A theoretical and computational framework for studying creep crack growth

Transcription

1 It J Fract DOI.7/s IUTAM BALTIMORE A theoretical ad computatioal framework for studyig creep crack growth Elsiddig Elmukashfi Ala C. F. Cocks Received: 2 March 27 / Accepted: 29 Jue 27 The Author(s) 27. This article is a ope access publicatio Abstract I this study, crack growth uder steady state creep coditios is aalysed. A theoretical framework is itroduced i which the costitutive behaviour of the bulk material is described by power-law creep. A ew class of damage zoe models is proposed to model the fracture process ahead of a crack tip, such that the costitutive relatio is described by a tractio-separatio rate law. I particular, simple critical displacemet, empirical Kachaov type damage ad micromechaical based iterface models are used. Usig the path idepedecy property of the C -itegral ad dimesioal aalysis, aalytical models are developed for pure mode-i steady-state crack growth i a double catilever beam specime (DCB) subjected to costat pure bedig momet. A computatioal framework is the implemeted usig the Fiite Elemet method. The aalytical models are calibrated agaist detailed Fiite Elemet models. The theoretical framework gives the fudametal form of the model ad oly a sigle quatity Ĉ k eeds to be determied from the Fiite Elemet aalysis i terms of a dimesioless quatity φ, which is the ratio of geometric ad material legth scales. Further, the validity of the framework is examied by ivestigatig the crack E. Elmukashfi (B) A. C. F. Cocks Departmet of Egieerig Sciece, Uiversity of Oxford, Park Road, OX 3PJ Oxford, UK elsiddig.elmukashfi@eg.ox.ac.uk A. C. F. Cocks ala.cocks@eg.ox.ac.uk growth respose i the limits of small ad large φ,for which aalytical expressio ca be obtaied. We also demostrate how parameters withi the models ca be obtaied from creep deformatio, creep rupture ad crack growth experimets. Keywords Creep Crack C*-itegral Damage zoe model Tractio-separatio rate law (TSRL) Double catilever beam (DCB) Dimesioless aalysis Nomeclature 2 l The spacig betwee two adjacet pores β A material parameter of the expoetial damage law δ c The critical ormal displacemet jump i the damage zoe at the crack tip δ f The ormal displacemet jump at failure i the crack tip δ i The displacemet jump vector across the damage zoe (i =, 2, 3) δ The separatio rate at the referece tractio T δ m The maximum ormal displacemet jump rate vector i the crack tip δ i The displacemet jump rate vector across the damage zoe (i =, 2, 3) ε The strai-rate at the referece stress σ

2 E. Elmukashfi, A. C. F. Cocks ȧ Ĉ k λ ( ) cr ( ) el ω a The steady state crack velocity The separatio history fuctio of model k The characteristic geometric legth scale The creep compoet of the quatity ( ) The elastic compoet of the quatity ( ) A scalar damage parameter The dimesioless steady state crack velocity φ The ratio of geometric to material legth scales σ The referece stress σ e The vo Mises equivalet stress σ ij The Cauchy stress tesor ε ij The egieerig strai tesor a The crack legth C The rate of the J-itegral C s The separatio history fuctio of the simple model E Youg s modulus f The curret area fractio of the pores f The iitial area fractio of the pores f c The coalescece area fractio of the pores h The curret height of a pore h The iitial height of the pores m The rate sesitivity expoet of the damage zoe The rate sesitivity parameter of the bulk material i The uit ormal vector (i =, 2, 3) s ij The deviatoric part of Cauchy stress tesor T i The tractio vector (i =, 2, 3) T The referece tractio of the damage zoe u i The displacemet vector (i =, 2, 3) x i The Cartesia material ad spatial coordiates (i =, 2, 3) Itroductio At elevated temperature, creep crack growth (CCG) is oe of the most commo failure mechaisms i may egieerig applicatios, e.g. structural compoets, similar ad dissimilar metal welds etc. This problem has received much attetio over the last forty years due to the importace i desigig structures with high itegrity ad safety. Hece, developig aalytical models for steady-state crack growth which ca be calibrated agaist detailed Fiite Elemet models is of great iterest. Further, a assessmet of the effect of differet material parameters ad damage developmet processes o the crack growth behaviour ca be provided usig such models. Studyig creep crack growth has a log history i the literature. A major feature of these studies is the developmet of a parameter that characterizes the crack tip fields as well as crack propagatio. Uder steady state creep coditios, the so called C -itegral (Lades ad Begley 976; Nikbi et al. 976; Ohji et al. 976) (i.e. the creep J-itegral, Rice 978) ca be used to characterize the crack tip fields ad creep crack growth. It provides descriptios of the strai-rate ad stress sigularities at the crack tip ad a correlatio of experimetal crack growth rate data (Taira et al. 979; Riedel ad Rice 98). Moreover, the C -itegral is path idepedet for cotours i which the material properties oly vary i the directio perpedicular to the directio of crack growth withi the family of cotours cosidered. Riedel ad Rice (98) studied the trasitio from short-time elastic to log-time creep behaviour assumig that primary creep is egligible (small-scale creep coditios). They itroduced a parameter C(t) that describes the strai, strai-rate ad stress fields withi a creep zoe that forms about the crack tip. Their aalysis also provides a characteristic time for the trasitio to the steady state stress field (i.e. the time for C(t) to equal C ). Later, Ehlers ad Riedel (98) proposed a relatio betwee C(t) ad C. Saxea (986) proposed a ew parameter C t which ca be measured easily i compariso with C(t). Bassai et al. (988) compared these two parameters ad cocluded that C t characterizes crack growth rates much better tha C(t). Further, the C(t) parameter is foud to be more suitable for characterizig a statioary crack ad C t is related to a rapidly propagatig crack. I the primary creep regime, Riedel (98) suggested a ew parameter Ch as a aalogy to the C -itegral. Further, Leug ad McDowell (99) icluded the primary creep effects i the estimatio of the C t parameter. To this ed, the C, C(t), C t ad Ch parameters are geerally accepted ad widely used i studyig creep crack growth.

3 A theoretical ad computatioal framework Uder creep coditios, cracks i polycrystallie materials advace as a result of the growth of damage ahead of the crack tip (geerally i the form of discrete voids or microcracks, which form primarily at grai boudaries). I the viciity of a macroscopic primary crack tip, secodary micro-cracks are formed as a result of itesive void growth ad coalescece ad/or a accumulatio ad growth of micro-cracks. These secodary cracks propagate ad coalesce creatig the ew crack surfaces, allowig the primary macroscopic crack to advace alog a iterface or itercoected grai-boudaries. The growth of damage ca ifluece the costitutive properties of the material ad therefore the details of the ear crack tip stress ad strairate fields. Early models of creep crack growth either assumed that the stress (e.g. Riedel 98; Tvergaard 984) or strai-rate field (e.g. Cocks ad Ashby 98; Nikbi et al. 984) is the same as that for the udamaged material ad used either empirical or mechaistic damage growth laws to determie the crack growth rate. I the strai based models the critical damage at the crack tip is expressed i terms of a material ductility (strai to failure) which is a fuctio of the local stress state. Extesios of this approach withi a fiite elemet framework (employig models i which the costitutive relatioships for deformatio are ot iflueced by the presece of damage) have bee udertake by Nikbi et al. (976, 984), Yatomi ad Nikbi (24). Studies of the ifluece of damage o the ature of the crack tip fields ad crack growth process where damage iflueces the deformatio respose have bee udertake by Riedel (987) ad Bassai ad Hawk (99) for empirical Kachaov (Kachaov 958; Rabotov 969) type cotiuum damage mechaics models. More recetly, the full iteractio betwee deformatio ad damage developmet ad how this iflueces the crack growth process has bee modelled directly usig the fiite elemet method, usig both mechaistic ad empirical models for the growth of damage (e.g. Ock ad va der Giesse 998; We ad Sha-Tug 24). I each of the above refereced studies damage developmet ad its ifluece o crack growth is modelled as a cotiuum process. Aother method of modellig crack propagatio is through the use of iterface cohesive or damage zoe models. Iterface damage zoe models of this type provide a couplig betwee the local separatio rate across a iterface ad bulk deformatio processes, i.e. they itroduce a physically meaigful legth scale that is related to the dissipative mechaisms resposible for damage developmet. A damage zoe model of this type describes the fracture process i the viciity of the crack tip as a gradual surface separatio process, such that the ormal ad shear tractios at the iterface resist separatio ad relative slidig. The cohesive/damage zoe modellig approach has its origis i the pioeerig work of Dugdale (96) ad Bareblatt (962). The first use of cohesive zoe models i a fiite elemet eviromet was udertake by Hillerborg et al. (976). Several models have bee proposed i the literature, wherei a variety of materials ad applicatios have bee successfully ivestigated (Camacho ad Ortiz 996; Elmukashfi ad Kroo 24; Hui et al. 992; Kauss 993; Needlema 987, 99; Rahul-Kumar et al. 999; Rice ad Wag 989; Tvergaard 99; Xu ad Needlema 993). Rate-depedet ad rateidepedet models as well as physically based ad pheomeological models have bee employed. However, to the authors kowledge, apart from the work of Ock ad va der Giesse (998), va der Giesse ad Tvergaard (994), Thouless et al. (983) ad Yu et al. (22) damage zoe type models have ot bee used to study the developmet of creep damage ad/or creep crack growth. I this paper, a theoretical ad computatioal framework for creep crack growth is preseted i which we assume that all the damage is cocetrated i a arrow zoe directly ahead of the growig crack tip. The objective is to model crack propagatio i materials that exhibit steady state creep behaviour outside of the damage zoe ad to ivestigate the effect of differet material parameters, forms of damage zoe costitutive law ad damage developmet processes o the crack growth behaviour. A theoretical framework is iitially itroduced i which the costitutive behaviour of the bulk material is described by powerlaw creep. A ew class of damage zoe model is proposed to model the fracture process such that the costitutive relatio is described by a tractio-separatio rate law. More specifically, three differet models, i.e. a simple critical displacemet model, Kachaov type empirical models ad a micromechaical based iterface model are ivestigated, which mirror the types of models employed i the cotiuum models of creep crack growth described above. We follow the recet approach of Wag et al. (26), who studied cleavage failure i creepig polymers, i which we keep the

4 E. Elmukashfi, A. C. F. Cocks material descriptios ad geometric cofiguratios as simple as possible to explore the relatioship betwee the form of the costitutive model, material parameters ad crack growth. With this i mid, we cocetrate iitially o the behaviour of a double catilever beam specime (DCB) of ifiite legth subjected to a costat pure bedig momet, i which C remais costat as the crack grows ad the crack growth rate evetually achieves a steady-state. We cocetrate o the behaviour i the steady state. By ivokig the path idepedece of the C -itegral ad choosig cotours i the far field ad surroudig the damage zoe we demostrate how a simple aalytical expressio for the crack growth rate ca be obtaied i terms of C, damage zoe material parameters ad a dimesioless scalig parameter that is a fuctio of the ratio of characteristic geometric ad material legth scales, that ca be determied usig the fiite elemet method. The theoretical framework is preseted i Sect. 2. The aalysis of creep crack growth i the double catilever beam specime ad the fiite elemet implemetatio of the damage zoe model are described i Sect. 3, with the crack growth results for the differet iterface models preseted ad discussed i Sect Theoretical framework for creep crack 2. Backgroud Cosider a solid cotaiig a statioary crack that is subjected to a costat load. The solid is assumed to exhibit elastic behaviour, together with primary, secodary ad tertiary creep. Creep deformatio evolves with icreasig time ad this evolutio ca be divided ito differet distict stages. These stages have bee described ad evaluated by Bassai ad Hawk (99). Iitially, a small-scale creep zoe, i.e. small i compariso with the physical characteristic legth of the body, is formed i the viciity of the crack tip. I this stage, the material deforms by primary creep iside the creep zoe ad remais elastic elsewhere. Followig the developmet of the primary creep zoe, a secodary (steady-state) creep zoe develops as a smaller regio iside the primary creep zoe. Thereafter, the primary ad secodary creep zoes cotiue to expad at the cost of the elastic ad primary zoes, respectively. Durig this process, damage accumulates i the crack tip regio, which may lead to crack propagatio if a critical coditio is met. Hece, crack propagatio may take place at differet istats durig the evolutio of the ear tip stress ad strai-rate fields. The crack propagatio scearios are characterized by the ature of the crack tip fields at these istats: (i) the small-scale creep zoe is formed surrouded by the elastic medium, (ii) the primary creep zoe is large eough but remais surrouded by the elastic medium, (iii) the secodary creep zoe is formed iside the primary creep zoe but both zoes remai surrouded by the elastic medium, (iv) the secodary creep zoe is expadig iside the primary creep zoe which domiates, (v) the secodary creep zoe domiates. This study cocers crack propagatio i creepig materials uder steady state coditios (type v). I this case, the C -itegral ca be used to characterize the creep crack growth behaviour. Note also, that as a crack grows i a elastic/creepig material, i the absece of damage, a zoe develops ahead of the crack tip i which the stresses are determied by the elastic ad creep properties of the material (Hui ad Riedel 98) who s size is a fuctio of the crack velocity. For steady state behaviour the size of this zoe must be small compared to the size of the crack tip damage process zoe. Uder these coditios, the path idepedet property ca be used to obtai a direct relatioship betwee the far field loadig ad the fracture process parameters. I the followig sub-sectios we describe the costitutive relatioships for the bulk cotiuum respose ad itroduce a umber of differet models to describe the respose ahead of the crack tip withi the damage zoe. We cocetrate o mode I crack growth ad oly preset relatioships for the opeig mode, although a descriptio of the shear respose is also required for the computatioal studies preseted later. We cosider the crack growth process i Sect. 3, where we use the path idepedece of the C -itegral to relate the ear crack tip damagig processes to the far field loadig. I the theoretical models preseted below we cocetrate o steady state crack growth, where creep domiates the material respose, i.e. we eed ot cosider the elastic respose. Similarly we do ot eed to cosider ay elastic/reversible cotributios to the deformatio withi the damage zoe.

5 A theoretical ad computatioal framework 2.2 Creep deformatio ad characterizatio of the remote field Cosider a body cotaiig a crack ad subjected to a costat far field loadig, see Fig.. A commo Cartesia coordiate system for the referece ad deformed cofiguratios x i, i =, 2, 3, is assumed. The bulk material is assumed to exhibit steady-state creep behaviour ad is defied by the costitutive law ε ij = φ σ ij = 3 2 ε ( σe σ ) sij σ e, () where σ ij is Cauchy s stress tesor, ε ij is the strai rate tesor, s ij = σ ij 3 σ kkδ ij is the stress deviator, 3 σ e = 2 s ijs ij is the vo Mises equivalet stress, σ is a referece stress, ε is the strai-rate at the referece stress ad is the rate sesitivity parameter. φ is the stress potetial φ = ( ) + + ε σe σ. (2) σ The eergy dissipatio rate, Ḋ, is give by Ḋ = φ + ψ, (3) where ψ is the dual rate potetial ψ = + ε σ ( εe ε ) +, (4) 2 ad ε e = 3 ε ij ε ij. Crack propagatio is assumed to be determied by a iterface model such that the propagatio takes place alog a fictitious iterface surface, Γ it. As the crack advaces separatio occurs alog the iterface to create two surfaces. Hece, a material poit alog the iterface is defied by the two ormal vectors i ad i +, where i = i +, i.e. the iitially itact material poit splits ito two poits with uit ormals actig opposite to each other ad ito the material o either side of the iterface. The displacemet-rate jump across the damage zoe surface ad the correspodig tractios are defied by the vectors δ i = u i + ui, where u i + ad ui are the displacemet rates either side of the iterface, ad T i + = σ ij + j, T i = σ ij j. I order to aalyse the problem the C -itegral is determied o the ier ad the outer cotours i Fig.. The C -itegral is defied as C = Γ W dx 2 T i u i x ds, (5) where Γ is a arbitrary cotour aroud the tip of the crack with uit outward ormal i, T i = σ ij j is the tractio o ds ad u i is the displacemet rate. The C - itegral i the outer path, Cout, is determied by the far field loadig. We cosider the situatio where a damage zoe exteds alog the x axis directly ahead of the crack tip, see Fig. 2. The first term of Eq. (5) the vaishes, sice dx 2 =. The cotributio of the damage regio to the C -itegral alog the ier path, Ci, is the evaluated as C i = = = = x 2 δ m i Γ i L L T i u i ds x σ ij i σ ij + i u i x 2 dx + δ x dx σ ij + i d δ i = Fracture zoe x δ m i L σ ij + i u + i x dx T + i d δ i, (6) + i i Γ out Γ i Γ it Fig. The schematic of iterface crack model i creepig solid. The figure illustrates the defiitio of the iterface surface Γ it. ier path Γ i ad the outer path Γ out

6 E. Elmukashfi, A. C. F. Cocks 2.3 Damage zoe models for creep crack growth δ m δ δ f Γ i δ l it (a) x 2 x 3 T T σ c x I this sectio, we preset models for the damage zoe. We limit our descriptio to mode I loadig. More geeral relatioship for mode II ad mixed mode loadig are described elsewhere (Elmukashfi ad Cocks 27). For each of the models here we assume that the relatioship betwee δ ad T ca be expressed i the form of a power-law. Further, these models are capable of predictig similar damagig processes ad crack growth behaviour through the appropriate selectio of material parameters, see Appedix B Simple critical displacemet model δ f l it δ, x (b) Fig. 2 The cohesive zoe for pure mode I crack propagatio: a schematic of the cohesive zoe; ad b the ormal tractioseparatio (T δ ) ad the separatio rate-separatio ( δ δ ) distributio alog the cohesive zoe. l it is the legth of the cohesive zoe, δ c is the critical displacemet, δf is the displacemet at failure, ad σ c is the cohesive stregth where δ m is the opeig rate at the tip of the crack. I order to simplify the relatioships used i subsequet aalysis we omit the superscript + fromthe tractio. Usig the path-idepedece property of C (ie, Cout = Ci ) provides a relatioship betwee the far field loadig ad the behaviour withi the damage zoe. I order to complete the aalysis we eed a costitutive relatioship for the damage zoe that relates δ i to T i. I this paper we cocetrate o mode I loadig ad therefore oly eed to cosider the compoets of tractio ad displacemet-rate ormal to the damage zoe. The δm Ci = T d δ. (7) I the followig sub-sectio we preset a umber of differet costitutive relatioships for the damage zoe respose. δ c The simplest form of tractio-separatio rate law is defied by a direct power-law relatioship betwee the ormal tractio T ad the separatio rate δ. Further, damage does ot ifluece the opeig rate ad failure is achieved whe δ achieves a critical separatio δ f. Thus, the costitutive model takes the followig form ( ) m T δ = δ, (8) T where T is a referece tractio (equivalet to of Eq. (), δ is the separatio rate at this tractio ad m is a expoet which ca have a differet value to i Eq. (). It should be oted that the tractio becomes zero whe the separatio exceeds the critical value, i.e. T = if δ δ f Kachaov damage type empirical model I this model damage is assumed to ifluece the costitutive respose ad a sigle scalar damage parameter ω is itroduced to icorporate the effect of damage. The damage parameter is assumed to evolve mootoically from to, i.e. from a udamaged to a fully damaged state. Followig Kachaov (958) ad Lemaitre ad Chaboche (994) we assume that the separatio rate is a fuctio of a effective tractio T, which is related to the ormal tractio T ad ω by T = T ω. (9)

7 A theoretical ad computatioal framework The costitutive respose is the simply obtaied by replacig T by T i Eq. (8)togive ( ) T m δ = δ. () ( ω) T The damage is assumed to be determied by the ormal separatio δ, i.e. ω = ω (δ ), ad the damage evolutio law may take differet forms depedig o the damage mechaism(s). I this study, we propose two differet damage models, amely liear ad expoetial models. The differet evolutio laws are: The liear damage model: δ δ c ω = δ f if δ δ δc c, () if δ <δ c. The expoetial damage model: [ ( δ δ c )] exp β ω = δ f if δ δ δc c, if δ <δ c. (2) where δ c is the separatio at which damage iitiates (for separatios less tha this value ω = ad the costitutive respose is give by Eq. ()), as before δ f is the separatio at failure ad β is a material parameter. It should be oted that for the expoetial damage law the tractio does ot ecessarily decrease smoothly to zero at failure but a abrupt respose may result. Figure 3 below shows the tractio-separatio law for differet separatio rates for the case of liear ad expoetial damage laws Micromechaical based model This model is based o the creep extesio of Yalcikaya ad Cocks (25) micromechaical damage zoe model for ductile fracture described by Cocks et al. (27), which are both derived from the creep cavitatio model of Cocks ad Ashby (98). These models are based o the growth of a array of pores idealized as cyliders. The relatio betwee the macroscopic tractio ad separatio ad the microscopic stress ad strai is the obtaied usig classical boudig theorems. (a) T/T [-] (b) T/T [-] δ / δ δ / δ δ /δ f [-] δ /δ f [-] Fig. 3 The Kachaov damage type model tractio-separatio rate law: a the liear damage model ad b the expoetial damage model. The rate sesitivity expoet is take to be m = 9 The radius ad height of the pores at a give istat are deoted as r ad h, respectively, ad the mea spacig is 2l, see Fig. 4. Thus, the pores are characterized by their area fractio i the plae of the cavitated zoe, i.e. by f = (r/l) 2. Further, the represetative volume elemet is assumed to be fully costraied i the radial directio ad the deformatio is oly cotrolled by the ormal separatio, i.e. l = cost. ad ḣ = δ.this simplificatio of the void profile captures the major features of the evolvig geometry (such as area fractio of pores ad pore aspect ratio) while allowig simple aalytical expressios for the evolutio of damage to be derived. More geeral forms of model are discussed by Cocks et al. (27) this is the simplest form of model of this class ad it is directly equivalet i form to classical rate depedet cohesive zoe models, icludig those described above. The resultig expressio for the

8 E. Elmukashfi, A. C. F. Cocks 2l T,δ h 2a 2l (a) Fig. 4 The micromechaical represetatio of the creep damage by pore growth: a pores of radius r ad spacig 2l o a grai boudary subjected to microscopic stress state σ ij ad the deformatio is cotrolled by steady-state creep ad b a idealizatio (b) of a pore as a cylider with height h ad diameter equal to the pore to pore spacig 2l ad the macroscopic ormal tractio T ad separatio δ opeig rate is 5. ( ) m T δ = δ, (3) ḡ ḡ T where ḡ = g ( f )/g ad [ g = ( f ) 2 + ( 3 l f ) 2 ] 2. (4) T/T [-] δ / δ ad g is a parameter that ca be used to provide a similar rupture time to the Kachaov models uder the same stress level ad the same value of the material parameters δ, T ad δ f (see Appedix B ). The matrix material is icompressible, therefore the total rate of chage i volume is equal to the rate of chage i pore volume. Hece, the pore area fractio evolves at a rate δ [mm] Fig. 5 The micromechaical based model tractio-separatio rate law. The iitial pore area fractio ad height are assumed to be f =. ad h =. mm The tractio-separatio rate relatio for differet separatio rates is illustrated i Fig. 5. f = δ h ( f ). (5) The iitial pore area fractio ad height are assumed to be f ad h, respectively. Further, the pores are assumed to coalesce ad reach a complete failure whe f = f c. The direct itegratio of Eq. (5) givesthe separatio at failure: f c h [ ] ( f ) d f = f h dh δf = h. f c f h (6) 3 Creep crack growth i a double catilever beam I this sectio, pure mode-i creep crack growth i the DCB specime show i Fig. 6 is aalysed. The legth ad height, of the specime are deoted by L ad 2H, respectively, ad the crack legth is deoted by a. Each arm of the specime is subjected to a costat momet M per uit depth. We assume that the overall legth L ad that the height of each arm H a. Uder these coditios C, remais costat as the crack grows, thus a steady state is evetually achieved i which the crack growth rate is costat. We focus o this steady state respose. The objective is to obtai a

9 A theoretical ad computatioal framework M a M x 2 x x 3 L Fig. 6 The schematic of the double catilever beam specime 2H itegral of Eq. (5) or by determiig the rate of chage of the rate aalogue of the total potetial eergy,, with crack legth, where = V ψ dv S T i u i ds. (7) M M x 2 x 3 Γ out Fig. 7 The defiitio of the ier path Γ i ad the outer path Γ out that are used to evaluate the C -itegral i the double catilever beam specime mathematical descriptio for the relatioship betwee the far field loadig ad the local damage developmet withi the damage zoe ad the crack growth rate uder both plae stress ad plae strai coditios. I order to determie the relatioship betwee the loadig ad fracture parameters, the path idepedece of the C -itegral is used as discussed i Sect. 2. The C -itegral is evaluated alog the outer ad ier paths idicated by the dashed lies ad differet colours i Fig. 7. Equatig the values of C determied from these two paths provides a relatioship betwee the crack-tip opeig rate ad the applied load. There is a sigle characteristic geometric legth scale for this problem, which we take as λ = H/2, ad i the steady state the separatio rate withi the damage zoe ca be expressed as a fuctio of x /λ, itegratig this fuctio as a elemet is covected towards the crack tip as the crack grows at costat velocity, allows the crack growth rate to be determied. We do ot kow the form of this fuctio a priori, but to determie the crack growth rate we oly eed to determie a sigle quatity the resultig itegral, which ca be determied from a sigle piece of iformatio from a fiite elemet aalysis of the problem. The details of this process are give below. Γ i 4 3 x We adopt the secod of these approaches here. For a elemet of beam uder bedig the curvature rate is give by κ = 2 ε H ( ) 4M σ H 2 = 2 ε ( ηm H η M ), (8) where M = 2 σ H 2, ad η = for plae stress ad 3/2 for plae strai. For the DCB specime of Fig. 7 Cout = a M [ = 2 + [ = ε H M 2 ε H M = 2 2 ε + H M ( ) + ] κ H M θ 2 ε a ( ) + ] κ H M κ 2 ε ( ) ηm +. (9) M The factor of 2 arises because there are 2 beams ad θ is the rotatio rate at the ed of oe of the beams. If we defie a referece stress, such that σ = 2 + 4ηM 2 H 2, (2) ad C out = f () ε σ λ, (2) 3. The C -itegral i the outer path Γ out C i the outer path ca be determied i a umber of differet ways for example by direct evaluatio of the 4 where f () = ad λ = H/2 isthe (2 + )( + ) characteristic legth scale for the DCB specime.

10 E. Elmukashfi, A. C. F. Cocks 3.2 The C -itegral i the ier path Γ i Usig the defiitio i Eq. (7), the C -itegral i the ier path Γ i ca be writte as δm Ci = T d δ = δ m ( ) δ m d k T d δ, (22) δ where the fuctio d k depeds o the form of iterface model adopted, with k idicatig the model, i.e. s simple, kl Kachaov liear, ke Kachaov expoetial ad m micromechaical models. For each of these models d k is give by d s =, d ke ad d ke = ω ad d m = g m+ m. (23) Apart for the simple model the itegral requires a kowledge of the stress history experieced by each material poit i the damage zoe, which is ot kow a priori. For the simple model the itegral of Eq. (22) ca be readily determied: δm Ci = ( ) δ m T d δ = 2m ( δ δ m + δ m ) m+ m T. δ (24) For all the remaiig models d k, ad the resultig itegral is therefore less tha or equal to that give by Eq. (24). We assume that the itegral for each model ca be approximated by C i = α k δ m ( ) δ m T d δ δ = α k 2m m + δ T ( δ m ) m+ m, (25) δ where the subscript k agai idetifies the model ad α k falls i the rage α k. 3.3 The crack tip opeig displacemet rate Equatig the values of C give by the ier ad outer cotours, i.e. Eqs. (2) ad (25), allows the crack tip opeig displacemet rate δ m to be expressed as a fuctio of the applied loadig. Equatig T to σ gives δ m ( ) m q (, m) = φ m+ δ, (26) α k where φ = ε λ = ε H is the ratio of geometric to δ 2 δ material legth scales for the problem ad g (, m) = f () m + 2m = 2 (m + ), which for = m (2 + )( + ) m reduces to q (, ) = q () = The aalysis of a steadily propagatig crack As oted earlier, uder a costat applied momet the crack velocity will, after a iitial trasiet, achieve a steady state, i which it reaches a costat value ȧ.we cosider a coordiate system that moves with the crack tip. A material elemet such as P i Fig. 8a the moves alog the x -directio at a rate (Cocks ad Ashby 98) dx dt = ȧ. (27) The are two characteristic legth scales i this problem, the geometric legth scale λ ad the material legth scale δ / ε. We ca therefore write the separatiorateitheform ( δ = δ m Λ x ) k = δ m λ Λ k ( x ), (28) where Λ k is a dimesioless fuctio that depeds o the iterface model whose detailed form depeds o φ, the ratio of geometric ad material legth scales, as show i Fig. 8b. For each of the models described i Sect. 2, failure of a elemet occurs whe the separatio across the damage zoe reaches a critical value, δ f. Itegratig the displacemet rate as a elemet is covected towards the crack tip gives t f δ f = = δ m λ ȧ δ dt = δ dx ȧ Λ k ( x ) d x = δ m λ ȧ C k (φ,, m), (29)

11 A theoretical ad computatioal framework Fig. 8 The schematics of a steadily propagatig crack i viscous solid: a a material poit P at distace x from the movig crack tip ad b the defiitio of the Λ k ad C k dimesioless fuctios for poit P x 2 x 3 x ȧ P δ / δ m C k Λ k x /x r (a) (b) wherewehaveusedeq.(27) to substitute for dt ad Eq. (28) to substitute for δ. The dimesioless fuctio C k (φ,, m) is oly a fuctio of φ,, m ad the detailed form of model for the damage zoe. Substitutig for δ m usig Eq. (24) gives the steady state crack velocity. a = ȧ m = q (, m) ε λ = q (, m) m m+ φ φ m+ m+ δ f m+ δ f C k (φ,, m) α m m+ k Ĉ k (φ,, m). (3) where δ f = δf /λ. We ca express this relatioship i a umber of differet forms. A alterative form that ca be used to provide some isight ito the material respose is: ȧ = A m+ λ δ f [ m + 2m C ] m m+ Ĉk (φ,, m) (3) where A = δ /σ m is a material costat for the damage zoe [see Eq. (8)]. The form of this equatio might suggest that the crack growth rate is a fuctio of C, for a give value of δ f. This is oly true if Ĉ k is oly a fuctio of ad m or φ is costat for the rage of coditios of iterest. Note that for the DCB specime of Fig. 7 ad m = φ = B A λ = B A H 2 (32) where B = ε /σ m is a material property. The for a series of experimets i which the geometry is kept costat we would expect ȧ to be proportioal to C m+ m, but the costat of proportioality could be differet for a differet choice of beam height H. For more geeral cracked geometries σ, ε ad λ chage as a crack grows ad therefore φ also chages. This eeds to be take ito accout i ay model ad descriptio of the crack growth process. We cosider this feature of the respose further below. I order to determie the crack growth rate we eed to evaluate the quatity Ĉ k. We ca determie this usig the fiite elemet method. We eed ot determie the distributios d k ad Λ k ahead of the crack tip. We ca determie directly by equatig the umerically determied crack velocity with the predictio of Eq. (3). 3.5 Numerical implemetatio of the goverig equatios The iitial-boudary value problem described i Sect. 3 is umerically solved usig the FE (Fiite Elemet) code ABAQUS (Abaqus 26). A oliear quasistatic aalysis is used for the iitial loadig, ad a oliear visco aalysis is used for the creep crack propagatio aalysis. I the visco aalysis implicit time itegratio is used to solve the FE equatios ad mixed implicit/explicit itegratio is used for the itegratio of the creep ad damage zoe equatios. The FE aalysis requires the solutio for a elastic/creep costitutive law i the bulk a elastic-rate depedet opeig model for the damage zoe. Elastic costitutive compoets have bee added to the costitutive relatioships of Eqs. (), (8), () ad (3), with the values of the elastic compoets chose to have limited ifluece o the computed results. The geometry of the double catilever beam (DCB) specime show i Fig. 6 is discretised, ad a typical fiite elemet mesh is show i Fig. 9. Oly oe half of the specime is aalysed due to the symmetry of the problem. The dimesios are take as L = mm, H = mm, ad B = mm. The iitial crack is assumed to be a = 4 mm, ad the crack propagatio

12 E. Elmukashfi, A. C. F. Cocks Bulk elemets x 2 a x 3 Δa x Iterface elemets (a) (b) Fig. 9 The fiite elemet mesh of the double catilever beam specime: a the mesh of the whole geometry; ad b mesh details alog the middle of the specime where the cohesive elemets are iserted alog the crack propagatio path is studied over a legth of Δa = 5 mm. The 4-ode reduced itegratio biliear plae stress ad strai elemets (CPS4R ad CPE4R) are used i the discretisatio for plae stress ad strai coditios, respectively. A 4-ode two-dimesioal liear damage zoe elemet was implemeted i ABAQUS usig the user-defied subroutie UEL. The details of the Fiite Elemet implemetatio are provided i Appedix A. The Fiite Elemet model is divided ito two regios, i which the bulk ad damage zoe elemets are defied. The damage zoe elemets are iserted alog the crack propagatio path, i.e. alog a x L ad x 2 =, ad the bulk elemets are defied elsewhere. The top faces of the damage zoe elemets are attached to the bulk elemets, see Fig. 9b. The damage zoe elemets are modelled with zero iitial thickess such that the top ad bottom face odes coicide. The mesh has,634 elemets, of which,279 are bulk elemets ad 355 are damage zoe elemets. A uiform refied elemet regio is created adjacet to the crack ad its propagatio for cotrollig the iterface elemet legth l ite. A covergece study o the mesh refiemet was carried out for differet values of φ ad iterface parameters δ f =.4, β =., f =. ad f c =.9. We foud that a iterface elemet legth of l ite =. mmis ecessary to obtai coverged solutios for the rage φ [ 5 4]. The iterface stiffess K = K t = 6 MPa mm is selected such that the elastic deformatio is egligible (see Appedix A ). The umerical aalysis was performed for differet combiatios of the dimesioless parameters defied above to cofirm that the fuctioal form of Eq. (3)is valid. (The model parameters are chose i such a way that the dimesioless parameters are cotrolled.) The relative ormal separatio displacemet, Δu 2, betwee each pair of iitially coicidet odes i the iterface (x 2 = ) is computed ad recorded durig the aalysis. The crack tip positio, x tip, is defied by Δu 2 = δ f, ad the crack tip velocity is determied usig forward differecig as tip = dx tip dt = x p+ tip x p tip (33) tp Δt p v p where idices p ad p + deote variable values at istats t p ad t p+, respectively, ad Δt p = t p+ t p is the time icremet. Further, the steady crack velocity, ȧ, is computed by takig the average velocity over the steady propagatio period. 4 Results ad discussio 4. The crack growth Several aalyses have bee performed for differet combiatios of the dimesioless parameters ad damage zoe properties. The crack tip positio, trasiet ad steady state crack propagatio velocity have bee obtaied for all the combiatios. For the simple iterface model the parameters = m = 9, φ =. ad δ f =.4 are used to illustrate the differet results. We first describe the results uder plae stress coditios. Figure i iv show the distributio of the effective creep strai ε e cr at four differet istats ad crack velocities. At t = the creep deformatio is zero everywhere ad elastic deformatio prevails. As time passes

13 A theoretical ad computatioal framework Fig. The Distributio of the effective creep strai ε e cr for the parameters = m = 9, φ =. ad δ f =.4: a the deformed DCBspecimeattime t = 8h;adbthe propagatig crack tip at differet istats of time x 2 x 3 (a) x x2 [mm] 2.5 x2 [mm] x [mm] x [mm] (i) t =. hr (ii) t =4. hr x2 [mm] 2.5 x2 [mm] x [mm] x [mm] (iii) t =6. hr (iv)t =8. hr (b) ε cr e [-] (t > ) creep deformatio evolves i the bulk, iitially primarily i the viciity of the crack tip, as well as damage alog the iterface, leadig evetually to crack growth whe the critical opeig is achieved at the crack tip. Figure a, b show the crack tip positio ad velocity as fuctios of time, respectively. The plots show that the crack starts to propagate slowly ad accelerates to a high velocity ad after a short time ( h) a lower steady state velocity is achieved. I this case a steady velocity of ȧ =.3 mm/h is obtaied. The result shows that the trasiet velocity is higher tha the steady state velocity suggestig that the stress at the crack tip is iitially high due to the elastic deformatio ad as the crack advaces the creep deformatio domiates where the stress relaxes leadig to a slower propagatio rate. Additioally the damage ahead of the crack tip is fully developed durig both trasiet ad steady propagatio. The other sceario is whe the damage is ot fully developed durig the trasiet stage which may lead to a slower propagatio before reachig a steady state where a fully developed damage zoe is achieved. 4.2 The C s -fuctio It proves istructive to cocetrate iitially o the respose for the simple damage zoe model of Sect The Fiite Elemet aalysis is used to determie the steady crack velocity ȧ ad the for a give set of iput parameters the C s -fuctio ca be evaluated from Eq. (3):

14 E. Elmukashfi, A. C. F. Cocks Fig. Crack propagatio results for = m = 9, φ =. ad δ f =.4: a crack tip positio x tip versus time t;adb crack tip velocity v tip versus time t (a) 6 5 (b).4.3 ȧ =.3 mm/hr xtip [mm] vtip [mm/hr] t [hr] t [hr] Ĉ k = a δ f φ m+ g (, m) m+ m. (34) The appropriateess of the dimesioless aalysis has bee examied usig the same set of dimesioless parameters with differet model parameters, e.g. the same value of φ with differet combiatios of ε, H ad δ. 4.3 The physical limits ad the validity of the framework Before evaluatig the computatioal results i detail it is istructive to examie the respose i the limits of small ad large φ. The first extreme is whe the iterface is very stiff i compariso with the bulk material (the bulk material creeps faster tha the iterface, i.e. ε δ /λ ad φ ). The other extreme occurs whe the iterface creeps faster tha the bulk material, i.e. the iterface is very compliat ( ε δ /λ ad φ ). I this aalysis we cosider the simple damage zoe model of Sect Whe a iterface is very stiff i compariso with the bulk material the deformatio alog the iterface is egligible ad it does ot ifluece the stress state i the body. The tractios see by the damage zoe are determied by the stress distributio i the bulk material, ad ca be expressed i terms of the C -itegral (provided the damage zoe is small compared to the regio i which the HRR field domiates; Hutchiso 968; Rice ad Rosegre 968). The HRR stress field is defied as σ ij = σ [ C ε σ I r ] + σ ij (,θ), (35) where I is a itegratio costat that depeds o ad σ ij is a dimesioless fuctio of ad θ. The values of these parameters are give for the cases of plae stress ad plae strai coditios by Hutchiso (968). It follows that the ormal tractio alog the iterface is give by T = σ [ C ε σ I r ] + σ θ (, ), (36) I this aalysis, we limit ourself to the case of T = σ ad m =. Therefore the opeig separatio rate for the simple model is evaluated from Eq. (8)as δ = δ [ C ε σ I r ] + σ θ (, ). (37) The critical opeig separatio is determied by itegratig the separatio rate, i a similar way to i Eq. (8), as

15 A theoretical ad computatioal framework δ f = δ dx r c ȧ = = ( + ) δ ȧ [ C δ dṙ a ε σ I ] + r + c σ θ (, ). (38) where r = x at θ = ad r c is the size of the fracture zoe which is very small i the case of stiff iterface (r c ). Rearragemet of Eq. (38) gives the dimesioless velocity as a = ( + ) r + c φ δ f [ f () I ] + σ θ (, ), (39) where r c = r c /λ. By comparig this equatio with Eq. (3), the C s -fuctio for the case stiff iterface becomes C s = ( + ) r + c [ 2 + φ I ] + σ θ (, ). (4) For the case of = m = 9 ad θ =, the itegratio costats are I ad σ θ (9, ) σ θ (3, ).2 for the case of plae stress ad I ad σ θ (9, ) σ θ (3, ) 2.6 for the case of plae strai (Hutchiso 968). Thus, the C s -fuctios for the cases of plae stress ad plae strai coditios are C s = 45.6 r c. φ.9 ad C s = 7. 4 r c. φ.9, respectively. The other limit is whe the iterface is too compliat i compariso with the bulk material which ca be regarded as rigid. Hece, i the case of a ifiite DCB specime, the equilibrium betwee the applied momet ad the tractio alog the ifiite damage zoe suggests that the tractio will ted to zero ad there will be o crack propagatio. o the other had, whe the specime is fiite a o zero tractio alog the fiite damage zoe. The deformatio alog the damage zoe ca directly be related to the agular deflectio at the ed of the beam. Thus, the separatio at the crack tip is obtaied as δ f = 2 (L a) θ = 2 W θ, (4) where W = L a is the legth of remaiig ligamet durig steady state propagatio. The opeig displacemet at the tip of the propagatig crack is costat ad equal to the critical value δ f =, therefore δ =, ad ȧ = W θ θ. (42) Similarly, the separatio at the crack tip ca be writte i this form δ m = 2 W θ. (43) The separatio rate i the damage zoe is give by [ δ = x ] δ m W. (44) Now the balace by the iteral ad exteral work rates gives 2 M θ = W T δ dx = W ( ) + δ δ T dx. (45) δ Itroducig Eq. (43) ad the opeig separatio rate for the simple model i Eq. (8) we obtai the separatio at the crack tip as δ m = δ [ 2 + ] M σ W 2. (46) The crack velocity is determied from Eqs. (4), (42), (43) ad (46) ad usig the defiitio of σ i Eq. (2) as δ ȧ = W 2 δ f [ ] 2. (47) η Scalig of Eq. (47) gives the dimesioless velocity as a = φ W 2 δ f [ ] 2 (48) η By comparig this equatio with Eq. (3), the C s fuctio for the case compliat iterface becomes C s = q + W 2 [ ] 2 φ +. (49) η

16 E. Elmukashfi, A. C. F. Cocks W is computed from the fiite elemet aalysis as the remaiig ligamet legth whe a crack reaches a steady state propagatio. Hece, for the case of = m = 9 ad usig the computatioally obtaied average value W.8, the C s -fuctios for plae stress ad plae strai coditios are C s =.5 5 φ.9 ad C s = φ.9, respectively. Aother limitatio comes from the time scale of the crack propagatio as metioed i Sect. 2.. C represets the ear crack tip field whe a crack propagates slowly. As the crack velocity icreases elastic deformatio becomes icreasigly importat i the viciity of the crack tip ad a zoe i which both elastic ad creep deformatio determies the respose becomes icreasigly sigificat. If this zoe becomes comparable i size to the damage zoe, the C ca o loger be used as a parameter for characterizatio of the ear tip filed ad damage growth process. Cocks ad Julia (99) studied this limit ad proposed coditios for the domiace of C. They demostrate that C cotrols crack growth provided the followig coditio is satisfied a = f () + Z () σ /E r + 2 c (5) where E is Youg s modulus ad Z () = ( ) I +. Usig this coditio we derive a coditio for C s fuctio by comparig Eq. (5) with Eq. (3) as C s 2 + f () + Z () r + 2 c δ f σ /E φ +. (5) This expressio implies that for particular values of δ f ad σ /E there is a maximum velocity for which C is a valid measure. Thus, for the case of = m = 9, the valid C s -fuctio for plae stress ad plae strai coditios are C s r.2 c δ f r.2 c δ f σ /E φ. ad C s σ /E φ., respectively. Elasticity is oly relevat i the computatioal models ad this relatioship ca be used to assess whether the coditios employed i the FE models are cosistet with the assumptios of the aalytical model preseted i Sect. 2. We eed to be careful, however, whe usig this expressio. It is derived from aalyses i which damage developmet is assumed to ot ifluece the ear tip fields. As illustrated above, the size of the damage zoe icreases with decreasig φ ad for small φ the ear tip fields give by the classical cotiuum aalysis are o loger valid. The relatioship of Eq. (49) is therefore oly valid i the limit of large φ where the developmet of damage has limited effect o the crack tip fields. It is also importat to emphasise here that although, the HRR field is o loger valid for small φ, C is still a valid parameter for characterizig crack growth. I order to evaluate the proposed framework, C s has bee determied from (34)forφ i the rage [, 5 ] ad compared with the limitig results preseted above. The rate sesitivity parameters are take to be = m = 9. Figure 2a, b show the relatioship betwee C s ad φ for plae stress ad strai coditios, respectively. Over the rage of the data, the results ca be fit usig two separate power-law relatios. Uder plae stress coditios this relatio is C s =.45 φ.6 over the rage of values φ [, 8 2 ] ad C s =.9 φ.67 for the rage φ [8 2, 3 ], see the dashed lies i Fig. 2a. The trasitio betwee the power law relatios occurs over the rage 4 φ. For a give value of φ, C s lies betwee the two limitig values. The power-law fit for high values of φ is slightly shallower tha that for the stiff limit described above, idicatig that respose teds to this limit for values of φ i excess of 6. I this limit the rate of deformatio i the damage zoe becomes very small compared to that i the surroudig matrix, which determies the stress distributio ahead of the crack tip ad therefore the rate of growth of damage. There is o evidece of the data mergig to the limitig result for low values of φ, but the values of φ required to reach this limit are much lower tha values we would expect from physical argumets (i this limit the material legth scale is sigificatly greater that the geometric legth scale i practice we would expect ay characteristic material legth scale to be less tha the geometric legth scale for the cracked body, i.e. we would expect φ to be greater tha ). The power-law rage for φ greater tha 8 2 is therefore more represetative of the physical behaviour of egieerig compoets, so we cocetrate o the relatio for this regime here. Substitutig this relatioship ito Eq. (3) gives the dimesioless velocity

17 A theoretical ad computatioal framework (a) Cs [-] (b) Cs [-] 2 δ f FE Results 2 Fit φ φ C limit φ [-] 2 δ f FE Results 2 Fit φ φ C limit φ [-] Fig. 2 The relatio betwee C s -fuctio ad φ parameter ad the physical limits i the case of = m = 9: a plae stress coditios; b plae strai coditios. The red ad blue lies represet the compliat ad stiff limits, respectively, the gree dash-dot lies represet the C validity limit for differet dimesioless separatio at failure δ f,adthedashed lies show the power law fit where we have substituted for C usig Eq. (2)toprovide a relatioship i terms of the referece stress σ. Figure 2a also shows a series of lies below which elastic effects ca be igored for σ /E = 8 6 (as used i the computatios) ad differet values of critical crack tip opeig displacemet, i.e. below which iequality (5) is satisfied. As oted earlier this relatioship is oly valid for large values of φ (say greater tha 8 2 ). I this regime the computatioal results lie below this series of lies, idicatig that the theoretical structure preseted i Sect. 4 provides a valid framework for modellig the crack growth behavior. We ca repeat the aalysis for plae strai coditios, see Fig. 2b. I the case of plae strai we agai fid that the results ca be fit usig two power-law relatioships: C s =.55 φ.5 over the rage of values φ [, 2 ] ad C s =.9 φ.29 for the rage φ [ 2, 4 ], see the dashed lies i Fig. 2b. Further, the trasitio betwee the power law relatios takes place i the rage 4 φ. The latter relatio gives a dimesioless crack growth rate of a = φ.39 δ f. (54) The compariso betwee the FE results ad the physical limits is also show i Fig. 2b, which are agai bouded by the physical limits of Eqs. (4) ad (49), with the results asymptotig to Eq. (4) at large values of φ. The large limit φ gives a faster crack growth rate i plae strai tha plae stress (i.e. C s is larger for a give value of φ ) due to the higher stress levels ahead of a plae strai crack. The differece i slope betwee this limit ad the computatioal results is greater tha that observed for plae stress, but the value of φ where the two curves meet is about two orders of magitude higher. As for plae stress, the results lie i a regime where elastic effects ca be igored. a =.8 2 φ.77 δ f. (52) or ito Eq. (3), the velocity i terms of C : ȧ = 9. 2 A.77 λ.33 δ f B.67 [.56 C ].9 =.9 2 A.32 B.23 λ.23 σ 9 δ f. (53) 4.4 The effect of damage model I order to ivestigate the effect of the detailed form of the damage zoe model o the crack growth respose, Ĉ k has bee determied for each of the differet damage zoe models described i Sect. 2. The parameters employed for these models are m = 9, δ f =.2 mm, β =., h =.2 mm, f =. ad f c =.5. It should be oted that δ is kept costat for all models.