descrbe rreversble behavour a robust way, sce the mesh becomes creasgly complex. Fally, t may be useful to vestgate to possblty to develop dfferet costtutve relatos. Sce the curret model already exhbts damage evoluto, ad therefore damage ca t be descrbed drectly, t may be worthwhle to try to develop a model whch s t based o the Xu ad Needlema model. Such a model could use a eergy potetal to descrbe the elastc respose alogsde ather potetal whch descrbes damage growth, smlar to cotuum damage mechacs. Furthermore, the fluece of the elemet sze may be vestgated, the elemet wdth should be less tha oe teth of the characterstc opeg, accordg to Tomar et al. (4). 5

27 Bblography Alfa, G. ad Crsfeld, M.A. (). Fte elemet terface models for the delamato aalyss of lamated compostes: mechacal ad computatoal ssues. Iteratoal Joural for Numercal Methods Egeerg, 5, Bareblatt, G. (96). The mathematcal theory of equlbrum cracks brttle fracture. Advaces Appled Mechacs, 7, va de Bosch, M.J., Schreurs, P.J.G. ad Geers, M.G.D. (4). Backgroud ad mplemetato of a cohesve zoe elemet. Iteral report. Chadra, N., L, H., Shet, C. ad Ghoem, H. (). Some ssues the applcato of cohesve zoe models for metal-ceramc terfaces. Iteratoal Joural of Solds ad Structures, 9, Daves, G.A.O. (). Bechmarks for composte delamato. NAFEMS publcato R84. Dugdale, D. (96). Yeldg of steel sheets cotag slts. Joural of the Mechacs ad Physcs of Solds, 8, 4. Goçalves, J., de Moura, M., de Castro, P. ad Marques, A. (). Iterface elemet cludg pot-to-surface costrats for three-dmesoal problems wth damage propagato. Egeerg Computatos, 7, Ortz, M. ad Padolf, A. (999). Fte-deformato rreversble cohesve elemets for threedmesoal crack-propagato aalyss. Iteratoal Joural for Numercal Methods Egeerg, 44, Schellekes, J.C.J. ad de Borst, R. (99). O the umercal tegrato of terface elemets. Iteratoal Joural for Numercal Methods Egeerg, 6, va der Slus, O. (4). Cotuum mechacal falure models a lterature overvew. Phlps CFT, CTB59-4-4, Iteral Phlps report. Tomar, V., Zha, J. ad Zhou, M. (4). Bouds for elemet sze a varable stffess cohesve fte elemet model. Iteratoal Joural for Numercal Methods Egeerg, 6, Xu, X. ad Needlema, A. (994). Numercal smulatos of fast crack growth brttle solds. Joural of the Mechacs ad Physcs of Solds, 4,

28 Ackwledgemets I would lke to thak Olaf va der Slus from Phlps Appled Techloges (or PAT, formerly kw as Phlps CFT) for hs supervso ad for helpg me out wth the project may tmes durg my tershp there. Furthermore I would lke to thak Ro Peerlgs from the uversty for hs supervso ad crtcal commet durg my tershp ad for afterwards helpg me wth the fshg touch o the fal mplemetato. 7

29 Appedx A Dervato of taget stffesses loadg stffesses The tagets for loadg follow by straghtforward dfferetato of equatos (.) ad (.4) as T = φ ( exp δ δ + q ( exp r ) ( ( δ ( t δ t δ ) ( ) exp t δt + )) ( )) r + δ δ δ (A.) T = φ ( exp ) { δ δ δ δt + q ( r δt exp t δt ( exp t )( r δ δ t ) + ) } (A.) = φ ( δt exp ) ( exp t δ t δ δ ) ( ( r q ) ) q r δ (A.) = φ δ t ( exp ) ( q + r q δ r δ )( t δ t ) ( exp t δ t ) (A.4) 8

30 coupled stffesses for uloadg To obta the stffesses case of uloadg usg the coupled mplemetato wth scalg the followg uloadg tractos are used T ul = eff T (, t ) T ul t = eff T t (, t ) (A.5) (A.6) wth = eff t = eff (A.7) (A.8) T ul ad T ul t are the uloadg tractos the rmal ad tagetal drecto respectvely. ad Tt ul ad ca be derved as The varato of eff s eeded for the dfferetato of T ul eff = + t (A.9) eff δ eff = δ + δ (A.) δ eff = eff δ + eff δ The varato of the tracto rmal drecto, T ul, ca be derved as δt ul = = = δ eff T + eff T ( ) δ eff δ eff + eff ( ) + eff T δ t eff δ eff eff ( ) T T eff T eff δ eff + T t δ + T δ t ( )( ) T T eff T eff δ + δ + t eff eff (A.) = + T ( + δ + T δ t T + t eff ( eff eff T + T T eff t eff T t eff ) δ + T ) δ (A.) 9

31 The varato of the tracto shear drecto: δt ul t = = δ eff T t + eff ( ) δ eff δ eff + eff ) δ δ eff eff ( + eff t ( T t eff eff t eff )( eff δ + eff δ ) + = + ( + δ + δ t T t + t eff ( eff eff T t + eff t eff t eff ) δ + ) δ (A.) The taget stffesses for uloadg the become T ul = T + t T eff eff T eff t (A.4) T ul = T + T eff eff t eff T (A.5) T ul t = eff T t + t eff eff t (A.6) T ul t = eff T t + eff t eff (A.7)