THE cohesive zone, CZ, in a material is the area between

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1 Inegral Equaions in Cohesive Zones Modelling of Fracure in Hisory Dependen Maerials Layal Hakim, Sergey Mikhailov Absrac A cohesive zone model of crack propagaion in linear visco-elasic maerials wih non-linear hisory-dependen rupure crierion is presened. The viscoelaiciy is described by a linear Volerra inegral operaor in ime. The sresses on he cohesive zone saisfy he hisory dependen rupure crierion, given by a non-linear Abel-ype inegral operaor. The crack sars propagaing when he crack ip opening reaches a prescribed criical value. A numerical algorihm for compuing he evoluion of he crack and cohesive zone in ime is discussed along wih some numerical resuls. Index Terms Keywords: Abel inegral equaion, viscoelsiciy, cohesive zone, hisory dependen fracure, nonlinear fracure I. INTRODUCTION THE cohesive zone, CZ, in a maerial is he area beween wo separaing bu sill sufficienly close surfaces ahead of he crack ip, see he shaded region in Figure 1. The cohesive forces presen a he cohesive zones pull he cohesive zone faces ogeher. The exernal load applied o he body, on he conrary, causes he crack faces and CZ faces o move furher apar and he crack o propagae. ˆσ ˆδ ĉ â â ĉ Fig. 1. Cohesive zone Our aim is o find he ime evoluion of he CZ before he crack sars propagaing, he delay ime, afer which he crack will sar o propagae, and furher ime evoluion of he crack and he CZ. When he crack propagaes, he cohesive forces vanish a he poins where he cohesive zone opening reaches a criical value and hese poins become he crack surface poins, while he new maerial poins, where he hisory-dependen normalised equivalen sress reaches a criical value, join he cohesive zone. So, he CZ is pracically aached o he crack ip ahead of he crack and moves wih Manuscrip received 6h of March, 213; revised 31s of March, 213. Deails of auhors: Boh auhors are currenly a he Deparmen of Mahemaical Sciences in Brunel universiy London, UK; s: layal.hakim@brunel.ac.uk; sergey.mikhailov@brunel.ac.uk q q ˆδ ˆσ he crack, keeping he normalised equivalen sress finie in he body. One of he mos popular CZ models for elaso-perfecly plasic maerials is he Dugdale-Leonov-Panasyuk DLP) ) model, see [1], [2]. In he DLP model, he maximal normal sress in he cohesive zones is consan and equals o he maerial yield sress, σ = σ y. This model, and several of is modificaions, have been widely used in nonlinear fracure mechanics. Anoher popular CZ model is he Barenbla 1962) model. The 3 main componens needed o implemen CZ models of he DLP-ype are: i) he consiuive equaions in he bulk of he maerial; ii) he consiuive equaions in he cohesive zone; iii) he crieria for he cohesive zone o break and he crack o propagae. II. PROBLEM FORMULATION The model presened in his paper is an exension of he DLP model o linear visco-elasic maerials wih nonlinear hisory-dependen consiuive equaions in he cohesive zone. To his end, we will replace he DLP cohesive zone sress condiion, σ = σ y, wih he condiion where Λˆσ; ˆ) = β bσ β Λˆσ; ˆ) = 1, 1) ˆ ˆσˆτ) β ˆ ˆτ) β b 1 dˆτ ) 1 β is he normalised hisory-dependen equivalen sress, ˆσ is he maximum of he principal sresses, and ˆ denoes ime. The parameers σ and b are maerial consans in he assumed power-ype relaion ) b ˆσ ˆ ˆσ) = beween he rupure ime ˆ ˆσ) and he consan uniaxial ensile sress applied o a body wihou cracks. The parameer β is a maerial consan in he nonlinear accumulaion rule for durabiliy under variable load, see [3]. Noe ha relaions 1)-2) were implemened in [4] and [5] o solve a similar crack propagaion problem wihou cohesive zone; i.e. i was assumed ha when condiion 1) is reached a a poin, his poin becomes par of he crack. However, such approach appeared o be inapplicable for b 2. In his paper, a cohesive zone approach is developed insead, in order o cover he larger range of b values relevan o srucural maerials. In he CZ approach, when condiion 1) is reached a a poin, his poin becomes par of he cohesive zone. Le he problem geomery be as in Figure 1, i.e, he crack occupies he inerval [ âˆ), âˆ)] and he cohesive zone σ 2) ISBN: ISSN: Prin); ISSN: Online) WCE 213

2 occupies he inervals [ ĉˆ), âˆ)] and [âˆ), ĉˆ)] in an infinie plane loaded a infiniy by racion ˆq in he direcion normal o he crack, which is consan in ˆx, applied a he ime ˆ = and kep consan in ime hereafer. The iniial CZ ip coordinae and crack ip coordinae are prescribed, ĉ) = â) = â, while he funcions ĉˆ) and âˆ) for ime ˆ > are o be found. The cohesive zone condiion 1)-2) a a poin ˆx on he cohesive zone can be rewrien as ˆ ˆ cˆx) ˆσ β ˆx, ˆτ) ˆ ˆτ) 1 β b dˆτ = bσβ β ˆ cˆx) ˆσ β ˆx, ˆτ) dˆτ, 3) ˆ ˆτ) 1 β b for ˆ ˆ c ˆx) and âˆ) ˆx ĉˆ). Here, ˆ c ˆx) denoes he ime when he poin ˆx becomes par of he cohesive zone. Equaion 3) is an inhomogeneous nonlinear Volerra inegral equaion of he Abel ype wih unknown funcion ˆσˆx, ˆ) for ˆ ˆ c ˆx). We will firs consider he case when he bulk of he maerial is linearly elasic and hen conver he obained soluion o he case of linear visco-elasic maerials using he so-called Volerra principle. Applying he resuls by Muskhelishvili see [6], Secion 12), we have for he sresses ahead of he cohesive zone in he elasic maerial, ˆσˆx, ˆ) = ˆx ˆq 2 ˆx 2 ĉ 2 π ˆ) ĉˆ) âˆ) ĉ 2 ˆ) ˆξ 2 ˆx 2 ˆξ ˆσˆξ, ˆ)dˆξ, 4) 2 for ˆ ˆ c ˆx) and ˆx > ĉˆ). As one can see from 4), ˆσˆx, ˆ) has generally a square roo singulariy as ˆx ends o ĉ. A sufficien condiion for he normalised equivalen sress, Λ, o be bounded a he cohesive zone ip is ha he sress inensiy facor, ˆK, is zero a he cohesive zone ip. Muliplying he sress in equaion 4) by ˆx ĉˆ) and aking he limi as ˆx ends o ĉˆ) yields ĉˆ) ˆKˆ) = ˆq 2 ĉˆ) ˆσˆξ, ˆ) 2 π ĉ 2 ˆ) ˆξ dˆξ. 2 âˆ) To simplify he equaions, we will employ he normalisaions = ˆ ˆ, x = ˆx â, a) = â ˆ ) â, c) = ĉ ˆ ), σx, ) = ˆσx â, ˆ ), â ˆq Kc, ) = ˆKc â, ˆ ) ˆq. 5) â ) b ˆq Here ˆ = ˆ ˆq) = denoes he fracure ime for σ an infinie plane wihou a crack under he same load, ˆq. Thus, afer he normalisaion, we sae he following principle equaions for he considered problem: a) he cohesive zone condiion 1) in he form cx) σx, τ) β dτ = b cx) τ) 1 β b β σ β x, τ) dτ 6) τ) 1 β b for a) x c); b) he expression for he sress ahead of he cohesive zone: σx, ) = 1 2 π c) a) x x2 c 2 ) ) c2 ) ξ 2 x 2 ξ 2 σξ, )dξ for x > c); 7) c) he zero sress inensiy facor, Kc, ) =, where c) 2c) c) σξ, ) Kc, ) = dξ. 8) 2 π c2 ) ξ2 a) III. COHESIVE ZONE GROWTH FOR A STATIONARY CRACK In his secion we will consider he saionary sage, before he crack sars propagaing, i.e., a) = a) = 1, and hus only he cohesive zone grows wih ime. Our aim is o find he cohesive zone ip posiion c) and he crack ip opening. A. Numerical Mehod Le us inroduce a ime mesh wih nodes i = ih, for i =, 1, 2, 3,...n, where h is he ime-sep size. A each ime sep i, we use he secan mehod o find he roos, c i ) = c i, of he equaion Kc i, i ) =, as follows: 1) Take 2 iniial approximaions, c i ) 1 and c i ) 2, for c i ). 2) Obain K 1 = Kc i ) 1, i ) and K 2 = Kc i ) 2, i ) using equaion 8). Noe ha: In order o evaluae he inegral in 8), ci) a) 1 c2 i ) ξ 2 σξ, i)dξ, 9) we linearly inerpolae σξ, i ) on he cohesive zone beween ξ = c k ) and ξ = c k+1 ), where k =, 1, 2,...i 1. On he oher hand, o find σξ, i ), a each ξ = c k ), we use he Abel inegral equaion 6). Firs, we evaluae he inegral k σ β c k ), τ) τ) β b 1 dτ in he righ hand side of equaion 6) by piecewise linearly inerpolaing he funcion σ β c k ), τ) beween τ = j and τ = j+1 for j =, 1, 2,...k 1. Then we use he analyical soluion of a generalised Abel ype inegral equaion o solve he inegral equaion 6). To his end, in urn, we need o find σ β c k ), j ) for j < k from equaion 7) since c k ) > c j )), where he inegral cj) c2 ) ξ 2 c k ) 2 ξ 2 σξ, j)dξ a) is calculaed similarly o inegral 9). This means we piecewise linearly inerpolae σξ, j ) beween ξ = c m ) and ξ = c m+1 ) for m =, 1,...j 1. 3) Find he nex approximaion for c i using c i ) 3 = K 2c i ) 1 K 1 c i ) 2 K 2 K 1 ISBN: ISSN: Prin); ISSN: Online) WCE 213

3 4) If c i ) 3 c i ) 1 < ɛ or c i ) 3 c i ) 2 < ɛ allocae c i ) = c 3 and go o he sep = i+1 ; oherwise, go o he nex iem. Here ɛ is some olerance. 5) Taking he new c i ) 2 as c i ) 3 reurn o iem 2 unil convergence is reached. We have used ε = 1 8 as he olerance value. All programming was implemened in MATLAB. Using his scheme, we obained he evoluion of he cohesive zone ip posiion as well as he sress disribuion on he cohesive zone. c) log scale) β=2 β=1 β=1/2 B. Numerical Resuls The graphs presened in Fig. 2 show he resuls obained for a saionary crack wih b = 4 and β = b/2 = 2 for various mesh sizes. Fig. 4. crack) CZ ip posiion vs ime for b = 4 and differen β non-propagaing 1.93 c) log scale) n=25 n=5 n=1 n=2 n=4 n=8 c) log scale) β=2 β=1 β=1/ Fig CZ ip posiion vs ime for b = 4, β = 2, and differen meshes Fig. 5. CZ ip posiion vs ime for b = 4 and differen β non-propagaing crack), zoomed The graph in Fig. 3 is a closer look a he graphs in Fig. 2. Figure 6 shows he sress behaviour wih respec o ime a he poin x = c.6), i.e., c x) =.6. c) log scale) n=25 n=5 n=1 n=2 n=4 n=8 σc.6,) β=2 β=1 β=1/ Fig. 3. zoomed CZ ip posiion vs ime for b = 4, β = 2 and differen meshes, Fig. 6. The sress σc.6), ) vs ime for b = 4, n = 5 The graphs in Figs. 4 and 5 show he resuls obained for 3 differen values of he parameer β. C. Crack Tip Opening in he Elasic Case Using he represenaions by Muskhelishvili see [6], Secion 12), i can be deduced ha he normal displacemen ISBN: ISSN: Prin); ISSN: Online) WCE 213

4 jump a a crack+cz shore poin ˆx is where [û e ˆx, ˆ)] = [û q) e ˆx, ˆ)] + [û σ) e ˆx, ˆ)] [û q) e ˆx, ˆ)] = 4ˆq1 ν2 ) ĉˆ) 2 ˆx 2, E ) [û σ) e ˆx, ˆ)] = 41 ν2 ) ĉˆ) σˆξ, ˆ)Γˆx, πe ˆξ; ĉˆ))dˆξ, âˆ) Γˆx, ˆξ; ĉ) = ln 2ĉ2 ˆξ 2 ˆx 2 2 ĉ 2 ˆx 2 )ĉ 2 ˆξ 2 ) 2ĉ 2 ˆξ. 2 ˆx ĉ 2 ˆx 2 )ĉ 2 ˆξ 2 ) In he above expressions, E and ν denoe Young s modulus of elasiciy and Poisson s raio, respecively. We denoe he displacemen jump a he crack ip, ˆx = â, for he elasic maerial by ˆδ e ˆ) := [û e âˆ), ˆ)] = 1 ν2 ) 4ˆq ĉ 2 ˆ) â 2 ˆ) + 4 π ĉˆ) âˆ) E ˆσˆξ, ˆ)Γâˆ), ˆξ, ĉˆ))dˆξ ). 1) and call i he crack ip opening. Using he normalisaion [u e x, )] = E ) [û e ˆx â, ˆ ] â ˆq 1 ν 2, δ e ) = E ˆδ ˆ ) ) â ˆq1 ν 2 ), 11) we obain [u e x, )] = 4 c 2 ) x 2 + δ e ) = [u e a), )]. 4 π c) a) σξ, )Γx, ξ; c))dξ, 12) D. Crack Tip Opening in he Viscoelasic Case To obain he crack ip opening in he visco-elasic case, we will implemen he so-called Volerra principle, according o which we have o replace he elasic consans E and ν in he elasic soluion by he corresponding viscoelasic operaors, o arrive a he viscoelasic soluion. Alhough his approach does no always bring a viscoelasic soluion for he problems wih moving boundaries, i is possible o show, cf. [7], ha his approach leads o a viscoelasic soluion for he plane symmeric problems wih a sraigh propagaing crack. This means ha for he viscoelasic problem we can direcly use he resuls by Muskhelishvili for he sress represenaion given in equaion 4) since hey do no include he elasic consans a all. For simpliciy, we will consider he viscoelasic maerial wih consan purely elasic) Poisson s raio. Then, o obain he crack opening in he viscoelasic case, we have o replace 1/E in 1) by he second kind Volerra inegral operaor E 1 defined as E 1ˆσ ) ˆ ) ) ˆσ ˆ ˆ = J ˆ ˆτ ) σ ˆτ) dˆτ, E ISBN: ISSN: Prin); ISSN: Online) where he creep funcion J is known and J is is derivaive. Hence he viscoelasic crack ip opening becomes ˆδ v ˆ ) = [û v âˆ), ˆ)] = E 1 E [û e âˆ), )] ) ˆ ) = ˆδ e ˆ ) ˆ E J ˆ ˆτ ) ) [û e âˆ), ˆτ)]dˆτ. 13) In our numerical examples we use he creep funcion of a sandard linear solid of he Kelvin-Voig ype), namely J ˆ ˆτ ) = e E 1 η ˆ ˆτ) ), 14) E E 1 J ˆ ˆτ ) = 1 E η e 1 η ˆ ˆτ). Here, η and E 1 are maerial consans, η/e 1 being he relaxaion ime and η he viscosiy of he polymer. Such viscoelasic models saisfacorily model some polymers, e.g. PMMA also known as plexiglas). For J in he form 14), equaion 13) becomes ˆδ v ˆ ) = [û v âˆ), ˆ)] = ˆδ e ˆ ) + E ) ˆ e E 1 η ˆ ˆτ) [ûe âˆ), ˆτ)]dˆτ. 15) η Employing he normalised parameers δ v ) = E ) ˆδ v ˆ â ˆq 1 ν 2 ), A = E ˆ, A 1 = E 1ˆ, 16) η η equaion 15) reduces o he following expression for he normalised crack ip opening in he viscoelasic case, δ v ) = [u v a), )] = δ e ) + A ca)) e A1 τ) [u e a), τ)]dτ ), 17) where he lower limi of he inegral is replaced wih c a)) since [u e x, τ)] = when τ c x). In he numerical examples we used values A = and A 1 = The graphs in Figs. 7-9 show he saionary crack ip opening evoluion for b = 4 and differen β in he elasic and viscoelasic cases. Crack ip opening Fig β=b/2 β=b/4 β=b/ Crack ip opening vs ime for b = 4, elasic case WCE 213

5 Crack ip opening Fig. 8. Crack ip opening β=b/2 β=b/4 β=b/ Crack ip opening vs ime for b = 4, viscoelasic case elasic case viscoelasic case Fig. 9. Crack ip opening vs ime for b = 4, β = 1 IV. CRACK PROPAGATION We have, so far, assumed ha he crack is saionary and only he cohesive zone is growing ahead of he crack. However, he crack will sar o propagae when he crack ip opening ˆδ reaches a criical value ˆδ c, where ˆδ c is considered as a maerial consan. The crack and cohesive zone will no necessarily grow a he same rae. The ime insan, when he crack ip opening reaches a criical value, will be referred o as he fracure delay ime and denoed by ˆ d. Similar o 5), 11) and 16), we employ he following normalised parameers, d = ˆ d ˆδ c E, δ c = ˆ â ˆq1 ν 2 ). 18) The aim now is o find d, he crack ip coordinae a) and he CZ ip coordinae c) for > d. A. Numerical Mehod Considering uniformly spaced ime seps, he crack ip opening δ i ) saisfies equaion δ e i ) = δ c, i d. 19) for he purely elasic case and equaion δ v i ) = δ c, i d. 2) for he viscoelasic case, where δ v i ) is given by 17). We use he secan mehod o solve equaion 19) in he elasic case) or 2) in he viscoelasic case) for a i ). To do his, we need o know c i ) a each ieraion. I is obained using he secan mehod o solve he equaion Kc i ), i ) = for c i ), where he sress inensiy facor Kc, ) is given by 8). Noe ha we choose previous cohesive zone ip posiions, c m ), as iniial approximaions for a i ) wihin he secan algorihm. The advanage of doing his is ha we already know he sress hisory a hese previous poins since hey were compued in he previous ime seps. Noe ha c a i ) he ime insan when a i became par of he cohesive zone) is unknown for cases when a i c m ). Thus, we linearly inerpolae δ e c a i ), a i ) beween δ e m, a i ) and δ e m+1, a i ) where c m < a i < c m+1. During implemenaion of he algorihm, we come across he sep, i, where a i will exceed c i 1, and for decreasing cohesive zone lengh we will have a i > c i 1 in all he seps which follow. Thus, for hese seps, only 1 previous value of c namely c i 1 ) can be aken as an iniial approximaion of a i. To his end, we will modify he algorihm by fixing a i and compuing he corresponding i and c i by solving equaion 19) in he elasic case) or 2) in he viscoelasic case) seing he crack ip opening displacemen equal o he criical crack ip opening) and Kc i, i ) = seing he sress inensiy facor o ) respecively. B. Numerical resuls We used in our calculaions he value δ c = 1.13 for he normalised criical crack ip opening, cf. Appendix. Le h = 1/5 be he sep size in he ime mesh. For β = b/4 = 1, we have 51 and 48 ime seps before crack growh begins in he elasic and he viscoelasic cases, respecively, while d =.12 for he elasic case and d =.96 for he viscoelasic case. The graphs in Figs. 1 and 11 show coordinaes of he crack ip and he cohesive zone ip for boh he elasic and viscoelasic cases. lengh log scale) 1 1 Crack ip coordinae elasic) Crack ip coordinae viscoelasic) CZ ip coordinae elasic) CZ ip coordinae viscoelasic) Fig. 1. Lengh vs. ime for b = 4, β = 1 The graphs in Fig. 12 show he behaviour of he cohesive zone lengh wih ime. The graphs in Fig. 13 show he evoluion of CZ ip coordinae in ime for 3 cases of β. ISBN: ISSN: Prin); ISSN: Online) WCE 213

6 lengh log scale) Crack ip coordinae elasic) Crack ip coordinae viscoelasic) CZ ip coordinae elasic) CZ ip coordinae viscoelasic) Fig. 11. Lengh vs. ime for b = 4, β = 1 cz lengh purely elasic case viscoelasic case Fig. 12. CZ lengh vs. ime for b = 4, β = 1 ip opening. Moreover, as β becomes smaller, he crack ip opening increases more slowly wih ime. The obained graph of he crack ip opening displacemen for he viscoelasic case demonsraes ha he crack opens a a higher rae han in he elasic case. As a consequence, he fracure delay ime d is longer for he elasic case han for he viscoelasic one: for b = 4 and β = 1, we obained d =.12 for he elasic case and d =.96 for he viscoelasic case. For he growing crack sage, > d, we can see from Figure 1, ha he crack growh rae increases, while he cohesive zone lengh decreases wih ime. The ime, when he cohesive zone lengh becomes seems o coincide wih he ime when he crack lengh becomes infinie and can be associaed wih he complee fracure of he body. The graph of he cohesive zone lengh for he viscoelasic case indicaes ha he fracure ime is slighly smaller han ha for he purely elasic case. For example, for b = 4 and β = 1, he normalised fracure ime is r =.188 for he elasic case and r =.186 for he viscoelasic one. I can be seen from Figure 13 ha as β decreases, he fracure ime also decreases. APPENDIX To give an idea on he parameer scales, we provide here some maerial parameers for PMMA from [8] pages ), [9], and [1]. Poisson s raio ν =.35; Young s modulus of elasiciy E = 31 MPa; relaxaion ime η/e 1 = s; viscosiy η = MPa s; criical crack ip opening: ˆδ c =.16mm. Then by 18), δ c = 1.13 for ˆq =5MPa. Some experimenal daa on he saic creep rupure under ensile sress for PMMA a he room emperaure were repored in [11]. Fiing hese daa o he power-ype durabiliy curve ˆ ˆσ) = ˆσ/σ ) b will give values for b, σ, ˆ ˆq), A and A 1. c) log scale) 1 1 β=b/2 elasic) β=b/2 viscoelasic) β=b/4 elasic) β=b/4 viscoelasic) β=b/8 elasic) β=b/8 viscoelasic) Fig. 13. CZ ip coordinae vs. ime for b = 4 V. CONCLUSIONS The soluion converges as he mesh becomes finer. For he saionary crack sage, < d, he cohesive zone lengh is smaller for smaller β, for he same ime insan. We can see from Figure 6 ha he sress reaches a sharp maximum in coordinae a he cohesive zone ip and monoonically decreases wih he disance from he ip. As expeced, we have an increase, wih ime, of he crack REFERENCES [1] Dugdale D.S., Yielding of seel shees conaining slis, J. Mech. Phys. Solids, Vol. 8, 1-14, 196. [2] Leonov M.Ya., Panasyuk V.V., Developmen of he smalles cracks in he solid, Applied Mechanics Prikladnaya Mekhanika), Vol. 5, No. 4, , [3] Mikhailov S. E., Namesnikova I. V. Hisory-sensiive accumulaion rules for life-ime predicion under variable loading, Archive of Applied Mechanics, Vol. 81, , 211. [4] Mikhailov S.E., Namesnikova I.V., Local and non-local approaches o creep crack iniiaion and propagaion, Proceedings of he 9h Inernaional Conference on he Mechanical Behaviour of Maerials, Geneva, Swizerland, 23. [5] Hakim L, Mikhailov S.E., Nonlinear Abel ype inegral equaion in modelling creep crack propagaion, Inegral Mehods in Science and Engineering: Compuaional and Analyic Aspecs, ediors: C. Consanda C. and P. Harris, Springer, , 211. [6] Muskhelishvili N.I., Some Basic Problems of he Mahemaical Theory of Elasiciy, Noordhoff Inernaional Publishing, The Neherlands, [7] Rabonov IU. N. Elemens of herediary solid mechanics, Mir Publishers, Russia, [8] Mark J.E., Polymer Daa Handbook, Oxford Universiy Press, New York, [9] Ro H.W., Ding Y, Lee H.J., Hines D.R., Jones R.L., Lin E.K., Karim A., Wu W.L., Soles C.L., Evidence for inernal sresses induced by nanoimprin lihography, Journal of Vacuum Science and Technology B, Vol. 24, No. 6, 26. [1] Coerell B., Fracure and life, World Scienific, Singapore, 21. [11] McKenna G.B., Crissman J. M., A Reduced Variable Approach o Relaing Creep and Creep Rupure in PMMA, MRS Proceedings, Vol. 79, , ISBN: ISSN: Prin); ISSN: Online) WCE 213