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Application of Nitriding Simulation

Here, simulated results of nitrogen concentrations, distortions and residual stresses in nitrided stainless steels are introduced for demonstrating some functions of MUSIMAP(1).


For developing the functions, the following review was performed.

"Review of Researches for Evaluating Nitriding Simulation Models", 145 pages, in Japanese.

You can see the contents and references in English on the linked PDF file.

Nitrogen Concentration in Nitrided Austenite Stainless Steels

It has been well recognized that the expansion austenite layer is produced in austenite stainless steels during nitriding at the temperature around 723 K. Christiansen et al. measured and simulated distributions of nitrogen concentration in AISI 316 stainless steel specimens after gas nitriding at 718 K for 22 h (2). The measurement of nitrogen concentration profile was reported only in the case under the nitriding potential KN = 1.41. Their simulation was executed by a program based on the finite difference method with the solubility product model for describing the trapped nitrogen atoms.

Experimental and simulated results by Christiansen et al. (2) were used for verifying some functions of MUSIMAP. The region of the specimen from the surface to 0.04 mm in depth was modeled by 80 two-dimensional finite elements with the same size. The surface concentration was assumed to change with time to the equilibrium value as specified by Christiansen et al. Also the same diffusion coefficient of nitrogen as Christiansen et al. was used for this simulation, which depends strongly on the nitrogen concentration.

Figure 1 shows simulated distributions of nitrogen concentration by MUSIMAP and Christiansen et al. (2) under the three stages of nitriding potential KN, 0.293, 2.49 and , as referred to as N1, N2 and N3, respectively. Also, the measured profile at KN = 1.41 is depicted. The concentration yN is the fraction of sites in the interstitial sub-lattice, which are occupied by nitrogen atoms.

Figure 1 Distribution of nitrogen concentration


















Simulated concentration by MUSIMAP are specified as the legend symbols, “M” and “M+I”, which correspond to only mobile: “M” and the addition of mobile and immobile: “I” nitrogen atoms, respectively. “M+I” curves correspond to the simulated results by Christiansen et al. Concentration changes of mobile and immobile nitrogen atoms simulated by MUSIMAP under the N3 condition are depicted in Figure 2 separately for clarifying the generation of their profiles.


Figure 2 Distribution of nitrogen concentration










                                                                                                      (a) Mobile nitrogen                (b) Immobile nitrogen   

Simulated residual stress distributions

Christiansen and Somers(3) measured residual stress distributions in gas nitrided disks, 13 mm in diameter and 2 mm in thickness, made from the AISI 316 stainless steel. Nitriding was performed at 718 K for 22 h under the same conditions of nitriding potentials, as N1, N2 and N3, specified in the previous section. Stress distributions along the depth direction were measured by X ray at the surfaces after removing its layer serially by diamond paste polishing.

The above nitriding processes were simulated by MUSIMAP for its verification. The finite element model was created using 99 elements for the two-dimensional generalized plane strain problem, which was arranged to the one half of the thickness in the disk. Finer mesh divisions were set at the region producing the expanded austenite. The temperature in the model was changed uniformly in a cycle between room temperature and 718 K. The same type of diffusion conditions as described in the previous section were adopted in this simulation.


Although elastic, plastic, and creep phenomena are able to be taken into account during the simulation, a few material characteristics data of the expanded austenite had been published. In this simulation, therefore, Youngs modulus and Poissons ratio were extracted from ordinary stainless steel(4). The diffusion expansion coefficient of the expanded austenite was set to 0.164 mm/mm/ yN based on the data of lattice constant depending on the site fraction of nitrogen sublattice of AISI 316 at 718K(5).


In this simulation, creep phenomena were expressed using the Nortons equation. Parameters in the equation were set by trial and error. Plastic behaviors during nitriding were not considered in this modeling, because of the simplicity.

Figure 3 Residual stress distributions


















Figure 3 shows residual stress distributions along the depth direction, which were obtained by MUSIMAP, with the measured results by Christiansen and Somers. The stress value corresponds to the component perpendicular to the depth direction. Hypothetical simulated results not considered with creep effects are included in Figure 3 for reference. Simulated stress distributions with creep effects show a similar tendency to the measurements by Christiansen and Somers, while higher compressive stresses are depicted near the surface in the cases without creep.


In addition, simulated total, elastic, diffusion and creep strain distributions after nitriding under the condition N2 and N3 are shown in Figure 4. It is clear that the distribution of elastic strain has a similar tendency to stress as shown in Figure 3, because of their inherent relationship. Also the elastic strain is expressed by adding diffusion strain and creep strain and changing its sign, based on the equilibrium condition of strains, because the total strain is a small horizontal line. This shows how the residual stress distributions are affected by the diffusion and creep strains.


Figure 4 Strain distributions










Plate Bending during Low-Energy Implantation with Nitrogen

Sienz et al.performed in situ measurements of bending in the stainless steel plate, which was nitrided by the low-energy implantation with nitrogen from one side of the plate(6). Plate specimens, 10×30 mm2 in size and 1 mm in thickness, were prepared from the X5CrNi18.10 stainless steel, containing 18 mass% Cr and 10 mass% Ni. The radius of curvature due to bending of the plate was measured during nitriding for 63 min at 673 K and also until 90 min after stopping nitrogen bombardment. The nitrogen distribution along the depth direction of the specimen was investigated after nitriding for 63 min.

The above experiment was simulated by MUSIMAP for its verification. The hatched two-dimensional region in the plate specimen, 5 mm in length, as shown in Figure 5, was modeled by 495 generalized plane strain elements, which was created by the 99 and 5 divisions in the Y and Z directions, respectively. The surface part of the region, where expanded austenite was produced, was divided finer by elements in the Y direction. A boundary condition for the nitrogen diffusion was specified to only the surface, Y=0. The nodes in the model on the Y axis were restrained in the Z direction because of symmetry.

Figure 5 Analyzing region of plate











The same technique shown in the foregoing sections was used for describing the phenomena of both the nitrogen diffusion and expanded austenite formation, although the boundary condition of the implantation was not the same as the gas nitriding. Stress/strain characteristic data was set in the same way as the previous section; except for the referenced temperature is 673 K and a creep parameter that was set by trial and error.

Figure 6 represents the simulated distribution changes of nitrogen concentration during the experiment by MUSIMAP, and the measured results at 63 min by Sienz et al., along the Y axis. A discrepancy is appeared between simulated and measured profiles, because of few condition data. The nitrogen concentration at 90 min shows a redistribution induced by stopping nitrogen bombardment.


Figure 6 Nitrogen concentration distribution changes

















Bending curvature changes in the specimen simulated by considering with and without creep effects are compared to experiments as shown in Figure 7. A curvature value can be calculated from the gradient of a total strain slope as described later. The curvature change considered with creep show a similar tendency to the experimental result. On the other hand, the curvature without creep is increased more significantly in absolute value and is kept at the horizontal level after stopping nitrogen bombardment.

Figure 7 Curvature changes















Figure 8 depicts distribution changes of the normal stress in the Z direction along the Y axis, which were obtained by considering with creep effects. The compressive stress near the surface is decreased in absolute value, while the width of its region is increased as time progresses.

Figure 8 Simulated stress distribution changes















Distribution changes of elastic, diffusion, creep and total strains in the Z direction along the Y axis are plotted as shown in Figure 9 for explaining the generation of the stress and bending. Strains have to satisfy the equilibrium condition of strains, and also total strain has to keep a linear distribution during bending as shown in Figure 9 (d), because of the inherent characteristics of a long object. A gradient of a total strain slope is proportional to a curvature of bending based on the geometric consideration.

Figure 9 Simulated strain distribution changes















                                         (a) Diffusion               (b) Elastic            (c) Creep                                  (d) Total

It is clear that distribution changes of elastic strain shown in Figure 9 (b) have a similar tendency to stress shown in Figure 8. Therefore, the behavior of stress can be explained by elastic strain changes, which are expressed by adding the diffusion strain and creep strain, and changing its sign, based on the equilibrium condition of strains, because the total strain is very small. The decrease of curvature in absolute value from 63 to 90 min corresponds to the drop of a total strain gradient as shown in Figure 9 (d). This can be explained from increasing the width of the negative creep strain region as shown in Figure 9(c) between 63 and 90 min.



(1) Arimoto, K., Ikuta, F., Yamanaka, S. and Funatani, K., 2008, “Development of Simulation Tool for Predicting Distortion and Residual Stress in Nitrided Parts”, 2nd International Conference on Distortion Engineering, Bremen, Germany, 17-19, Sep., 2008, pp. 461-469.

(2) Christiansen, T., Dahl, K. V. and Somers, M. A. J., 2006, “Simulation of Nitrogen Concentration Depth Profiles in Low Temperature Nitrided Stainless Steel”, Defect and Diffusion Forum, Vols. 258-260, pp. 378-382.

(3) Christiansen, T. and Somers, M. A. J., 2004, “Simultaneous Determination of Stress and Composition Profiles in Expanded Austenite Obtained by Low Temperature Gaseous Thermochemical Treatment of AISI 316” in “Low Temperature Surface Hardening of Stainless Steel”, Ph. D. Thesis, TechnicalUniversity of Denmark, pp. 217-245.

(4) Date, E. H. F.: Elastic Properties of Steels. JISI, Vol. 207, 1969, pp. 988-991.

(5) Christiansen, T. and Somers, M. A. J., 2006, “Controlled Dissolution of Colossal Quantities of Nitrogen in Stainless Steel”, Metall. and Mater. Trans. A, Vol. 37, pp. 675-682.

(6) Sienz, S., Mandl, S. and Rauschenbach, B., 2002, “In Situ Stress Measurements during Low-Energy Nitriding of Stainless Steel”, Surface & Coatings Technology, Vol. 156, pp. 185-189.