ISSN 10283358, Doklady Physics, 2014, Vol. 59, No. 11, pp. 534–538. © Pleiades Publishing, Ltd., 2014. Original Russian Text © D.A. Indeitsev, E.V. Osipova, V.A. Polyanskiy, 2014, published in Doklady Akademii Nauk, 2014, Vol. 459, No. 3, pp. 294–297.
MECHANICS
A Statistical Model of HydrogenInduced Fracture of Metals Corresponding Member of the RAS D. A. Indeitseva, b,
E. V. Osipovaa, and V. A. Polyanskiya Received June 25, 2014
Abstract—The fracture process of metals due to hydrogen embrittlement is described theoretically as a first order phase transition. The fractured and unfractured phases are in equilibrium at the instant of fracture and are described by the equality of stresses and thermodynamic potentials. In the context of this approach, the dependences of the fracture stress on the molar hydrogen concentration and on the fracture deformation are calculated. The dependence on the molar hydrogen concentration turned out to be close to the power depen dence, while the dependence on the fracture deformation is very close to linear. Not only the qualitative but also the quantitative correspondence of this model to the experimental results is shown. DOI: 10.1134/S1028335814110093
It is known well that many metals including hard ened steels are subjected to hydrogen embrittlement, and this effect is enhanced substantially as the mechanical load increases [1–3]. The nature of this phenomenon is clarified incompletely. We can distin guish several main approaches to its quantitative mod eling: the account for the influence of hydrogen on the buildup and motion of the dislocations, the account for the influence of hydrogen on the development of cracks, and the account for the internal pressure of hydrogen in metal; as well as various physical approaches. The motion and formation of dislocations lead to the phenomenon of hydrogenenhanced local plasticity (HELP model) [4] due to the very high dis location concentration. Later, the authors of [5, 6], based on physical notions on the interaction potentials of hydrogen with dislocations, proposed the determin ing equations, which model the local variations in the material rheology in the crack mouth. However, the calculations performed in [5] show that the substantial variations in the stress–strain curve occur at local hydrogen concentrations of about 9000 ppm, which is a very high concentration for most metals. The implicit power dependence on the local hydrogen concentration, which is impossible to be measured
a
Institute of Problems in Machine Science, Russian Academy of Sciences, Vasil’evskii Ostrov, Bol’shoi pr. 61, St. Petersburg, 199178 Russia b St. Petersburg State Polytechnic University, ul. Politekhnicheskaya 29, St. Petersburg, 195251 Russia email:
[email protected]
directly, is put into the determining equations, i.e., the parameters of the law (including the power index) can be evaluated only indirectly, which can lead to consid erable errors. To verify the model, the authors of [7] computed the local plasticity when considering the crack with a spherical tip. Local hydrogen concentra tions, which are attained during modeling, are higher than the average initial ones by a factor of approxi mately 100, but the average concentrations are usually about 1 ppm. Their increase even by a factor of 100 will not lead to digits on the order of 9000 ppm (we bear in mind steels). Thus, verification is attained only quali tatively. The second source of possible errors is the modified Fick law [8], which explicitly includes the temperature dependence of the influence of coeffi cients of the stress field but does not include the expo nential temperature dependence of diffusivity. A whole series of indefinitenesses, which are written by the authors of the model, should be mentioned, par ticularly, a nonlinear dependence of the internal potential on the magnitude of stresses and hydrogen concentration occurs, and since we consider local concentrations exceeding all those observed in prac tice, all nonlinearities will play an important role. The hydrogenenhanced decohesion model (HEDE) is similar to HELP [9]. The difference is in that a decrease in the formation energies of free frac ture surfaces with an increase in the local hydrogen concentration is taken into account in HEDE. The authors of [10] noted that the HELP approach requires tremendous computing resources when solv ing any applied problem; therefore, the only output is
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the use of the continual model of the dislocation development; however, such replacement is often inadequate, and the authors propose to use the criteria of submicrocrack growth; i.e., they reduce all hydro gen problems to modeling the crack development and lowering the crack resistance. In this study, a quantitative model, which describes the dependence of the fracture stress of metals with hydrogen dissolved in them on the hydrogen concen tration, is developed for the first time. This model is based on the fact that the destruction process under the load is a firstorder phase transition [11–14], which is either negated or not taken into account in models [4–10]. Correspondingly, at the instant of fracture, the fractured and unfractured metal phases are in equilibrium. This equilibrium state is described by the equality of thermodynamic potentials of each phase and by the equality of elastic stresses. This hypothesis is sufficient in order to describe the fracture process quantitatively. The found dependence of the fracture stress on the hydrogen concentration is com pared with the experimental data, and their excellent correspondence to one another is shown. Thus, following [11], let us consider the simplest model of hydrogenated metal with the total number N0 of chemical bonds, where NH of them are broken, i.e., atomic hydrogen is already builtin inside them. We will consider that all NH broken bonds and N0 – NH unbroken bonds are identical and, consequently, indistinguishable. In other words, we assume that either one hydrogen atom or no hydrogen atoms can be in each bond. The statistic weight of this system, i.e., the total number of various combinations, is N0 Z = . N H! ( N 0 – N H )!
(1)
The equilibrium value of NH with the sample tension with the initial size L0 to value L, where L – L0 = εL0 and ε is the relative deformation (elongation) of the sample, corresponds to the maximal value of Z with the fulfillment of conservation laws of NH and elastic deformation energy Eel. In this study, similarly to [11], we used the simplest model with the linear depen dence of Eel on NH: E el = constN H ε + E 0 ( ε ),
(2)
where E0(ε) is the arbitrary function of ε. In essence, dependence (2) comprises the first summands of the corresponding Taylor series. To take into account the conservation laws, we will seek the maximum of the auxiliary function rather than that of the Z function itself: DOKLADY PHYSICS
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σ, 103 MPa 8
6
4
2 2
1 10
0
20
30
40
50 ε, %
Fig. 1. Dependence of stress σ on deformation ε deter mined by formula (5) at the following constants: E0 = 200 GPa, EH = 4 GPa, HE = 0.1 wppm, and εc = 1%. The dashed part of the line corresponds to the stable and meta stable states, and the solid part corresponds to the stable elongation of the sample. Fracture of the sample occurs from point 1 to point 2 with identical thermodynamic parameters (8).
ε U = ln Z + N H + λ 0 N H , εc
(3)
–1
where ε c and λ0 are two Lagrangian multipliers that correspond to two conservation laws. Equating partial ∂U derivative to zero, we will derive the equilibrium ∂N H distribution of hydrogen by chemical bonds of metal N0 – NH ( ε ) 1 , = ε–μ N0 1 + exp εc
(4)
N where μ = εcln ⎛⎝ 0 – 1⎞⎠ is the analog of the Fermi N H0 level in metal and NH0 = NH(0) is the number of broken bonds at ε = 0. Experimenters express the hydrogen concentration in metal in wppm (weight parts per mil lion), i.e., in millionth parts of the weight of N H0 1 components. In other words, = 10–6HE, μ = N0 W –εcln(10–6HEW), where W ≈ 56 is the atomic weight of steel and HE is the concentration of the incorporated hydrogen in wppm. Here, the unit is neglected com N pared with 0 ~ 104. N H0 The coincidence of formula (4) with the Fermi dis tribution is associated with the fact that the filling
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INDEITSEV et al. Φ, 104 J 4
σF, 103 MPa 4
3 2 2 0 1
2 1
−2
0
10
20
30
40
50 ε, %
1
2
3
4 5 HD, wppm
Fig. 3. Dependence of fracture stress σF on the molar con centration of hydrogen HD at the following constants: E0 = 200 GPa, EH = 4 GPa, and εc = 1%. The curve corresponds to the theoretical dependence determined by set (9), and points correspond to the experimental data [3] recom puted in terms of the local stress and local concentration.
Fig. 2. Dependence of thermodynamic potential Φ on deformation ε determined by formula (8) at the following constants: E0 = 200 GPa, EH = 4 GPa, HE = 0.1 wppm, and εc = 1%. Fracture of the sample occurs from point 1 to point 2 with identical stresses σ (5).
numbers of hydrogen in chemical bonds can accept the values of either 0 or 1, which ensures the complete analogy of this system with the electron distribution by energies in metals. The relation between stress σ and relative deformation (elongation) in a twophase medium is known to have the form [12, 13] Eε σ ( ε ) = 0 , E NH ( ε ) 1 + ⎛ 0 – 1⎞ ⎝ EH ⎠ N0
0
For a quantitative description of the fracture pro cess, let us introduce the free energy of the system ε
F(ε) =
∫ σ ( x ) dx
(7)
0
and the thermodynamic potential [15] (5)
where E0 is the Young modulus of pure metal, i.e., without hydrogen; and EH is the Young modulus of metal, all bonds of which are occupied by hydrogen. Substituting equilibrium distribution (4) here, we derive
Φ ( ε ) = F ( ε ) – εσ ( ε ).
(8)
Dependence Φ(ε) is presented in Fig. 2. According to the thermodynamics, the unfractured phase of metal, which is characterized by elongation ε1, and the frac tured phase with elongation ε2 will be in equilibrium only with the equality of stresses in them and their thermodynamic potentials [15], i.e., σ ( ε 1 ) = σ ( ε 2 ),
Eε σ ( ε ) = 0 . E 0 /E H – 1 1 + 6 –1 –1 1 + 10 H E W exp ( – ε/ε c )
(6)
Dependence σ(ε) is looplike (Fig. 1) and thereby describes the fracture process in hydrogenated metals under the effect of the load as the firstorder phase transition from the phase where hydrogen is almost not builtin into the chemical bonds of metal, into the phase where, on the contrary, almost all bonds are occupied by hydrogen.
(9)
Φ ( ε 1 ) = Φ ( ε 2 ).
Then the equilibrium fracture process will be the first order phase transition from ε1 into ε2 (ε2 > ε1), which are unambiguously determined by set (9) (see Figs. 1, 2). To compare the theoretical results derived from (9) and the experimental data, it should primarily be noted that dependence σ(ε) (6) is determined by the concentration of the builtin hydrogen HE, while the total concentration of hydrogen HD (builtin and not builtin), which is able to diffuse in metal, is deter mined experimentally. This concentration HD can be DOKLADY PHYSICS
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computed from distribution (4) taken at ε = ε1 since this information is the maximum possible with the conservation of the uniformity of the system. In other words, almost all mobile hydrogen is builtin into chemical bonds at ε = ε1, i.e.,
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σF, 103 MPa 3.0 2.5 2.0
NH ( ε1 ) 6 H D = × 10 N0 W
1.5
6
(10)
10 = . ε 1 ( H E )⎞ 6 –1 W + 10 H E exp ⎛ – ⎝ εc ⎠
1.0 0.5
Thus, solving set (9), we can find the dependence of maximally possible elongation ε1, which corre sponds to the fracture, and fracture stress σF = σ(ε1) on HE; then, substituting ε1(HE) into formula (10), to determine the unambiguous relation between HE and HD; after which, to determine finally dependence σF(HD) = σ(ε1(HE(HD))). Figure 3 shows the computa tions of this dependence σF on HD at the following val ues of constants: E0 = 200 GPa, EH = 4 GPa, and εc = 0.01 (i.e., 1%). The experimental results of study [3] on the measurement of σF(HD) for two samples of hardened steel AISI 4135 with various notches are also presented there. Various notches were necessary in order to calculate local fracture stress σF and local hydrogen concentration HD in the vicinity of the notch apex. Due to this, the measured dependence σF(HD) does not depend either on the sample or on the notch and has a universal character. It is seen from Fig. 3 that the proposed theoretical model of the fracture due to the hydrogen embrittlement completely describes the experimental results in the error limits. The parame ters of computations EH and εc were selected from the best coincidence with the experimental results. It is noteworthy that the computed dependence σF(HD) is actually close to the power one, which was noted in various experiments [2–6], and corresponds at these
0
0.5
1.0
1.5 ε1, %
Fig. 4. Dependence of fracture stress σF on the fracture deformation ε1 at the following constants: E0 = 200 GPa, EH = 4 GPa, and εc = 1%. The line shows the theoretical dependence, and the points show to experimental data [3] corresponding to the samples with various notches.
this case. The closeness of this dependence to the lin ear one just indicates that the fracture is a firstorder phase transition (Fig. 1) far from nonlinear effects of σ(ε). Thus, the fracture process of metals due to hydro gen embrittlement is described theoretically as a first order phase transition in this article. The fractured and unfractured phases are in equilibrium at the instant of fracture, and, consequently, they are described by the equality of stresses and thermodynamic potentials (9). In the context of this approach, the dependences of the fracture stress on the hydrogen concentration and on the fracture deformation are calculated. Not only the qualitative but also the quantitative correspon dence of this model to the experimental results is shown.
– 0.14
constants to dependence σF ≈ 1900 H D [3]. It is noteworthy that approximately half of the hydrogen atoms is incorporated into the chemical bonds of metal in this case. Figure 4 shows the dependence of fracture stress σF on deformation ε1, which corresponds to the onset of fracture, for two samples of hardened steel AISI 4135 with various notches. It is seen that this dependence is very close to linear, which also completely corresponds to the experimental data [3] (recalculation for the local stress and local deformation was not performed in this case; therefore, the results for these notches are different). The slope angle corresponds to the Young modulus E0, which was selected equal to 200 MPa in DOKLADY PHYSICS
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Translated by N. Korovin
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