SELFSIMILAR SOLUTIONS FOR STRESS DRIVEN MATERIAL DISSOLUTION

SELFSIMILAR SOLUTIONS FOR STRESS DRIVEN MATERIAL DISSOLUTION P. Ståhle nd A. P. Jivkov Solid Mechnics, Mlmö University SE205 06 Mlmö, Sweden pers@ts.mh.se Abstrct During corrosive dissolution of metl ions from body surfce, n oxide compound is produced. This compound forms protective film tht reduces the dissolution rte. When frction of millimetre depth is dissolved the dissolution rte become insignificnt. However, repeted loding will dmge the film with continued dissolution s result. In connection with this threshold strin is ssumed to exist. This pper proposes model where electrochemicl processes nd the mechnicl lod work together in forming corrosion pit. The rtio between the threshold strin nd the remotely pplied strin is shown to control the shpe of the pit. For smll pplied strins crcks re formed. A crck evolving from surfce irregulrity is studied. The growth rte of the crck is determined by the dissolution rte t the crck tip. No crck growth criterion is needed. The growing crck is itself creting conditions for strin concentrtion, which leds to high crck growth rte. The model simultes how dissolution forms pit tht grows to become crck in single continuous process. For smll lods the crck growth rte is independent of pplied lod. Introduction As much s 25% of ll ccidents in process industry reported to the Swedish Plnt Inspectorte re cused by stress corrosion crck growth, Bergmn et l. [1]. The sitution is believed to be similr in ll industrilised countries. Becuse crck growth occurs t low lod the crck is usully not known before the ccident, which contribute to the severity of the problem. Figure 1 shows the tip of trnsgrnulr corrosion crck in nucler power plnt feed-wter pipe. In stress corrosion, loss of toms to the environment leds to crck growth. This is dissolution process tht strts if bre metl surfce is exposed to ggressive environments (cf. Roberge [2]). Fortuntely, pssivting process compete with the dissolution. Thus, n impermeble film of minly metl oxides nd hydroxides formed by dissolved metl reduce the dissolution rte. Even though the thickness of this film is typiclly not more thn 1 to 4 nm it reduces the rte of dissolution severl orders of mgnitude. An intct protective film increses the lives of the structurl members tremendously. When frction of millimetre depth is dissolved the dissolution rte become insignificnt. However, repeted chnges of the electrochemicl conditions, or cyclic mechnicl lod my dmge the film, which leds to (comprtively) rpid mteril loss. The filure of the film due to mechnicl loding is the bsis for widely used hypothesis for stress corrosion crcking, where the crck dvnce is believed to be the result of metl dissolution, loclised t the surfce experiencing the highest strins, s in Perkins [3] nd Turnbull [4]. Recent experimentl reports by Kussmul et l. [5] nd by Heldt nd Seifert [6] show tht ctive loding in terms of either

FIGURE 1. Trnsgrnulr corrosion crck with crck length 7 mm nd notch rdius ρ 5 µm. monotoniclly incresing or ftigue lod is n essentil prerequisite for continuous corrosion crck growth. The rte of the dissolution s function of the strin rte is not known. However, the dissolution rte is believed to be incresing with incresing strin rte. In the present report simple liner reltion between dissolution rte nd strin rte is ssumed. In ddition to this, threshold strin is introduced, below which no dmge occurs. The proposed model ws erlier used to study nucletion of crcks from surfce pits in elstic mterils, nd in elstic-plstic mterils (cf. Jivkov nd Ståhle [7] nd Jivkov [8]). The results of these works show tht the model leds to strin nd dissolution loclistion, resulting in the nucletion of crck. The present study formultes set of governing equtions tht suggest self-similr growth of the pit (or crck). A plusible solution for low lod consisting of crck with prcticlly prllel surfces nd hlf-circulr tip is exmined. This solution is compred with corresponding finite element results. Model Consider lrge body with stright trction free edge. At lrge distnces unixil stress is pplied prllel with the edge. The mteril is ssumed to be liner elstic with the modulus of elsticity E nd Poisson's rtio ν. Plne conditions re invoked. Here plne stress is selected but the result for plne strin is redily found by replcing E with E/(1-ν 2 ) nd ν with ν/(1-ν). Stresses re defined vi trctions s T α = n β σ βα, where T α is the trction vector cting on surfce with the outwrd norml n β. The stresses re ssumed to be in equilibrium, stisfying the eqution σ αβ,β = 0. Strins re ssumed to be smll nd defined by (1) (1b)

ε αβ = 1 2 (u α,β + u β,α ). (1c) Stresses nd strins re linerly relted through Hooke's lw s follows σ αβ = E 1+ ν ε αβ + νe 1 ν 2 δ αβε γγ. Indices in (1) to (1d) ssume the vlues 1 nd 2. The boundry conditions re the lod t lrge distnces T 1 = 0 nd T 2 = σ, t x 1 2 + x 2 2, (2) nd long the trction free edge T 1 = T 2 = 0, t x 1 = 0. (2b) In ddition to this, mteril is removed t rte υ proportionl to the stretch ε of the surfce (1d) υ = R d (ε ε f ) = R d σ σ f E, for ε - ε f > 0 nd υ = 0 for ε - ε f 0 (3) where R d is rte prmeter. A threshold strin ε f is introduced in (3). The stresses σ f = Eε f nd σ = Eε. The evolving surfce is ssumed to remin trction free. By introducing non-dimensionl trctions, stresses nd dissolution rte s follows T ˆ i =T i /σ f, σ ˆ =σ /σ f, σ ˆ =σ /σ f, σ ˆ ij =σ ij /σ f nd υ ˆ = υ /(R d ε f ). (4) Displcements nd strin become obsolete nd (1c) nd (1d) is reduced to ˆ σ αα,ββ = 0 The requirement is tht the stresses should fulfil the eqution of equilibrium, i.e., ˆ T α = n β ˆ σ αβ, where ˆ σ αβ,β = 0. (7) The boundry conditions (2) my be written on non-dimensionl form s follows T ˆ 1 = 0 nd T ˆ 2 = σ ˆ, t x 2 1 + x 2 2, (8) T ˆ 1 = T ˆ 2 = 0, t x 1 = 0. (8b) Mteril is removed t the rte expressed s υ ˆ =( σ ˆ 1) Obvious now is tht the results for given initil geometry depend only on lod, i.e., onσ ˆ. With given stress boundry conditions lso the mteril prmeters E nd ν become obsolete. Thus, the problem is defined by the initil geometry nd the equtions (6) nd (7) with the boundry conditions (8) nd evolution of the body in ccordnce with (9). (6) (9) Anlysis In mny cses no relevnt length prmeters exist, e.g., when pit initite nd develops from smll indenttion or scrtch in stright edge. During initition the geometry of the initil

indenttion is relevnt. However, when the pit hs grown severl times the liner extent of the indenttion no other length thn the extent of the pit should influence the continued growth. If the sitution immeditely fter initition is excluded the solution must be selfsimilr, i.e. ll lengths of the solution scle with ech other or, e.g., the pit depth. One impliction is tht the dissolution rte t ll points of the pit surfce hve to fulfil the selfsimilrity condition υ = r υ tip sinβ, (10) where r is the rdius from the pit mouth nd β is the ngle between the norml to the surfce nd the rdius vector. Erlier clcultions with the present model show tht slender crck shped pit develops for σ ˆ =1 [8]. It is expected tht strins re very smll on the flnks of the pit. From the pit mouth t the trction free edge of the specimen A (see Fig. 2) to point close to the pit tip the strins my be smller thn the strin threshold ε f. It is redily seen from (10) tht β hve to vnish on the prts where υ vnishes becuse ε < ε f. Thus, one conclude tht these prts re stright nd rdilly directed. The tip of the pit ws observed to be very close to hlf circle (cf. [8]). Selfsimilrty hs the mening tht the rtio /ρ hs to ssume constnt vlue, where is the pit depth nd ρ is the tip rdius. This llows us to exmine model cse for pit boundries consisting of stright edges forming n ngle α with the centre plne of the pit (see Fig. 3). For smll ngles α the geometry is regrded to be slot with lmost prllel edges nd tip region. This my be used to determine the strin t the tip of the pit. From Td et l. [9] we obtin ε tip = 2.24 σ E ρ = 2.24 ˆ σ ε f ρ. (11) Figure 4 shows strin distribution for pit tips with the shpe of circulr sector. At the connection between the circulr segment nd the stright prt of the crck surfce, i.e. t the curve distnce Γ = (2/π)ρ from the pit tip, strin equls 0.24ε tip. The dissolution rte υ should vnish on the stright crck surfces nd therefore ε f = 0.24ε tip. (12) This provides us with enough informtion to compute the ngle α for smll lods. Eliminting ε f nd ε tip from (11) nd (12) gives ) b) A ε<ε f ε=ε f r ε>ε f P 2ρ P r υ β υ=υ tip notch surfce FIGURE 2. ) A point on the surfce of developing pit, b) the projection of rdil dissolution rte υ=υ tip projected on the norml to the surfce

) b) ε=ε f ε<ε f ε=ε f 2ρ ε>ε f FIGURE 3. ) Typicl pit for smll remote lods, b) pit tip. The ngle between the pit surfces is α = ρ/ α = ρ = 0.29 2 σ ˆ (13) for smll pit ngles α, i.e. when lod is smll. The pit growth rte is for this smll lod pproximtion given by (3) nd (12) s υ tip = 3.17 R d ε f, (14) nd is s observed independent of remote lod. Numericl nlysis A solution for ˆ σ = 1 is been exmined in the following. In the numericl nlyses below ν = 0.3. A finite element method is employed for computtion of the boundry vlue problem depicted in (6) to (9). The commercil code ABAQUS [10] is used. A mesh generting progrm written by Jivkov [11] is used. The, from necessity, finite mesh ws chosen qudrtic nd so tht the growing pit mesh never becme lrger thn 0.02 of the mesh side. Pit growth ws initited from smll hlf circulr indent in the trction free edge. The liner extent of the indent ws less thn 0.01 of the finl pit depth. Figure 5 shows resulting crck like pit. The selfsimilr growth is indicted by the obvious increse of crck opening with incresing crck length. It should be noted tht the crck surfces does not chnge when the tip hs pssed. Therefore the finl seprtion of the crck surfces displys the width of the tip for ll shorter crck lengths. The ner tip region is shown in Fig. 5b. The rtio /ρ is round 100, which is very high s compred with the theoreticl result. Figure 8 shows the incresing crck tip width s the crck is growing. The sctter for longer crcks indictes somewht unstble development of the crck. Discussion The selfsimilrity of the growing pit emnte from the evolution lw (9). It is worth noting tht the specifiction (9) does not involve ny length scle. This is in contrst with crck growth bsed on common crck growth criterions. These normlly dd length scle tht serves s reference for crck length. The selfsimilrity leds to pit growth rte tht is independent of the lod. In perspective of the crck like pits tht is observed for σ = Eε f nd

ε tip ρ Γ ε f 0 (π/2)ρ FIGURE 4. Strin versus distnce Γ long the notch tip surfce. The strin is ε f exctly where the curved section meets the stright sides of the notch. Γ ssumed to pper for σ Eε f, one note tht the width of the crck-tip, however smll, still remins proportionl to the crck length. This gurntees tht the locl lod in the vicinity of the crck-tip remin constnt. This provide possibility to estimte the threshold stress for rel crcks such s the one shown in Fig. 1. The informtion provided with the figure is tht the crck length is round 7 mm nd the crck tip rdius ρ is round 5 µm (Lgerström [12]). Following (13) this tell us tht the remote strin is only round 0.1 of the threshold strin. A perhps better estimte is provided by the numericl clcultion. The suggestion is tht (13) is replced with ρ/ = 0.01 ˆ σ 2. Thus, the remote strin should be round 0.5 of the threshold strin. Conclusions In the exmined model for corrosive dissolution controlled by liner reltion between surfce strin nd dissolution rte, the filure strin of so clled pssivting film threshold strin for filure ply n importnt role. The model leds to surfce instbility nd formtion of crck like pit if the remote strin is the sme s the threshold strin. The pit is observed to grow while preserving its over ll shpe. This selfsimilr growth imply tht the crck tip speed is constnt nd independent of the remote lod nd crck length. The result offers possibility to determine the threshold strin by observtion of the crck geometry. The rtio between the remote strin nd the threshold strin is proportionl to the rtio between the crck tip width nd the crck length. The nlysis reveled n unexplined discrepncy between the proportionlity fctor s determined from n pproximte nlyses nd finite element simultion. References 1. Bergmn, M., Brickstd, M., Nilsson, F. A procedure for estimtion of pipe brek probbilities due to IGSCC. Interntionl Journl of Pressure Vessels nd Piping, 74(3), pp. 239-248, 1997. 2. Roberge, P.R. Hndbook of Corrosion Engineering. McGrw-Hill, New York, 2000.

3. Prkins, R.N. Current understnding of stress-corrosion crcking. Journl of Metls, 44(12), pp. 12-19, 1992. 4. Turnbull, A. Modelling of environment ssisted crcking. Corrosion Science, 34(6), pp. 921-960, 1993. 5. Kussmul, K., Blind, D., Läpple, V. New observtions on the crck growth rte of low lloy nucler grde ferritic steels under constnt ctive lod in oxygented hightemperture wter. Nucler Engineering & Design, 168(1-3), pp. 53-75, 1997. y/ 0.375 0.25 0.125 crck mouth crck tip 0-0.125-0.25 body edge -0.375 y/ 0.06 0 0.25 0.5 0.75 1 x/ 0.04 0.02 body crck surfce 0 crck -0.02-0.04-0.06-0.15-0.1-0.05 0 0.05 x/-1 FIGURE 6. ) Crck shpe for σ ˆ = 1. b) ner tip region showing somewht unstble development of the shpe.

6. Heldt, J., Seifert, H.P. Stress corrosion crcking of low-lloy, rector-pressure-vessel steels in oxygented, high-temperture wter. Nucler Engineering & Design, 206(1), pp. 57-89, 2001. 7. Jivkov, A.P. nd Ståhle, P., Strin-driven corrosion crck growth. A pilot study of intergrnulr stress corrosion crcking. Engineering Frcture Mechnics, 69(18), pp. 2095-2111, 2002. 8. Jivkov, A.P. Evolution of ftigue crck corrosion from surfce irregulrities. Theoreticl & Applied Frcture Mechnics, 40(1), pp. 45-54, 2003. 9. Td, H., Pris, P.C., Irwin, G.R. The stress nlysis of crcks hndbook, 3d Ed. ASME Pres: New York, 2000. 10. ABAQUS User s Mnul, Version 6.3, Hibbitt, Krlsson & Sorensen Inc., 2002. 11. Jivkov, A.P., 2000. DIGITAL MATERIA - finite strin bsed finite element progrm. Reserch report MUMAT2000:2, Mlmö University Mterils Science, Mlmö, Sweden. 12. Lgerström, J. Privte communiction, 2003. 0.02 Normlised crck-tip opening 0.015 0.01 0.005 0 0 0.25 0.5 0.75 1 Normlised crcklength FIGURE 8. Crck tip width versus crck length, for different crck lengths, normlized with lrgest computed crck lentgth,.