on July 17, 2018 Acomprehensiveanalysis rspa ...rspa.royalsocietypublishing.org/content/royprsa/471/2178/20140703... · supplemented by new results to deﬁne a comprehensive analysis

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ResearchCite this article: Lee D, Huang Y, AchenbachJD. 2015 A comprehensive analysis of thegrowth rate of stress corrosion cracks. Proc. R.Soc. A 471: 20140703.http://dx.doi.org/10.1098/rspa.2014.0703

Received: 17 September 2014Accepted: 17 April 2015

Subject Areas:mechanical engineering

Keywords:stress corrosion cracking, anodic dissolution,hydrogen embrittlement

Author for correspondence:J. D. Achenbache-mail: [email protected]

A comprehensive analysisof the growth rate of stresscorrosion cracksD. Lee, Y. Huang and J. D. Achenbach

Department of Mechanical Engineering, McCormick School ofEngineering and Applied Science, Northwestern University,Evanston, IL 60208, USA

This paper is a synthesis of earlier resultssupplemented by new results to define acomprehensive analysis of the growth rate of stresscorrosion cracking (SCC). Two mechanisms, anodicdissolution (AD) and hydrogen embrittlement (HE),have been considered to calculate the SCC growthrate of AA 7050-T6 for a surface-breaking crack withblunt tip in an aqueous environment. The relativecontributions of each mechanism and their mutualinteractions have been quantitatively assessed.Results show that AD provides critical conditions forHE, which explains in part a stepwise propagationof the crack. Finally, the total crack growth ratedue to the combined effects of AD and HE hasbeen determined, and numerical results have beencompared with experimental data, and a calculationof the crack growth rate for a practical configurationhas been presented.

1. IntroductionStress corrosion cracking (SCC) is an important form offracture, because it can cause significant deteriorationof the integrity of structures under mechanical loadingconditions that would be quite safe without thepresence of a corrosive environment. SCC is one of themost frequent failure mechanisms in the aeronautical,chemical, nuclear and offshore industry. In particular,high-strength 7XXX-series aluminium alloys presenta continuing problem for the aeronautical industry.Significant efforts have been devoted to understandingSCC, but the prediction of the SCC rate is still achallenging subject because of the complex nature ofthe synergistic interactions among the environment, thematerials and the stress that involve electrochemical andmechanical processes.

2015 The Author(s) Published by the Royal Society. All rights reserved.

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The system of interest in this paper is a surface-breaking crack in a structural aluminiumalloy (AA 7050) in an aqueous environment, where anodic dissolution (AD) and hydrogenembrittlement (HE) are considered to be the principal mechanisms of damage [1–3]. Thissystem is chosen because it widely occurs in the aeronautical industry and local chemistrymeasurement data are available [3]. To develop a comprehensive crack growth model, manyfactors need to be considered. Among the most important factors are: (i) the relationshipbetween the bulk environment and the crack tip environment; (ii) the AD rate expected fromthe electrochemical kinetics for a given crack chemistry and the overpotential for hydrogenproduction; (iii) the relationship between the overpotential from hydrogen production and theinternal hydrogen, or the hydrogen discharge current density and the hydrogen permeationcurrent density; and (iv) stress-assisted and trapping-affected hydrogen diffusion ahead of thecrack tip.

The link between the crack chemistry and the bulk chemistry has been established by thetransport equation, which is important because different transport rates of each species cancause a crack chemistry which is quite different from the bulk environment [3–5]. Ions in anelectrolyte move in response to an electric field (a process called migration) and also in responseto concentration gradients (diffusion) and bulk fluid motion (convection) [6]. As SCC has nocyclic motion of the crack wake surface, as in fatigue [7,8], the convection is assumed to benegligible. The Nernst–Planck equation with the assumption of electroneutrality has been appliedto evaluate the chemistry and electrochemistry at the crack tip [9–12]. At ambient temperature,concentrated solutions will most likely arise because of high solubility of metal cations [12].To consider the interaction of ions [13,14] in a concentrated solution, the modified form of theequation that features an activity term in it has been used in this work [6,11]. The geometry ofthe crack has an important role in crack chemistry [7,8]. The geometrical effects can be capturedby using the modified one-dimensional model proposed by Turnbull [12] or the two-dimensionalmodel which has been used in this work. Chemical reactions have been treated by the widely usedassumption of local equilibrium of chemical species [15]. Local equilibrium has been obtainedwith the well-established geochemical reaction code PHREEQC, which has the ability to solvethermodynamic equilibrium in a mixed aqueous/solid-phase system with homogeneous andheterogeneous reactions [16]. It is assumed that the crack solution is fully deaerated becauseof rapid depletion of O2 during the oxygen reduction reaction. On the surface near the crackmouth, deaeration affects crack chemistry and electrochemistry [12], which has, however, notbeen considered here. Two other factors, the formation of hydrogen bubbles and the precipitationof solute species, have been introduced to adjust the transport of species, and they jointly resultin an effective diffusivity and mobility [17–19].

The boundary electrochemical kinetics is treated using the Tafel equation. The dependenceof electrochemical kinetics on the crack chemistry has been obtained based on the bare metalkinetics experiment conducted by Ford et al. [20]. Gangloff et al. [21] have obtained the relationshipbetween the overpotential for hydrogen production and the internal hydrogen for a Ni-basedsuperalloy. Similarly, we have related the hydrogen flux at the crack tip to the hydrogen dischargecurrent density based on the relation that the permeation current density is proportional to thesquare root of the discharge current density, as proposed by Iyer et al. [22].

The reported hydrogen diffusivity shows a large variability due to the effect of trapping atgrain boundaries, dislocations and vacancies. According to Young & Scully [23], vacancies havethe most influence on the effective diffusivity while grain boundaries have minimal influencein aluminium and its alloys. In this work, dislocations and vacancies have been consideredin hydrogen transport analysis. The effect of local plasticity on the density of trap sites hasalso been considered [24]. To model the crack tip stress and strain fields, a blunted crack tiprelated to J2 plastic deformation [25–27] has been considered in the paper. It is well knownthat hydrostatic stress in front of the crack dilatates the lattice, which attracts hydrogen atoms.Hydrogen-enhanced decohesion has been considered as a mechanism for crack growth due to HEwith the model proposed by Katz et al. [28]. Their model provides a reasonable prediction of thetoughness degradation due to hydrogen localization. The total crack growth rate is determined



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based on the superposition model which combines Faraday’s law for AD and a fracture mechanicsmodel for HE which has been derived from the fracture criterion.

The purpose of this paper is to present a computational framework to determine the SCCgrowth rate. In future work, the results will be used as a critical building block for the prognosisof SCC.

2. Mathematical model of stress corrosion cracking

(a) Coupling procedureBefore describing the mathematical models for each mechanism, it is of interest to draw a bigpicture which shows how the problem has been solved. As it is a coupled multi-physics andmulti-mechanism problem, multiple software packages have been used. For the calculation ofcrack chemistry, PHREEQC (interfaced by IPhreeqc) and COMSOL have been used, and stress-assisted and trapping-affected hydrogen diffusion has been solved by ABAQUS and its usersubroutines. To overcome the difficulties related to stiff equations in the discretized time domainof the numerical model, the two simultaneous and dependent processes that are multi-componenttransport and reaction are separated using the sequential non-iterative approach, also called theoperator splitting or time-splitting approach [15].

Figure 1 illustrates the program structure for the coupling of COMSOL, ABAQUS and IPhreeqcusing Matlab as the controlling application and interface for the three programs. The first columnand the third column describe the calculation of crack chemistry and the crack growth ratedue to AD. At the beginning of the simulation, the program initializes COMSOL and IPhreeqccomputations and specifies boundary conditions. Precise concentration values for initial andboundary conditions in COMSOL have been determined by preliminary PHREEQC speciationcalculations. After the initialization procedure, computations enter the iteration loop of the splitoperator, where COMSOL and IPhreeqc procedures alternate for the calculation of transport(TRANSPORT OF SPECIES) and reactions (BALANCE ci AND GET pH). The second columnindicates the procedure for the hydrogen transport and the crack growth rate due to HE. It isassumed that the crack is opened before exposure to the aqueous environment. Initial openingis simulated by fully coupled deformation and hydrogen transport analysis, and subsequenthydrogen transport analysis has been conducted under different boundary conditions based onthe crack tip hydrogen flux NH calculated from crack chemistry analysis. Further discussion willbe provided in the model verification section.

(b) Crack chemistry(i) Governing equations

The modified form of the Nernst–Planck equation for moderately dilute electrolytes has beenused to simulate mass transport inside a crack [6]. We have

Ni = −Di∇ci − Dici∇ ln γi,n − ziuiFci∇φs, (2.1)

where

γi,n = γi

γzi/znn

. (2.2)

In these expressions, Ni is the flux of species i, Di is the diffusivity of species i and ci is theconcentration of species i. Also γi and γn are the activity coefficient of species i and a chosen ionicspecies n, respectively, zi and zn are the valences of species i and n, ui is the mobility of species i,



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START

INITIAL CONDITIONS

SET INITIALTIMEt = 0

ANODIC DISSOLUTION

TRANSPORT OF SPECIESNERNST–PLANCK EQUATION

CALCULATE fs

BOUNDARY REACTIONSRELATE fs TO itip

( )AD BY FARADAY’S LAW ( )HE FROM FRACTUREMECHANICS

HYDROGENEMBRITTLEMENT

NH FROM ADCALCULATION

HYDROGEN DIFFUSION

EFFECTIVE FRACTUREWORK

TOTAL CRACK GROWTHRATE

t = ttarget?

END

BALANCE ci AND GET pHIN SOLUTION

UPDATE TX PROPERTIESDi, ui

UPDATE INITIAL CONDITIONt = (1 + n)Dt

dtda

dtda

Figure 1. Flowchart for the coupling procedure,where t is the time,φs is the potential of the solution, itip is the crack tip currentdensity,NH is the hydrogen permeation flux through the surface of the crack, ci is the concentration,Di is the diffusion coefficientand ui is the mobility of species i.

F is Faraday’s constant and φs is the difference in the electrode potential, E, in the electrolyticsolution (φs = E(x = 0) − E(x)). The current density can be written as

i = F∑

i

ziNi = F

(∑i

−ziDi(∇ci + ci∇ln γi,n) − ∇φs∑

i

(zi)2uiFci

)(2.3)

and the conductivity of the electrolyte is defined as

κ0 = F∑

i

(zi)2uiFci. (2.4)

In the case of an aqueous environment, the area near the tip is filled with gas bubbles generatedfrom an electrochemical reduction reaction such as a hydrogen evolution reaction (HER), as



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observed by Singh et al. [29]. Thus, the crack solution can be viewed as a porous medium. To solvethe mass transfer in a porous medium, averaging needs to be performed in a volume elementwithin the electrode [6,19,30,31]. The averaged flux of species i can be expressed by using thevolume fraction of solution, ε, with equation (2.1). Let ci be the solution-phase concentration ofspecies i, averaged over the pores. The superficial concentration is thus εci. Then

Ni = −Di,eff∇(εci) − Di,effci∇(ε ln γi,n) − ziui,effFci∇(εφs), (2.5)

where Di,eff is the effective diffusion coefficient of species i, and ui,eff is the effective mobilityof species i. In equation (2.5), the volume fraction is inside the gradient operator, because it isthe function of the position inside the crack [31]. The effective conductivity of the electrolyte isdefined as

κeff = εF∑

i

(zi)2ui,effFci. (2.6)

It is assumed that gas bubbles are homogeneously distributed, so that the empirical Bruggemanformula can be used to relate the conductivity and diffusivity to the porosity [17–19],

κeff = κ0(1 − f )1.5 = κ0ε1.5, (2.7)

where κ0 is the conductivity without bubbles, and f is the volume fraction of bubbles insidethe electrolyte, ε = 1 − f . According to equations (2.6) and (2.7), the bulk values of diffusioncoefficients and mobilities are multiplied in the paper by ε0.5 to yield Di,eff and ui,eff valuescorrected for tortuosity caused by hydrogen bubbles. The actual tortuosity and constriction ofa crack are not considered in this work, but it is expected that they have a finite effect onmass transfer.

From the conservation of species and charge, we obtain

∂(εci)∂t

= −∇ · (−Di,eff∇(εci) − Di,effci∇(ε ln γi,n) − ziui,effFci∇(εφs)) + Ri,tot, (2.8)

where

Ri,tot =∑

mRi,m + εRi,src. (2.9)

As no net charge is accumulated, produced or consumed in the bulk electrolyte, we can get

∇ · i = ∇ ·(

F∑

i

−ziDi,eff(∇(εci) + ci∇(ε ln γi,n)) − κeffε−1∇(εφs)

)= 0. (2.10)

In this work, the double layers (i.e. the whole array of charged species and oriented dipolesexisting at the metal–solution interface) are assumed to be thin, leaving the pore space to be filledwith neutral solution, and Poisson’s equation can be replace by electroneutrality (

∑ni=1 zici = 0)

without accounting for the background charge [32]. With the electroneutrality assumption whichreduces the degree of freedom by 1, a total of n coupled equations (equations (2.8)–(2.10)) hasbeen solved for ci and φs.

To get the distribution of the volume fraction of gas bubbles, the crack is divided into severalvolume elements, and the fraction of hydrogen bubbles is the ratio of the volume of hydrogenbubbles and the volume element. It is assumed that the distribution of hydrogen bubbles followsthe normal distribution function:

f = 12

⎡⎣1 + erf

⎛⎝x − μx√

2σ 2x

⎞⎠⎤⎦ , (2.11)

where x is the coordinate along the crack, μx is the mean of the normal distribution function,which is adjusted to match the volume of hydrogen bubbles, and σ 2

x is the variance of thedistribution.



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(ii) Activity coefficient model and transport properties of electrolyte solutions

The calculation of the activity coefficient has been done by PHREEQC (figure 1 ‘BALANCE ciAND GET pH IN SOLUTION’). As mentioned earlier, it is expected that the solution in the crack isrelatively concentrated, so that the deviation from ideal may be significant. Specific ion interactiontheory (SIT) is used to get an activity coefficient, which accounts for binary interactions of anionsand cations with interaction parameters that are specific for individual ions in chloride or othermedia [33]. It is known that SIT is valid up to ionic strength of 3.5 mol kg−1. The general SITequation for a single ion, i, can be written as

log γi = −z2i D +

∑j

εij(I)mj, (2.12)

where D, the Debye–Hückel term, equals

D = 0.509√

I

1 + 1.5√

I. (2.13)

In these expressions, zi is the charge of ion i, and mj is the molality of major electrolyte ion j, whichis of opposite charge to ion i. Interaction parameters, εij(I), refer to the interaction between ion iand major electrolyte ion j at ionic strength I =∑

miz2i /2.

Transport properties in the governing equations depend on composition and concentration.To take the dependence into account, they are updated in every loop (figure 1 ‘UPDATETX PROPERTIES’). As the three properties, namely diffusivity, mobility and conductivity(or conductance), are interrelated, only conductance has been calculated in this work. TheTurq–Blum–Bernard–Kunz (TBBK) conductance model [13] is used to get the compositionand concentration-dependent conductance. In the TBBK model, the Fuoss–Onsager continuityequations were solved directly by a Green’s function technique using the mean sphericalapproximation (MSA) pair distribution functions for the unrestricted primitive model (differentionic sizes). The single electrolyte solution consists of two types of free ions, cation (1) and anion(2), with total conductance given by

ΛF = λF1 + λF

2, (2.14)

where

λFi = λ0

i

(1 + δvel

i

v0i

)(1 + δX

X

)(2.15)

is the ith ion conductance. In the above equation, λ0i is the limiting equivalent conductance at

infinite dilution, δveli /v0

i is the free ion electrophoretic velocity effect and δX/X is the free ionrelaxation force correction.

Equation (2.15) is valid for systems containing only a single cation and a single anion. Sharyginet al. [14] reviewed the literature on mixture models and found several different recommendationsfor essentially the same model although for different properties and with varying refinements.The consensus expression of the conductivity for a solution with a molar ionic strength I is

κ(I) = ceq∑M

∑X

fMfX[λM(X)(I) + λX(M)(I)] = ceq∑M

∑X

fMfXΛMX(I), (2.16)

where ceq =∑cMzM =∑

cX|zX| is the equivalent concentration, the sums are over all positivespecies M and all negative species X with ΛMX calculated at molar ionic strength I. The equivalentfractions of the species in the solution are given by fM = cMzM/ceq and fX = cX|zX|/ceq. Theaverage conductance of the Mth cation (λM) is obtained by averaging over all anions that existin the mixture, i.e.

λM =∑

X

fXλM(X)(I). (2.17)



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Similarly, the average conductance of the Xth anion, λX, is obtained by averaging over allcations, i.e.

λX =∑M

fMλX(M)(I). (2.18)

Ionic mobilities are related to ionic equivalent conductances by

ui = λi

|zi|F2 . (2.19)

Ionic diffusivities can also be calculated with the aid of the Nernst–Einstein relation,

Di = uiRT. (2.20)

The (simple) MSA theory has been found to be in agreement with experimental values fordissociated electrolyte solutions up to 2 mol l−1 [34]. The only unknown parameters are thediameters of the ions. The hydrated diameters of ions have been used in this work.

(c) Hydrogen transportHydrogen is considered to reside at either normal interstitial lattice sites (NILSs) or trappingsites, e.g. grain boundaries, dislocations and vacancies. Young & Scully [23] observed the effect oftrapping on diffusivity, in which the large influence of vacancies at relatively low concentrationsis significant (e.g. 1 × 10−11 vacancies per lattice site lowers the diffusivity by almost a factor of10 at 25◦C), while large amounts of cold work (more than 10 × 1011 lines cm−2) are needed fordislocation traps to appreciably affect the effective diffusivity. No evidence of hydrogen trappingwas found at grain boundaries in either the pure aluminium or the Al–Zn–Mg alloy. Accordingto first-principle calculations done by Lu & Kaxiras [35], a large amount of H atoms (up to 12)can be trapped at a single vacancy, which overcompensates the energy cost to form the vacancydefect. Therefore, current work only considers hydrogen trapping at dislocations and vacancies.According to Oriani’s theory [36], the trapping site occupancies for dislocations θ

(d)T and for

vacancies θ(v)T are always in equilibrium with the lattice site occupancy,

θ(j)T

1 − θ(j)T

= θL

1 − θLexp

(W(j)

BRT

), (2.21)

where the superscript j denotes d for dislocation or v for vacancy, W(j)B is the binding energy of

hydrogen to the corresponding traps, and R is the universal gas constant equal to 8.31 J K−1 mol−1.The hydrogen concentration in the NILSs is given by cHL = βNLθL, where NL = NA/VM denotesthe number of solvent atoms per unit volume, VM is the molar volume of the host lattice,NA is the Avogadro constant and β is the number of interstitial sites per solvent atom. The

hydrogen concentration in trapping sites is calculated as c(j)HT = α(j)N(j)

T θ(j)T , where α(j) denotes the

number of trapping sites per trap of type (j), and N(j)T is the corresponding trap density. The trap

densities at dislocations, N(d)T , and vacancies, N(v)

T , increase with plastic straining. The hydrogentransport equation accounting for stress-driven diffusion and trapping is described by Sofronis &McMeeking [25]

DH

Deff

∂cHL

∂t= DH∇2cHL − ∇ ·

(DHVH

RTcHL∇p

)−⎛⎝∑

j

α(j)θ(j)T

∂N(j)T

∂εp

⎞⎠ ∂εp

∂t, (2.22)

where VH is the partial molar volume of the hydrogen atoms in solid solution, p = Tr(σ )/3 isthe hydrostatic stress, εp is the effective plastic strain and Deff is an effective diffusion coefficientrelated to the NILS diffusion coefficient DH by

DH

Deff= 1 +

∑j

∂c(j)HT

∂cHL. (2.23)



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(d) Fracture criterion and crack growth rate modelIn a purely brittle material, a crack will propagate when the stress intensity factor exceeds theintrinsic Griffith toughness, given by

KIc = kIG =√

2γ E(1 − ν2)

, (2.24)

where 2γ is the required energy for the creation of new fracture surfaces, E is Young’s modulusand ν is Poisson’s ratio.

In systems where dislocations easily nucleate and propagate, the resulting plastic zone screensthe applied mechanical load. When a dislocation nucleates, its stress field increases the criticalvalue KIc. By superposition, Lin & Thomson [37] have obtained the stress intensity factor at thecrack tip in the presence of a dislocation ‘cloud’. Based on their result, Huang & Gerberich [38]have proposed the relationship between KIc and kIG which is given by

KIc = 1β ′ exp

(k2

IGα′′σys

), (2.25)

where σys is the yield stress, and β ′ and α′′ are constants from a computer simulation of thedislocation structure around the crack tip.

To determine the reduction of the intrinsic fracture resistance due to H-decohesion, Katz et al.[28] coupled H localization with the dislocation model of the stressed crack tip to predict thethreshold of KIc for hydrogen environment-assisted cracking as

KHIc = 1

β ′ exp

[(kIG − αcH)2

α′′σys

], (2.26)

where cH = cHL +∑c(j)

HT is the diffusible H concentration adjacent to (but not along) the crackpath, and α is a parameter related to the effectiveness of H decohesion. At the crack tip, hydrogenconcentration lowers KIc, which facilitates the crack to propagate. The corresponding fracturecriterion is

KI ≥ KHIc . (2.27)

The crack growth rate due to HE is derived from the fracture criterion. A surface-breakingcrack of depth a(t), under uniform distant tensile loading, σ , is considered. In linear elastic fracturemechanics, the stress intensity factor of a surface-breaking crack is known as

KI = 1.12σ√

πa. (2.28)

Relative to a fixed coordinate x at the crack mouth, the coordinate moving along with the crack tipmay be expressed as x = a(t) + ξ and the time is defined as τ = t + η. A similar transformation hasbeen used by Choi & Chudnovsky [39]. The crack grows when the stress intensity factor reachesthe hydrogen-affected critical stress intensity factor, i.e.

KI = KHIc . (2.29)

The crack growth rate can be obtained by taking the derivative of both sides of equation (2.29)with respect to time,

dKI

dt= dKI

dadadt

(2.30)

anddKH

Icdt

= ∂KHIc

∂η

dη

dt+ ∂KH

Ic∂ξ

∂ξ

∂adadt

= −∂KHIc

∂η− ∂KH

Ic∂ξ

dadt

. (2.31)

Equating equations (2.30) and (2.31) by using equation (2.29) yields the crack growth rate due toHE as (

dadt

)HE

= − ∂KHIc/∂η

dKI/da + ∂KHIc/∂ξ

∣∣∣∣∣ξ=0

. (2.32)



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The crack growth rate due to AD can be calculated from Faraday’s law as

(dadt

)AD

=∑

j

i(j)a(tip)Mj

zjFρj, (2.33)

where for each alloying element j, i(j)a(tip) is the anodic current density at the crack tip, Mj is themolar mass, zj is the valence and ρj is the density.

3. Comparison of model predictions with experimental data

(a) ElectrolyteAt the start of the simulation, the solution contains 0.5 M Na2CrO4 + 0.05 M NaCl. The dissolutionat the crack tip introduces alloying elements which undergo hydrolysis as follows:

Al 3+ + H2O −−⇀↽−− AlOH 2+ + H+, log K = −4.997, (3.1)

Al 3+ + 2 H2O −−⇀↽−− Al(OH)+2 + 2 H+, log K = −10.094, (3.2)

Al 3+ + 3 H2O −−⇀↽−− Al(OH)3 + 3 H+, log K = −16.791, (3.3)

Al 3+ + 4 H2O −−⇀↽−− Al(OH)+4 + 4 H+, log K = −22.688 (3.4)

and Zn 2+ + H2O −−⇀↽−− ZnOH+ + H+, log K = −8.997. (3.5)

Chromium(VI) is also extensively hydrolysed in water, giving only neutral or anionic species,including chromate (CrO 2 –

4 ), dichromate (Cr2O 2 –7 ) and hydrogen chromate (HCrO –

4 ) [40]. Theeffect of chromate in this reaction is that of a buffer. The consumption of hydrogen during theconversion of a chromate ion into a dichromate or a hydrogen chromate ion helps to maintainthe pH,

2 CrO 2−4 + 2 H+ −−⇀↽−− Cr2O 2−

7 + H2O, log K = 14.56 (3.6)

and

CrO 2−4 + H+ −−⇀↽−− HCrO−

4 , log K = 6.51. (3.7)

The third reaction is self-ionization of water [41],

H2O −−⇀↽−− H+ + OH−, log K = −13.997. (3.8)

It is assumed that every reaction in the solution remains at equilibrium, which is justifiablebecause the reactions are rapid, except for the precipitation of Al(OH)3, which is active if thesolution becomes supersaturated with aluminium hydroxide [40]. In PHREEQC, this can beconsidered by adjusting the saturation index.

The reactions assumed to occur at the metal/solution interface are

Al(s) −−⇀↽−− Al 3+(aq) + 3 e−, (3.9)

Mg(s) −−⇀↽−− Mg 2+(aq) + 2 e− (3.10)

and Zn(s) −−⇀↽−− Zn 2+(aq) + 2 e−. (3.11)

These reactions are assumed to occur at relative rates proportional to the atomic composition ofthe AA 7050; thus, the anodic current has been assumed to consist of 0.95ia of Al 3+, 0.025ia ofMg 2+ and 0.025ia of Zn 2+.



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Table 1. Values for parameters used in the mathematical model.

Ecorr (V) icorr (A m−2) irp (A m−2) βa (V dec−1) βrp (V dec−1) βc (V dec−1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−0.86 1 × 102 1 × 10−3−0.25×pH 0.12 0.72 −0.28. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(b) Electrochemical kineticsMany protective anodic films are oxides and hydroxides whose dissolution depends upon thehydrogen ion concentration, and whose rate follows the Freundlich adsorption equation [42],

log ip = log k − npH, (3.12)

where k is a constant, and n is related to the valency of the cations: n = 0.5, 0.33, 0.25 formonovalent, divalent and trivalent ions, respectively. Equation (3.12) is used to get the crack wallcurrent density. Due to passivity, AA 7050 shows a faster dissolution rate over the breakdownpotential [43]. If the surface film is fully removed, the rate converges to the bare surface reactionrate [20]. In this paper, it is assumed that the tip is fully exposed to the environment, becausethe passive film is broken by plastic deformation and low pH. The data from Ford et al. [20] donot show much variation in the bare surface corrosion potential, Ecorr, and the current densityat corrosion potential, icorr, with respect to pH. But the current density depends on pH due torepassivation, so that an equation similar to equation (3.12) is used to describe the dependence ofthe current density on the pH at the crack tip. The experimental data can be represented by

ia = icorr exp{2.3/βa(E − Ecorr)}1 + icorr exp{2.3/βa(E − Ecorr)}/irp exp{2.3/βrp(E − Erp)} , (3.13)

where Ecorr and icorr are the corrosion potential and the corrosion current density on bare surface,respectively, Erp and irp are on repassivating surface, and βa and βrp are the Tafel slopes for anodicreactions on bare surface and repassivating surface, respectively. irp is the function of pH as shownin equation (3.12). The electrode potential, E, can be calculated from

E = φm − �φm,film − φs, (3.14)

where φm is the potential of AA 7050-T6, and φm,film is the potential over the film. The cathodiccurrent density is

ic = icorr exp{2.3/βc(E − Ecorr)}1 + icorr exp{2.3/βc(E − Ecorr)}/icl

, (3.15)

where icl is the diffusion-limited hydrogen evolution current density which can be estimated from

icl = nFDHcH

δ, (3.16)

where n is the charge number of the cell reaction and δ is the diffusion layer thickness. Thenecessary parameters (shown in table 1) to construct the polarization curve have been obtainedfrom the literature [1,20,43] where aluminium specimens are exposed to NaCl solution withvarious pH.

When the crack pops, it is quickly covered by a passive film, which causes an exponentialdecrease in current density. Therefore, it is more realistic to use transient reaction kinetics or time-averaged current density with the rate of production of bare metal [44]. According to the resultobtained from this work, however, using bare surface kinetics is reasonable, because HE will onlybe activated when the crack tip chemistry reaches a critical condition. More importantly, the timeinterval between each pop is decreasing due to increasing hydrogen flux. As a result, the time toform the passive film is very limited.



11


...................................................

(c) Boundary conditionAt the start of the simulation, oxygen is assumed to be depleted in the crack. The concentrationoutside the crack maintains the bulk concentration. Boundary reactions follow the electrochemicalkinetics presented in the previous section. The dissolution of alloy elements has been obtainedfrom the anodic current density, which is the sum of the contributions of each element. Asthe material is assumed to be homogeneous, the contribution ratio of each element is thesame throughout the simulation. At the crack wall, the current density is three orders lowerthan at the the crack tip. The hydrogen discharge current density has also been obtained fromthe polarization curve. For stress-assisted hydrogen diffusion simulation, the flux boundarycondition is used. The boundary conditions that are used in the simulations of HE are eitherthe Dirichlet condition or a prescribed concentration boundary concentration [26]. Based on theirstudy, Turnbull et al. [45] support the use of the flux boundary condition, especially in materialswith high hydrogen diffusivity.

The rate constants for hydrogen adsorption, absorption, discharge (H+ + e – + M −→ MHads)and recombination (2 Hads −→ H2) can be obtained from a knowledge of the steady-statehydrogen permeation current, cathodic charging current, hydrogen overvoltage and hydrogendiffusivity [22]. When the steady-state permeation current i∞ is proportional to the squareroot of the charging current, the HER is generally considered to follow a coupled discharge–recombination mechanism. Following this argument, hydrogen flux at the crack tip may then bedescribed as

NH = Fμ√

ic, (3.17)

where F is Faraday constant and μ is the proportionality constant. In this work, μ is obtainedfrom a calibration which matches the calculated crack growth rate with the experimental crackgrowth rate. In addition, the flux boundary conditions are chosen so that the resulting hydrogenconcentration near the crack tip is within the experimentally measured concentration of hydrogenat the crack wake (0.007 to 0.30 wppm) [46].

(d) Results(i) Crack chemistry

The crack chemistry analysis has been worked out using a two-dimensional model with thegeometry shown in figure 2a, where a = 17 mm is the crack length, hm = 150 µm is the crackmouth opening displacement and ht = 2.4 µm is the crack tip opening displacement whichcorresponds to the applied stress intensity factor KI = 12.5 MPa. ht has been obtained fromtransport–mechanical analysis using the geometry shown in figure 2b. A detailed explanation isgiven in §3d(ii). The governing equations are solved using COMSOL and PHREEQC, as explainedin §2a. The two-dimensional model can effectively capture the dependence of crack chemistry ongeometry, because the change in geometry changes the volume of the reaction vessels, whicheventually modify the concentration of each species. The domain is discretized by quadratictriangular elements.

Figure 3a shows the changes in pH, [Cl – ] and [CrO 2 –4 ] at the crack tip with the applied

potential, EApp = −0.495 VSCE. After an initial pH drop, a gradual decrease in pH has beenobserved, which is followed by a sharp drop when the chromate (CrO 2 –

4 ) is depleted completely.The concentration of chloride ion increases throughout the simulation. During the transition tosteady state, chromate is completely converted to dichromate (Cr2O 2 –

7 ). The delay in acidificationis likely to be related to the buffering capacity of chromate solutions at near-neutral conditionsvia the hydrogen-consuming reactions (equations (3.6) and (3.7)). When chromate is completelyconverted to dichromate or hydrogen chromate (HCrO –

4 ), the pH starts to drop quickly dueto the lack of a base. The ingress of chloride ion also contributes to the decrease in pHbecause, when combined with hydrogen ions, it produces hydrochloric acid, which can lowerthe pH significantly.



12


...................................................

a

(a)

xL

flux = 0or

flux = constant

cHL = c0HL

ui = KI/E L/2p fi (v,q)

h0t /2

(b)s

cH,fmci,fs

hm ht

Figure 2. (a) A surface-breaking crack under an aqueous environment has been considered. Through chemical analysis, theconcentration of each species ci and the potential of solution φs have been calculated. The concentration of hydrogen cH hasbeen obtained using the two-dimensionalmodel shown in (b). The potential ofmetalφm is constant. (Online version in colour.)

2

4

6

8

10

time (s)

pH

0 50 100 150 200 250 300 3500

200

400

600

800

conc

entr

atio

n (m

olm

–3)

[Cl–]pH

[CrO2–]4

(a)

0 50 100 150 200 250 300−0.80

−0.75

−0.70

−0.65

−0.60

−0.55

−0.50

−0.45

−0.40

time (s)

E (

V v

ersu

s SC

E)

fMax = 0.4

fMax = 0.6

fMax = 0.8

(b)

Figure 3. (a) The concentration of chloride, chromate ion and calculated pHwith fMax = 0.8 and (b) potential drop at the cracktip versus time with various fMax values (EApp = −0.495 VSCE). (Online version in colour.)

The electrode potential within the local cell is one of the key parameters. In figure 3b, thecrack tip potential shows a two-stage behaviour, the first one at 100 s is caused by the rapiddrop of pH, which increases the current density sharply. The second drop, which shows agradual decrease, is caused by the evolution of hydrogen bubbles and the precipitation of gibbsite(Al(OH)3). This is consistent with experimentally observed data that a constriction formed byhydrogen bubbles and solid corrosion products results in a significant potential drop [47–49]. Thecalculated conductivity of an electrolyte solution ranges from 8.5 to 22 S m−1. The conductivitymeasured from concentrated (but not saturated) crack solutions ranges from 8 to 16 S m−1 [50].The difference is caused by the fact that a change in viscosity at higher concentrations entailschanging ionic mobility, which is a manifestation of altered solvation structure around ionsand short-range interactions between ions [51]. Due to the formation of hydrogen bubbles, theeffective conductivity at the crack tip is as low as 2 S m−1. Bubbles adhering to an electrode surfaceinsulate a fraction of the surface that becomes inactive, and the actual current density is increased[52]. This means that the net flux of metal ions is the same as the one without bubbles. As aresult of the decreased conductivity, the potential near the crack tip drops. It is also observed



13


...................................................

−5 0 5 10 15 20−0.85

−0.80

−0.75

−0.70

−0.65

−0.60

−0.55

−0.50

−0.45(a) (b)

distance from crack mouth (mm)

E (

V v

ersu

s SC

E)

this workCooper & Kelly [3]

0 50 100 150 200 250 300 350−100

0

100

200

300

400

500

600

700

800

time (s)

curr

ent d

ensi

ty (

Am

–2)

anodiccathodic

Figure 4. (a) The distribution of potential of the crack solution after 350 s of simulation (EApp = −0.495 VSCE, fMax = 0.8)and (b) the anodic and cathodic current densities versus time (EApp = −0.495 VSCE, fMax = 0.8). (Online version in colour.)

that the crack tip potential is a function of the fraction of hydrogen bubbles. As the fraction ofcoverage increases, the potential decreases. Figure 4a shows the distribution of crack potential,which matches well with the experimental result. The discrepancies are caused by the distributionof hydrogen bubbles and gibbsite.

Figure 4b shows the crack tip current density versus time. The anodic current density showsa sharp increase at 100 s, which corresponds to the drop in pH. Then it gradually decreases until150 s. As the crack growth rate due to AD is proportional to the anodic current density, the shapeof the curve will be the same for the crack growth rate by AD. Its contribution is diminishing withtime. The cathodic current density keeps increasing until it reaches a steady-state condition. Withthe condition of low pH and potential calculated from the crack tip chemistry, the HER also occursactively. As depicted in the cathodic polarization curve (equation (3.15)), significant amounts ofcathodic current density can flow at the bare surface of the aluminium alloy. So from the resultof the simulation, it is expected that the contribution of HE on the crack growth rate at lower pHcannot be ignored.

The limitation of the current AD simulation is that it does not consider the effect of crackpropagation. Displacements during propagation affect the gas accumulation process and increaseconvective transport within the crack [53]. The distribution of hydrogen bubbles also requiresfurther study. The transport of bubbles is affected by other phases, aqueous and solid, and surfaceroughness inside the crack. The nature of this process is probabilistic, therefore a probabilisticconsideration will be introduced in future work.

(ii) Hydrogen transport to the region in front of the crack tip

Hydrogen transport analysis has been carried out using the small-scale yielding approach formode I opening of the crack, as shown in figure 2b. The initial crack tip opening displacement is h0

t ,and the far-field radius of the model is taken as L(= 30 000 h0

t ). A coupled transport–mechanicalanalysis is performed using the general-purpose finite-element modelling code ABAQUS basedon the method proposed by Oh et al. [54]. A user subroutine, UMAT, has been used to calculatethe effective plastic strain, the dilatational hydrogen-induced deformation and the hydrostaticstress. The material is assumed to obey the von Mises yield criterion with an associated flowrule. Another user subroutine, UMATHT, accepts these inputs to solve the governing equation(equation (2.22)). The 4-node plane-strain element, CPE4T, has been used. The number of elementswas taken as 475, and the number of nodes as 516. The crack tip is modelled as a smooth notch toprovide a good approximation of smooth blunting of a sharp crack at a higher deformation level.

The hydrogen transport analysis consists of multiple steps. It is assumed that the crackis opened before exposure to the aqueous environment. In step 1, hydrogen inside the



14


...................................................

coupledtransport–mechanical

analysis(step 1)

transportanalysis(step 2)

KI ≥ KHIc

growth ratecalculation

update initialconditions

yes

no

(a)

0.175 0.180 0.185 0.190 0.195 0.200 0.2052.60

2.65

2.70

2.75

2.80

2.85

2.90

2.95

3.00

(kIG−acH)2

ln(K

H Ic)

experimentlinear model

(b)

Figure 5. (a) Flowchart for hydrogen transport analysis and (b) the plot of KHIc versus the hydrogen-reduced Griffith toughnessusing total hydrogen concentration, cH, from computer experiments,α and kIG to show the linearity validating equation (2.26).(Online version in colour.)

computational domain is in equilibrium with atmospheric hydrogen. Coupled hydrogentransport–mechanical analysis under zero flux boundary conditions has been carried out. Instep 2, the crack is filled with solution, which causes electrochemical reactions on the cracksurfaces. Constant flux boundary conditions based on the crack chemistry have been consideredduring this step. After each step, except step 1, propagation of the crack is checked. If thecrack has propagated, step 2 will be repeated with the step 1 result as the initial conditions. Ifthe crack has not propagated, step 2 will also be repeated with the current result as the initialconditions until the fracture criterion is satisfied. The flowchart shown in figure 5a describes thisprocess.

In step 1, a far-field boundary condition of the displacement, ui, corresponding to an elasticKI-field is applied incrementally over the outer layer of elements,

ui = KI

E

√L

2πfi(ν, θ ), (3.18)

where Young’s modulus is E = 71.7 GPa, Poisson’s ratio is ν = 0.33 and θ is the angle of a pointon the boundary. The target stress intensity factor is KI = 12.5 MPa

√m. After loading, the crack

tip opening displacement, ht, is three times greater than the initial value, h0t . This satisfies the

condition that an asymptotic solution, which is independent of h0t , requires blunting to three

times the initial notch radius [55]. The uniaxial stress–strain law is given by the Ramberg–Osgoodrelationship of the form ε = σ/E + 0.002 × (σ/σY)n, where the yield stress σY is equal to 530 MPa,and the Ramber–Osgood parameter is n = 10.

The lattice diffusion coefficient of hydrogen at 300 K is DH = 2 × 10−11 m2 s−1. The trapbinding energy is taken as 27.3 kJ mol−1 for dislocations and 68.6 kJ mol−1 for vacancies [23].The interstitial hydrogen expands the lattice isotropically and its partial molar volume insolution is VH = 1.7 × 10−6 m3 mol−1. The molar volume of aluminium is VM = 10.00 cm3 mol−1.The uniform hydrogen concentration in the unstressed lattice, c0 = 1.00 × 103 atoms m−3 inequilibrium with the ambient environment, is used as the initial condition. The parameter α istaken to be equal to 1. The parameter, β, is set equal to 1, which corresponds to a maximumNILS concentration of hydrogen atoms per solvent lattice atom. The trap density at dislocationsis expressed as a function of the dislocation density, ρ(d), and the lattice parameter, a = 4.05 ×10−10 m: N(d)

T = √2ρ(d)/a [26]. For the dislocation density in line length per cubic metre, we follow

Detemple et al. [24], who have used the one-parameter approach of Kocks [56],

dρ(d)

dεp= b0

√ρ(d) − b1(ε̇p, T)ρ(d), (3.19)



15


...................................................

where b0 is a constant, and b1 is the annihilation coefficient which depends on strain rate andtemperature. However, b1 can be assumed to be constant for aluminium in the temperature rangebetween 200 and 600 K. Equation (3.19) means that the dislocations are formed by the appliedstress, σ , using σ ∝

√ρ(d), and that they disappear through a kinetic reaction of first order. The

solution of equation (3.19) can be written as

ρ(d)(εp) = ρ(d)eq

⎡⎢⎣1 −

√ρ

(d)eq −

√ρ

(d)0√

ρ(d)eq

exp(

−12

b1εp

)⎤⎥⎦2

, (3.20)

where ρ(d)eq = 3.4 × 1016 m−2 is the equilibrium dislocation density and ρ

(d)0 = 5 × 1012 m−2 denotes

the initial dislocation density at εp = 0. The total vacancy concentration is the sum of theconcentration in thermal equilibrium, cv,eq, and the excess concentration, cex

v : cv = cv,eq + cexv . The

vacancy concentration in thermal equilibrium reported by Seeger et al. [57] has been used,

cv,eq = 23 exp(−0.66 eV

kT

), (3.21)

where k is the Boltzmann constant equal to 1.38 J K−1. Detemple et al. [24] have reported thatas high as 10% of the total vacancy concentration is caused by plastic deformation, and theyhave proposed a model to predict the excess vacancy concentration. Due to the high bindingenergy at vacancy trap sites and the high excess vacancy concentration even with low strain rate(1 × 10−4 s−1), the determination of the equilibrium concentration using Oriani’s theory wasimpossible. In this work, only the vacancy concentration with the thermal equilibrium has beenused, and the excess vacancy concentration will be considered in future work.

To use the criterion described in §2, it is required to get the parameters β ′, α′′ and α. Gangloff[58] has used experimental data for high-strength steel and superalloys, and obtained a goodfit for the two parameters. In this work, the parameters obtained from the experiment done byJang et al. [59] using high-strength aluminium (AA 8090) have been used. Their experiment wasdesigned to observe the effect of microstructure on the critical stress intensity factor. Their resultsshow that the decrease in the threshold stress intensity factor is caused by the population of trapsites adjacent to the crack path at different heat treatment conditions. To obtain the total hydrogenconcentration at the crack tip, computer experiments have been done at different charging timesfrom 12 to 48 h. The result has been ploted in figure 5b. There are three unknowns, and only oneset of experimental data is available. It has been assumed that α′ and β ′′ are of the same order asfor steel (α′′ = 0.0002 MPa m, β ′ = 0.2(MPa

√m)−1). From the curve fitting shown in figure 5b, β ′ is

determined as 0.25(MPa√

m)−1, and α′′ is 0.00025 MPa m, while α is obtained as 40 MPa√

m whichis 80 times higher than α for steel. This is caused by the fact that we do not consider the excessvacancy concentration, which has high binding energy. In addition, pile-up of dislocations nearobstacles such as grain boundaries or particles results in a heterogeneous distribution of trap sites.Therefore, 80 can be viewed as a correction factor for vacancy and dislocation concentrations.

For a different loading time, the normalized hydrostatic stress σkk/σ0 and the hydrogenconcentration cHL/c0

HL, and the normalized total concentration cH/c0HL with the number of

dislocation trap sites N(d)T /NL are plotted, respectively, in figure 6a,b on the axis of symmetry

(θ = 0) ahead of the crack tip at the end of loading (ht = 3h0t ). It shows the hydrogen distribution

in the NILSs, which is largely affected by loading time. For a short loading time (10 and 100 s), thehydrogen concentration near the tip is below the initial concentration due to the formation of trapsites. Because of slow transport of hydrogen in aluminium, it takes time to reach the steady state.When the steady state is reached, the maximum concentration is obtained at the location wherethe hydrostatic stress has a maximum. The point where the maximum hydrostatic stress is locatedis in front of the crack tip. The reason is that the large strains at the crack tip cause the crack toblunt, which reduces the stress triaxiality locally [60]. The total hydrogen concentration in both theNILSs and the trap sites is shown in figure 6b. Severe plastic deformation at the crack tip causesthe formation of dislocation trap sites. The number of dislocation trap sites is saturated near the



16


...................................................

2

4(a) (b)

x/ht x/ht

c HL

/c0 H

L

c H/c

0 HL

0 2 4 6 8 10

N(d

) /NL

1

2

steady state

s kk/3

s 0

skk/3s0

100

200

300×10–12

0 2 4 6 8 10−1

0

1

23

T

t = 10 st = 102 s

steady-stateNT

(d )/NL

t = 10 st = 102 s

Figure 6. Plot of (a) hydrogen concentration cHL/c0HL in NILSs and hydrostatic stress σkk/σ0 and (b) total hydrogenconcentration cH/c0 versus x/ht for AA 7050 (σ0 = 530 MPa) at a blunting crack tip upon the completion of loading underzero flux boundary conditions. (Online version in colour.)

0 200 400 600 800 10000.4

0.5

0.6

0.7

0.8

0.9

1.0

time (s)

KH Ic

/KIc

flux = 1013 atom mm–2 s–1

flux = 1015 atom mm–2 s–1

(a)

0 2 4 6 8 100.6

0.8

1.0

x/ht

0 2 4 6 8 100

0.5

1.0×10−3

conc

entr

atio

n (a

tom

fra

c H

)

(b)

KHIc/KIc

cH

Figure 7. Plot of the normalized hydrogen-affected critical stress intensity factor at the crack tip (a) versus time and(b) versus normalized distance ahead of the crack at t = 1000 s under constant flux boundary conditions (flux= 1 ×1013 atom mm−2 s−1). (Online version in colour.)

crack tip. As the hydrogen concentration is in equilibrium with the concentration in the NILSs,the location of the maximum concentration is not at the crack tip where the NILS concentrationis lower.

In the steps other than step 1, the crack surfaces are exposed to an aqueous environment,which causes electrochemical reactions. In these steps, the flux boundary condition described in§3c is applied at the crack tip. Figure 7a shows KH

Ic/KIc at the crack tip versus time. The criticalvalue of the stress intensity factor KIc decreases with the increase in the crack tip hydrogenconcentration, as expressed by equation (2.26). When the fracture work at the crack tip satisfiesKI ≥ KH

Ic , the crack starts to grow. As the hydrogen concentration decreases with distance fromthe crack tip, the resistance to crack propagation increases as the crack propagates (figure 7b).Therefore, the crack will stop eventually. At that point, an intermediate stress intensity factorcan be expressed by KH

Ic < KI < KIc. After the crack has stopped, hydrogen diffusion to the areaahead of the crack tip reduces KIc again, and crack propagation resumes. This provides a partialexplanation of discontinuous crack propagation. During the process, the crack tip electrochemicalkinetics are affected by the formation of a new surface and film. In this work, however, theseeffects have not been considered. Experimentally, the discontinuous crack growth behaviour is



17


...................................................

0 2 4 6 8 1010−7

10−6

10−5

10−4

10−3

10−6

10−5

10−4

10−3

pH ic(A m–2)

crac

k gr

owth

rat

e (m

ms–1

) (da/dt)AD

(da/dt)HECooper & Kelly [3]

(a) (b)

101.5 101.6 101.7

Figure 8. SCC growth rate (a) versus pH and (b) versus cathodic current density. (Online version in colour.)

evidenced by crack arrest marks, stepwise changes in the direct current potential drop or straingauge measurements [61].

The limitation of the current hydrogen diffusion analysis is that it does not consider theexcess vacancy concentration due to plastic deformation. Parameters obtained from AA 8090experimental data for hydrogen-affected fracture toughness may not be applicable because Al–Lialloys have higher solubility and different types of trap sites. It is also required to use moreaccurate boundary conditions, which should be justified either theoretically or empirically. Inaddition, Gangloff [62] has stated that the consideration of strain gradient plasticity is necessary.

(e) Crack growth rate and comparison with experimental resultThe crack growth rate due to AD and HE has been calculated. The contribution from AD isobtained by Faraday’s law, and the crack growth rate for HE is obtained from fracture mechanics.The crack growth rate calculation shows that initially there is no contribution of HE because oflow cathodic current density. After the crack tip, pH becomes low enough to activate the HER, andthe contribution of HE to the crack growth rate can be observed. Figure 8a shows the calculatedcrack growth rate with respect to pH. The calculated (da/dt)AD well represents the crack growthrate for pH > 3, and for pH lower than that (da/dt)HE presents a better fit. This indicates that thetwo different mechanisms act dominantly in different pH regions. The reason for the dominanceof HE at lower pH is that the HER is more likely at lower pH and lower potential. For highhydrogen flux, the time to get the critical fracture condition is shorter than for low hydrogen flux.As the critical stress intensity factor KH

Ic is the function of the crack tip hydrogen concentration,the numerator of the HE crack growth rate model (equation (2.32)), i.e. the time derivative of KH

Ic ,is higher for high hydrogen flux. The change in the gradient of KH

Ic , which is the denominator, isless sensitive to the change in the H flux. Thus, the higher crack growth rate has been calculatedwith high hydrogen flux. Interestingly, the crack growth rate is independent of the applied stressintensity factor KI, because dKI/da is several orders lower than ∂KH

Ic/∂ξ . This corresponds to theexperimentally observed KI-independent stage II crack growth.

Figure 8b shows the linear relationship between the cathodic current density, ic, and the crackgrowth rate on a log–log plot. As it has been assumed that HER follows a coupled discharge–recombination mechanism, the steady-state permeation current density is proportional to thesquare root of the charging current density, as expressed in equation (3.17). Thus, the crack growthdue to HE is a function of permeation current density i∞, which is equivalent to the flux ofhydrogen multiplied by Faraday’s constant. The following relationship holds in this work:

log(

dadt

)HE

∝ log i∞ ∝ log ic. (3.22)



18


...................................................

Under crack tip conditions of low pH and low potential, the cathodic current density is limitedby diffusion in double layers, and the limiting current density is a function of pH,

log ic ≈ log icl ∝ pH. (3.23)

As a result, the logarithm of the crack growth rate due to HE shows a linear dependence on pH.At higher pH, however, the crack tip flux of hydrogen is not enough due to the low HER rateand the removal of material by AD, so that it is difficult to obtain a critical concentration nearthe tip. The total crack growth rate is dominated by either HE or AD at the extremes of pH.There may be an effect of stress on the AD rates [63], but this has not been considered. Furtherinvestigation is required for the crack growth rate near the transition point; however, the authorsbelieve that the two mechanisms are equally important in that region because of faster diffusionthrough the grain boundary [64] and the existence of phases which require less work to fracture,as modelled by Novak et al. [27]. Thus, the total crack growth rate may be represented by a simplesuperposition law as (

dadt

)tot

=(

dadt

)AD

+(

dadt

)HE

. (3.24)

4. Model application to predict stress corrosion cracking growth rateThe model developed in §2 is now applied to a realistic surface-breaking crack. The crackgeometry is almost the same as in figure 2a except for the crack length, which is a = 1.7 mm, andthe crack mouth opening displacement, which is hm = 17.2 µm. It is possible that this crack lengthwill go undetected during manufacturing. Normally, the length of a crack detectable duringmanufacturing is longer than 1.9 mm using penetrant inspection only. It is assumed that the crackis exposed to an aqueous solution containing 0.1 M NaCl. The external loading is taken to resultin a stress intensity factor of 12.5 MPa

√m. The expected chemical reactions at the crack tip are

the same as in §3 except the one related to chromate (CrO 2 –4 ). Thus it is expected that the change

in pH is more drastic than in the case discussed earlier in this paper. The same electrochemicalkinetics and boundary conditions are used for the simulation.

Figure 3a shows the changes in pH, [Cl – ] at the crack tip at EApp = −0.860 VSCE versus time.As there is no buffering species, the pH drops quickly as soon as the [Al 3+] is introduced tothe solution. The precipitation of gibbsite is also important in pH control. Without precipitation,a higher pH ∼ 5 is calculated with the same conditions. In experiments, so-called ‘mud-crack’,which is caused by the precipitation of solute species, is observed [2]. The low pH andcathodic potential are favourable for the production of hydrogen bubbles, thus the fraction ofhydrogen bubbles at the crack tip soon reaches the saturation point, fMax. Thus, the effect of themaximum volume fraction of hydrogen bubbles at the crack tip is not as significant as in §3. Theconcentration of chloride is stable after reaching its highest point. The current density at the cracktip is shown in figure 9b. At more anodic potential (EApp = −0.495 VSCE), the simulation yields alower pH with higher potential at the crack tip, which causes higher anodic current density, butthe cathodic current density is relatively constant. As a result, a faster crack growth rate has beencalculated. At more cathodic potential (EApp = −1.200 VSCE), the crack tip cannot reach criticalconditions.

As shown in table 2, the crack growth rate varies with the applied potential. The steady-stategrowth rate is 1.83 × 10−8 mm s−1 (EApp = −1.200 VSCE), and at a higher anodic applied potential(EApp = −0.860 VSCE), the crack growth rate is 5.58 × 10−6 mm s−1. If the crack is continuouslyexposed to electrolyte solution, the crack will reach an inspectable crack length (2.5 mm) in 240days (EApp = −1.200 VSCE). This result agrees with the data from Nguyen et al. [2], who havealso observed a faster crack growth at more anodic applied potentials. Their observation of thedeposition of corrosion product, as well as hydrogen bubbles, indicates that the gaseous and solidphases may have affected the transport of species.



19


...................................................

0 50 100 150 200 250 3001.0

4.5

8.0

11.5

15.0

time (s)

pH

0

1500

3000

4500

6000

conc

entr

atio

n (m

olm

–3)

(a)

0 50 100 150 200 250 300−100−50

050

100150200250300350

time (s)

curr

ent d

ensi

ty (

Am

–2) anodic

cathodic

(b)

pH[Cl–]

Figure 9. (a) The concentration of chloride ion and calculated pH, and (b) current density at the crack tip versus time (EApp =−0.860 VSCE, fMax = 0.8). (Online version in colour.)

Table 2. Crack growth rate and pH at the tip of the stress corrosion crack.

EApp (VSCE) crack growth rate (mm s−1) pH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−0.495 1.11 × 10−5 3.16. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−0.860 5.58 × 10−6 3.70. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−1.200 1.83 × 10−8 8.86. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5. ConclusionA comprehensive SCC growth rate model has been developed. The model includes both ADand HE. For AD, the modified form of the Nernst–Planck equation with the effect of hydrogenbubbles and precipitates has been applied to simulate the transport of moderately dilute species.The coupling of PHREEQC through the sequential non-iterative approach makes the reactivetransport simulation easier. For HE, a simple but elegant form of the crack growth rate equationhas been derived, and the stress-assisted hydrogen diffusion with effective diffusivity has beenused to obtain the distribution of hydrogen atoms which lowers cohesive energy. The mostimportant finding from this work is that both AD and HE are important for crack growth rateprediction, and affect each other in the process of SCC.

The model shows good agreement with experimental data, as discussed in §3. A two-stagebehaviour of potential drop has been observed. The crack tip pH is much lower than in bulksolution. Hydrogen bubbles readily form and accumulate within the crack during localizedcorrosion. These crack tip conditions promote hydrogen diffusion in the region in front of thecrack tip, therefore both AD and HE contribute to the crack growth rate. The crack growthrate due to the combined effects of AD and HE has been determined, and numerical results forcomparison with experimental data, as well as a calculation of the crack growth rate for a practicalconfiguration, are presented.

The result also shows that the distribution of phases that impede the transport of electrolytespecies is important for crack growth rate prediction. However, the calculation of the distributionof phases is a difficult task because of its probabilistic nature. The tortuosity of SCC makes it morecomplicated. Therefore, it is necessary to analyse this with a probabilistic technique, and this willbe future work.

Data Accessibility. There are no data to report in this manuscriptAuthors’ contributions. J.D.A. proposed the problem and supervised the research. D.L. formulated and workedout the details, and wrote the first draft. Y.H. supervised the research.



20


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Competing interests. All authors have declared that no competing interests exist

Acknowledgements. The authors thank Professor Yun-Jae Kim at Korea University for providing the Abaqus usersubroutines. They also wish to thank the anonymous reviewers for their constructive comments.

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