41111
Delft University of Technology
Ship Hydromechanics Laboratory
Library
Mekelweg 2, 2628 CD Delft
The Netherlands
Phone: +31 15 2786873 - Fax: +31 15 2781836
Hydrogen Transport in Steel
Stellingen behorende bij het proefschrift 'Numerieke ModeHering van Waterstoftransport in Staid'
1 Bij de vorming van vallen geeft het model voor waterstoftransport van Sofronis en McMeeking niet langer een correcte waterstofbalans.
(dit proefschrift,hoofdstuk 3)
2 Zolang er geen drukopbouw plaatsvindt, kan de vorming van vallen voor waterstof gezien worden als een gunstig aspect bij waterstofverbrossing.
(dit proefschrift)
3 Bij waterstofverbrossing gaat men er vaak ten onrechte vanuit dat dit alleen optreedt in procesinstallaties onder hoge-waterstofdrukcondities.
4 De 'A-klasse' auto van Mercedes Benz laat zien dat het omzetten van model naar toepassing niet zonder een goede test kan.
5 In de behandeling van de darmziekte colitis ulcerosa zou meer aandacht moeten worden gegeven aan een goede voeding daar medicijnen erop gericht zijn om het
afweersysteem negatief te beinvloeden.
6 De gr, ootste vraag naar kloneren van mensen zal voortkomen uit de wens van paren om ook indien een van heiden onvruchtbaar is een biologisch eigen kind te krijgen.
7 Het feit dat veel mensen 'je' zeggen waar ze 'ik' bedoelen suggereen de onderliggende bedoeling 'als jou dit overkwam, zou je ook zo reageren', en komt vaak voon uit het zoeken naar bevestiging dat hun reacties normaal zijn.
8 Eenvoud siert met alleen de mens maar oak de wetenschap.
9 Bij hoger opgeleiden komt dementie minder voor omdat zij hun hersenen meer zouden gebruiken. flit suggereert dat we niet alleen regelmatig oefeningen moeten doen om het lichaam maar ook om de hersenen in conditie te houden.
10 Trouwen is promotie in het leven.
11
Numerical Modelling of
Hydrogen Transport in Steel
Numerieke Modellering van
Waterstoftrarisport in Staal
PROEFSCHRIFT'
ter verlcrijging van de grand van doctor aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus Prof. it, KF. Wakker
in het openbaar te verdedigen ten overstaan van een comniissie,, door het College voor Promoties aangewezen,
OP dinsdag 24 maart I 998te 13.30 uur
door
Alfonsus Hendricus Maria KROM
materiaalkundig ingenieur geboren te Amsterdam
Prof. dr. ir. A. Bakker.
.u.
Samenstelling promotiecommistel Rector Magnificus
Prof. dr. ir. A. Bakker Prof. dr. ft. E. van der Giessen
Prof. dr. ir. E.J. Mittemeijer Prof. dr. G. den Ouden Prof. dr. W. Wei Prof. dr. J.H.W. de Wit
R.W.J. Koers
Published and distributed by:,
Delft University Press
Mekelweg4 2628 CD Delft The Netherlands Telephone: +31 15 783254 Fax: +31 15 7816611 E-mail: dup@dup.tudellt.til ISBN 90-407-16474IVCIP MUG! 84!
Copyright CD 199,8 by A.H.M. Krom
Al! rights reserved. No part of the material protected by this copyright notice may bereproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without permission from the publisher: Delft University Press.
Printed in The Netherlands
vooditter
Technische Universiteit Delft, promotor Technische Universiteit Delft
Max-Planck-Institut fiir Metallforschung, Stuttgart Technische Universiteit Delft
Universiteit Twente Technische Universiteit Delft
Shell Research and Technology Centre, Amsterdam;
Contents
1
Introduction
11.1 Hydrogen 1
1.2 Theories of hydrogen embrittlement of steel 2
1.3 Why modelling? 4
References 5
2 Hydrogen trapping in steel
72.1 Introduction 7
2.2 Hydrogen diffusion with trapping 9
Determination of trap parameters 15
2.4 The equilibrium between hydrogen in lattice sites and in trap sites 20
2.5 Conclusions 26
26
References
3 Hydrogen transport in a plastically deforming body
293.1 Introduction 29
3.2 Hydrogen transport equation 29
3.3 Conclusions
References 33
Appendix 34
4 Hydrogen transport near a blunting crack tip
394.1 Introduction 39
4.2 Transient solutions for the hydrogen distributions 39
4.3 Steady-state solutions for the hydrogen distributions 49
4.4 Discussion of results 57
4.5 Conclusions 59
References 60
5 Hydrogen transport in tensile specimens
635.1 Introduction 63
5.2 Experimental data 64
5.3 Mesh and material parameters 68
5.4 Results 70 5.5 Discussion of results 75 5.6 Conclusions 76 References 77 a . .
6 Hydrogen transport in pipeline walls with an internal crack
796.1 Introduction 79
6.2 Numerical procedure 81
6.3 Mesh and material parameters 86
6.4 Results and discussion 88
6.5 Conclusions 99 References 100
Summary
101Samenvatting
103Dankwoord
105Curriculum vitae
1061 Introduction
1.1 Hydrogen
Our present-day economy is based on coal, oil and gas. Such fossil energy will be less
available in the future, however, as production of these fuels becomes economically
unfeasible or in some cases as reserves are exhausted. The sun on the other hand provides us with energy every single day. This energy can be used to produce hydrogen gas out of water. Thus, a future economy based on hydrogen would seem to be a possible attractive alternative to the use of fossil energy [1.1]. Hydrogen has the advantage of being clean,
since its use produces only water. The storage of hydrogen gas raises problems,
however, as it is difficult to compress and also highly inflammable. Conversion to
methane gas is an option which eliminates some of these problems.
The use of hydrogen also presents another problem: hydrogen embrittlement.
Hydrogen degrades metals and alloys, especially iron and steel. This has already caused major failures in the oil and gas industry 11.21. Commercial metals and alloys come into contact with hydrogen throughout all phases of use. The first contact is in the liquid state
during production, refining and remelting. During these processes molten metals can absorb hydrogen in solution. Metals prepared by electrodeposition, for example hard
chromium plating, often contain large quantities of hydrogen. Hydrogen is also formed during welding processes. In service the metal may absorb hydrogen by a free corrosion
reaction or as a consequence of an imposed potential during cathodic protection. The
results of such interaction between hydrogen and metals are usually undesirable. In some instances, however, hydrogen may indeed be useful, for example as a temporary alloying element in titanium alloys, enhancing ductility at high temperatures [1.3].
Since hydrogen is the smallest and lightest atom, it is reasonable to expect that it will diffuse through metals more easily than any other element. This is borne out by the jump frequency: at 300 K an iron atom changes position once in 100 years and a carbon atom every 10 s, while a hydrogen atom jumps at a rate of 1012 s-1 [1.4]. Many problems
caused by hydrogen are due to the fact that the solubility of the hydrogen in metals drops
sharply at the solidification point and decreases further in the solid state. The sudden
decrease in solubility on solidification may cause porosity in castings and weldings due to the formation of hydrogen gas molecules. The high diffusivity of hydrogen and the low solubility in steel makes it extremely difficult to conduct experiments and to interpret the results.
1.2 Theories of hydrogen embrittlement of steel
Figure 1.1 illustrates the origin of the term 'hydrogen embrittlemene. When a tensile test is conducted at room temperature in air the strain at fracture is virtually independent of
strain rate. This is not the case when the tensile test is carried out in a
hydrogen-containing environment. In such tests the strain at fracture, a measure of the ductility, decreases as the strain rate decreases, while the yield stress and the strain hardening are
unchanged. Apparently hydrogen embrittles steel, as is also shown by the fracture surface. On a macroscopic scale the tensile specimen appears to be broken in a brittle
manner. However, on a microscopic scale the fracture mechanism is still ductile, i.e. void nucleation, growth and coalescence occur.
Fig. 1.1 True tensile strain at fracture InAdA versus temperature T and strain rate t for steel [1.51
a) annealed, b) charged cathodically for 1 h in 4 % sulphuric acid (blisters were observed).
To the left of curve i in Fig. I.3b the steel behaves like uncharged steel. On surface c embrittlement increases as the temperature increases, while on surface d the embrittlement
1 Introduction
A more appropriate term for the behaviour observed is 'hydrogen-enhanced fracture'
or 'hydrogen-assisted fracture'. Note that in commercial metals hydrogen does not influence the yield stress or the strain hardening but for very pure iron (99.999 % Fe) Kimura [1.6] found that hydrogen causes softening in the temperature region from
170 K up to room temperature and hardening outside this region.
As experiments on hydrogen are difficult to conduct and are dependent on charging
conditions, temperature, the particular alloy concerned and the microstructure, many
theories have been developed. Almost all theories share the belief that hydrogen in its
dissociated, dissolved state is responsible for embrittlement, quite independent of the initial source of the hydrogen, as long as kinetic harriers do not prevent the entry of hydrogen. A summary of the sequential steps of hydrogen introduction, transport, accumulation and influence on fracture, as mentioned in literature, is shown in
Figure 1.2.
lattice boundariesgrain
CLEAVA GE
incoherent precipitates
dislocation tangles
voidsand coherent Pores precipitates
Fig. 1.2 Summary of hydrogen processes as mentioned in literature: sources leading to hydrogen in solution, transport by diffusion or dislocation atmospheres leading to microstructural accumulations, and finally the various resultant fracture processes [1.7].
Over the years certain theories relating to steel at ambient temperatures have survived: the pressure theory [1.8], the decohesion theory [1.9] and the hydrogen-enhanced local plasticity theory (HELP theory) [1.10]. One of the first theories is the pressure theory. At
high charging conditions recombination of hydrogen atoms at internal interfaces may result in high pressures, which may reduce the stress required to initiate voids and increase the growth rate of voids. Embrittlement, however, is also observed at low hydrogen pressures. A second theory is the decohesion theory, which is based on the
assumption that dissolved hydrogen lowers the cohesion between metal atoms. A problem with this theory is that hydrogen concentration in the lattice, in the order of ppms, is too
low to have any influence on the cohesive strength between the metal atoms. The decohesion theory is also applicable to interfaces which may have a higher hydrogen
concentration due to trapping. Owing to the higher concentration hydrogen could decrease the stress to initiate dislocations and voids. The latest theory, the HELP theory assumes that hydrogen enhances shear localisation by shielding interactions of dislocations. Tabata
and Birnbaum [1.11] have shown with in-situ electron microscopy experiments that
dislocations start to move when hydrogen is introduced into the specimen. The effect of hydrogen occurs in the later stages of ductile fracture. i.e. during growth and coalescence of voids. Nowadays the overall opinion is that no single mechanism is applicable and that a combination of mechanisms is active in the embrittlement mechanism.
A mechanism occurring at higher temperatures (> 150 °C) is hydrogen attack:
hydrogen atoms react with carbides to form methane bubbles along grain boundaries. In
metals which have a tendency to form hydrides the hydrogen-related phase change mechanism is applicable. Hydrides are generally brittle, and cracking is by a brittle cleavage mode.
Why modelling?
Although literature provides a considerable amount of information, the embrittlement mechanism is still not precisely known. Over the years many overviews of the problem
have been given
[1.12,1.13,1.14,1.15,1.16]. Knowledge of hydrogen transport behaviour in steel is important for the understanding of the kinetics of hydrogenembrittlement. Traps may play an important role [1.17,1.18]. It is necessary to know the
hydrogen distribution and to know how this depends on the hydrostatic stress and
trapping. Surprisingly, less attention has been paid to numerical modelling, as shown in an overview by Turnbull [1.19]. The subject of this thesis is the modelling of hydrogen transport phenomena in steel. It will provide greater insight into the hydrogen transport process and therefore into the embrittlement mechanism itself. In Chapter 2 a model will be derived for hydrogen trapping in steel due to plastic deformations. This model serves
ill Introduction
as a basis for a hydrogen transport model in Chapter 3, which also incorporates the effect of hydrostatic stress and strain rate. Chapter 3 also presents afinite element formulation'. In Chapter 4 the model derived in Chapter 3 will be used toinvestigate the transient hydrogen distribution near a blunting crack tip using the finite element method. Analytical solutions for the hydrogen distribution at a blunting crack tip will also be derived and.
compared with the finite element results. The tensile test, already discussed in the
Section 1.2, will be modelled in Chapter 5, using the hydrogen transport model as derived in Chapter 3. The numerical results will be compared with experimental results found in literature. Finally, in Chapter 6 the pressure theory will be used in order to
model hydrogen transport through pipelines or vessel walls with an internal crack. Crack growth will be simulated in which the driving force is provided by the hydrogen pressure
in the crack. The degradation mechanism under these conditions is termed hydrogen,
induced cracking (HIC).,
References
![1.1] Verhoeven, B., `De waterstofeconomie', De Ingenieur 17 (1996) 6-10..
[1.2] Timmins, PP., Solutions to Hydrogen Attack in Steels, AMS International USA (1997)
IF .31
Froes, F.H., Eliezer, D., and Nelson, HG., 'Hydrogen effects in titanium'
Hydrogen effects in Materials, eds. A.W. Thompson and N.R. Moody (1996)1
719-733.
114.4] Shewmon, P., Diffusion in solids, TMS Warrendale USA,2nd edition (1989)i. 14 .5] Toh, T., and Baldwin, W.M., 'Ductility of steel with varying concentrations of
hydrogen', Stress Corrosion Cracking and Embrittlement, ed. W.D. Robertson,, Wiley New York (1956) 176-186.
14.6] Kimura, H., 'Mechanical Properties of high purity iron and effects of solutes on them', ISIJ International 34 (1994) 225-233.
104.7] Thompson, A.W., and Bernstein, 1.M., in Advances in Corrosion Science and Technology, eds. R.W. Staehle and M. Fontana. 7 (1980).
11:8] Zapffe, C.A., and Sims, CE., 'Hydrogen embrittlement, internal stress and defects in steel', Transactions of the AIME 145 (1941) 225-271.
[1.9] Oriani, R.A., 'A mechanistic theory of hydrogen embrittlement of steels', Berichte der Bunsengesellschaft fiir Physikalische Chemie 76 (1972) 848-857.
[1.10] Beachem, CD., 'A new model for hydrogen-assisted cracking',, Metallurgical
Transactions 3 (1972)437-451.
r1. 111g Tabata, T., and Birnbaum, H.K. 'Direct observations of ,the effect of hydrogen on the behavior of dislocations in iron', Scripta Metallurgica 17 (1983) 947-950.
[1.12] Troiano, A.R., 'The role of hydrogen and other interstitials in the mechanical behavior of metals', Transactions of the ASM 52 (1960) 54-80.
[4.13] Bernstein, I.M., 'The role of hydrogen in the embrittlement of iron and' steel'.
Materials Science and Engineering 6 (1970) 1-19.
[01.14] Hirth, 1.P., 'Effects of hydrogen on the properties of iron and steel', Metallurgical
Transactions All (1980) 861-890.
K.151 Bimbaum, 'Mechanisms of hydrogen-related fracture of metals',
-
Environment-induced Cracking of Metals, eds. R.P. Gangloff and M.B. Ives(1990) 21-30.
[tl.16] Bernstein. I.M., 'The role of hydrogen: is the story any clearer?, Hydrogen
effects in Materials, eds. A.W. Thompson' and N.R. Moody (1996)3-11. H.K.,
[1.17] Pressouyre, G.M., and Bernstein, I.M., 'An example of the effect of hydrogen trapping on hydrogen embrittlement', Metallurgical Transactions A 12 (1981)
835-844.
[1.18] Park, 1.-G., and Thompson, A.W., 'Hydrogen-assisted ductile fracture in spheroidized 1520 steel: part I axisymmetric tension', Metallurgical Transactions A 21 (1990) 465-477.
[1.19] Turnbull, A., 'Modelling of environment assisted cracking', Corrosion Science
A lattice site is taken to mean an interstitial site.
2 Hydrogen Trapping in Steel
2.1
Introduction
In this chapter a model is derived which describes the role of trapping during hydrogen diffusion in steel. Hydrogen is not homogeneously distributed in steel, as it would be in a perfect iron crystal. Hydrogen will be found not only in the lattice sites' but also in atomic and microstructural imperfections such as vacancies, solute atoms, dislocations, grain boundaries, second-phase particles, and voids. The generic term for this phenomenon is
trapping [2.1]. A general model of trapping will first be derived. It will be then demonstrated that this model can be simplified by assuming equilibrium between
hydrogen in lattice sites and in trap sites. Experimental data for trap parameters are also discussed. In the following chapter the hydrogen diffusion model will be incorporated in
a hydrogen transport equation.
In voids or cracks, atomic hydrogen can recombine to form hydrogen molecules.
According to Sieverts' law [2.2] the hydrogen concentration in lattice sites CL depends on the pressure pH, of the hydrogen molecules in the void:
Cj. = kvpiT2 (2.1)
where k is the solubility of the hydrogen. Darken and Smith [2.3] were among the first to observe trapping of hydrogen other then void trapping. They observed that the hydrogen
diffusivity increases with increasing hydrogen concentration, whereas it should be
independent of the hydrogen concentration. This effect could not arise from void trapping since, by Sieverts' law, the trapped hydrogen should be an increasing fraction of the total hydrogen concentration when the hydrogen concentration in lattice sites is increased. Consequently the diffusivity should decrease with increasing hydrogen concentration in lattice sites, which is in contrast to what was observed.
The two prominent effects of trapping are to increase the apparent hydrogen solubility and to decrease the apparent diffusivity. When steel is equilibrated against a fixed external
chemical potential of hydrogen, it will absorb hydrogen to the solubility limit of he
lattice, while additional hydrogen will occupy the traps. Equilibrium is established when
the chemical potentials of the hydrogen distributions in lattice and trap sites and the
external chemical potential become equal. Thus the apparent solubility, or total hydrogen concentration, may be significantly higher than the lattice solubility. A trapped hydrogen
atom must acquire an energy substantially larger than the lattice migration energy to escape the trap. Hence, the mean residence time of a diffusing hydrogen atom is
considerably longer in a trap site than in a lattice site. Consequently, in the presence of trapping the apparent diffusivity will be less than the lattice diffusivity. This could be the
reason why the reported diffusion constants for hydrogen in iron differ by orders of
magnitude [2.4].
At present the trapping phenomena in steels are not well characterised and the connection with hydrogen emhrittlement is not clear. Despite this, it is believed that
trapping plays an important role in hydrogen embrittlement. It has been difficult,
however, to obtain unambiguous experimental information on the role of traps, partly on account of the small hydrogen concentrations involved [2.1,2.5].
2.0
0.5
0 500 1000 1500
DISTANCE AHEAD OF NOTCH (Am) Fig. 2.1 a) Modified wedge-opening loading specimen (dimensions in mm, thickness 8 mm)
b) Distribution of hydrogen ahead of the crack tip in the direction of maximum hydrostatic
stress. C, is the hydrogen concentration far from the crack tip. Its value is not reported [2.61.
Experimental evidence has been found only recently that hydrogen is not homogeneously distributed in steel. Gao et al. [2.6] investigated the hydrogen distribution in the vicinity of a crack tip in steel under mixed mode I/11 loading using mass spectrometry and Sun et al. [2.7] investigated the hydrogen distribution around a crack tip in an fcc single crystal. Their results show two peaks in the hydrogen distribution ahead
of a notch: one peak is in the immediate vicinity of the notch, corresponding to the
location of maximum equivalent strain, and another peak is located at some distance from the notch corresponding to the location of maximum hydrostatic stress. The latter effect
will not be regarded here in terms of trapping, since it influences only the hydrogen
diffusion through lattice sites. In Chapters 3 and 4 the effect of the hydrostatic stress will be considered. 4 A 0" 3.5 01 I, o=45. O.! 1 I *.00. 3 0 o 2
2 Hydrogen trapping in steel
2.2 Hydrogen diffusion with trapping
A trap model is followed as presented by McNabb and Foster [2.81 and Oriani [2.91. The model is based on saturable traps such as dislocation cores and grain boundaries. A saturable trap is a trap of which the physical nature is such that it can accommodate only a finite number of atoms [2.10]. In contrast, a nonsaturable trap can accommodate an
infinite number of atoms. An example of a nonsaturable trap is a void, since by Sieverts'
law the amount of hydrogen in the void (pressure) increases as the hydrogen lattice concentration increases.
In the following derivations a lattice site is denoted by the subscript L and a trap site by the subscript T. The chemical potentials of the hydrogen in lattice sites and in trap sites
can be expressed using a Fermi-Dirac distribution, since both types of sites can
accommodate only a finite number of hydrogen atoms and it is assumed that there are no interactions between the occupied sites [2.9,2.11]. The chemical potential of the hydrogen in lattice sites is1-LL =11, + RT In = + RT IneL
(I - at)
0t. (2.2)
where ut is the chemical potential of the hydrogen in lattice sites in a reference state. R the universal gas constant. T the absolute temperature, and 0 the occupancy of lattice sites:
0 = CL/NL where CL is the hydrogen concentration in lattice sites and N, the number of
lattice sites. Since the solubility of hydrogen atoms in steel is very low, 0, « 1, which
justifies the above simplification.
The chemical potential of hydrogen in trap sites is given by:
pT = RT In 8T
(1 - OT)
where 4 the chemical potential of hydrogen in trap sites in a reference state and OT the occupancy of trap sites: OT = Ci./Nr where CT is the hydrogen concentration in trap sites and NT the number of trap sites, i.e. the number of trap sites per unit volume.
The number of lattice sites is related to the atoms constituting the host lattice:
N Al3p
NL
with13the number of interstitial sites per atom, see Table 2.1, NA Avogadro's number,
i.e. 6.022x1023 mol , p the density of the metal, and At. the atomic weight of atoms constituting the host lattice. In bcc and fcc metals, hydrogen can occupy either tetrahedral
(2.3)
(2.4)
or octahedral site positions. In the case of a-iron indirect evidence indicates tetrahedral site occupancy of hydrogen at room temperature [2.12.2.13], in which case 13 is 6. The
density of iron at 293 K is 7.87x103 kg/m3 and its
atomic weight is 55.8x10-3 kg/mol [2.14]. Hence Eq. 2.4 gives: NL = 5.1x1029 m-3.Table 2.1 The number of interstitial sites per atom for a bcc and a fcclattice
An estimate of the trap density can be made when only dislocation cores are considered
as trap sites. For example, in case of edge dislocations with <I 1 1> slip directions on {110} planes in a bcc lattice, assuming one trap per atomic plane intersected by a
dislocation, the trap density is
Pdis 42
NT
="
= Pdis bcc "bccwhere pdis is the dislocation density and db is the distance between two ( 110 ) planes
given by: db = aN2, where
is the lattice constant. The lattice constant for a-iron is 0.29 nm [2.14] and the dislocation density varies from 1010 M-2 in the annealed state to 1016 m-2 in the cold-worked state. Hence Eq. 2.5 gives: NT 5x1019 to 5x1026 m-3. A higher estimate of the trap density can be achieved by increasing the number of trap sitesper intersecting plane. For example Chou and Li [2.15] assumed five traps per
intersecting plane. Experimental measurements of NT will be discussed in Section 2.4.
In the trap model it is assumed that traps are isolated. i.e. that they do not form an
extended network, and that transport between traps is by lattice diffusion. The hydrogen diffusion along grain boundaries and dislocations is not considered important as these can
be considered as strong traps for hydrogen. The hydrogen transport by moving
dislocations is similarly not considered important, since dislocations cannot move throughgrain boundaries. Hence it is questionable whether there is a net flux of hydrogen by moving dislocations in a polycrystalline metal [2.1,2.16]. The hydrogen flux through lattice sites JL is related to the hydrogen concentration CL and the gradient of the chemical potential of the hydrogen in lattice sites as follows [2.17]:
(2.5)
lattice site
number of sites per unit
cell
number of atoms per unit
cell (3 bcc octahedral 6 2 3 bcc tetrahedral 12
/
6 fcc octahedral 4 4 li fcc tetrahedral 8 4 2 = H H2 Hydrogen trapping in steel
(2.10)
I This is not the case when plastic straining occurs. The effect of plastic straining will he considered in
the next chapter.
JL = MLCLV4tL (2.6)
where ML is the mobility of the hydrogen atoms. Substitution of the derivative of Eq. 2.2 in Eq. 2.6 gives Fick's first law for steady-state diffusion:
J1, = DLVCl (2.7)
where ID, = MLRT is the diffusion coefficient which follows an Arrhenius relation:
DL = Doe-QT. with ID, the diffusion pre-exponential factor and Q the activation enthalpy
[2.17]. According to Oriani [2.9] and Johnson and Lin [2.10] the time-dependent
diffusion of hydrogen with trapping may be written as
CL aCT
- DLV-C, = 0 (2.8)
Eq. 2.8 is a modification of Fick's second law: acLiat = DLV2C. In situations where
the hydrogen concentration in trap sites is a function only of the hydrogen concentration in lattice sites', Eq. 2.8 can be rewritten as
act. DL
-3T (1 + acTiacov
CL = 0 (2.9)The exact form of aCTMCL will depend on the trapping mechanism. By comparison with the standard form of the diffusion equation, an effective diffusivity may be defined as
DL
(1 + acyaco
In many cases Defy Will be concentration-dependent, and it will not usually be possible to
obtain solutions of the diffusion equation [2.5]. Nevertheless, some useful physical
insights can be obtained from examination of Eq. 2.10.
saturable traps
In constructing a relation between the two hydrogen distributions, the following trapping reaction is considered [2.181:
[H], 15 'HT (2.11)
where [H] is the hydrogen atom in a lattice site and [HIT is the hydrogen atom in a trap site.
The number of hydrogen atoms moving from lattice sites to trap sites is proportional to: the oscillation frequency v of the hydrogen atom, which is usually taken to be equal to the Debye frequency, i.e. 1013 s.1 [2.19,2.20];
the probability that an oscillation is successful, which is equal to e-E011. [2.19,2.20], where EQ is an activation energy;
the number of hydrogen atoms in lattice sites, i.e. the concentration;
the probability that a neighbouring site is a trap site, which is equal to the number of unoccupied trap sites divided by the total number of unoccupied sites.
Hence, the derivative of the concentration of hydrogen atoms moving from lattice sites to trap sites with respect to time can be written as
NT(1-0T)
= KCL (2.12)
NL(1-00 NT(1-0/.)
where K is the frequency of hydrogen atoms jumping from lattice sites to trap sites: K
= veT
-E,ar
,in which ELT is the activation energy for moving from a lattice site to a trapsite. Referring to Figure 2.2, this activation energy is equal to the activation energy for moving from a lattice site to an adjacent lattice site and hence to the activation energy for normal lattice diffusion: ELT = ELL = ED'
metal energy 1-12(ads) H(ods) acT diffusion Es
Fig. 2.2 Schematic view of energy relations in hydrogen-metal equilibria, including trapping
trapping untrapping
AET
+
2 Hydrogen trapping in steel
Similarly, the number of hydrogen atoms moving from trap sites to lattice sites is proportional to:
the oscillation frequency v of the hydrogen atom; the probability that an oscillation is successful;
the number of hydrogen atoms in trap sites, i.e. the concentration;
the probability that a neighbouring site is a lattice site, which is equal to one, since it is assumed that there is no transport between trap sites.
Hence, the derivative of the concentration of hydrogen atoms moving from trap sites to lattice sites with respect to time can be written as:
ac,
-"a7 = -ACT
where A is the frequency of hydrogen atoms jumping from trap sites to lattice sites: =Ve-ETL1RT,in which ErL is the activation energy for moving from a trap site to a lattice
site. Referring to Figure 2.2, this activation energy is the same as the activation energy for moving from a lattice site to an adjacent lattice site, ED, minus the trap binding energy AEr: ETL = ED - AET. The derivative of the hydrogen concentration in trap sites with
respect to time is found by adding Eq. 2.13 to Eq. 2.12. First the previous two equations can be simplified by taking 0 L <<1, since the solubility of hydrogen atoms in steel is very low and NT «NL, as shown previously. This results in
aCT
KOLNT( -0T) - 'ACT (2.14)
When the number of trap sites is constant, using ar = CT/NT, we can write
act
de,.77-=
Eq. 2.14 can then be rewritten as
aeT
77. Ko, (i-e)
- AN.This is the equation derived by McNabb and Foster [2.81, except that Ell, is replaced by CL. The term 1/NL was 'merged' in the parameter IC, as a result of which in their equation the unit of this parameter differs from that of the parameter A. From a dimensional point of view this is not correct.
In the case of equilibrium the time-dependent variable is zero and Eq. 2.16 yields:
OT K
e
= KT OL(1-0T) (2.13) (2.15) (2.16) (2.17)where KT is termed as the trap equilibrium constant.
Another derivation of Eq. 2.17 is given by Oriani [2.9]. He assumed that there is
local equilibrium between the hydrogen in lattice sites and the hydrogen in trap sites. This means that the chemical potentials must be equal: tIL = 1.t.T. Equalising Eq. 2.2 and
Eq. 2.3 results in Eq. 2.17, where AEI- = ,(,).
The hydrogen concentration in trap sites as a function of the hydrogen concentration in lattice sites is obtained by substitution of OL = CL/NL and OT = CT/NT in Eq. 2.17:
CT
( 1 +
-KTCL
If the trap density and temperature remain constant, the partial derivative of the hydrogen concentration in trap sites with respect to the hydrogen concentration in lattice sites, using
Eq. 2.18, is
acT c T( -13T)
-
Cl.Substituting in Eq. 2.10 we obtain for the effective diffusivity
CL
NT
Dr
DLCL +CT(1-0T)
When the occupancy of trap sites by hydrogen is also low (OT «1), Eq. 2.17 reduces to
CT= KT NL CLNT (2.21)
Hence, in the regime of low trap occupancy, saturable traps are equivalent to linear,
nonsaturable craps. Using Eq. 2.20 the effective diffusivity now becomes
Deff DL1 + KTNT/NL (2.22)
Since Eq. 2.22 is independent of the hydrogen concentration, Oriani [2.9] used it to determine trap parameters in the regime of low trap occupancy.
void traps
The surface of a void can trap atomic hydrogen but the void itself can trap molecular
hydrogen. At the surface of the void the following absorption reaction (inverse trapping reaction) takes place:
(2.18)
(2.19)
(2.20)
2 Hydrogen trapping in steel
N 447,
N
pH,
Eq. 2.25 rearranges to
CL = k (2.26)
'where k contains all constants of Eq. 2.25. Experimental data show that k follows an
Arrhenius relation: k = Icoe-miaT, with lc o the solution pre-exponential factor and AH, the
enthalpy of solution, for iron 5.0x1023 m4Pa1 and 28.6 IcEmol respectively [2.21]. At
low hydrogen pressures the hydrogen gas can be regarded as an ideal gas and the fugacity
may be replaced by the pressure p, which leads to Eq. 2.1. In Chapter 6 the fugacity will be related to the pressure. For molecular trapping in voids Johnson and Lin [2.10] derive for the effective diffusivity:
DL ell
( 2ak 1-474)
where cc contains model parameters and 'physical constants. Johnson J2.5] analysed
diffusion data for hydrogen using Eq. 2.27. However, the nature of the experiments was
such that trends with lattice concentration or temperature could not be examined.
2.3 Determination of trap parameters
Diffusion data are usually obtained from permeation tests on a thin plane sheet or a' cylindrical tube.The concentrationsof the diffusing atoms are maintained at C, andt C2 on
(2.24)
(2.25)
(2 .274)
3 H2Z-4 iH1L (2.23)
where the brackets mean that the 'hydrogen is absorbed in lattice sites. If we assume equilibrium between the hydrogen gas in the void and the hydrogen in lattice sites, the 'chemical potential of the hydrogen molecules in the void must be equal to the chemical potential of hydrogen in lattice sites: uL - uH2 = 0. The chemical potential. of the
hydrogen gas is:
412.
pH2 + FCC In
PH2
where 42 is the chemical' potential in a referencestate, the fugacity of the hydrogen. gas and p2 reference pressure. Equalising Eq. 2.2 and Eq. 2.24:
010
- VIZT CL e 2 2 + = 41.12 a Lthe opposite sides of the sample. The diffusion constant can then be determined using the thickness of the sample, the concentrations and the measured flux.
without trapping with trapping ei4 without trapping with trapping
The diffusion constantDL can be determined using the lag time tL, see Figure 2.3, and the sample thicknessdmwithout knowing the concentrations:
d2
DL= 61.L (2.28)
The trap parameters can be deduced from the lattice lagtime ELand apparent lag timetrfor various conditions and their dependence on the hydrogen concentration. Approximate
solutions for the permeation curves with trapping were obtained by Caskey and
Pillinger [2.22], who applied the finite difference method to the equations of McNabband Foster [2.8]. A solution for the time lag with trapping tT was shown for sheet
membrane permeation boundary conditions [2.8]:
tL 1 + c)111(1 c) 34r 64/ 6xv c (2.29) with
This limit can be derived usIng Taylor expansion for In() + c) resulting in:
tTAL= I + tir - 2Nic + ).
NTK CT
111= N
X CL(1 -OT)
and:
C°L C°L ar 8T }
NLX NL0L(1 - OT) (1 - OT) input surface
whereC°Lis hydrogen concentration at the input surface. The limiting cases of Eq. 2.29
are dilute occupancy of traps (w, << 1) and near saturation of traps (w, >> 1,
Sir 1). For the first limit Eq. 2.29 yields'
//
time time
Fig. 2.3 Definition of parameters obtained by a permeation test.
K
C
a
-2 Hydrogen trapping in steel
and for the second limit
3w 3NT
= 1 +
EL Co
When hydrogen gas is used for charging the specimen. C°L can be replaced by Sieverts'
law:
3N.1.==
3N
1 +
tL k
This equation was used by Kumnick and Johnson [2.23] to measure the trap density and binding energy in very pure iron (zone-refined Ferrovac E). They measured the lag time at
various hydrogen pressures and at various deformation levels. The experimental data
were analysed for trap densities and binding energies through plots of (t-ritL - 1) versus
JINN. The slope at high pressures was determined through which the trap density is
known. Using this trap density, a fit was made through all data points by changing the trap binding energy. This was done for several deformation levels. It was found that the
trap density increases sharply with plastic strain at low plastic strains levels and then
increases more gradually with further straining, see Figure 2.4.
0 0.4 0.8 1.2 1.6
equivalent plastic strain ci,
Fig. 2.4 The number of trap sites in a-iron as a function of equivalent plastic strain along with the fit
log NT = 23.3 - 2.33e'4.05P. (2.30) (2.31) (2.32) 24 23 22
oce Kumnick and Johnson
2 A
20
A fit close to their observations, see also Figure 2.4, is
log NT = 23.3 -1.33e-4.oE (2.33)
Kumnick and Johnson estimated a trap binding energy of -60±5 kEmol, independent of temperature within the range of 288 to 343 K and independent of plastic deformations within the range of 0 to 80 %. The physical picture that may be inferred from these observations is one of deep and isolated traps with a single binding energy and a
concentration which is dependent on deformation level. The variation in the trap density with deformations is similar to the variation in quantities such as yield strength, hardness, and dislocation density with deformation level. This implies that the traps are associated
with the imperfection structure, probably dislocations and perhaps point defects,
developed during plastic deformation [2.23]. Other trap sites may be newly-created
microvoid surfaces. In the annealed state the measured trap density, NT is approximately 1021 m-3, can be identified with the dislocation density, since according to Eq. 2.5 the
trap density varies between 5x1019 - 5x1021 m-3. On the other hand, in the heavily
deformed state the measured trap density is approximately 1023 m-3, while according to Eq. 2.5 the trap density is 5x1025 m-3. This can be explained by the fact that not all
dislocations are edge dislocations, as assumed in Eq. 2.5. The trap binding energy seems very high but Hirth 12.12] argued that this is easily possible, as there are even stronger traps, see Table 2.2. Strong traps are for example dislocation cores and grain boundaries, while weak traps are voids and the stress field of dislocations. Riecke and Bohnenkamp [2.24] carried out the same analysis as Kumnick and Johnson. They found for iron a trap binding energy of -56 kJ/mol and a trap density of 1.9x1023 m.3 at 60 % deformation, approximately the same values as found by Kumnick and Johnson.
Perng et al. [2.25] investigated trapping in austenitic stainless steel by measuring the
hydrogen flux, from which the diffusivity and the permeation were deduced. The gas
permeation experiments were performed in the temperature range 100 to 350 °C and in
the hydrogen pressure range of 0.1
to 30 kPa. They observed that the amount of
hydrogen trapped increases rapidly at first and then less rapidly with increasing deformation. However, the estimated increase in the trap density, from 5.0x1026
-5.0x1027 m-3, was not large as compared with Kumnick and Johnson. This was probably
because the number of traps such as carbide interfaces was already large, so that the increase due to
plasticity could not be detected. In
ferritic stainless steel Pernget al, [2.26] did not observe a significant increase in the effective hydrogen diffusivity due to the increase in hydrogen pressure, probably because the hydrogen pressure range
was not large enough to detect any trapping [2.26]. Kumnick and Johnson varied the
2 Hydrogen trapping in steel
number of traps such as carbide interfaces could already be large, so that the increase in trap sites could not be detected.
Table 2.2 Trap binding energies AE, of several hydrogen traps in c4-iron [2.121.
enthalpy of solution
-varies with inverse distance from dislocation 3
enthalpy of adsorption
Xie and Hirth [2.271 examined hydrogen trapping in spheroidised carbon steel by
cathodic charging permeation tests. They found that trapping increases with increasing
deformation. Since the specimens were charged cathodically, an analysis such as that
carried out by Kumnick and Johnson could not be performed.
Oriani [2.9] used Eq. 2.22 in order to analyse the existing experimental data and
observed the following trend: the trap density increases with increasing deformations, carbon content and the strength level of the steel. He found a surprisingly uniform trap binding energy of -27 kJ/mol and a trap density of the order of 1025 111-3 for undeformed steel. Considering Eq. 2.5 with a dislocation density of 1012 m-2 gives a trap density of
21 -3
5.0x10 m . Omni s trap density is therefore too large to be due to dislocations alone.
For undeformed steel a dislocation density of the order of 10151ines/m2 would be needed in order to justify this. Oriani concluded that interfaces and micro cracks are involved in trapping. It is, however, doubtful whether Oriani determined dislocations as trap sites, since the measured trap binding energy is approximately the same as the heat of solution but of opposite sign. In addition Eq. 2.22, used by Oriani, is similar to Eq. 2.27, which can be used for void trapping. Furthermore, the hydrogen concentrations were also orders
of magnitude higher than
the hydrogen concentrations used by Kumnick andJohnson [2.23], diminishing the effect of dislocation trapping and increasing the chance
trap site AE.i
(kJ/mol)
perfect lattice 0
void' -28.6
stress field dislocation2 -20 - 0
dislocation core -60
free surfaces' -71
grain boundary -74
of hydrogen damage (voids). Hence it is very reasonable to assume, that Oriani
investigated void traps.
2.4 the equilibrium between hydrogen tin lattice sites and in trap sites
It will be investigated in this section whether equilibrium can be assumed between the hydrogen in lattice sites and in trap sites by using the experimental data of the previous
section. Eq. 2.14 can be solved analytically when the total 'hydrogen concentration,
CL + CT, and the trap density are held constant during time. Eq. 2.14 then becomes,
acT
-OF=KiKcT - use!. + for ++ mio CT + KC,D(NTI
0.34)
which has the following solution:
-
e
C'T 2r3 + it 2r3 r3 r3 2r3Cm+ r2 - r, r, I),-where Cm is the hydrogen concentration in trap sites at t = 0 and
+ KNT + AN1,12 - 4K2C10(NT
Jr2 = IsylKL, + KNT +1,N_
1
K'
fl3.7.21,-7
Eq. 2.35 with CTo = 0 is plotted in Figure 2.5 using the data of Kumnick and Johnson, i.e. a low trap density and a high trap binding energy, and in Figure 2.6 using Oriani's
data, i.e. a high trap density and a low trap binding energy. Oriani's data are used for
comparison purposes, despite the fact that these data are believed to reflect void traps. The vibration frequency is as usual taken as equal to the Debye frequency, i.e. = 1013 s* In the first case equilibrium is found within 4 ms and in the second case within 0.2 us. It is
found that by increasing the trap density, the temperature or the total hydrogen
concentration, the time to reach equilibrium decreases, while changing the trap binding energy has a small influence on the time to reach equilibrium. The trap binding energy influences the ratio of the hydrogen concentrations in the equilibrium situation.C
-(2.35)
r1
'2 Hydrogen trapping in steel
time t [ms]
Fig. 2.5 Plot of Eq. 2.35 using NT = 2x1021 ma, AE T = -60 klimol, Cm=3x1011hm4.,
NIL = 5.1x1019 10-3, T = 293 K, = 6.9 klimol, v = lx1013 , cro =0
l' 4 f
0.2 0.3 0.4
time tlyst
05
Fig. 2.6 Plot of Eq. 2.35 using = 2x 1023 ma, ART = -27 kJ/mol, = 3x 1015 ma,
NIL = 5.1x 1029 ma, T =293 K, ED = 6.9 kJ/molt v = hole ii, = 0
'The above analytical solution of Eq. 2.34 corresponds to the situation in which at rit= 0 a number of trap sites are suddenly present. It is more appropriate to follow the
hydrogen distribution in time starting from an equilibrium situation and then changing the trap density. The equilibrium hydrogen concentration in trap sites, Crsq, can be calculated'
d 0.8
ED 0I 0.6
OA L.-Q. 0.2 0z
3 5 0.8 0.6 0.4 0.2 4 6 0.1 CL CTusing Eq. 2.18 with CL=C1 - CT. This results in
+ C + NT)Cr.eq+ = 0 (2.36)
This quadratic equation of the hydrogen concentration in trap sites has one correct
solution, given by 0.1 s' 1 s' 0.95 0.9 10 s' 0.85 0 0.05 0.1 0.15 02 tdeildt
Fig. 2,7 Plot of Eq. 2.35 with NT depending on the strain, using ,6,E, = -60 kJ/mol, C. = 3x1021 m.3, NL = 5.1x m 3, T = 293 K, E, = 6.9 kl/mol, v = lx I 013 S
Now that it has been demonstrated that equilibrium can be assumed, the effect of
trapping on the effective diffusion constant will be considered. In Figure 2.8 the effective diffusion constant is shown according to Eq. 2.20 using a high binding energy. As the
hydrogen concentration in lattice sites increases the effective diffusion constant also
increases from zero up to the lattice diffusion constant. This behaviour was observed by
Darken and Smith (2.3]. Thus the effect of trapping becomes less significant as the
hydrogen concentration in lattice sites increases. Ultimately there will be no effect at all, since all trap sites are filled. When the trap density is increased, the effective diffusion constant becomes the lattice diffusion constant at higher hydrogen concentrations in lattice
CT.eq =T Cm, +
NL NT - +Clot + NT).- - 4NTCtot) (2.37)The other solution is not valid, as it results in a negative hydrogen concentration in lattice sites. Since the trap densityNT is time-dependent, Eq. 2.34 can only be solved
numerically. ForNT = NT(EP)= N1(t) Eq. 2.33 is used.
2 Hydrogen trapping in steel
sites. In Figure 2.7 the hydrogen concentration in trap sites as a function of time is shown for various plastic strain rates in the case of strong trapping. The time axis is normalised by the strain rate and the concentration axis by the equilibrium hydrogen
concentration in trap sites. The analysis is started in an equilibrium situation. As trap sites
are created, hydrogen will move from lattice sites to trap sites in order to reach
equilibrium. When the strain rate is less than 0.1 s- I the filling of trap sites may follow the creation of trap sites. With increasing strain rate the hydrogen concentration deviates from the equilibrium concentration, as less time becomes available for filling the newly-created traps. However, this deviation is a maximum of 0.13 for a strain rate of 10 s.I. At a certain time the trap creation rate decreases down to zero and the hydrogen concentration in trap sites becomes equal to the equilibrium hydrogen concentration in trap sites. For weak trapping, .6,ET = -27 ki/mol, there is no effect of the plastic strain rate for strain rates lower than 10 s-1. In tensile tests the average strain rate is around 0.1 - 1 ill. The
strain rate in tensile tests conducted in a hydrogen environment is even lower. Thus local equilibrium can be assumed between the hydrogen concentration in lattice sites and trap sites as long as the strain rate is less than 1 s- I. In the above solutions the total hydrogen concentration was constant. When the total hydrogen concentration may increase as is the case when the hydrogen concentration is prescribed on the boundaries, the time to reach equilibrium will be shorter, since more hydrogen is available.
19 20 21 22 23 24
log CL ([m-31)
Fig. 2.8 Ratio DID, in the case of strong trapping as a function of the hydrogen concentration in lattice sites for various trap densities: I = 293 K and Aar= -60 Ici/mol
1.0 0.8 0.6 N= 10 in NT=10m NT= 10'm ' 0.4 0.2 0.0
--22 23 24 25 26 27
log CL ([m-3])
Fig. 2.9 Ratio De/DL in the case of weak trapping as a function of the hydrogen concentration in lattice sites for various trap densities: T = 293 K and AE,. = -27 kJ/mol
The same trends can be seen when the trap binding energy is low, see Figure 2.9.
However, the effective diffusion constant will not now become zero as is the case with a high trap binding energy. Moreover, the effective diffusion constant becomes the lattice
diffusion constant at higher hydrogen concentrations. Note that in Figure 2.9 the
hydrogen concentrations in lattice sites are so high, that the equilibrium pressure will result in the formation of voids in steel, see Chapter 6. The effect of temperature for various trap densities is shown in Figures 2.10 and 2.11 according to Eq. 2.20 in the
temperature range where the trap parameters are valid, see Section 2.3. The hydrogen concentration in lattice sites is in equilibrium at a constant hydrogen pressure of 0.1 MPa.
By increasing the temperature, the solubility of hydrogen increases, since hydrogen dissolves endothermally. Consequently the hydrogen concentration in lattice sites increases as the temperature increases. Moreover, the equilibrium situation, see Eq. 2.18,
shifts to the hydrogen in lattice sites as the temperature increases, since this is an
exothermic reaction. Nevertheless, the effect of temperature on the effective diffusion constant is small when trap sites have a high trap binding energy, see Figure 2.10. For weak trapping the effect of the temperature is much greater, see Figure 2.11.
I .0 N, = IQ rif' NT = 10 M 0.8 0.6 NT=IC in., 0.4 0.2 0.0
250 300 350 400 temperatureT1K1
Fig. 2.10 Ratio fici,/1), in the case of strong trapping (MT= -60 kJ/mol) as a function of the temperature
for various trap densities. The hydrogen concentration in lattice sites is in equilibrium at 0.1 MPa: CL = k0eA"RTNip with lc, = 4.9x1023 m.3Pa.112 and AHs = 28.6 kJ/inol 12.211
temperature T JR]
Fig. 2.11 Ratio DC/DL in the case of weak trapping (AE.i. = -27 kJ/mol) as a function of the temperature for various trap densities. The hydrogen concentration in lattice sites is in equilibrium at
0.1 MPa: koeali,./RT /pwith k, = 4.9x 10- m 3Pa and Alis = 28.6 kJ/mol 12.21]
2Hydrogen trapping in steel
1 0.75 0.5 0.25 0 =1012 Nt=HY' mo NT= 1 ON 250 300 350 400
as Conclusions
In steel various traps exist for hydrogen atoms. Trapping enhances the solubility of
hydrogen but decreases the diffusivity. Traps are, for example, dislocations and voids. In order to investigate the effect of traps on the hydrogen distribution in steel, a model for trapping was derived which resembles the McNabb and Foster model [2.8]. Assuming
equilibrium between the hydrogen in lattice sites and in trap sites Oriani's trap
model [2.9] is obtained.
Experimental data show an increasing trap density as the amount of plastic deformation increases. These traps can be identified as dislocations and other imperfections created by plastic deformations. Experimentally it is difficult to determine the number of traps as a function of plastic strain. Only the experimental data of Kumnick and Johnson [2.26] obtained from pure iron can be used in order to obtain a relation for the number of traps as a function of equivalent plastic strain.
Modelling of traps in iron and steel can be done on the basis of local equilibrium
between hydrogen in lattice sites and hydrogen in trap sites as long as strain rates are less than 1 s-1. This has the advantage that the rate of the hydrogen concentration in trap
sites can be expressed as a function of the hydrogen concentration in lattice sites. The effect of traps on the diffusion depends strongly on the trap binding energy and the trap density with respect to hydrogen concentration in lattice sites.
References
[2.1) Turnbull, A., 'Modelling of environment assisted cracking', Corrosion Science 34 (1993) 921-960.
[2.2] Sieverts, A., and Krumbhaar, W., 'Ober die Losigkeit von Gasen in Metallen und Legierungen', Berichte der Deutschen Chemischen Gesellschaft 40 (1910) 893-900.
[2.3] Darken, L.S., and Smit, R.P., 'Behavior of hydrogen in steel during and after
immersion in acid', Corrosion 5 (1949) 1-16.
[2.4] VOW, J., and Alefeld, G.. 'Hydrogen diffusion in metals', Diffusion in Solids,
Recent Developments, eds. A.S. Nowick and J.J. Burton, Academic Press New York (1975) 231-302.
[2.5] Johnson, H.H., 'Hydrogen in iron', Metallurgical Transactions A 19 (1988) 2371-2387.
[2.6]
Gao, H., Cao, W., Fanf, C., and Rios, E.R. de los,
'Analysis of crack tip hydrogen distribution under MI mixed mode loads', Fatigue and Fracture ofEngineering Materials and Structures 17 (1994) 12/3-/220.
[2.7] Sun, S., Shiozawa, K., Gu, J., and Chen, N., 'Investigation of deformation field and hydrogen partition around crack tip in fcc single crystal', Metallurgical and Materials Transactions 26 (1995) 731-739.
[2.8] McNabb, A. and Foster. P.K., 'A new analysis of the diffusion of hydrogen in
iron and ferritic steels', Transactions of the AIME 227 (1963) 618-627.
[2.9] Oriani, R. A., 'The diffusion and trapping of hydrogen in steel', Acta Metallurgica 18 (1970) 147-157.
2 Hydrogen trapping in steel
[2.10] Johnson, H.H.,
and Lin, R.W.,
'Hydrogen deuterium trapping in iron',Hydrogen Effects in Metal, eds. I.M. Bernstein
and A.W. Thompson,
Metallurgical Society of AIME, New York (1981) 3-23.[2.11] McLellan, R.B.. 'Thermodynamics and diffusion behaviour of interstitial solute atoms in non-perfect solvent crystals, Ada Metallurgica 27 (1979) 1655-1663. [2.12] Hirth, J.P., 'Effects of hydrogen on the properties of iron and steel', Metallurgical
Transactions A 11 (1980) 861-890.
[2.13] Kiuchi, K., and McLellan, R.B., 'The solubility and diffusivity of hydrogen in
well-annealed and deformed iron', Ada Metalurgica 31 (1983) 961-984.
[2.14] Smithells Metals Reference Book, eds. E.A.
Brandis and G.B.
Brooks, Butterworths, London, 7'1 edition (1992).[2.15] Chou, Y.T., and Li, J.C.M., 'Chemical interaction between interstitial solutes and dislocations', The interactions between dislocations and point defects, ed. B. L. Eyre (1968) 105-117.
[2.16] Zalcroczymski, T., 'The effect of straining on the transport of hydrogen in iron, nickel, stainless steel', Corrosion 41 (1985) 485-489.
[2.17] Porter, D.A., and Easterling, ICE., Phase Transformations in Metals and Alloys, Van Nostrand Reinhold,Wokingliam, England (1981).
[2.18] Pressouyre, G.M., 'Trap theory of hydrogen embrittlement', Acta Metallurgica
28 895-911.
[2.19] Shewmon, P., Diffusion in solids, TMS Warrendale USA, second edition (1989). [2.201 Kirkaldy, J.S., and Young. D.J., Diffusion in the condensed state, The Institute
of Metals, London (1987).
[2.21] Quick, N.R., and Johnson, H.H., 'Hydrogen and deuterium in iron, 49-506 °C',
Acta Metallurgica 26 (1978) 903-907.
[2.22] Caskey, G.R., and, Pillinger, W.L., 'Effect
of trapping on
hydrogen permeation', Metallurgical Transactions A 6 (1975). 467-476.[2.23] Kumnick, A.J., and Johnson, H.H., 'Deep trapping states for hydrogen in
deformed iron', Acta Metallurgica 28 (1980) 33-39.[2.24] Riecke, E., and Bohnenkamp, K., 'Uber den EinfluB von Gitterstorstellen in
Eisen auf die Wasserstofdiffusion', Zeitschrflfür Metallkunde 75 (1984) 76-81.
[2.25] Perng, T.-P.,
and Altstetter, CJ., 'Effects of deformations on hydrogen
permeation in austenitic stainless steels', Acta Metallurgica 34 (1986) 1771-1781. [2.26] Perng. T.-P.. Johnson, M., and Altstetter. C.J.. 'Influence of plastic deformations on hydrogen diffusion and permeation in stainless steels', AdaMetallurgica 37 (1989) 3393-3397.
[2.27] Xie, S.X., and Hirth, J.P., 'Permeation of hydrogen, trapping, and damage in
34 Introduction
lb this chapter a hydrogen transport model is derived for a plastically deforming body. The model is based on local equilibrium between the hydrogen in lattice sites and in trap
sites as discussed in the previous chapter. The effect of the hydrostatic stress on the
hydrogen diffusion will also be considered. The hydrogen transport model is based on that used by Sofronis and McMeeking [3.1]. It will be shown that their model does not provide a correct hydrogen balance. A modification is proposed which includes a factor depending on the plastic strain rate. This modified hydrogen transport model provides a
correct hydrogen balance. In order to model the effects of the hydrostatic stress and hydrogen trapping due to plasticity on the hydrogen distribution in a body, a coupled
diffusion elastic-plastic stress analysis is necessary. The finite element method is a very good tool for this purpose. To the author's knowledge, Sofronis and McMeeking [3.1] are the only researchers who have so far carried out coupled diffusion elastic-plastic stress finite element analyses incorporating the effect of hydrostatic stress and trapping. Toribio et al. {3.21 also carried out coupled finite element analyses on notched bars but without the trapping of hydrogen,
3.2 Hydrogen transport iequatiore
The hydrogen transport model of Sofronis and McMeeking [3.1]i is followed, which is based on the equilibrium trap theory as described in Chapter 2. Note that 'the subscript refers to lattice sites and the subscript T to trap sites. It is assumed that traps are isolated, i.e. that they do not form an extended network. Hence, hydrogen transport between trap sites is by lattice diffusion. Moreover, only one kind of trap is considered, namely one which is saturable and reversible, such as a dislocation core. Considering a body with
volume V and surface S, mass conservation requires that the rate of change of total hydrogen inside V is equal to the flux through S:
a
DT {CL + CT} AV+Si in dS = 0 t(3.1) V
where amt is the partial derivative with respect to time, CL is the hydrogen concentration in lattice sites, CT is the hydrogen concentration in trap sites, n is the outward-pointing ,unit normal vector and J the hydrogen flux, defined as
-MLCLVIAL
l3.2)
L
Numerical modelling of hydrogen transport in steel
where ML is the mobility of the hydrogen in lattice sites and p.L the chemical potential of the hydrogen in the lattice sites. There is no direct evidence available indicating which
type of interstitial site is occupied by the hydrogen atom in a-iron. Whether hydrogen atoms occupy tetrahedral or octahedral sites in a-iron, both sites are too small for the
hydrogen atom. Hence, the dissolved hydrogen atom causes a volume increase in a-iron. The chemical potential of hydrogen in regions of a tensile hydrostatic stress is therefore lower. As a consequence, a hydrogen flux is generated towards these regions in order to
lower the chemical potential. The opposite effect occurs of course in the regions of
compressive hydrostatic stresses. The chemical potential of hydrogen in lattice sites in a body under stress, neglecting higher-order terms of the hydrostatic stress ah (first-order approach [3.3.3.4]), is
C,
= p,L + RT1n - ah
L
where .t the chemical potential at a reference state, V14 the partial molar volume of
hydrogen, i.e. 2.0x10-6 m3 for a-iron at 293 K [3.5], and ah the hydrostatic stress:
.1.
=1
3WxX -yy as)* Substitution of Eq. 3.3 in Eq. 3.2 results inDLCLVH J = -DLVCL +
RT C5h
where D, is the concentration independent lattice diffusivity: DL = MLRT. Substitution of
Eq. 3.4 in Eq. 3.1 gives
DCL
ac,
DLCLVH771 dV + f {(-DLVC,
RT Vall) n} dS =
V
Using the divergence theorem we find
aCL ac,LD cor,
-V (DLVCL) +-vs (RT Vah)} dV = 0
V
Since this equation holds for an arbitrary volume V, the integrand must vanish or
+ V (DLVCL) + V (DLCI-Fix-7RT 0),) =0
acL ac,
In the case of equilibrium the hydrogen concentration in trap sites can be expressed as a function of the hydrogen concentration in lattice sites. see Eq. 2.18
(3.3) (3.4) (3.5) (3.6) (3.7) is +
NIT(E
CT 1
1 +
KTOL
where KT is the trap equilibrium constant: KT = e-4IRT with AET the trap binding energy, R the universal gas constant, i.e. 8.3144 J/mol/K. and T the absolute temperature. As
shown in Chapter 2 the number of trap sites depends on the deformation level. Hence, the number of trap sites can be expressed as a function of equivalent plastic strain Ep. Since the number of lattice sites is a constant and the temperature will be kept constant,
the partial derivative of the hydrogen concentration in trap sites with respect to time becomes
ac,
CTaCL acT dNITac,71-- ;777+
ITCLT depThe partial derivative of the hydrogen concentration in trap sites with respect to the
hydrogen concentration in lattice sites, using Eq. 3.8, is
ac,.
cT( -eT)
CL
The partial derivative of the hydrogen concentration in trap sites with respect to the
number of trap sites, using CT = ()TNT, is
acT
TNT = T (3.11)
Substituting Eqs. 3.10 and 3.11 in Eq. 3.9, the partial derivative of the hydrogen concentration in trap sites with respect to time is
acT cr(1-eT)ac, dN aE
-CL 77+
6T dejj
Finally, using Eq. 3.12, Eq. 3.7 becomesCL
cr(l
-eT)ac,LD
c,cf.
dNTac-a= v (D,,vc,) +
RT eT dE 0 (3.13)
The last term in Eq. 3.13 is called the strain rate factor. This was not taken into account in the model of Sofronis and McMeeking [3.1]. The variational and finite element form of Eq. 3.13 can be found in the Appendix.
(3.8)
(3.9)
(3.10)
(3.12)
Numerical modelling of hydrogen transport in steel
The effect of the strain rate factor in the case of an insulated, uniformly-stressed body In order to demonstrate the effect of the strain rate factor, a volume element is considered in an insulated plastically deforming body which is uniformly stressed and has a uniform starting hydrogen concentration. This is a problem without hydrogen diffusion. Firstly,
the gradients of the hydrostatic stress are zero, since the body is uniformly stressed. Secondly, the total hydrogen concentration, C. = CL + CT, must remain constant
during the loading history, since the body is
insulated. Hence, thereis only a
redistribution of hydrogen between lattice and trap sites. In this case the solution of
Eq. 3.13 results in Eq. 3.8, which can now be rewritten as
NL
-(KT+Clot+ NTT+ NTCtot=
This quadratic equation of the hydrogen concentration in trap sites has one correct
solution, given by
The other solution is not valid, as it results in a negative hydrogen concentration in lattice sites. Using Eq. 3.15 together with Eq. 2.33, the hydrogen concentration in lattice and
in trap sites as a function of equivalent plastic strain is shown in Fig. 3.1.
2.5 1.5 I NL
C =
C +
KT tot 1 0.5 0 NL1+ Ctot
+ NT)2- 4NTC,,)CT PEA without strain rate factor
CTanalytical = CT PEA with strain ratefactor PEA without strain rate factor
CLanalytical =C, PEA with strain rate factor
0.1 0.15 02
(3.14)
(3.15)
Fig. 3.1 Hydrogen concentration in lattice and in trap sites as a function of equivalent plastic strain Man insulated, uniformly-stressed body according to the analytical solution (3.15), thePEA result of the hydrogen transport model of Sofronis and McMeeking andthePEA result of the modified
hydrogen transport model.
Ci.
is the total hydrogen concentration atEl,= 0. Atcp = 0,= 2.08x10 m1 and C, = 8.42x le rn.3.
The hydrogen concentration in trap sites increases with increasing plastic strain, while the hydrogen concentration in lattice sites decreases with increasing plastic strain. While the number of trap sites is still increasing, these sites are no longer filled, due to the fact that
the lattice sites are almost empty. Since there is equilibrium between the hydrogen in lattice sites and the hydrogen in trap sites, the hydrogen concentration in lattice sites
cannot become zero. Figure 3.1 also shows the finite element results of one-element calculations based on the hydrogen transport model and on the modified hydrogen
transport model. In the case of the hydrogen transport model the hydrogen concentration in lattice sites remains constant, while the hydrogen concentration in trap sites increases until the number of trap sites becomes constant. At 0.8 plastic strain, the total hydrogen concentration is 60 times higher than the initial total hydrogen concentration. Since the
body is insulated, this means that hydrogen is created. In the case of the modified hydrogen transport model, i.e. with the strain rate factor, the finite element result
coincides with the analytical solutions, i.e. Eq. 3.15. Thus the modified hydrogen
transport model provides a correct hydrogen balance.
3.3 Conclusions
A hydrogen transport model is derived which is based on the hydrogen transport model of Sofronis and McMeeking [3.1). The model describes hydrogen transport in a plastically deforming body, including the effect of trapping and the effect of hydrostatic stress. It
was found that the model of Sofronis and McMeeking does not provide a correct
hydrogen balance. A modification of the hydrogen transport model is proposed in order to
provide a correct hydrogen balance. This is demonstrated in an insulated,
uniformly-stressed body. The hydrogen transport model is modified by including a strain rate factor.
It can be concluded that when the strain rate factor is not included in the hydrogen
transport model, hydrogen is created.
References
[3.1] Sofronis, P., and McMeeking, RN., 'Numerical analysis of hydrogen transport near a blunting crack tip', Journal of the Mechanics and Physics of Solids 37
(1989) 317-350.
[3.2] Toribio, J., Valiente, A., Cortes, R., and Caballero, L., 'Modelling hydrogen embrittlement in 316L austenitic stainless steel for the first wall of the Next
European Torus', Fusion Engineering and Design 29 (1995) 442-447.
[3.3] Leeuwen, H.P. van, A Quantitative Analysis of Hydrogen-induced Cracking,
Ph.D. thesis, Delft University of Technology, The Netherlands (1976).
[3.4] Li, J.C.M., Oriani, R.A., and Darken, L.S., 'The thermodynamics of stressed
solids', Zeitschrift fiir Physikalische Chernie Neue Folge 49 (1966) 271-290. [3.5] Mirth, J.P., 'Effects of hydrogen on the properties of iron and steel', Metallurgical
Transactions All(1980) 861-890.
Numerical modelling of hydrogen transport in steel
Appendix
Variational form of the hydrogen transport equation
Consider a body with volume V bounded by a surface S which consists of a part Se
where the hydrogen concentration is prescribed Cb and a part So where the flux through the surface is prescribed by a value 0 such that
J
= 0 (3A1))The parts of the surface S are such that: S = Se L.) S. Multiplying Eq. 3.13 by an
arbitrary function C which is differentiable and integrating over the volume of the body gives where IS "is CL+ C/(11 41+) D CL However CV (DLVCL)= V-(CDLVCL) -VC-(DLVCOF (3A.1), and DLCL:QH D CLV DLCLVH
'CV RT (C LRT Vel^- RT dhi (3A.4)
Hence Eq. 3A.2 becomes
J{C D V .(CIDLyCL)F+ (DLVCL) + V (C
RI
Vryhy DLCLC//4 dNT IC OD or - V (DLVCL) + V ( RT VG)+IeT clEP DLCLV dNT de VC RT"Val) + Carr-sal dV =0
"P ti`
dV rI0I (3A.2), (3A.5) aE L V =V Vcrh) -VCApplying the divergence theorem to the second and fourth integral, we find
DLicjill aNT ac
Sic D -a7+ VC (DLVCL ) RT Vah + dV
u`p
D,C,
+ f(- CDLVCL + C "Vah) dS = 0 (3A.6)
C may be considered to be an admissible, arbitrary variation of SCL of CL. Since the
concentration is fixed onSc
SC= 0 on Sc (3A.7)
Substitution of Eqs. 3A.1 and 3A.7 in Eq. 3A.6 gives
1( SCL D -- +VoCLDLVCL - VC D1 CL"RT -Vah+
dIV, de
) dV + f8CLO dS = 0 uCLGT dep dt
Sc)
Finite element form of the hydrogen transport equation
For the finite element formulation we define the interpolation matrix [Al, which transforms nodal point variables to local element variables, for example for the hydrogen concentration
= BA] [CU (3A.9a)
where [CL] is the vector of nodal point concentrations. Matrix [A] is assumed to be time-independent, in which case we can also write
= [A][CL] (3A.9b)
Gradients are related to the nodal point values by the gradient matrix [B]. For the gradient of the concentration
VCL = [13][CL] (3A.9c)
Similarly for the hydrostatic stress
=(Mich] (3A.9d)
(3A.8)
ac,
COT
Numerical modelling of hydrogen transport in steel
Using Eqs. 3A.9a to d. Eq. 3A.8 can be written in matrix form:
T T DLVH
J'{ [8CL]T[Ailb*[Aild.L] [15Cjr[B]rDLEBI[CLI - [oC.7/ 1 [B] BliahllANCL1+
and [1(2] = -f [B]TDLVTzrr [B][ah][A] dV and [F] = [F1] + [F2] given by [Ft] = -
f
OdS elements so and T T dNT aEKJ' [A] OT c-TEr7 atP) dV + i[SCL]T[AITOdS = 0
so
Since this equation must hold for any admissible variation [5C1.], it can be written as
f{[A]TD.[A][ L] + [1311DL[B][CL] T
DLCTHT
el-dNT de -1131 RTABircsdiAircLi + [Al. idv
f [A110 dS= 0 (3A.11) de d so Or [M][dd + [K][CL] = [F]where [M] is the concentration capacity matrix, given by
[M] = [A]TD*[A] dV
V
and [K] is the diffusivity matrix [K] = [K1] + [K2] given by
[K11 = [B]ibL[B] dV
(3A.10)
(3A.12)
+
