Probabilistic Analysis for Estimating the Hydrogen Desorption Time from Steel Wire Rods Using Monte Carlo Simulation

2.1 Deterministic analysis

The commercial grade steel used in this study was produced as a hot-rolled and controlled-cooled wire rod with 11mm diameter. Its chemical composition (in weight percent) is mentioned in Table 1.

The modeling of hydrogen desorption is established that the passage of hydrogen through 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 [12].

Table 1. Chemical composition of steel [11]

The origin of the lattice diffusion of hydrogen in a metal is the hydrogen concentration gradient between the surface and the inside of the metal created by absorption of hydrogen. The reticular diffusion of hydrogen consists of a series of jumps from one interstitial site to the next one. These jumps require activation energy for crossing 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, the stress gradients, temperature or existence of an electric field which can also be the cause of hydrogen diffusion [13-14].

2.2 Mathematic model

Fick’s first law describes the change in concentration over time as the following:

$J=-D \nabla \phi$   (1)

where, J is the diffusive flux of hydrogen atoms in $\mathrm{mol} / \mathrm{m}^{2} \mathrm{s}$ is the diffusion coefficient in $\mathrm{m}^{2} / \mathrm{s}, \nabla$ is the gradient operator, for many material problems it is not the flux that is

as important as the concentration at any given time [15].

The second Fick's law [15] gives a relation which describes the variation of the concentration gradient according to time and is expressed by Equation 2:

$\frac{\partial C_{h y d}}{\partial t}=-D \nabla^{2} C_{h y d}$    (2)

where, C_hyd is the concentration gradient in $\mathrm{mol} / \mathrm{m}^{3}$.

By reducing it in cylindrical coordinates form, we obtain:

$\frac{\partial C_{hyd}}{\partial t}=D \frac{\partial^{2} C_{\operatorname{hyd}}}{\partial r^{2}}$    (3)

By considering the following initial and boundary conditions:

$C(r, 0)=C_{0}$ for $0 \leq r \leq R$ and $t=0$   (4)

$\frac{\partial C}{\partial r}=0 \quad$ for $\quad r=0$ and $\quad t>0$   (5)

$C(R, t)=\operatorname{Coo}$ for $r=R$ and $t>0$   (6)

where, r is radius and R is radius of wire road

The exact solution of Eq. (2), considering conditions calculated by Eqns. (4)-(6), is as follows [15]:

$\frac{c(r, t)-c_{0}}{c_{\infty}-c_{0}}=1-\frac{2}{R} \sum_{n=1}^{\infty} \frac{\exp \left(-D \alpha_{n}^{2} t\right) J_{0}\left(\alpha_{n}^{2} r\right)}{\alpha_{n}^{2} J_{1}\left(\alpha_{n}^{2} R\right)}$    (7)

$\alpha_{n}=\frac{(4 n-1) \pi}{4 R}=\frac{(4 n-1) \pi}{2 d}$    (8)

Provided the $\alpha_{\mathrm{n}}$ are positive roots of Bessel function and d is the diameter.

Using only the first term of this series and considering an effective diffusion coefficient De for trap-controlled diffusion and $C_{t}$ mean hydrogen concentration at time $t$ and $C_{0}$ is concentration at time $t=0$ lead to the following equation:

$\frac{C_{t}-C_{\infty}}{C_{0}-C_{\infty}}=\frac{64}{9 \pi^{2}} \exp \left[-\left(\frac{3 \pi}{2 d}\right) D_{e} \mathrm{t}\right]$   (9)

$C_{t}=C_{\infty}+0.72\left(C_{0}-C_{\infty}\right) \exp \left(\frac{-22.2 D_{e} t}{\mathrm{d}^{2}}\right)$   (10)

By assuming a linear relation between “Z” where is the reduction in area and concentration:

$Z=Z_{0}+E\left(C_{0}-C_{t}\right)$     (11)

By replacing Eq. (11) in Eq. (10) we obtain:

$Z=Z_{0}+E\left(C_{0}-C_{\infty}\right)\left[\left(1-0.72 \exp \left(\frac{-22.2 D e t}{\mathrm{d}^{2}}\right)\right)\right]$   (12)

2.3 Probabilistic analysis

Direct Simulation Method of Monte Carlo consists of conducting a large number of simulation $N_{s}$ (prints) of the random variables of the studied problem for each simulation. The state function is calculated and there are simulations leading to failure of the $N_{s f}$ structure. The failure probability $P_{f}$ is then estimated by the ratio between the numbers of simulations $N_{s}$ leading to failure and the total number of $N_{s}$ prints:

$P_{f}=\frac{N_{s f}}{N_{s}}$   (13)

where, N is the total number of simulations. This failure probability estimator can also be written as follows [16-17]:

$P_{f}=\frac{1}{N_{S}} \sum_{1}^{N} I[G(X)]$     (14)

where, I (G(x)) is the domain of the lack of safety (failure), it is equal to 1 in unsafe areas and it is equal to 0 in the safe zone.

$I[G(x)]=\left\{\begin{array}{l}{1 \text { if } G(x) \leq 0} \\ {0 \text { if } G(x)\rangle 0}\end{array}\right\}$     (15)

In this work a probabilistic analysis is performed on the reduction in area of a heterogeneous steel rod used in pre-stressed concrete whose properties are spatially variable by determining the mean values of the concentration and diffusion coefficient and coefficient of variation (COV). In this study the random fields of these properties were discretized into a finite number of random variables using the Karhunen-Loeve expansion (KL). The illustrative values used for the statistical moments of the different random variables are those commonly encountered in practice [11]. These are given in Tables 2 and 3.

Table 2. Initial values of hydrogen content, C₀, tensile strength and reduction in area, Z, of the studied steel

Table 3. Input model data

The accuracy of the estimated failure probability can be measured by calculating its coefficient of variation as follows:

$\partial P_{f}=\sqrt{\frac{1-P_{f}}{P_{f} N_{s}}}$     (16)

In the literature, we can find several methods of discretization proposed by the researchers, which consist of breaking down the initial fields into optimal complete deterministic functions [18-19]. To do this, the discretization of a section involves a subdivision of the sample structure into small pieces of random field in the direction of the x-axis and the y-axis, as shown in Figure 1. But, the discretization with more elements provides more accurate results but also requires more computing time.

The transition from the representation of the continuous random field to a limited number of random variables is necessary to introduce the uncertainty of the properties of the materials into a computation model. To do this, we must choose a discretization method. In this study, the Karhunen-Loève expansion method (K-L) was adopted for the sample discretization.

1.png

Figure 1. Dimensional discretization model

where, n is the number of elements along the x-axis which is equal to 10 elements with a pitch n equal to 40 mm and m is the number of elements following the y-axis which is equal to 10 elements with a pitch m equal to 9.5 mm .The random field was discretized using the expansion of Karhnuen-Loeve (KL). According to this method, it provides a good approximation of the random field H ($\mathrm{X}, \theta$) [20-23].

$H(X, \theta)=\mu+\sum_{i=1}^{\infty} H_{i}(X) \xi_{i}(\theta)$    (17)

To introduce uncertainty of material properties in a computational model it is necessary to move from the representation of continuous random field H (X) to a limited number of random variables. To do this one must choose a discretization method. These approaches are various and can be divided into three categories: point method, the discretization average method and series expansion method. In this paper we used the expansion Karhunen-Loève (KL) which belongs to the series expansion method. The KL expansion provides a characterization of the second-moment of a random process in terms of deterministic orthogonal functions and non-correlated random variables as a result [24].

The approximate random field is defined by truncating the ordered series by taking the value M instead of $\infty$:

$H(X, \theta) \rightarrow H(H, \theta)=\mu+\sum_{i=1}^{\infty} H_{i}(X) \xi_{i}(\theta)$   (18)

$H_{i}(X)=\sqrt{\lambda_{i}} \varphi_{i}(x)$   (19)

With:

$\xi_{i}(\theta):$ Uncorrelated Random variable, $\mu=0$ et $\sigma^{2}=1$

θ: indicates the random nature of the corresponding amount

X: spatial coordinates in the physical space

Eigen vectors and eigen values of autocorrelation function.

In our case we took M equal to 200; the choice of the number M of terms depends on the desired accuracy of the problem being treated

In this study the different values of the distance of horizontal and vertical autocorrelation were studied and analyzed.

The limit state function used to calculate the failure probability is defined as follows:

$G=Z_{\mathrm{lim}}-Z$    (20)

where, $Z_{\lim }$ = 30 %.

It should be emphasized here that this study makes use Monte Carlo Simulation (MCS) methodology to compute the Probability Distribution Function (PDF) of the system response or the failure probability.

This Paper aims at presenting a probabilistic analysis at serviceability limit state of reduction in area of hydrogen desorption phenomenon in the steel rod for pre-stressed concrete C-Mn with a spatially varying effective diffusion coefficient (De) and concentration (C_∞) using Monte Carlo simulation method. The deterministic model was based on the first according second Fick’s laws conventionally describe the lattice diffusion of hydrogen implemented in MATLAB software. The main conclusions of the present paper can be summarized as follows:

(1) A parametric study has shown that the variability of reduction in area Z was mainly induced by effective diffusion coefficient (De). Concentration parameter (C_∞) has a minor effect in response variability and this parameter can be considered as deterministic. The strategy of detecting the most influential parameters that significantly reduces the computation time of the probabilistic analysis by software simulation. One of the main practical obstacles to these methods is the very high computation time of numerical models combined with the large number of simulations to these models necessary for the application of probabilistic methods hence the value of the computational time optimization study which has reduced the time and computational cost for the present study and for the future perspectives. It has also the advantage of reducing the cost of experimental investigation of new similar projects because the less-influential parameters do not need a thorough experimental investigation on their variability’s

(2) The fields (De and C_∞) were discretized into a finite number of random variables using the Karhunen-Loève expansion. Therefore the input uncertain parameters that have a significant impact in the variability of this response are the effective diffusion (De) should be thoroughly investigated in practice. In fact that in a reduced cross section where the autocorrelation distances are much smaller we note that a heterogeneity of material which results by the variability of diffusion coefficient. The most likely explanation is that in a great heterogeneity in area there are more microstructural defects at room temperature that facilitate diffusion of hydrogen. In contrast to a larger section the material tends to homogeneity involving less defects leading to the slowing down of hydrogen diffusion.

(3) The standard deviation of the system response increases with the increase of the Coefficient of Variation of inputs random fields. Concerning the mean value it remains quite constant in diffusion parameter.

(4) Finally, the validation of our obtained results is made by comparing them with those obtained by Carneiro’s [11]. It should be noted that the optimal storage time deduced from the deterministic approach by Carneiro is greater than that its around 98hours for storage time to reach a required reduction in area. For probabilistic approach and especially for a failure probability equal to 10^-3 which means a great precision and the required reduction in area is around 30 % is reached at 62h for storage time. As a result we obtain a gain equal to 36hours of storage time and this duration ensures obtaining the ductility required for the use of this material. This will guarantee a saving of storage time and consequently an economic gain for large quantities of this steel. However, it is desirable to involve probabilistic methods in feasibility studies for more reliable and economic decision-making. Also these approaches contribute in the decrease of the cost of the future experimental studies.

Recommendations for future studies are needed to reduce the calculation time and the coefficient of variation, it is recommended to use in similar studies the subset simulation method as an alternative to the Monte Carlo Simulation (MCS) method. Also the method of descritization EOLE can be used of interest to increase the precision of the results.