By 3D-FEM analysis for hydrogen diffusion, the heat transfer, stress-strain analysis and transient microstructure distribution must be coupled. This FEM formulation leads to asymmetric equations. Other than that the formulation is straight forward. The value of temperature at the nodes is obtained by solving the associated energy problem. The microstructure is evaluated at the nodes by solving the associated ordinary differential and algebraic equations for the evolution of austenite grain size and the decomposition of austenite into ferrite, pearlite, bainite and martensite. The value of у/ at the nodes is obtained by solving the associated stress problem. Given these values, the value of hydrogen at the nodes is obtained by solving the solute diffusion problem.

For each time step, the variations of the temperature, stress, strain and the volumetric fractions of different phases should be obtained, so that, the physical-chemical “constants” in every phase, such as activation coefficient, diffusion coefficient, transfer coefficient, solubility and so on, of every element can be calculated. The concentration of the “effective, diffusible” hydrogen, c* may be obtained through the following mass diffusion equation, adopted from [3 and 24]:

[*]{c*} + [/?]|-{C*} = {F} (6-32)

at

where matrix [KJ, [R] are related to the activation у and diffusion D coefficients; {c*} is the concentration matrix; {F} is the hydrogen flow matrix. The obtained c* multiplied by /? makes c, the actual hydrogen concentration, where P=l/y. According Dubios [26] 0 = 1 +є pl where £pl is the plastic strain produced in welding.

Goldak et al [15] refers to the computational analysis of the Slit test and compares the results to experimental data. The 3D transient temperature, microstructure, stress and strain fields in the Slit Test are computed. In addition, the 2D transient hydrogen concentration field is computed on the central cross-section. The hydrogen diffusion model assumes the weld pool has a prescribed hydrogen level. In addition the hydrogen flux due to gradients in the stress field is included. The 3D transient temperature, microstructure and stress are coupled to the 2D hydrogen field.

A mesh for a hydrogen diffusion simulation is shown in Figure 6­24.

Figure 6-25: Hydrostatic stress distribution at temperature 100°C

The compressive stress at the bottom of the weld tends to repel the hydrogen. The formation of two peaks in the hydrogen distribution near the center of the weld also appears to be associated with the hydrostatic stress distribution, Figure 6-25.

Figure 6-27: At the point with maximum hydrogen levels at 50°C in slit test [15], the effective plastic strain (xl), effective plastic strain rate (x2) and evolution of hydrogen (x3) ppm are plotted vs. log (time s)

The numerical study by Streitenberger and Koch sheds some light on stress-driven diffusion processes of solute near stress concentrators and should be useful for the interpretation of dynamic embrittlement and impurity induced fracture processes at elevated temperatures.

Figure 6-29: Concentration distribution (left) and lines of constant concentration (right) in the vicinity of a crack tip in the opening mode for a drift parameter Q=0.5 and after the reduced time t =0.01 in the limit of pure drift, adopted from Streitenberger and Koch [25].

In Figures 6-29 and 6-30 the calculational power and accuracy by Streitenberger et al [25] numerical scheme is tested for the limiting cases of pure drift and pure random diffusion, respectively, in the vicinity of a mode-I loaded crack tip.

O J

Figure 6-30: As for Figure 6-29 but in the limit of pure random diffusion Q=0, adopted from Streitenberger and Koch [25]

In the Figures 6-29 and 6-30 a three dimensional plot of the solute concentration field and the lines of constant concentration after 10000 time steps are displayed. As illustrated, the typical features of the pure-drift approximation, namely the discontinuity in the solute concentration separating the depleted from the non­depleted region on both sides of the crack faces and the shape of the characteristics, are well reproduced by the Streitberger/Koch [25] numerical solution.