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Metal Science and Heat Treatment

, Volume 58, Issue 5–6, pp 293–298 | Cite as

Nomograms to Determine the Controlling Factors in Vacuum-Carburizing Regimes

Article
First Online:

A method based on computer evaluation of mechanical properties and a mathematical model of vacuum carburizing are used for creating two nomograms, i.e., (1 ) for determining the parameters that the carburized layers of gears of steel 16Kh3NVFMB-Sh must have to obtain the required service properties and (2 ) for determining the values that the factors in periodic carburizing regimes must have to ensure that the layers have the prescribed parameters. The nomograms are used to determine the factors for two gears that are to undergo vacuum carburizing.

Key words

carburizing heat-resistant steels resistance to contact fatigue resistance to bending fatigue mathematical model nomogram 

Introduction

The well-known advantages of vacuum carburizing account for its wide use in industry as a substitute for the gas carburizing processes that have traditionally been used to harden machine parts made of high-strength steels. Among these parts are toothed wheels for gas-turbine aircraft engines made abroad and at leading Russian plants [1, 2]. The fact that the results obtained by saturation are consistently reproducible is an important advantage of the given process.

One of the distinguishing features of vacuum carburizing is that there are several independent controlling factors (the total duration of the process; the number of cycles, each of which consists of an active saturation stage and a passive diffusional equalization stage; the ratio of the stages of the cycle), and these factors determine the resulting level of the properties. In addition, vacuum carburizing regimes can be periodic (with a constant ratio of the stages) or aperiodic. Different sets of factors correspond to different concentration profiles for the saturation of the hardened layers with carbon, different distributions of the content of carbide phases based on cementite and high-melting carbides, different average diameters for their constituent particles, and different changes in hardness through the thickness of the layer. The main service properties of carburized layers of toothed wheels — resistance to contact fatigue, resistance to fatigue in bending, and resistance to seizing — can be calculated with an accuracy that is acceptable for practical purposes by using the information just mentioned. The relationships alluded to above were employed in developing a method of planning vacuum carburizing regimes with the use of application programs [3]. In addition to this use of the programs, they can help in automatically constructing nomograms [4]. The nomograms can then be used to identify the factors in standard regimes for industrial vacuum carburizing. The studies [5, 6] solved the problem of constructing nomograms for determining the controlling factors in gas carburizing and gas carbonitriding with a regulated carbon potential (or carbon and nitrogen potential). The factors were determined without allowance for their effect on the parameters of the layer and the service properties.

The goal of the present investigation is to construct nomograms for determining the factors in vacuum carburizing as a function of the required service properties.

Method of Constructing Nomograms

Nomograms for vacuum carburizing were constructed by using results calculated with application programs written in the language Object Pascal in the integrated development environment Embarcadero Delphi XE5 [7, 8]. The factors in vacuum carburizing were calculated for steel 16Kh3NVFMB-Sh at a process temperature of 940°C, which was adopted as a conditionally constant factor in the process. Steel 16Kh3NVFMB-III has the chemical composition, wt.%: 0.14 – 0.19 C; 2.6 – 3.0 Cr; 1.0 – 1.5 Ni; 0.4 – 0.6 Mo; 1.0 – 1.5 Mn; 1.0 – 1.4 W; 0.6 – 0.8 Si; 0.35 – 0.55 V; 0.1 – 0.2 Nb.

The following was chosen as the criterion of resistance to fatigue:
$$ {\upsigma}_{\mathrm{cr}}{S}_H\le {\sigma}_{H \lim }, $$
(1)

where σ Hlim is the contact fatigue limit in bending, with the method [9] being used to determine it at each point of the hardened layer in relation to its saturation and the morphology of the carbide phases; σcr is the corrected stress in the contact region; S H = 1.25 is the safety factor.

The equivalent corrected contact stresses were calculated based on the fatigue strength criterion used by M. M. Saverin [10]:
$$ \begin{array}{l}{\upsigma}_{\mathrm{cr}}=\frac{1}{\sqrt{2}}\left[{\left({\upsigma}_x-{\upsigma}_y\right)}^2+{\left({\upsigma}_y-{\upsigma}_z\right)}^2+{\left({\upsigma}_z-{\upsigma}_x\right)}^2+\right.\hfill \\ {}{\left.\kern6em 6\left({\uptau}_{xy}^2+{\uptau}_{yz}^2+{\uptau}_{zx}^2\right)\right]}^{1/2},\hfill \end{array} $$
(2)
where σi and τ ij are the normal stresses and shear stresses determined at each point of the diffusion layer by established methods as a function of the coefficient of sliding friction for the working surfaces of the teeth and the maximum normal stresses:
$$ {\upsigma}_{z \max }=\frac{2p}{\uppi b}, $$
(3)

where p is the distributed load in the gearing, N/mm; b is half the width of the contact area, mm.

Fatigue life in bending is expressed by the following inequality:
$$ {\upsigma}_{\mathrm{b}}{S}_F\le {\upsigma}_{F \lim }, $$
(4)

where σ Flim is the fatigue limit in bending and is determined by the method in [11]; σb is the maximum bending stress; S F is the safety factor (for aviation-grade toothed wheels, S F = 2.0 – 2.2).

The maximum bending stresses were determined from the formula [12]:
$$ {\upsigma}_{\mathrm{b}}=\frac{p}{\uppi\;m\;{y}_{\mathrm{f}}}, $$
(5)

where m is the modulus of the toothed wheel; y f is the form factor of the teeth and is calculated from parametric formulas.

The required effective thickness of the layer h eff , mm, was evaluated by using the empirical formula:
$$ {h}_{\mathrm{eff}}=0.28\pm 0.2. $$
(6)

The effect of the hardness of the treated surface on resistance to seizing was determined by the method in [13, 14].

The following mathematical model [15] was used to determine the controlling factors which ensure attainment of the required parameters for the diffusion layer and thus guarantee that the product will have the specified service properties:
$$ \begin{array}{l}\frac{\partial {C}_{\mathrm{C}}}{\partial t}=\frac{\partial }{\partial x}\left({D}_{\mathrm{C}}^{\mathrm{C}}\frac{\partial {C}_{\mathrm{C}}}{\partial x}\right)+\frac{\partial }{\partial x}\left({D}_{\mathrm{C}\mathrm{r}}^{\mathrm{C}}\frac{\partial {C}_{\mathrm{C}\mathrm{r}}}{\partial x}\right)-\frac{\partial {C}_{\mathrm{C}}^{\mathrm{c}}}{\partial t};\hfill \\ {}\frac{\partial {C}_{\mathrm{C}\mathrm{r}}}{\partial t}=\frac{\partial }{\partial x}\left({D}_{\mathrm{C}}^{\mathrm{C}\mathrm{r}}\frac{\partial {C}_{\mathrm{C}}}{\partial x}\right)+\frac{\partial }{\partial x}\left({D}_{\mathrm{C}\mathrm{r}}^{\mathrm{C}\mathrm{r}}\frac{\partial {C}_{\mathrm{C}\mathrm{r}}}{\partial x}\right)-\frac{\partial {C}_{\mathrm{C}\mathrm{r}}^{\mathrm{c}}}{\partial t},\hfill \end{array} $$
(7)

where t is the time of the process; x is the distance from the surface; C C and C Cr are the concentrations of carbon and chromium in the solid solution, respectively; C C c and C Cr c are the concentrations of carbon and chromium in the carbides; D C C and D Cr Cr are the known diffusion coefficients of carbon and chromium in austenite, the values of these coefficients depending on the temperature of the process and the concentrations of the saturating and alloying elements; D C Cr and D Cr C represent values taken from the literature data for the coefficients that characterize the mutual effect of the carbon concentration gradient and the chromium concentration gradient.

The boundary conditions and initial conditions of system of differential equations (7) are as follows:

a) at t = 0
$$ {C}_{\mathrm{C}}={C}_{\mathrm{C}}^0; $$
b) at the stage of diffusion saturation by carbon (the active stage)
$$ \begin{array}{l}{D}_{\mathrm{C}}^{\mathrm{C}}\frac{\partial {C}_{\mathrm{C}}\left(x=0\right)}{\partial x}+{D}_{\mathrm{C}}^{\mathrm{C}\mathrm{r}}\frac{\partial {C}_{\mathrm{C}\mathrm{r}}\left(x=0\right)}{\partial x}+\hfill \\ {}\kern1em \left(1-{P}_{\mathrm{c}}\right){\upbeta}_{\mathrm{C}}\left({\uppi}_{\mathrm{C}}-{C}_{\mathrm{C}}\left(x=0\right)\right),\hfill \end{array} $$
(8)

where πC is the carbon potential; βC is the mass-transfer coefficient; P c is the percentage of surface carbides not participating in the mass transfer of C;

c) at the stage of diffusional equalization (the passive stage)
$$ {\displaystyle \underset{0}{\overset{\infty }{\int }}\left({C}_{\mathrm{C}}+{C}_{\mathrm{C}}^{\mathrm{c}}\right)\mathrm{d}x=\mathrm{const};} $$
(9)

d) during the entire time of the process at x = ∞ : C C = C C 0 , where C C 0 is the initial concentration of carbon in the steel.

Results of Nomogram Construction and Discussion

As was noted above, vacuum carburizing regimes can be periodic or aperiodic. Thus, the number of possible vacuum carburizing nomograms is nearly limitless. In this investigation, we analyzed periodic regimes in which the length of a single cycle is 10 – 25 min.

The following nomograms were developed to determine the parameters required of diffusion layers:

Nomogram 1, to determine the values for the effective thickness of the layer and the saturation of the surface by carbon (C sfc ) which ensure attainment of the prescribed contact fatigue limit (Fig. 1).
Fig. 1

Nomograms for determining the parameters of the diffusion layer of steel 16Kh3NVFMB-Sh (before removal of the grinding allowance) after carburizing done to ensure attainment of the prescribed contact fatigue limit σ Hlim (m is the modulus of the toothed-gear transmission): a) for determining the effective thickness of the diffusion layer h eff (the numbers next to the curves); b ) for determining the concentration of carbon on the surface C sfc (the numbers next to the curves).

Nomogram 2, to determine the values for the effective thickness of the layer and the saturation of the surface by carbon (C sfc) that ensure attainment of the prescribed fatigue limit in bending (Fig. 2).
Fig. 2

Nomograms for determining the parameters of the diffusion layer of steel 16Kh3NVFMB-Sh after carburizing (before removal of the grinding allowance) performed to ensure attainment of the prescribed fatigue limit in bending σ Flim: a) allowable range of layer thickness h eff (m is the modulus of the toothed-gear transmission); b ) dependence of σ Flim on the concentration of carbon on the surface (the numbers next to the curves represent the effective thickness of the diffusion layer).

When using nomograms 1 and 2, it is assumed that the removal of 0.15 – 0.20 mm of the allowance by grinding will be done as is required for high-precision aviation-grade toothed wheels [16].

Nomogram 3, for determining the surface hardness that is required to ensure the prescribed seizing resistance for the given values of sliding speed and the maximum normal stresses on the contact surface (Fig. 3).
Fig. 3

Nomograms for determining the parameters of the diffusion layer of steel 16Kh3NVFMB-Sh after carburizing (before removal of the grinding allowance) performed to ensure attainment of the prescribed degree of resistance to seizing (the numbers next to the curves represent the maximum allowable normal stresses on the contact area): HV g) surface hardness determined with allowance for the removal of the grinding tolerance; v sl) sliding speed.

Use of this nomogram requires determination of the corresponding degree of saturation of the surface by carbon (C s ). That is obtained from the surface-hardness nomogram (Vickers hardness) by means of the formula:
$$ HV={p}_{\mathrm{c}\mathrm{em}}H{V}_{\mathrm{c}\mathrm{em}}+{p}_{\mathrm{c}}H{V}_{\mathrm{c}}+\left(1-\left({p}_{\mathrm{c}\mathrm{em}}+{p}_{\mathrm{c}}\right)\right)H{V}_{\mathrm{m}}\left({C}_{\mathrm{s}}\right), $$
(10)
where HV cem ≈ 1200 and HV c ≈ 1900 are respectively the Vickers hardness of cementite alloyed with about 5% Cr [17] and the Vickers hardness of the high-melting carbides (averaged with allowance for the data in [18] and the chemical composition of steel 16Kh3NVFMB-Sh); HV m (C s) is the dependence of the hardness of martensite on C s; p cem ≈ (C s − 0.8)/5.9, p c ≈ 0.015 are the volumetric contents particles of cementite and the high-melting carbides, respectively. By approximating the data in [19] with the use of a polynomial, the hardness of martensite can be determined from the formula:
$$ H{V}_{\mathrm{m}}\approx 60+1097{C}_{\mathrm{s}}-426{C}_{\mathrm{s}}^2. $$
(11)
The above data was used to determine the dependence of hardness on the carbon content of steel 16Kh3NVFMB-Sh (Fig. 4).
Fig. 4

Dependence of hardness HV on the carbon content of the surface of steel 16Kh3NVFMB-Sh after carburizing.

The data in [20] indicates that a hardness of 800 HV (64 HRC ) is nearly the maximum level that can be obtained in Cr – Ni – Mo steels. Vacuum carburizing can produce a somewhat higher surface hardness (about 65 – 66 HRC, or 820 – 860 HV ) when steel 16Kh3NVFMB-Sh is alloyed with tungsten.

The following nomograms reflect the relationship between the parameters of the diffusion layer and the controlling factors in vacuum carburizing.

Nomogram 4, for determining total carburizing time and the ratio of the stages of the cycles as functions of the required size of the diffusion layer and the concentration of carbon on the surface (Fig. 5).
Fig. 5

Nomograms for determining the controlling factors in the vacuum carburizing of steel 16Kh3NVFMB-Sh (the numbers next to the curves represent the fraction of the active-saturation stages) at 940°C: a) to ensure the required h eff; b ) to ensure the required C sfc.

Nomogram 5, for determining total carburizing time and the stage ratio as functions of the required size of the active carbide zone. In addition to high-melting carbides of W, Mo, and Cr, this zone contains alloyed cementite; the cementite is alloyed mainly with Cr and Mn (Fig. 6).
Fig. 6

Nomogram to determine the values that the controlling factors in the vacuum carburizing of steel 16Kh3NVFMB-Sh at 940°C must have to ensure the prescribed size of the active carbide zone h a (the numbers next to the curves represent the fraction of the active saturation stages).

Nomograms 1 – 5 were used to develop vacuum carburizing regimes for the following cases:

I — toothed-gear transmission No. 1 operating with elastohydrodynamic lubrication; vacuum carburizing ensures a contact fatigue limit of at least 1800 MPa and a minimum of 950 MPa for fatigue limit in bending with a transmission modulus equal to 4 mm;

II — high-speed toothed-gear transmission No. 2; vacuum carburizing ensures resistance to seizing with values of 1500 MPa for the maximum normal stress on the contact area, 50 m/sec for sliding speed, at least 0.30 for the size of the active carbide zone (after removal of the allowance, a guaranteed value of 0.10 – 0.15 mm for the size of the zone hardened with high-melting carbides and alloyed cementite); the modulus of the transmission is 3.5 mm.

For transmission No. 1, we used nomograms 2 to determine the requirements for the diffusion layer with respect to the fatigue limit in bending: 1.12 mm ≤ h eff ≤ 1.52 mm; 1.0% ≤ C sfc ≤ 1.4%. Nomograms 1 were used to determine the requirements on resistance to contact fatigue: h eff ≥ 1.30 mm; C sfc ≥ 1.35%. Thus, it is necessary to ensure that the diffusion layer has the following parameters: 1.30 mm ≤ h eff ≤ 1.55 mm and C sfc = 1.35 – 1.40% (before removal of the grinding allowance). Nomograms 4 were used to determine the duration of the process, which is 12 h; the ratio of the active saturation stages to the passive diffusional equalization stages is 1 : 9; there are 36 cycles of 20 min each.

We used nomogram 3 to evaluate the required surface hardness for toothed-gear transmission No. 2 and determined it to be 760 HV, which corresponds to a carbon content of 1.6% on the surface (after removal of the grinding allowance).

Nomogram 5 was used to determine the duration of the process, which is equal to 10 h with a stage ratio equal to 1 : 4 and 40 cycles of 15 min each. Nomogram 4 determined that 1.4 mm was the total size of the diffusion layer before the grinding operation, which is sufficient to ensure an adequate thickness for the ductile core: 1.0 mm ≤ h eff ≤ 1.4 mm (nomogram 2 ).

Conclusions

1. Computer application programs were used to develop nomograms for determining the controlling factors in the vacuum carburizing of steel 16Kh3NVFMB-Sh in the periodic regime. The nomograms reflect the dependence of the service properties of surface-hardened toothed wheels on the parameters of the diffusion layers, in addition to the dependence of layers’ parameters on the factors.

2. The nomograms were used to select variants for the regimes that are to be used in the vacuum carburizing of two toothed-wheel transmissions: transmission No. 1, the performance of which depends on the constituent material’s contact fatigue limit and fatigue limit in bending; transmission No. 2, whose performance is determined mainly by the material’s resistance to adhesive and abrasive wear and erosion in the presence of a sufficiently thick diffusion layer.

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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Moscow State Technical University im. N. E. BaumanaMoscowRussia

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