Numerical analysis of Hydrogen embrittlement of high strength steels using Monte Carlo method
Soumia Ourrad
Smart Structure Laboratory/DGRSDT, University Center of Ain Temouchent,, Algeria.
ourrad-soumia13@hotmail.fr Youcef Houmadi
Smart Structure Laboratory/DGRSDT, University Center of Ain Temouchent, Algeria.
houmadiyoucef@yahoo.fr
F. Javier Belzunce Materials Science Department, University of Oviedo, Gijón, Spain
belzunce@uniovi.es Abdelkader Ziadi
Smart Structure Laboratory/DGRSDT, University Center of Ain Temouchent, Algeria.
aekziadi@yahoo.com
Abstract— A probabilistic approach has been applied to hydrogen desorption phenomena in the wire rod for pre-stressed concrete. The phenomena was treated in a deterministic study by Carneiro in 2010 , this work aims to reflect uncertainty property of the material of a high carbon steel such as effective diffusion coefficient (De) and concentration parameters(C∞). A probabilistic simulation method of Monte Carlo was used to determine the contribution of each random variable on the variability of reduction in area in our case the limit state criteria required in the study is reduction in area parameter must be greater than or equal to 30% (Carneiro, 2010). Afterwards we study the influence of parameters that govern the phenomenon desorption hydrogen and dispersion of the parameters while optimizing calculative time.
Keywords— hydrogen embrittlement, the probabilistic simulation method of Monte Carlo, spatial variability.
I. INTRODUCTION
Increasing the lifetime of a structure while decreasing the time and maintenance budget represents a new approach in the context of sustainable development policy. This is especially true when the work is large scale, as is the case of some pre- stressed concrete structures. Steels commonly used in reinforced concrete are pre-stressing steel wire (C-Mn).
Although their elastic limit be high to ensure normal service life in the area of elasticity, resilience is less than that of ductile steel with lower yield strength, and are therefore more susceptible to defects such as cracks or notches [1].
Among the different pathologies, inherent in pre-stressing steel wires used, corrosion is usually the most worrying. It is manifested by two phenomena [2,3]:
(i) The stress corrosion induced by chlorides, (ii) Hydrogen embrittlement (HE).
There is a general agreement that hydrogen embrittlement (HE) plays an important role in the environmental cracking of prestressing steel wires. It mainly depends on the material (chemical composition, heat treatment), the environment or
the mechanical state. The factors that play an important role in hydrogen embrittlement phenomena are:
- Trapping of hydrogen: the structural defects (inclusion, carbides, dislocation clusters, grain boundaries),
- Stress state: the presence of areas where there is a stress concentration appearing to be essential for embrittlement by hydrogen phenomena occur,
- Plastic deformation: the hydrogen interactions - plastic deformation appear to play a fundamental role in phenomena of hydrogen embrittlement.
The present work aims to report on the variations of hydrogen content and reduction in area, with holding times at room temperature, of high-carbon wire rods used in the manufacture of steel rods for pre-stressed concrete applications. The uncertain parameters of the materials, as the effective diffusion coefficient (De) and the concentration (C ∞), are modelled by random fields to take into account the material spatial variability as the reduction in area Z (Table 1) . In such case, Monte Carlo Simulation (MCS) methodology is generally used based on the deterministic model of Carneiro [4].The commercial grade steel used in this study was produced as a hot-rolled and controlled-cooled wire rod of 11-mm diameter. Its chemical composition (in weight percent) was:
0.82% C, 0.77% Mn, 0.015% P, 0.004% S, 0.12% Si, 0.00218% Al, and 0.0040% N [4].
TABLE I. INITIAL VALUES OF HYDROGEN CONTENT, C0,TENSILE STRENGTH AND REDUCTION IN AREA, Z,OF THE STUDIED STEEL
Parameters Co (ppm) Tensile strength
(MPa) Z (%)
Measured
value 1.85 1192±15 20±2
Specified
value / 1130–1230 ≥30
II. MODELING OF HYDROGEN DESORPTION A. Deterministic Model
It is established that the passage of hydrogen through a steel is hindered by lattice imperfections, which tend to attract and bind it, thus rendering it immobile at temperatures where it should normally be able to diffuse readily [5].
There are two types of energy states for the hydrogen in the metal: firstly, hydrogen in solid solution also called diffusible hydrogen and secondly trapped hydrogen.
The origin of the lattice diffusion of hydrogen in a metal is the hydrogen concentration gradient between the surface and the core of metal, created by the absorption of hydrogen. The reticular diffusion of hydrogen consists of a series of jumps from one interstitial site to the next, jumps that require activation energy for passing the energy barrier between two sites. The first and second Fick's laws conventionally describe the lattice diffusion of hydrogen in the absence of any other driving force, stress gradients, temperature or existence of an electric field can also be the cause of the diffusion of hydrogen [6,7].
A non-linear regression analysis was used to relate the hydrogen content of the steel to its room temperature storage time, using Eq. (1):
P
tP
C
tP
1 2exp 3 (1)Where P1, P2, and P3are regression coefficients that are not directly linked to any kinetic model of hydrogen release. The value of the correlation coefficient close to one and the low values of the standard deviation of the regression coefficients indicate that Eq. (1) effectively describes the experimental data.
Fig. 1. Variation of the hydrogen content, Ct,with the storage time at room temperature.[1]
We tried to see influance adjustment parameres we introduce the experimental values of Carneiro in software and got the distance curve shown in Fig2. which gives a value of 93% which R² n is not far from the value found by Carneiro wich 96%.
The second Fick's law [8,4] gives a relation which describes the variation of the concentration gradient in function of time expressed by Equation 2:
C C
hyd
hyd D
t
2
(2) Reducing it in cylindrical coordinates, we obtain:
C r C
hyd D hydt 2
2
(3)Fig 2. Trend curve variation of the hydrogen content, in function of time storage of wire rod
Considering the following initial and boundary conditions:
C(r, 0) = Co for 0 ≤ r ≤ R_ and t = 0 (4) 0
0
0
for r and t
r
C (5)
0 )
,
( R t C for r R and t
C
(6)The exact solution of Eq. (2), considering conditions calculated by Eqs. (4)–(6), is as follows:
1 1
0 2 '
0
0 '
2 exp , 1
n n n
n n
R r t
D t
r C
J J C R
C C
(7)The
n coefficients are the positive roots of the Bessel function of the first kind and order zero; such roots can be approximated by the relation below:d n R
n
n 2
) 1 4 ( ' 4
) 1 4
(
(8)The mean hydrogen concentration, Ct, in the cylinder can be obtained by integrating Eq. (7) with a radius for a given time, t [8]:
1 2
2
2 4 1
) 2 / 1 4 exp 64
n n
Dt d n
C
CoCt C
(9)Using only the first term of this series and considering an effective diffusion coefficient, De, for trap-controlled diffusion, the following equation is reached:
Det
d C
Co C
Ct 2
2 2
exp 3 9
64
(10)
When traps are reversible (saturable), the effective diffusion coefficient of hydrogen, according to analysis, Considering an effective diffusion coefficient which takes account of microstructural defects (such as dislocations that serve as traps hydrogen expressed by (θ) which represents the fraction of the trapping sites occupied by hydrogen atoms[6]:
t t LL
C
LC D C
De 1 (11)
Where DL, CL, and Ctrepresent the lattice diffusion coefficient of hydrogen, the lattice concentration of hydrogen, and the mean hydrogen concentration in traps at time t, respectively.
Parameter t is the fraction of trap sites occupied by hydrogen atoms. Considering θt ≈1, then:
TD QH
aH
bR
1
ln (12)
Where R is the universal constant of gases, T is the absolute temperature, Ha is the activation energy for hydrogen trap- controlled diffusion, Hmis the activation energy for hydrogen diffusion in a perfect iron lattice, and Hbis the binding trap- hydrogen energy. Eq. (10) can be expressed as:
d
C Det Co C
Ct 0.72 exp 22 .22 (13)
This equation has the same form of Eq. (1). In this sense, C∞
is equal to the regression value of P1 in Eq. (1). This value (0.56 ppm, Fig. 1) agrees with the experimental value of C∞
(from 0.55 to 0.60 ppm). On the other hand, P2 (1.32 ppm) must be compared to the numerical value of 0.72(C0−C∞) in Eq. (13). This value is 0.91 ppm. The difference found may well be due to the use of only the first term of the series given by Eq. (9) in this analysis. Finally, by comparing Eqs. (1) and (13), one arrives at P3=−22.2De/d2. Considering a bar diameter of 11mm and using the regression value of P3 (−0.01531 h−1 or −4.25×10−6 s−1), De may be estimated to be equal to 2.3×10−5mm2s−1. [4]
An equation relating reduction in area, Z, to the storage time can be obtained by assuming a linear relation between Z and the hydrogen content:
) (Co Ct E
Zo
Z (14)
In this equation, Z0is the initial reduction in area of the wire rod measured just after cooling, E is a constant, and C0and Ct
have the same meanings as defined above.
Fig. 3 illustrates the means for obtaining the value of E due to a trend curve, which gives a correlation coefficient of 91%
who is much higher than 70% ,the result is judge reliable.
Using Eq. (13) together with Eq. (14), it can be shown that:
Zo E Co C
d
DetZ 2
2 . exp 22 72 . 0 1
(15) Z is equal to Z0 at t = 0, and it is equal to its saturation value when t→∞. The fact that Z = Z0+ 0.28F at t = 0 came from the use of only the first term within the series in Eq. (9).
t
Z
P
4 P
5 1 0.72 expP
3 (16)Fig. 3 . Reduction area Z vs hydrogen content ,
Fig. 4. Variation of the reduction of area, Z, with the storage time at room temperature.
The values obtained for R2and the standard deviations of P3, P4, and P5 (Fig. 4) indicate that the experimental data fit quite well to this equation. Adequacy of control parameters with measured values. The value of P4+ P5(1 - 0,72) is equal to 19,3%, which is close to the experimental to Zo(20%).
(P4+ P5) is equal to 37,8% which correspond exactly with the experimental value.
B. Probabilistic Model
Direct Simulation Method of Monte Carlo Monte Carlo is to conduct a large number of simulation Ns (prints) of the random variables of the problem studied for each simulation, the state function is calculated and there are simulations leading to failure of the Nsf structure. The probability Pf is then estimated by the ratio between the number of simulations Nsfleading to rupture and the total number of Ns prints [9]:
N N
P
Ssf
f (17)
Where N is the total number of simulations. This failure probability estimator (27) can be also written as follows: [9]
N
x G
sI
P N
s
f 1
1 (18)
Where I (G(x)) Is the domain of insecurity (failure), it is equal to 1 in areas of insecurity and it is equal to 0 in the security field:
0 0
0 1
x ifG
x x ifG
G I
In our case, the state function G (x) is in direct relation with reduction in area. An exhaustive study was made to determine the values of the concentration and diffusion coefficient in terms of average [4] and VOC (the coefficient of variation of the quantity is defined as the ratio between the standard deviation and the average of that size.
The accuracy of the estimated failure probability can be measured by calculating its coefficient of variation as follows:
N P P
P
f sf f
1
(19)
III. INFLUENCE OF SPACE VARIABILITY INTRINSIC PARAMETERS OF THE MATERIAL ON OBTAINING THE
CRITERIA REQUIRED
Discretization of a section involves a subdivision of the sample structure into small pieces of random field in the abscissa direction and in the direction of the ordinates as shown in Fig.5:
Fig5. discretization model of a cylindrical section
Where n is the number of elements along the x-axis and m is the number of elements following the y-axis. The random field was discretized using the expansion karhnuen-Loeve (KL).According to [9-13] this method provides a good approximation of the random field H (X,θ).
i
i i x
X
H
H
1
, (20)
The approximate random field is defined by truncating the ordered series (take the value M instead of ∞)
,
,
( ) ( )1
M
i H X i
H H X
H (21)
X i i X
H
i
(22)Avec
ξi (θ): Uncorrelated Random variable, =0 et 2=1 θ: indicates the random nature of the corresponding amount X: spatial coordinates in the physical space
An exponential covariance function for the case of a random field is given as follows:
y L y
L x y x
y x
x x y
C 1, 1, 1, 2 2exp 1 2 1 2 (23)
The autocorrelation function is:
L
y y L x y x
y x
x x y
2 1 2 1 1 2
1, 1, , exp
(24)
Lx and Ly are the horizontal and vertical distances respectively autocorrelation and (x1, y1) and (x2, y2) are the coordinates of two points in a given domain D to which the H process (X, θ) is defined.
According to [14], the random variable is the diffusion coefficient (De), the random field appropriate lognormal can be obtained by exponentiation of the Gaussian field of equation (20)
M
i i i
De X
x
H , exp
ln 1H
(25)In this case, the exponential covariance function is given by
y L y
L x y x
y x
x x y
C 1, 1, 1, 2 2exp 1 2 1 2 (26)
When the standard deviation and the average log from respectively are given by:
De
De De
2ln ln 1 22 (27)
1 2lnln ln
2 De
De
De
(
28)In this study, the different values of the distance of horizontal and vertical autocorrelation were studied and analyzed. The objective function used to calculate the probability of failure is defined as follows:
G=Zlim- Z où Z30%
The number of elements depends on the scale of fluctuation of the random variable. The distance between the centers of two consecutive elements is in the order of half-scale of fluctuation d.
The greater the number of elements in the discretization the closer to the accuracy of results and it involves a number of simulations.
Direct Simulation Method of Monte Carlo includes a considerable number of prints to assess the probability of failure.
A. Influence of autocorrelation distances
A parametric study was conducted to determine the effect of the variability of the material by the analysis of the influence
of the autocorrelation distances Lx and Ly of the random field (From) the likelihood of failure. Fig.6 shows the effect of the distance autocorrelation on the failure probability. for a case of an isotropic random field (ie d. Lx = Ly).
Fig. 6 Effect of autocorrelation distance on the probability of failure in the case of an isotropic random field.
Fig 7 shows that the increase distance autocorrelation increases the probability of failure and that the latter ceases to increase when the ratio of Lx and Ly is greater than or equal to 40 mm; This is because the random field tends to the case of a homogeneous material for large values of the autocorrelation distances autocorrelation For smaller distances, a heterogeneity of the material is obtained.
Fig.7. Effect of horizental autocorrelation distance on thefailure probability for different values.
Fig.8 shows that the probability of failure ceases increase when ≈100 mm Lx values; This observation can be explained as follows the height of the sample studied, which is equal to 95mm, so when Ly exceeds this value, the probability of failure is almost constant.
Note that the configurations used in this study correspond to the autocorrelation distances equal to 50 for the x-axis and 20-axis and y it will likewise for probabilistic calculations of our study.
B. Parametric study of Random Variables
A sensitivity analysis based on a parametric study of random variables in the interest to provide the contribution of each of these variables which are the effective diffusion coefficient (De) and the concentration at a given time (C∞) on the variability of the response of the system (Z) by an analysis of failure probability in function of time storing samples of the rods in use prestressed concrete in accordance with our case the failure states non-compliance the reduction area criterion must be greater or equal to 30%. The input model data are shown in Table II.
Fig. 8. Effect of vertical autocorrelation distance on thefailure probability for different values.
C. Parametric study of Random Variables
A sensitivity analysis based on a parametric study of random variables in the interest to provide the contribution of each of these variables which are the effective diffusion coefficient (De) and the concentration at a given time (C∞) on the variability of the response of the system (Z) by an analysis of failure probability in function of time storing samples of the rods in use prestressed concrete in accordance with our case the failure states non-compliance the reduction area criterion must be greater or equal to 30%. The input model data are shown in Table II.
TABLE II. INPUT MODEL DATA Parameters mean coefficient of variation
C∞ 0.56 0.06
De 0.08 0.13
The effect of the diffusion coefficient on failure probability is shown in Fig. 8. This figure shows that for COV (De) = 0.06 failure probability is maximum up to 28h unlike VOC (De) = 0.19 that failure probability begins to decline to 21h and COV (De) = 0.13 for which failure probability begins to decrease to 24. Also between 29 and 34 hours there is a convergence at the inflection point in failure probability of the different diffusion coefficients (De), 32 h after failure
probability for both COV (De) of 0.19 and 0.13 is reversed, and decreases more slowly.
It can be concluded that the variability of the diffusion coefficient such as random fields has a significant impact on failure probability, hence the importance of this study taking into account the spatial variability of material.
The results in Fig. 9 show that the decrease in the probability of failure for the three COV (C∞) is starting from 24h and that failure probability is zero to 43h. Finally, it can be concluded that the concentration of hydrogen has a low impact on failure probability..
Fig. 9. Time evolution of failure probability for different COV(De).
Fig. 10. Time evolution of failure probability for different COV (C∞) D. Calculative time Optimization
Using equation 22, the coefficient of variation COVpf of failure probability was calculated and shown in Fig. 11. It has been found that it decreases with increase in the number of achievements. It reaches a value of coefficient of variation of 25% when the number of simulation Ns is equal to 10,000 achievements. For this purpose it is necessary to associate
each time interval to a specific number of simulations as shown in table 3
Fig.11 Optimization of calculative time by the number of simulationdu temps
TABLE III. ADAPTATION TIME CALCULATED ON FUNCTION OF SIMULATION NUMBER
Intervalle de
temps Nombre de tirage d’échantillon
[20,50] 1000
[51,54] 6000
[56,62] 10000
E. Overview of the results in this study
Based on the results obtained, will be considered the diffusion parameter to evaluate the influence of the autocorrelation distances since it regulates the release of hydrogen within the metal.
In a reduced cross section where the autocorrelation distances are more weak we note heterogeneity of the material which results by variability of diffusion coefficient the most likely explanation is that in a great heterogeneity in area there is more microstructural defect that facilitate diffusion of the hydrogen, In contrast to a larger section the material tends to homogeneity it involves there are less defects, that's what slowed the diffusion of hydrogen..
The major disadvantage of the probabilistic approach to the direct simulation Monte Carlo method, is that it is very greedy in computation time, the solution to this dilemma is to adopted an optimal calculative time to the accuracy of results within a reasonable calculative cost.
IV. CONCLUSION
This study helped us to identify the fragility problems of steels by hydrogen desorption including the phenomenon of hydrogen by parametric studies of the intrinsic parameters and to define the objectives adulated that follow in subsequent tests through the program which enabled us to determine the
minimum storage life of samples, to allow the hydrogen to escape in the interest of not to compromise the ductility of the material for use in prestressed concrete.
Acknowledgment
This work was sponsored by the Ministry of high Education and scientific research.
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