Special fracture behavior of nanocrystalline metals driven by hydrogen

Materials Science & Engineering A 577 (2013) 105–113

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Materials Science & Engineering A journal homepage: www.elsevier.com/locate/msea

Special fracture behavior of nanocrystalline metals driven by hydrogen Shujuan Hu a, Jianqiu Zhou a,b,n, Shu Zhang a, Lu Wang a, Shuhong Dong a, Ying Wang a, Hongxi Liu a a b

Department of Mechanical Engineering, Nanjing University of Technology, Nanjing, Jiangsu Province 210009, China Department of Mechanical Engineering, Wuhan Institute of Technology, Wuhan, Hubei Province 430070, China

art ic l e i nf o

a b s t r a c t

Article history: Received 5 December 2012 Received in revised form 2 April 2013 Accepted 7 April 2013 Available online 11 April 2013

The embrittlement of conventional metallic systems by hydrogen is a well documented phenomenon. However, the precise role of hydrogen in this process for nanocrystalline materials is poorly informed and comprehensive theoretical models are not available yet. Here, a new model is proposed wherein hydrogen atoms accumulate before a nanocrack tip and interact with piled up dislocations ahead of dislocation free zone (DFZ) actively. The interaction between hydrogen atoms and dislocations can prevent dislocations emitting from nanocrack tip, and thus suppressing nanocrack tip blunting and ductile fracture while promoting brittle failure. In addition, the size of DFZ in nanograins was analyzed from the macro–micro fracture mechanics point of view and the relative computing method was derived. The dependence of maximum number of dislocations emitted from nanocrack tip on grain size with and without hydrogen in nanocrystalline Ni is clarified and compared. The results show that the introduction of hydrogen into nanocrystalline materials gives rise to a reduction in critical crack intensity factor more than 30% in contrast with hydrogen free case, and this special fracture behavior driven by hydrogen atoms is especially remarkable with the reduction of grain size. & 2013 Elsevier B.V. All rights reserved.

Keywords: Nanocrystalline metals Hydrogen embrittlement Brittle fracture Dislocation emission Dislocation free zone

1. Introduction The embrittlement of metallic systems by hydrogen is a kind of serious environmental failure that affects almost all metals and their alloys. With the development of technology and the establishment of low-carbon society, the usage of high-strength structural materials for lightweight construction and energy conservation becomes a necessity. Although material scientists try their best in developing alloys that integrate excellent tensile strength with prominent fracture toughness, hydrogen embrittlement still poses a widespread hazard to the fracture resistance of metallic materials. Moreover, with the depletion of fossil fuels, mankind is searching for additional alternative energy. Hydrogen is believed to be an appropriate substitute energy source and it is most likely that a “hydrogen-based economy” will be realized within the next several decades. In this particular case, largescale production, storage, transportation and utilization of hydrogen will become inevitable. However, detrimental problems that accompany the employment of hydrogen are substantial. Of vital significance is the impairing effect of hydrogen on the mechanical properties of materials, particularly high strength steels and nickel, where hydrogen uptake in the metal can induce premature, unexpected and potentially n Corresponding author at: Department of Mechanical Engineering, Nanjing University of Technology, Nanjing, Jiangsu Province 210009, China. Tel.: +86 25 83588706; fax: +86 25 83374190. E-mail addresses: [email protected], [email protected] (J. Zhou).

0921-5093/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.msea.2013.04.019

catastrophic failures [1]. The degradation in mechanical properties of metals due to hydrogen is often followed by a sharp transition in fracture mode from ductile fracture (i.e. strain controlled microvoid coalescence) to brittle fracture (i.e. stress controlled intergranular cracks propagation) [2–8], despite substantial plastic deformation may still occur in this embrittling process. For example, slip trace along the fracture path is observed in high-resolution fractography when embrittlement occurs in Ni and its alloys [3–5,9–12]. Ever since 1874, scores of scientific predecessors devote themselves to filling in this blank field. Although extensive experimental and theoretical studies have brought about various mechanisms concerning hydrogen-induced embrittlement, the conclusive predictions for hydrogen embrittlement that reflect the fundamental mechanisms have remained unavailable. One feasible reason that accounts for this embarrassment may be that each candidate is supported by its specific experimental observations, theoretical hypothesis and personal views. Among the diversified proposals, three candidates seem to be quite fascinating, namely (i) hydrogenenhanced decohesion (HEDE) [13–18], where the segregation of hydrogen atoms at internal interfaces is assumed to weaken the surface cohesive energy there, and facilitate more cleavage-like brittle failures consequently. This theoretical model was based upon several experimental observations [18–20]. (ii) Hydrogen-enhanced local plasticity (HELP) [21–22]. In the HELP mechanism, hydrogen atom is expected to decrease the motion barriers for dislocations, thereby aggravating the extent of plastic deformation that occurs

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in a relatively localized region adjacent to the fracture surface [23]. The fracture process is not so much a hydrogen-induced brittle failure as a highly localized plastic deformation. (iii) Hydride formation and cleavage [24–26]. This suggestion is viable under the specific circumstances where hydride is feasible to form, and this “new comer” is thought to provide a “low energy” fracture path for brittle failure [27]. Generally speaking, while each proposal has its unique merits, and while different proposals may be suitable for different materials, none of them have a solid proof to explain the existing phenomenon concerning hydrogen embrittlement for nanocrystalline materials comprehensively. Hence, more in-depth experimental and theoretical investigations are welcomed in this field. Here we propose a mechanism for hydrogen embrittlement in which we focus on the sharp transition from ductile fracture to brittle one. Particularly, our efforts are devoted to quantifying the aggravated effect of grain size on maximum number of dislocations emitted from a nanocrack tip under the circumstance where hydrogen atmosphere is formed in nanocrystalline materials. The distinguishing feature of this work is that by employing the present model, one can clarify the size dependent relationship between grain size and dislocation free zone (DFZ). In the framework of our theoretical model, it is assumed that hydrogen atoms accumulation around the rim of DFZ prevents dislocations emission from the nanocrack tip, and thus suppresses crack-tip blunting while impelling brittle failure. At the same time, whether the nanocrack propagates or not is considered to be of vital significance in the failure course of nanocrystalline materials. A detailed process can be seen in Fig. 1. Fig. 1(a) shows that a nanocrack becomes blunt due to dislocations emission without the invasion of hydrogen atoms. On the contrary, with the introduction of hydrogen atoms, a rather distinctive event will occur. As can be seen in Fig. 1(b), substantial hydrogen atoms accumulate against DFZ which just lie in the immediate front of the nanocrack. The existence of hydrogen atoms suppresses further dislocations

emission and prevents nanocrack from blunting. Consequently, it advances forward dramatically. The remainder of this paper is aimed at verifying the mechanism sketched above by means of micro-mechanics and numerical analysis in Ni–H system, and it is structured as follows: In Section 2.1, the stress field of DFZ around the pre-existing nanocrack in nanocrystalline materials was described. In Section 2.2, the classic Von Mises yield criterion and the Hall–Petch relationship were employed to predict the size dependence of DFZ upon grain size for a pre-cracked nanocrystalline bulk. In Section 2.3, the comprehensive effect of grain size and hydrogen atoms on dislocation nucleation and brittle fracture in Ni–H system at room temperature was studied. In Section 3, the developed model was discussed further on the basis of the results that we obtained. In Section 4, within the theoretical frame of this work, a more conclusive idea about the mechanism that we recommended is summarized.

2. Hydrogen-induced aggravated effects of grain size on crack blunting in nanocrystalline materials 2.1. The stress field around DFZ in nanocrystalline materials Both experiments and previous theoretical analyses [28–33] imply that brittle-cleavage process is driven by dislocation piled up against the DFZ, and a trigger point mainly locates at the tip of piled-up dislocations that are adjacent to a nanocrack tip. Consequently, the size of DFZ is of vital scientific importance in understanding the brittle versus ductile behavior caused by hydrogen in nanocrystalline materials. The formation of DFZ in nanocrystalline materials means that a square-root stress singularity in the immediate vicinity of the nanocrack tip is restored, and it is feasible to characterize the highly distorted and localized stress field near/at the nanocrack tip

Fig. 1. A schematic of the evolution of a pre-existing nanocrack which is subjected to a remote loading: (a) without the attendance of hydrogen atoms, the nanocrack is blunted by emission of dislocations from the crack tip, and (b) with the introduction of hydrogen atoms, they segregate near the nanocrack tip substantially and inhibit dislocation emission, favoring brittle fracture in return.

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with a stress intensity factor as well. Ohr [34] pointed out that without DFZ, the local stress intensity factor approaches zero and the crack will not propagate because there is no elastic energy release associated with the propagation. It was found that the prerequisite for the generation of dislocations from the nanocrack tip can be defined in terms of a local effective stress intensity factor K nI . The DFZ was shown to be a manifestation of an impediment to the generation of dislocations at the nanocrack tip. This impediment was also shown to be directly related to the local effective stress intensity factor for dislocation emission. Therefore, our following effort should be focused on determining the local effective stress intensity factor K nI around the nanocrack tip. As mentioned in Ref. [34], the condition for nanocrack tip deformation can be expressed by local effective stress intensity factor K nI . Once emitting, the dislocations will interact elastically with the nanocrack and change the magnitude of K nI in return. This change in K nI will influence the condition for nanocrack propagation as well as subsequent emission of dislocations from the nanocrack tip. Hence, to treat this problem systematically and accurately, one must calculate the stress field in terms of dislocations [35]. Next, let us consider a pre-cracked nanocrystalline bulk which is subjected to uniaxial tension as schematically shown in Fig. 2. If the local effective stress intensity factor that is close to the nanocrack tip is sufficiently large, just as demonstrated by Ovid'ko and Sheinerman [36], the nanocrack tip can induce plastic deformation through the emission of lattice dislocation from the nanocrack tip. Since grain boundaries (GBs) serve as obstacles for lattice dislocation slip, we suppose that the emitted dislocations are trapped at the neighboring GB. As a corollary, the hampered dislocations modify the stress field of the nanocrack in return. Following the same method proposed by Liu et al. [37], we will use the standard crack growth criterion based on the balance between the driving force related to the decrease in elastic energy and the hampering force relevant to the formation of a new free surface during the course of crack growth. The criterion can be written as [38] 1−ν 2 ðK þ K 2II Þ ¼ 2γ 2G I

ð1Þ

107

where v is Poisson's ratio, G is the shear modulus, γ is the specific energy of the new free surface, K I and K II are the intensity factors for normal stress (to the nanocrack line) and shear stress respectively. In this paper, we assume that the nanocrack's growth direction is perpendicular to the direction of the external loading; hence, the coefficients K I and K II can be given as ∧ d

K I ¼ K sI þ kI ;

∧ d

K II ¼ kII

ð2Þ

the stress intenwithin this theoretical delineation, K I represents ∧ ∧ d d sity factor for the external load s, while kI and kII are the stress intensity factors caused by the emitted dislocations (Fig. 2). As delineated before, the influence of dislocations on nanocrack propagation can be reasoned by the introduction of K nI . However, the nanocrack's growth direction needs to be perpendicular to the tensile load direction, so as to assure that the newly generated dislocations just change the magnitude of K nI compared with the none emitting case. Under such circumstances, the critical condition for nanocrack growth can be represented as [39]: K sI ¼ K nI ; according to formulas (1) and (2) and the critical condition K sI ¼ K nI , we can get the following expression for K nI : K nI ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4Gγ ^ d 2 ^ d −ðkII Þ −kI 1−ν

ð3Þ

d d In an attempt to calculate the stress intensity factors k^ I and k^ II , it should be noted that within the theoretical frame of this paper, the generation of the first dislocation is followed by the emission of further dislocations along the same slip plane. The newly emitted dislocations glide away from the nanocrack tip and come to rest on their equilibrium positions that are determined by the balance of the force exerted by the external stress (that facilitates dislocation slip) and the force subjected by the formerly emitted dislocations (that retards dislocation slip). As carefully elaborated in Ref. [37], we can calculate the dislocation-induced stress d d intensity factors k^ I and k^ II for ith dislocation as d

kIi ¼ −

3πDb sin θcosðθ=2Þ d πDb½cosðθ=2Þ þ 3cosð3θ=2Þ pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi ; kIIi ¼ − 2 2πr i 2πr i

Fig. 2. Nanocrack in a deformed nanocrystalline material bulk. The magnified inset highlights the emission of edge dislocations from the nanocrack tip.

ð4Þ

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where b denotes the Burgers vector magnitude of the newly generated dislocations, r i denotes the glide distance of the ith dislocation. In order to simplify the calculation, we treat r i as a constant value which is equal to the grain diameter d. θ is the angle made by the slip plane and the Cartesian coordinate axis Ox. If there are N dislocations emitted from the nanocrack tip along a same slip plane already, the stress intensity factors induced by all dislocations are introduced by Liu et al. [37]: ∧ d

N

∧ d

d

kI ¼ ∑ kIi ;

N

d

kII ¼ ∑ kIIi

i¼1

i¼1

ð5Þ

the following inequality: N

sKrθI ðr N ; θÞ þ sim rθ ðr N ; θÞ þ ∑ srθ ðr N ; r j ; θÞ 4 0 j¼2

ð9Þ

It is worthy to be noted that formula (9) only gives the necessary condition for dislocation emission, but not the sufficient condition. That is to say, the real conditions for dislocation emission are much more complicated than described in formula (9). The stresses sKrθI ðr N ; θÞ, sim rθ ðr N ; θÞ and srθ ðr N ; r j ; θÞ presented in formulas (8) and (9) can be calculated by the same method that is introduced by Lin and Thomson [44]. 2.3. The aggravated effect of grain size on dislocations emission caused by the presence of hydrogen atoms in nanocrystalline materials

2.2. The size dependence of dislocation free zone on grain size in nanocrystalline materials The purpose of the following part is to employ the combination of the classical Von Mises yield criterion and the Hall–Petch relationship to predict the size dependence of DFZ upon grain size for a pre-cracked nanocrystalline bulk, however, with two important modifications and one restriction that are (1) the original stress intensity factor in Von Mises yield criterion is a constant value that represents the stress field in the vicinity of the crack, whereas this value has nothing to do with the emission of dislocations from the crack tip. Hence, we choose the local effective stress intensity factor K nI as an alternative one, so that the stress field near the nanocrack tip is a variable in accordance with the emission events. (2) The material strength in this region follows the classical Hall–Petch relationship, namely, yield strength increases with decreasing grain size [40]. To obtain the correct size of the DFZ in question, we will treat the yield strength as alterable instead of constant. (3) As summarized in Refs. [40,41], only in the case of d ≥10 nm where the classical Hall–Petch description can be roughly used, we restrict grain sizes in our research ranging from 15 nm to 200 nm rigidly. After the complement and the amendment, one can finally obtain the formula that is suitable for calculating the size of DFZ in nanocrystalline grains, as follows:   K nI 3 2 r np ¼ sin θ þ ð1 þ cosθÞ ð6Þ n 2 4πsys 2

A hydrogen-induced ductile versus brittle response driven by hydrogen atoms that accumulate around DFZ can be envisaged as follows. In general, the generation of the first dislocation is accompanied by the generation of further dislocations along the same inclined slip plane subsequently. The newly emitted dislocations glide away from the nanocrack tip until they get to their equilibrium positions, and those equilibrium positions are determined by the balance of the stress subjected by the applied shear stress (which favors dislocation emitting) and the stress exerted by the formerly generated dislocations together with the “hydrogen atmosphere” (which impedes dislocation generation and gliding), just as shown in Fig. 3. As Rice and Thomson [43] stated, brittle failure is inhibited by dislocations emission for the intrinsic ductile materials. It is worthy to note that the above scenario is validated under the circumstances where lattice-mediated deformation is dominant. That is to say, the grain size is supposed to be large enough, the emitted dislocations can slip forward far enough from the nanocrack tip and do not hinder the generation and motion of new dislocations significantly unless the number of the emitted dislocations becomes saturated [36]. Only in this way can we hope to imagine that the dislocation emission along one inclined slip plane

where snys represents the variable yield stress given by [40] n

snys ¼ s0 þ kh d

ð7Þ

d is the average grain size (from 15 nm to 200 nm in this study), snys is the 0.2% yield strength (or hardness), s0 is the lattice friction stress to move individual dislocation (or the hardness of a single crystal specimen,d-∞), n is the grain size exponent (normally −1/2), and kh is a constant (often termed as the Hall–Petch slope and is material dependent), called Hall–Petch intensity parameter [40–42]. To calculate the size of DFZ, we will use the following calculation procedures. To begin with, the number N of dislocations that are emitted from a nanocrack tip along one inclined slip plane should be calculated by the Rice–Thompson criterion [43]. Following this criterion, we assume that the prerequisite for the emission of first dislocation could be met if the distance from the crack tip to the equilibrium position where the first emitted dislocation is located exceeds the dislocation core radius r 0 . Hence, the critical condition for the emission of the first dislocation is sKrθI ðr 1 ; θÞ þ sim rθ ðr 1 ; θÞr ¼ r0 4 0

ð8Þ

Moreover, if this criterion is valid, we will examine whether it is possible for the emission of the N th dislocation ðN ¼ 2; 3; …Þ by

Fig. 3. Hydrogen atoms (red balls) accumulate around DFZ and interact with the heavily inverse piled up dislocations, consequently, inhibiting the generation of dislocations from nanocrack tip and favoring brittle fracture. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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can arrest nanocrack growth and make the solid ductile. Following Refs. [36,37,45], owing to the repulsive force between dislocations that are arrested at the opposite grain boundary during former plastic deformation, the generation of even one dislocation will impede the generation of the following dislocations along the same inclined plane. Under such circumstances, dislocation emission from the nanocrack tip does not lead to efficient blunting. Consequently, the nanocrystalline bulk is inclined to behave in a brittle manner instead of a ductile one. Then, let us calculate the maximum number N max of dislocations generated from a pre-blunted nanocrack which lies in the interior of the grain along one inclined slip plane as a function of the grain size d, as schematically shown in Fig. 4. Before carrying out the calculating task, three aspects should be taken into consideration. Firstly, the calculating process will be carried out under the circumstance where N dislocations have already existed and are located at their equilibrium positions respectively. Hence, the already emitted dislocations will impose a stumbling force on the ðN þ 1Þth dislocation which is going to emit. Secondly, it is assumed that an opposite dislocation will be generated inside the nanocrack as a consequence of the generation of each dislocation from the nanocrack tip. In other words, if we try to construct a Burgers contour around the internal crack, we also gain a non-zero Burgers vector which can feature the crack in return. As a corollary, after the emission of ðN þ 1Þth dislocations, there will be a super dislocation formed inside the nanocrack of which the Burgers vector amounts to ðN þ 1Þb (see Fig. 4) [36]. Finally, it is of vital importance to note that the stress field in the vicinity of this super dislocation does not depend on where it located, as was noted in Refs. [36,41]. In other words, this internal nanocrack can be characterized by this super dislocation no matter where it located. Hence, the total stress subjected to the ðN þ 1Þth emitted dislocation can be projected onto the r-axis as follows: N

sðr; θÞ ¼ semit ðr; θÞ þ sim ðr; θÞ þ ∑ sdd ðr k ; r; θÞ k¼1

−ðN þ 1Þsdd ð0; r; θÞ−sh ðr; θÞ

109

the projection stress of the super dislocation inside the crack exerted on the ðN þ 1Þth generated dislocation onto the r-axis, and sh ðr; θÞ is the interactive stress between the ðN þ 1Þth emitted dislocation and hydrogen atoms that accumulate around DFZ. The detailed formulas for these stress components are listed in Appendix A. It is supposed that the generation of the ðN þ 1Þth (N ¼1, 2, 3, …) dislocation is possible within the interval r 0 o r ≤ d (r 0 is the dislocation core radius) if the following inequality could be validated, that is N

semit ðr; θÞ þ sim ðr; θÞ þ ∑ sdd ðr k ; r; θÞ k¼1 dd

−ðN þ 1Þs

ð0; r; θÞ−sh ðr; θÞ 40

ð11Þ

Based on the above mentioned theoretical preparation, the value of N m which represents maximum number of dislocations that emit along the same slip plane can be calculated by the following calculation procedures. First of all, we suppose there already exist a certain number of dislocations around the nanocrack tip. That is to say, the nanocrack has been blunted due to previous consecutive processes of plastic deformation already. Moreover, the emission of ðN þ 1Þth dislocation is so fast that the local effective stress intensity factor K nI (though it is closely related to the emission event) does not change in this interval. Similarly, the size of the DFZ remains unchanged during this course. However, a case worth mentioning is that the shape of DFZ translates in accordance with the generation of each new dislocation actually. Finally, we verify the validity of inequality (11) for the first dislocation. If this inequality is valid, we place the first dislocation at opposite GB and go on to verify the validity of inequality (11) for the emission of the second dislocation. If this inequality remains valid for the second dislocation, we calculate the equilibrium position for it and so for the emission of the third dislocation and so on. This procedure will be carried out for all the new dislocations and ends when inequality (11) is no longer valid.

ð10Þ

where semit ðr; θÞ is the projection stress imposed by the applied loading onto the r-axis, sim rθ ðr; θÞ is the projection of the image stress created by free boundary of crack surface onto the r-axis, sdd ðr k ; r; θÞ is the projection stress that the K th dislocation exerts on the ðN þ 1Þth dislocation onto the r-axis, −ðN þ 1Þsdd ð0; r; θÞ is

Fig. 4. Dislocations emit from a blunted nanocrack tip due to former plastic deformation.

3. Results and discussion 3.1. Effect of grain size on dislocation free zone for nanocrystalline metals By putting formulas (1)–(5) into usage, the local effective stress intensity factor K nI can be determined. And then, with the availability of formula (7), one can obtain the value of snys in accordance with the grain size as well. Depending on these calculating procedures, we have calculated r np which represents the size of DFZ as a function of grain size d for nanocrystalline Ni (Fig. 5). For definiteness and simplification, we put θ¼ 701 and b ¼0.25 nm in formula (4). The following typical parameters of nanocrystalline nickel (Ni) were also employed: G ¼73 Gpa, v ¼0.34, γ¼ 1.725 Jm−2. Fig. 5 schematically shows that r np significantly decreases in accordance with the reduction in grain size d. Particularly, an increase in grain size from 15 nm to 200 nm makes r np nearly 25 times larger. This means that, when the grain size reduces to a certain value, the emissive event along one inclined slip plane cannot contribute to significant crack blunting and so to the enhancement of the local effective stress intensity factor K nI next to the nanocrack tip. Moreover, it is reported that a GB may require half as many dislocations for equilibrium compared to no blockage [29]. It is further reported that the GB gains an increasing fraction as the grain sizes become smaller. Particularly, although there are about 50% of the atoms that locate in boundaries for nanocrystalline materials with an average grain size of 5 nm, it is not difficult to imagine that the smaller the grain size, the smaller the DFZ. Lastly, in accordance with the

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Fig. 6. Local effective stress intensity factor K nI vs. grain size d along one slip plane in nanocrystalline Ni.

Fig. 5. The size of dislocation free zone as a function of grain size d in nanocrystalline Ni.

downgrade of the grain size, snys takes on an ascending trend; however, as clearly elaborated in Fig. 6, with the reduction of grain size, the magnitude of the local effective stress intensity factor decreases consequently, this declining trend is particularly evident in the area where grain size is below 30 nm. Hence, by taking these two factors together, the decreasing size of DFZ can be accounted in a more potent way. 3.2. Influence of angle θ (between slip plane and the x-axis) on maximum value of r np in nanocrystalline materials On the basis of the above section, we can get the notion that r np obtains its maximum value in the first quartile clearly. Hence, by setting θ ranging from 01 to 901, we are able to clarify the relationship between θ and r np which is expressed in detail in Fig. 7. After analyzing the phenomenon described in Fig. 7 carefully, we acquire the knowledge that r np increases along with the increment in θ until it reaches its maximum value around 701 (exactly 70.531) and then it downgrades in a slow manner (this trend is especially evident for the bigger grain size). There are two causes available accounting for the observed phenomenon: Firstly, when the value of θ reaches its maximum value of 70.531, the stress intensity factor that is required for a single dislocation to be rested in an equilibrium position along the slip plane ahead of the nanocrack tip attains its minimum value [46]; putting it in another way, under the same stress conditions, the dislocation can glide further when the slip plane makes the angel θ around 70.531 with the x-axis. Secondly, local effective stress intensity factor K nI in the vicinity of the nanocrack tip achieves its maximum value under the condition of θ¼70.531 as elaborated by Liu et al. [37], judging from Eq. (6), we can get r np;max at θ¼70.531 as well. 3.3. Inhibiting effect of hydrogen on emission of dislocation from nanocrack tip With the circulated calculating procedure that is represented in detail in Section 2.3, we calculated the maximum number N m of the dislocations that are emitted along the same inclined slip plane as a function of grain size d for nanocrystalline Ni with and without hydrogen at room temperature and low strain rate.

Fig. 7. Effect of angle θ (between inclined slip plane and the x-axis) on the size of dislocation free zone.

The values of typical parameters for nanocrystalline bulk of Ni are chosen as follows: G ¼73 Gpa, v ¼ 0.34, γ¼ 1.725 Jm−2, r 0 ¼ 2nm, b¼ 0.25 nm, θ¼70.531. As can be seen from Fig. 8, when the hydrogen atmosphere is formed against DFZ, N m becomes especially size dependent in comparison with its counterparts under the circumstance where hydrogen atoms are not available. It also should be noted that with the invasion of hydrogen atoms on the interface between plastic zone PZ and DFZ, the emissive event from the nanocrack tip is strongly hampered, which is particularly evident when the grain size is smaller than 30 nm. It means that for nanoscale grains, once the hydrogen atmosphere is formed at the immediate vicinity of the nanocrack, crack blunting caused by emitting lattice dislocations along the same inclined slip plane becomes extravagantly hard and the crack can advance forward in a more feasible way. As a corollary, the effect of grain size on ductile vs brittle response featured by a inter-granular fracture that is followed by a dramatic reduction in ductility and toughness is more remarkable for nanocrystalline materials under the circumstance where the hydrogen atoms are introduced into the rim of the DFZ. Although the advancing rate of the nanocrack may be rather smaller, it will be lead to failure of the bulk by a brittle manner eventually.

S. Hu et al. / Materials Science & Engineering A 577 (2013) 105–113

111

rates in the vicinity of the crack in Ref. [2]. By introducing these two declarations into the theoretical framework of this paper, one can come to a verdict which is consistent with the phenomena illustrated in Fig. 8. 3.4. A remarkable decline in critical crack intensity factor caused by hydrogen in nanocrystalline materials

Fig. 8. The maximum number N m of edge dislocations that can be emitted from the crack tip along the same inclined slip plane as a function of grain size d in nanocrystalline Ni with and without the invasion of hydrogen atoms.

Additionally, when the enhancement of local effective stress intensity factor (due to increase in grain size and numbers of dislocations emitted from the crack tip consequently) meets the weakening effect (caused by the augmentative size of DFZ) of hydrogen atoms on the dislocation which is ready to emit, we hope much more dislocations could emit from the nanocrack tip and make the nanocrystalline bulk much more ductile. However, drawn from Fig. 8, we cannot get the conclusion as supposed either. This paradoxical phenomenon can be attributed to (i) increased local effective stress intensity that allows for further segregation of hydrogen atoms around the rim of DFZ, which can further suppress dislocation emission in return; (ii) local expansion (caused by the ingression of hydrogen atoms) that generates a high hydrostatic compression in the vicinity of the crack tip which will counteract the enhancement of the local effective stress partly; and (iii) in calculating these values, we take dislocation emission along one slip plane into consideration only. Apparently, allowing for dislocations emission along multiple slip planes in the course of crack advancement would increase the calculated values of dislocations for any given grain size and make the deteriorative effect of grain size to be caused by hydrogen atoms on dislocation emission (N max decreases with the reduction of grain size) even more dramatically. Moreover, since the inhibiting effect on dislocation emission in nanoscale can be accounted for by the size effect that prevents the activity of lattice dislocation sources (take Frank–Read sources as an example), it is also viable for us to relate the nanocrack tip (act as an dislocation source) with the DFZ which is equipped with “super hydrogen” from this point of view. With the decrease in grain size, the distance from the “dislocation source” (nanocrack tip) to “trigger point” (DFZ) shrinks accordingly. Hence, it is not surprising to find that the presentation of hydrogen atoms ahead of the DFZ aggravates the ductile vs brittle response of nanocrystalline materials in accordance with the reduction of grain size. Above all, for nanocrystalline metals, it is widely acknowledged that the jump rates of atoms along dislocations and grain boundaries are much higher than that in lattice [47]. Because of the higher diffusivity in these regions, they were often termed as “high-diffusivity paths” or “diffusion short circuits”. Therefore, we can draw an conclusion that the influence of grain-boundary diffusion is much smaller in coarse-grained than in nano-grained crystallines. Besides, Scully et al. have proved that the crack propagation rates directly correlate with hydrogen atoms diffusion

In order to characterize the effect of hydrogen and initial nanocrack tip curvature radii ρ0 on crack growth, one should 12 21 22 compare the value of K 11 IC , K IC , K IC and K IC . Thereinto, the first superscript 1 represents the case where hydrogen atoms are not available in the vicinity of the nanocrack tip. On the contrary, the first superscript 2 means that a sufficient hydrogen atmosphere is formed already. Moreover, the second superscript 1 characterizes the condition that the initial nanocrack tip curvature radii is ρ0 ¼ 0nm(a sharp crack), whereas, the second superscript 2 means that the curvature radii of the nanocrack tip that we take into consideration are deemed as ρ0 ¼ 5nm(a blunt crack). Previously, the maximum number of dislocations that can be emitted from the tip along the same inclined plane of a blunt nanocrack has been calculated with and without the invasion of hydrogen atoms. In doing so, we should do some assumptions as introduced by Ovid'ko et al. [36]. Firstly, the nanocrack has been blunted already because of the previous processes of dislocation emission and crack advancement. Secondly, the emission of dislocation is so fast that the local stress intensity factor K nI does not change during this emission interval. However, in this part, we will consider an otherwise condition. We suppose an initially sharp crack (ρ0 ¼ 0 nm) is blunted due to the emission of dislocations from the nanocrack tip along the same inclined slip plane. Moreover, we deem that the emission of dislocations demands some time intervals during which plastic deformation can proceed and the applied external loading can increase. Consequently, the local effective stress intensity factor K nI closely related to the applied load s is also supposed to be able to gain an enhancement during the emission course. Under such a scenario, the maximum number of emitted dislocations Nm corresponds to the situation where K nI ¼ K nI ðNÞ, i.e. K nI gradually approaches its critical value K nIC ðNÞ with each new generation of dislocation from the tip of the nanocrack. The enhancement in the critical stress intensity factor is caused by the emission of dislocations along the same inclined slip plane. As a corollary, after the emittion of the last dislocation, the critical stress intensity factor is K nIC ¼ K nIC ðN m Þ. Hence, it is also validated for us to calculate the critical crack intensity factor K nIC ðN m Þ by formula (3). Fig. 9 shows the critical crack intensity 12 21 22 factors K 11 IC , K IC , K IC and K IC as a function of grain size d; the superscript x means 11, 12, 21, 22. It proves that all K xIC increases with the increasing grain size d and thus enhances the toughness of the nanocrystalline materials markedly. On the contrary, when the grain sizes downscale to the smaller 22 scope, K 21 IC and K IC decrease dramatically (more than 30% ) thus making the nanocrystalline materials much more brittle compared 12 with their counterpart K 11 IC and K IC respectively. Judging from Fig. 9, one can get the notion that the accumulated hydrogen atoms against DFZ quicken the transition in fracture mode from ductile failure to brittle one in comparison with the case where nanocrystalline materials are not injured by hydrogen atoms. It is of vital importance to note that with the invasion of hydrogen atoms on DFZ, the repressive effect on the emission of dislocations from the nanocrack tip is much more stronger for the sharper crack, and this phenomenon is particularly evident when the grain is smaller than 30 nm. As clearly demonstrated in Fig. 9, the smaller the crack tip curvature radii, the smaller the critical crack intensity factor. That is, the sharper crack is more prone to behave

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S. Hu et al. / Materials Science & Engineering A 577 (2013) 105–113

(10502025, 10872087, 11272143), Natural Science Foundation of Jiangsu Province (BK2007528) and Natural Science Foundation of Hubei Province (Q20111501).

Appendix A Stress field of a blunt crack interacting with already emitted dislocation and hydrogen atoms The interaction between a blunt nanocrack, already emitted dislocations and hydrogen atoms has been explained in detail in the text. In this appendix, we supplement every stress components by the following complex potentials. The stress semit ðr; θÞ can be termed as semit ¼ bsrθ , where b is the rθ magnitude of the Burgers vector for each dislocation and srθ is the component of the stress field introduced by the external tensile loading s near the nanocrack. Hence, following Ref. [52], we can define srθ as follows: 0

Fig. 9. Critical crack intensity factor K xIC vs. grain size d in nanocrystalline Ni under various circumstances.

in a brittle manner with respect to the blunted ones. Apparently, the critical stress intensity factor decreases according to the decrease in curvature radii of the nanocrack tip until it becomes smaller than the critical value at which fracture toughness reaches a constant value. This agrees with the experimental fact that a smaller radius of curvature leads to a thicker nanohydride, consistent with the higher stress around a sharper nanocrack [48]. Hence, if the hydrogen atmosphere is formed before the tip of the nanocrack, the nanocrystalline materials will show a ductile vs. brittle transition with the reduction in grain size in a more effortless manner.

srθ ðr k ; r; θÞ ¼ Im½ðzφ} ðzÞ þ ψ ðzÞÞe2iθ ;

ðA1Þ

where φ and ψ are the complex potentials:   sR 1 ξ−ð2 þ mÞ φ¼ 4 ξ

! sR 1 ð1 þ mÞð1 þ mξ2 Þ ξ− − ðA3Þ 2 ξ ξðξ2 −mÞ pffiffiffiffiffiffi z ¼ x þ iy ¼ a þ reiθ ,i ¼ −1, the overbar represents the compffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi plex conjugate, R ¼ að a þ ρÞ=2,m ¼ ð a− ρÞ=ð a þ ρÞ, and ξ is  one of the two roots of the equation z ¼ Rðξ þ m=ξÞ, such that   ξ≥1. The image stress imposed by the surface of nanocrack on the ðN þ 1Þth dislocation can be given as [53]: ψ¼

sim ðr; θÞ ¼ b lim ½sim rθ ðr 0 ; r; θÞ

ðA4Þ

r a -r

4. Conclusions In summary, we have proposed a new mechanism of hydrogeninduced embrittlement for nanocrystalline metals in a typical situation where crack blunting and propagating processes are controlled by dislocations emission from the tip of the nanocrack, and the mechanism was demonstrated in the Ni–H system. DFZ is of vital significance in this embrittling mechanism. The accumulation of hydrogen atoms against the DFZ is found to suppress dislocation emission from nanocrack tip and encourage nanocrack propagation, thus driving a transition in fracture mode from ductile failure to brittle one. The TEM in-situ observation showed that subcracks and nanovoids were nucleated in this highly stressed area and then linked to the main crack, which will accelerate the brittle fracture process intensively [21,49,50,51]. From numerical computations, it can be found that when the hydrogen atmosphere is formed against DFZ, the critical crack intensity factor is much more sensitive to the grain size compared with their counterpart under the hydrogen free condition. Particularly, when the grains decrease from 100 to 15 nm, the reduction in critical crack intensity factor is more than 30%. It was also found that, for the same hydrogen atmosphere, the sharper cracks are more prone to hydrogen embrittlement, and this result matches well with the molecular dynamics simulations operated by Song and Curtin [48].

ðA2Þ

The stress sim rθ ðr 0 ; r; θÞ can be calculated through the employment of the potentials ϕim and ψ im as [54] 2iθ sim  rθ ðr 0 ; r; θÞ ¼ Im½ðzφ″im ðzÞ þ ψ′im ðzÞÞe

ðA5Þ

where φim ðzÞ and ψ im ðzÞ are specialized by the same manner in Ref. [36]:     m 1 φim ðzÞ ¼ 2Aln ξ−A ln ξ− −A ln ξ− ξd ξd 2

þA

ξd ð1 þ mξd Þ−ξd ðξ2d þ mÞ

ðA6Þ

2 ξd ξd ðξd −mÞðξ−1=ξd Þ

    m 1 ψ im ðzÞ ¼ 2A ln ξ−A ln ξ− −Aln ξ− ξd ξd 2

þA

ξd ðξ2d þ m3 Þ−mξd ðξd þ mÞ ξd ξd ðξ2d −mÞðξ−m=ξd Þ

−ξ

1 þ mξ2 dφim ; ξ2 −m dξ

ðA7Þ

The parameter named ξd is the solution  of the equation   zd ¼ Rðξd þ m=ξd Þ under the circumstance of ξd ≥1. Additionally, other related parameters in Eqs. (A6) and (A7) can be termed as z ¼ a þ r 0 eiθ zd ¼ a þ reiθ A ¼ Gbeiθ =½4πið1−νÞ

ðA8Þ

Acknowledgments

sdd ðr k ; r; θÞ stands for the stress exerted by the already emitted dislocation which is located at the point ðr k ; θÞ on the ðN þ 1Þth dislocation, and it can be described as

This work was supported by Key Project of Chinese Ministry of Education (211061), National Natural Science Foundation of China

sdd ðr k ; r; θÞ ¼ bsdrθ ðr k ; r; θÞ

ðA9Þ

S. Hu et al. / Materials Science & Engineering A 577 (2013) 105–113

Following Ref. [36], we get the expression of srθ ðr k ; r; θÞ by sdrθ ðr k ; r; θÞ ¼

Im½ðzφd00

ðzÞ þ

ψ d0 ðzÞÞe2iθ 

ðA10Þ

here φd ðzÞ ¼ Alnðz−zd Þ þ ϕim ðzÞ ψ d ðzÞ ¼ Alnðz−zd Þ−A

zd þ ψ im ðzÞ z−zd

z ¼ a þ reiθ ; zd ¼ a þ r k eiθ

ðA11Þ ðA12Þ ðA13Þ

Lastly, the stress sh ðr; θÞ is created by hydrogen atoms and it can be featured by sh ðr; θÞ ¼

2An ρðxr −xh Þ ½ðxr −xh Þ2 þ ρ2 2

ðA14Þ

where An is the constant which stands for the magnitude of mechanical interactions between the dislocation which is going to emit and the hydrogen atoms locating around the rim of DFZ. ρ is the range of the Cottrell atmospheric distance between an atom and a solute atom [55]. xr denotes the site of the ðN þ 1Þth dislocation, and xh signifies the site that is occupied by the super hydrogen; especially, we define it being equal to the size of DFZ, that is, xh ¼ r p . With the employment of the formulas from (A1)– (A14), we are able to calculate the resultant force sðr; θÞ which is subjected to the ðN þ 1Þth dislocation. Under the circumstances as schematically shown in Fig. (4), one is able to correlate the external load s with the effective stress intensity factor K nI as follows [56]: pffiffiffiffiffiffi ðA15Þ s ¼ K nI = πa References [1] R.S. Irani, MRS Bull. 27 (2002) 680–682. [2] R.L.S. Thomas, J.R. Scully, R.P. Gangloff, Metall. Mater. Trans. A 34 (2003) 327–344. [3] I. Moro, et al., Mater. Sci. Eng. A 527 (2010) 7252–7267. [4] D.H. Lassila, H.K. Birnbaum, Acta Metall. Mater. 34 (1986) 1237–1243. [5] H.J. Maier, H. Kaesche, Mater. Sci. Eng. A 117 (1989) 11–15. [6] Y. Lee, R.P. Gangloff, Metall. Mater. Trans. A. 38 (2007) 2174–2190. [7] R.P. Gangloff, in: I. Milne, R.O. Ritchie, B.L. Karihaloo (Eds.), Hydrogen Assisted Cracking of High Strength Alloys, vol. 6, Elsevier Science, New York, 2003, pp. 31–101. [8] A.P. Moona, R. Balasubramaniama, B. Pandab, Mater. Sci. Eng. A 527 (2010) 3259–3263. [9] J. Besson, L. Devillers-Guerville, A. Pineau, Eng. Fract. Mech. 67 (2000) 169–190.

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