Issue 35

G. Gobbi et alii, Frattura ed Integrità Strutturale, 35 (2016) 260-270; DOI: 10.3221/IGF-ESIS.35.30 265   0 exp b C C g RT     (5) where θ is the hydrogen coverage factor, 0 b g  is the variation of Gibbs free energy, R is the gas constant and T is the temperature. In turn, this term contributes to the calculation of k factor, by which the cohesive energy is reduced simulating the embrittlement effect [15]: 2 1 1.0467 0.1687 k        (6) This scaling factor is applied on both stress and displacement of the TSL curve reducing the entire area. All the relationships and parameters in Eqs. (4), (5) and (6) are calculated in the last analysis, for each increment, in the unique integration point of the continuum element. The parameter, k , is then transferred from the continuum to the corresponding cohesive elements by implementing three subroutines. These are the user subroutines to manage user- defined external databases and calculate model-independent history information (UEXTERNALDB), the user subroutine to redefine field variables at a material point (USDFLD), and the user subroutine to generate element output (UVARM). These subroutines work interactively each other in a Fortran common block. The performed experimental toughness test from which the model have been developed was in accordance with standards, and specimen dimensions ensured the definition not only of J -toughness but also of K -toughness. Therefore, at the specimen core, plane strain condition is likely to occur. On the contrary, at the external surfaces, the stress state would be more similar to plane stress condition. Since the model we are implementing is bi-dimensional, limitations in the analysis should be taken into account. For this reason, the model has been developed both in plane strain and plane stress conditions. Apart from the different type of continuum elements used to describe the material (CPE4 for plane strain and CPS4 for plane stress analysis), the model geometry, the applied displacement and boundary conditions and the followed calibration procedure are coincident. Even if the modelling steps are the same, the plane strain and plane stress models give different results, which will be critically discussed in the following section. R ESULTS AND DISCUSSION irst, a TSL calibration for cohesive elements is developed to find out the cohesive parameters that replicate the behavior of the material in absence of hydrogen for both plane stress and plane strain configurations model. Tab. 2 shows the resulting TSL parameters. The TSLs have a very small flat part, i.e. (δ N – δ 0 ); on the contrary, they present a long trend describing the damage to failure, i.e. (δ F – δ N ). This means that the TSL is more similar to a triangle than a trapezoid. The TSL plateau stress, σ 0 is between 1.75 and 1.85 the yielding stress. The area below the TSL plot corresponds to the separation work needed for crack initiation and propagation. Values of this area are deeply different between plane strain and plain stress cases, accounting for the stress combination at the separation surface. It should be mentioned that data in Tab. 2 are results of a trial and error procedure, and they correspond to the best combination of TSL parameters able to fit experimental data. Hydrogen-free model δ 0 [mm ] δ N [mm ] δ F [mm ] σ 0 [MPa ] Area [N/mm] Plane strain 0.009 0.015 0.320 1260 205.4 Plane stress 0.004 0.011 0.090 1323 28.5 Table 2 : Characteristic values for TSL parameters for the hydrogen-free model, for plane strain and plane stress models. Displacement and stress notation is referred to Fig. 1; the area is below the TSL curve. Now, considering the model that predicts the response of the material in presence of hydrogen, the first step of analysis evaluates the hydrostatic stress field. This is imported as predefined field in the second mass diffusion analysis. Here, the hydrostatic stress drives the flux of hydrogen computing the interstitial lattice sites content, C L . Fig. 3 shows C L F