FEM and multi-objective optimization of steel case hardening

Journal of Manufacturing Processes 17 (2015) 9–27

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Journal of Manufacturing Processes journal homepage: www.elsevier.com/locate/manpro

Technical Paper

FEM and multi-objective optimization of steel case hardening P. Cavaliere ∗ , A. Perrone, A. Silvello Department of Innovation Engineering, University of Salento, Via per Arnesano, Lecce I-73100, Italy

a r t i c l e

i n f o

Article history: Received 17 June 2013 Received in revised form 16 October 2014 Accepted 23 October 2014 Keywords: Case hardening Modeling Mechanical properties Microstructural evolution

a b s t r a c t Steel case hardening is a thermo-chemical process largely employed in the machine components production mainly to solve wear and fatigue damage in materials. The process is strongly influenced by many different variables such as material properties and processing parameters. In the present study, the influence of such parameters affecting the carburizing quality and efficiency was evaluated. The aim was to streamline the process by numerical–experimental analysis allowing for the definition of optimal conditions for the success of the process. The optimization software used is modeFRONTIER® (Esteco); a set of input parameters was defined (steel composition, carbon potential, carburizing time, etc.) and evaluated on the basis of an optimization algorithm carefully chosen for the multi-objective analysis. © 2014 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.

1. Introduction

1.2. Carburizing

1.1. Multiobjective optimization

Thermo-chemical processing such as carburizing is used in the manufacturing of steel components in order to improve surface properties such as resistance to fatigue, corrosion and wear. The reason for this surface modification is that advantageous changes in microstructure and distribution of residual stresses take place in the carbon rich surface layers after carburized quenching. Both the effect of cooling rate and chemical composition can be predicted on the microstructural evolution in steel during its heat treatment. However only a few data are available in the literature for microstructural simulation in case carburized hardened steels [4–9]. The surface hardening of steels has been the object of research and development for a long time and still continues to be a field of interesting research [10,11]. Not many papers are available in literature on the analysis on thermo-mechanical diffusion processes of carburizing based on the Fick’s laws model. There have been many theoretical studies on gas carburizing [12–15], which are mostly associated with one dimensional modeling. The effect of diffusion of carbon in steels is of fundamental importance in the increase of superficial strength due to martensite formation during quenching. In addition, a strong superficial compressive state is induced in the carburized components after the treatment. Fundamental importance is given to the availability of models capable of predicting the superficial layers dimensions and compositions. In [16], the authors described a mathematical model to predict the carbon and nitrogen contents as well as residual stresses and distortions after carburized quenching. As a general point of view, in fact, the hardening of steels during the carburizing process is due to the effect of carbon in solid solution (e.g. martensite formation [17]). The formation of carburized layers leads to the increase in strength and fatigue

Optimization analyses is achievable through integration with multiple calculation tools and explicable by effective postprocessing tools. The role of the optimization algorithm is to identify the solutions which lie on the trade-off Pareto Frontier. These solutions all have the characteristic that none of the objectives can be improved without prejudicing another. The use of mathematical and statistical tools to approximate, to analyze and to simulate complex real world systems is widely applied in many scientific domains. These kinds of interpolation and regression methodologies are also known as Response Surface Methods (RSMs), now becoming common even in engineering [1–3]. Once data have been obtained, whether from an optimization or DoE (Design of Experiment), or from data importation, the user can turn to the extensive post-processing to analyze the results. The software (modeFRONTIER® ) offers a wide-ranging toolbox, allowing the user to perform sophisticated statistical analysis and data visualization. DoEs can serve as the starting point for a subsequent optimization process, or as a database for response surface (RS) training, or for checking the response sensitivity of a candidate solution.

∗ Corresponding author. Tel.: +39 0832297357; fax: +39 0832297357. E-mail address: [email protected] (P. Cavaliere).

http://dx.doi.org/10.1016/j.jmapro.2014.10.005 1526-6125/© 2014 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.

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Nomenclature C Pot (carbon potential) Temp (carburizing temperature) Time (carburizing time) Diff T (diffusion temperature) Diff time (diffusion time) Oil (quenching oil identification) TQuench (quenching temperature) time Quench (quenching time) C xxx (carbon concentration at xxx millimeters from the surface) Mart xxx (percent of martensite at xxx millimeters from the surface) RA xxx (percent of retained austenite at xxx millimeters from the surface) Hv xxx (microhardness at xxx millimeters from the surface) sigZZ xxx (residual stresses at xxx millimeters from the surface)

properties due to the hardening coupled with a compressive state on the surface. The thermo-chemical phase of the treatment is the high temperature diffusion of carbon on a surface starting with a concentration of 0.1–0.2% and arriving to a concentration around 0.7–1.2%. This phase can be divided into two different reactions: (a) The reaction of the carbon with the low concentration surface. (b) The diffusion of the carbon in the component bulk. The carbon solubility limit increases in the austenitic phase and for this reason the temperatures employed during the treatment are 900–950 ◦ C. The carbon potential on the surface (Cs) depends on the temperature and the pressure. It remains constant during the treatment and at the same time the carbon diffuses in the bulk. The concentration profile depends on two fundamental factors: (a) The carbon concentration in the austenitic phase. (b) The alloy elements concentration influencing the carbon diffusion. The increase of carbon content in the case reduces the martensitic transformation starting temperature, leading to a delayed martensite transformation on the surface. This well-known phenomenon is the reason compressive residual stress is generated for carburized components [18,19]. In the present paper a broad range of processing conditions was studied. The mechanical properties of the surface layers depend on their chemical composition. In particular the carbon concentration can vary the presence of martensite, residual austenite and bainite presence. The residual stresses depend also on the carbon concentration being related to the martensite formation leading to compressive stresses on the surface. These processes always have complicated interactions among temperature, phase transformation, stress/strain considered with diffusion and distribution of carbon content [20,21]. Distortion provisional models have been presented in literature from results obtained by finite element calculations [22–29]. The carburizing process through the analysis performed by modeFRONTIER is summarized in the workflow of Fig. 1: Data flow (solid lines) and logic flow (dashed lines) have as their common node the computer node in which to introduce physical and mathematical functions representing the carburizing process. In the data flow all input parameters optimized in the numerical simulations are included:

• • • • • • • •

steel composition Carbon potential Carburizing temperature Diffusion temperature Diffusion time Cp (quenching oil’s specific heat) Oil temperature Quenching time

and the following outputs: • % carbon hardened layer as a function of distance from the surface of the sample • Residual stress from the sample surface until the thickness of the hardened layer • Hardness stress profile from the sample surface until the thickness of the hardened layer • Phase martensite layer cemented • Phase austenite-hardened layer At this stage the nodes that make up the logic flow of numerical analysis are defined. The first node is the DoE (the set of different designs, reproducing different possible working conditions, among which the most affective ones are highlighted). Therefore it means creating a set number of designs that will be used by the scheduler (the node where the best algorithm is introduced) for the optimization. In the present case an appropriate method of assessment was used: “Reduced Factorial”. Generally, in this kind of analysis, the heart of the optimization is represented by a series of equations of chemical and physical nature of a given resolution to get the desired output. In the present case, all this information is not clear, due to the complexity of the process so the decision was taken to employ the methodology of response surfaces. For each output variable to be minimized it is necessary to create a response surface. 2. Experimental procedure Compositional and martensite thickness measurements were performed through EDX in a Zeiss EVO40 SEM. Residual stresses and retained austenite measurements were performed through X-ray diffraction by using a Rigaku Ultima+ diffractometer by employing Hall-Williamson plotting. Microhardness was measured by employing a Vickers indenter with a 1000 gf load for 15 s. The database consists of 596 designs obtained from an experimental campaign. From the whole database two subsets were extracted: a training set of 585 designs and a validation set of 11 designs. In the validation phase, only the input remaining conditions were included in the RSM “trained” and the numerical calculated output was compared with the experimental output, measuring the  error. The database is built by introducing the input parameters, the corresponding output for each working condition experimentally analyzed and the physical correlations between the different conditions. Taken into account are the carbon potential of the furnace atmosphere, the furnace temperature; the carburizing time, the diffusion temperature, the diffusion time, the physical properties of the quenching media. In this case 3 kinds of oil were employed and the software associates the transmission coefficient of the oil [W/m2 * ◦ C] vs. surface temperature for each kind of oil for each experimental case, the quenching oil temperature and the quenching time. The heat transmission coefficient as a function of the steel surface temperature is shown in Fig. 2. All the results belonging to the optimization procedure were employed to obtain the equations governing the process and relating processing parameters to mechanical–microstructural behavior of steels. Such equations were used in an ANSYS model

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Fig. 1. Workflow of analysis.

to predict compositional and mechanical properties of carburized spur gears. 3. Results and discussion 3.1. Database and chemical–physical properties The experimental results are listed in a database capable to be analyzed in modeFRONTIER. The database consists of a listing of steel composition, carburizing time and temperature, carbon concentration in the carburizing atmosphere, diffusion time and temperature, quenching oil temperature, oil’s specific heat, carbon concentration at different distances from the surface, martensite percentage at different distances from the surface, hardness at

Fig. 2. Heat transmission coefficient as a function of the surface temperature of the steel.

different distances from the surface, residual stresses at different distances from the surface, microhardness and carbon concentration in steels are strongly dependent on the carbon potential. On the contrary, they are dependent each in its own way on the carburizing temperature as a function of steel composition and carburizing time. In the present study the employed RS was the Radial Basis Functions (RBF) that is a powerful tool for multivariate interpolation of scattered data. The first fundamental result of the analyses was the obtaining of the so called “correlation matrix” that allows us to immediately recognize how much the different variables are correlated with each other. The parameters are strongly correlated if the corresponding values in the table are distant from zero in a range between −1 and 1. If the value is 1 the parameters are directly correlated, while if the value is −1 the parameters are inversely correlated. An example for the present study is given in the following figure, from such matrix it is also possible to observe the different weight of all the parameters, the more the value differs from 0 the more it influences the corresponding variable (Fig. 3). Carbon concentration on the surface is strongly dependent on carbon potential, then on carburizing time and finally on carburizing temperature, an example of bubble graph for AISI1020, AISI8620, AISI 5115 and all the collected data is given in Fig. 4. The same behavior is observed for the carbon concentration at 0.5 mm from the surface with a stronger dependence on carburizing time. Surface hardness is strongly and directly dependent on carbon potential while it is inversely proportional to carburizing temperature (Fig. 5), hardness at 0.5 mm from the surface is also directly related to carburizing time. The hardness behavior confirms the linear correlation with carbon content. The hardness increase (with respect to the bulk value) is higher for low carbon content in the starting material. Surface residual stresses are strongly dependent on quenching temperature, while residual stresses at 0.5 mm are also dependent on carburizing time and temperature (Fig. 6).

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Fig. 3. Correlation matrix of the different input-outputs analyzed in the present study.

Fig. 4. Carbon concentration on the surface as a function of carbon potential and carburizing time for different steels, representative of all the collected data.

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Fig. 5. Surface microhardness as a function of carbon potential and carburizing temperature for different steels, representative of all the collected data.

Fig. 6. Residual stresses as a function of carburizing temperature and carburizing quenching temperature for different steels, representative of all the collected data.

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Fig. 7. Diffusivity of carbon in austenite, in AISI1020 and AISI8620.

All the results were used to analyze the material properties from the surface to the bulk in order to obtain the curves to be compared with those curves that will be available after the calculations. 3.2. Analytical modeling All the data were employed to obtain a generalized solution of Fick’s law: ∂C = D(∇ 2 C) ∂t

(1)

C is the carbon concentration, t is the diffusion time, the equation is capable of predicting the diffusion process and the carbon profile evolution in a very broad range of conditions (steel, carbon potential, temperature, time). Then (1) can be rewritten as:

 

∂C − → =∇ D ∂t

∂C − → ∂C − → ∂C − → i + j + k ∂x ∂y ∂z



(2)

where x, y, and z are the directions in the space, the contour conditions were: diffusion acts just in the surface orthogonal direction, carbon diffusivity D is independent on carbon concentration variation, In this way a simpler well known equation can be obtained: ∂C(x, t) ∂2 C(x, t) =D ∂t ∂x2

(3)

Carbon diffusivity has been considered as a function of carbon concentration in the material bulk and of temperature by

Fig. 8. Graphs employed to define Do and Qd for different ranges of temperature.

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Fig. 9. Numerical–experimental comparison between transformation curves of martensite and bainite for 16MnCr5 and 31CrMo12 steel.

Fig. 10. Correlation matrix between input parameters and martensite formation.

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Fig. 11. Correlation matrix between input parameters and microhardness values at different distances from the carburized surface (a); correlation matrix between input microhardness values at different distances from the carburized surface and the corresponding martensite content (b).

employing the correlation between such parameters for each kind of steel, as shown in Fig. 7. By employing such data for all the steel it is possible to develop a relationship for the diffusion coefficient:

In addition a detailed study has been conducted for the definition of the diffusion coefficient as a function of the alloying elements; the basic system to be employed is:

D = f (C, T, x, t)

⎧ D = D0 ∗ e−Qd /RT ⎪ ⎨

⎫ ⎪ ⎬

⎪ ⎩

⎪ ⎭

(4)

D is the diffusivity coefficient, T is the absolute temperature This function can be used in a numerical code, the diffusion coefficient at a certain step is calculated, from Fick’s equations the carbon concentration and then recalculate D for the next step is calculated.

D (%elements) + f D (%elements) D0 = D00 + flin sq

Qd =

Q Q Qd0 + flon (%elements) + fsq (%elements)

(5)

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modeFRONTIER my matching in closed loops the solutions with the experimental data. The dependence of diffusion parameters by processing conditions (time and temperature) was described through: Qd = Qd0 + (q1 ∗ t 2 + q2 ∗ t) + (q3 ∗ T 2 + q4 ∗ T ) 2

2

D0 = D00 + (d1 ∗ t + d2 ∗ t) + (d3 ∗ T + d4 ∗ T )

(8) (9)

In the same way all the coefficients were evaluated, for each condition, by employing ANSYS and modeFRONTIER calculations.The tempering conditions in different sections of the specimens were defined through a detailed digitalization of CCT diagrams. In this way the software is able to reconstruct the different quenching curves and match such curves with the data of the phases behavior listed in the output table. The different phases of the study were: digitalization of CCT diagram for each kind of steel, import of the data in the software, generation of the equations relating the quenching conditions to the phase transformations for each steel, numerical–experimental comparison. An example of transformation curves deriving from the study is shown in Fig. 9. The main parameter in this part of the study is the precise definition of martensite thickness through the following equation: pm (T ) = pm [1 − exp(−b(Ms − T ))]

(10)

with: Ms = 512 − 453C − 16.9Ni + 15Cr − 9.5Mo + 217(C)2 − 71.5(C)(Mn) − 67.6(C)(Cr) Ms is the martensite start temperature depending on the C, Ni, Cr, Mo and Mn Percentages. The correlation matrix for the martensite formation belonging to such approach is shown in Fig. 10. The hardness behavior was modeled by employing the following equations: HVphase i = HVphase i (bi0 C, bi1 Si, bi2 Mn, bi3 Ni, bi4 Cr, bi5 Mo, f (vR )) HVtot =

with: lim D0 ∗ e

T →+∞

= D0

(HVphase i ∗ %phase i)

i

Fig. 12. Training set employed for the definition of the control designs.

−Qd /RT

(6)

And with flin and fsq referring to the linear and square dependence on the alloying elements. Generally, in an Arrhenius type equation, D0 and Qd can be considered as constant in a small temperature range. R is the gas universal constant. In such a way it is possible to calculate the equation. Then the solution can be employed for the next step and the coefficient for the new temperature range recalculate. The followed procedure was to define the different coefficients in defined temperature range by fixing D0 and Qd respectively as shown in Fig. 8. The explicit equations for such dependences are given by:

(11)

where f(vR ) is a function of the cooling rate that can be calculated by the CCT diagrams. By correlating the different phase formation to the microhardness (HV) in different sections of the treated samples with the same instruments employed for the diffusion and martensite formation, it is possible to obtain a provisional instrument for the mechanical properties of case hardened steels. In Fig. 11 is shown the correlation matrix between the different input parameters and the hardness values at different distances from the surface and the correlation matrix between the hardness values in different sections as a function of the martensite content. Finally, the equations employed for the calculation of residual stresses are given by:  = f (ε) ε = εe + εth + εpc + εtp

(12)

where εe is the elastic component of the deformation, εpc is the plastic component, εth is function of the phase transformation by:

D = AA C ∗ (%C) + AA Si ∗ (%Si) + AA Mn(%Mn) + · · · flin 2 2 2 D = AA fsq sq C ∗ (%C) + AAsq Si ∗ (%Si) + AAsq Mn(%Mn) + · · ·

εth,i (T ) = ˛(T ) ∗ [T − T0 ]

Q flin = BA C ∗ (%C) + BA Si ∗ (%Si) + BA Mn(%Mn) + · · ·

εth (T ) =

Q fsq = BAsq C ∗ (%C)2 + BAsq Si ∗ (%Si)2 + BAsq Mn(%Mn)2 + · · ·

pi ∗ εth,i (T )

(13)

phases

(7) Si is the silicon percentage, Mn is the manganese percentage, AA, AAsq , BA, BAsq are constants. The defined system led to the definition of 38 independent coefficients solved by ANSYS (13.0) through

˛(T) is the expansion coefficient, T0 is the starting temperature. evaluating the volume change due to each phase transformation. εtp takes into account that when austenite transforms into martensite a micro residual stress is create, in the surrounding austenite

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Fig. 13. Results belonging to modeFRONTIER calculation for a control design of AISI1010.

Fig. 14. Results belonging to modeFRONTIER calculation for a control design of AISI1010.

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Fig. 15. Results belonging to modeFRONTIER calculation for a control design of AISI1020.

Fig. 16. Results belonging to modeFRONTIER calculation for a control design of 20CrMnTi.

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Fig. 17. Results belonging to modeFRONTIER calculation for a control design of Pyrowear53.

Table 1 MSE calculated for carburizing output. OUTPUT

MSE

MSE (medium)

−4

Mart 0.5 3 × 10 HV 0.5 C 0.5 C 0.75 C 1.5

4.4 1.2 × 10−3 5.6 × 10−4 1.5 × 10−4

RMSE

RMSE (medium) −2

−4

6.4 × 10

1.7 × 10 2.09761 3.4 × 10−2 2.3 × 10−2 1.2 × 10−2

2.5 × 10−2

matrix. All the results provided by modeFRONTIER can be employed to develop an analytical instrument to predict the diffusional and mechanical properties of carburized specimens. Tre main result of this phase is the possibility to obtain the dependence of microstructural and mechanical properties of the carburized materials as a function of all the employed input parameters. An example of such equations (Eqs. (14)–(17)), describing the dependences of carbon concentration, martensite phase, hardness and residual stresses at a thickness of 0.05 mm are reported in the Appendix. 3.3. Validation phase The reliability of the response surfaces was critically analyzed in this phase. The validation set analysis for the input parameters and for the alloying elements is shown in Fig. 12.

Fig. 18. Spur gear carburized in the present study.

In such a way it was possible to define a fixed number of control designs to train the response surfaces. The results in terms of microhardness, carbon profile, retained austenite and residual stresses of the control designs are shown in the Figs. 13–17. For all the validation designs an excellent agreement is recorded for the carbon and microhardness profiles; a greater difference between experimental and numerical results is, on the contrary

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Table 2 Experimental conditions employed to carburize spur gears. abs id

Steel

1 2 3 4

18NiCrMo5 AISI8620 18NiCrMo5 AISI8620

C Pot % 2.1 1 0.9 1.2

Temp ◦ C 950 975 925 940

Time h 17 9 2 5

noted for the retained austenite and residual stresses profiles. The Mean Square Error (MSE) applied on the points of measurement of the specific size of output was employed for the error calculation. Expressing the discrepancy between experimental and numerical values as follows: yi = yexp,i − ynum,i

MSE =

i

5 i=1

Diff time h

Oil –

Tquench ◦ C

Time quench h

900 850 850 850

0.5 0.5 0.5 0.5

1 2 2 2

300 65 170 25

3 2 3 2

In Table 1 shows the summary of MSE calculations on the outputs relating to the carburizing process. The results belong to the calculation of the difference between numerical and experimental data. To give them a physical sense, that is attributable to the units of measure of the question, was sufficient to calculate root mean square error (RMSE).

(18)

the Mean Square Error (MSE) for the outputs representative of a point is equal to: 5 2

(y )

Diff T ◦ C

(19)

3.4. FEM calculations By coupling ANSYS calculations and modeFRONTIER correlations with the experimental data the solutions for each condition were obtained. Actually, the final analysis was the inspection of different conditions performed on spur gears. The input conditions

Fig. 19. Comparison between calculated and experimental results for design 1 of 18NiCrMo5 spur gear.

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Fig. 20. Comparison between calculated and experimental results for design 1 of AISI8620 spur gear.

are given in Table 2. The designs 1 and 2 are new designs while the conditions of designs 3 and 4 were also investigated for the round bars and repeated on the spur gears for comparison. For each condition 3 different spur gears were produced (Fig. 18) and carbon, microhardness, residual stresses, retained austenite profiles, were measured at root and pitch. With the same order of the previous table the results are shown in the Pictures 19–22, In all the case can be observed an excellent agreement between numerical and experimental data for carbon and microhardness profile. The agreement for the residual stresses calculation appears strongly dependent on the analyzed steel. Note that the “num” curves in the graphs above come from meta-models which were trained on cylindrical samples. The good correlation between “num”, “root” and “pitch” for the carbon percentage and the microhardness demonstrates that the carbon diffusion is not sensitive to the geometry while it has an important role during the quenching and the subsequent residual stresses. 3.5. Optimization In the optimization study they were identified different objective functions to be related to mechanical properties to be

optimized under physical constraints. The identified mechanical properties were hardness and residual stresses. By taking a look to the hardness behavior it was decided to maximize the bulk hardness at different distance from the surface and the surface hardness. In Fig. 23 it is plotted the convergence of the two objective functions with the number of simulation runs, from a computational point of view it can be affirmed that the obj1 referring to the maximization of bulk hardness is much more faster than obj2 referring just to the surface hardness. By looking for a good compromise between the two functions, a Pareto analysis was performed and the results shown in Fig. 24. The results, as a function of the carburizing time confirm (identifying specific designs) that to obtain high bulk hardness it is necessary to employ longer carburizing times, such necessity is lower if just superficial hardness is required. The Pareto curve (Fig. 24b) identifies those designs with a good compromise between superficial and bulk hardness. At the end of the analyses the hardness profiles as a function of the number of runs were identified (Fig. 25), and for those profiles they were obtained the processing parameters to be employed for the achievement of the desired mechanical properties (Fig. 26); in black the Pareto designs processing parameters are underlined.

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Fig. 21. Comparison between calculated and experimental results for design 3 of 18NiCrMo5 spur gear.

Fig. 22. Comparison between calculated and experimental results for design 4 of AISI8620 spur gear.

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Fig. 23. Convergence of the chosen objective functions for the surface and bulk hardness of the carburized steel.

Fig. 24. Pareto analyses of the two chosen objective functions.

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Fig. 25. Different hardness profiles as a function of the computational runs.

Fig. 26. Processing parameters chart for the optimized hardness profiles.

4. Conclusions In the present paper were integrated experimental data, finite element calculations, and multi-objective optimization in order to develop a model capable of predicting mechanical properties, microstructural evolution and phase transformations during steel case hardening as a function of chemical compositions of steel and processing parameters. From the experimental results the precise mapping of carbon concentration, microhardness, residual stresses, microstructural evolution as a function of different processing parameters for various steels was obtained. The results were summarized in a database useful for the further analyses, in particular to obtain the data to be compared with the numerical ones. From the analyses of the correlation matrix it was calculated that, at the end of the thermo-chemical treatment, carbon concentration on the surface is mainly dependent on carbon potential with a factor four with respect to carburizing time and carburizing temperature. The dependence on carburizing time tends to increase in the material bulk. Surface hardness is also strongly dependent on carbon potential even if it decreases as carburizing temperature increases. Surface residual stresses are governed by the quenching temperature. Bulk residual stresses are dependent on carburizing time and temperature in a reasonable way. All the data were employed

for the calculation and optimization through modeFRONTIER and ANSYS in order to develop an analytical instrument capable of predicting the microstructure and mechanical properties of steels during case hardening in a broad range of conditions. In particular, a generalized solution of the Fick’s law was obtained and the equations relating microstructural and mechanical properties (carbon concentration, martensite percentage, microhardness and residual stresses) of different steels as a function of carburizing parameters were calculated. The calculations led to the definition of 38 independent coefficients solved by ANSYS through modeFRONTIER my matching in closed loops the solutions with the experimental data. Diffusion coefficients and activation energy were calculated as a function of steel composition, carburizing time and temperature in a discrete way. The tempering conditions were obtained from the digitalization of CCT diagrams of each studied steel. In this way, carbon concentration, hardness, martensite percentage and residual stresses were calculated as a function of input parameters and steel composition (Appendix). In the validation phase, the correlation between experimental and numerical results (obtained by the previous described procedure) was analyzed for selected control design belonging to AISI 1010, AISI 1020, 20CrMnTi and Pyrowear53 steel. The results showed an excellent agreement for the carbon concentration and hardness, a more pronounced shift of

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numerical data from experimental ones can be underlined for the residual stresses. The validation of the developed model gave good results especially for the carbon profile, microstructural evolution and microhardness for simple and complex component geometries. The analyses of the results, belonging to the simulations performed on the spur gears, leads to the conclusions that an excellent agreement between numerical and experimental data for carbon and microhardness profile is underlined also in the case of very complex geometries. The agreement for the residual stresses calculation appears strongly dependent on the analyzed steel. At the end of the study they were identified different designs optimized to achieve the maximization of surface and bulk hardness of the carburized steel identifying that to obtain high bulk hardness it is necessary to employ longer carburizing times, such necessity is lower if just superficial hardness is required.

Hv 0.5 = 1707.3635144786174 + (((((((((p Si − 10)/(C Pot + p Al)) + (Diff T ∗ p S))/(C Pot − p Al)) + (((((C Pot − p Al) ∗ p S) + ((p V ∗ p Fe) + (Diff T ∗ p S))) + (p Cr ∗ ((p Mo ∗ 10)/C Pot))) + (((C Pot − p Fe) + p Cr) + (C Pot − p Mn)))) + (((p Cr + ((p Cu − TQuench) ∗ p Ti)) + ((p S + p Ni) + (p Co + ((p Cu − TQuench) ∗ p P)))) + ((p Cu − TQuench) ∗ p S))) + ((((p S + p Ni) + ((((p Mo − 10)/C Pot)/(C Pot + p Al)) + (C Pot − p Al)))/ (C Pot − p Al)) + ((((((C Pot − 10)/C Pot) + (Diff T ∗ p Al)) + (Diff T ∗ p S))/(C Pot − p Al)) + (((p Si − 10)/(C Pot + p Al))

Acknowledgments

+ ((p V ∗ (p V ∗ (C Pot − Time))) ∗ (C Pot − p Al)))))) The authors gratefully acknowledge ENGINSOFT S.p.A. for the technical support and the PUGLIA REGION for the foundlings provided to the project S.T.A.R.-EXD (Simulation Technology Aeronautic Research-Extended Data) in the program action F.E.S.R. sul P.O. Regione Puglia 2007–2013, Asse I-Linea 1.1 – Azione1.1.2. Aiuti agli Investimenti in Ricerca per le PMI”.

+ ((((((p Cr + ((p Cu − TQuench) ∗ p P)) + (((p Mo − 10)/ (time Quench + p Al)) + ((p Cu − TQuench) ∗ p P)))/ (C Pot − p Al)) + (((((p Si − 10)/(C Pot + p Al)) + (Diff T ∗ p S))/ (C Pot − p Al)) + (p Cr + ((p Cu − Temp) ∗ p P)))) + ((((p Cu − TQuench) ∗ p P) + (p Cr + (((p V ∗ p Fe)

Appendix.

+ (Diff T ∗ p S)) ∗ p P))) + ((((p Si − 10)/(Time − p Al)) Example of the calculated carbon percentage, martensite percentage, hardness and residual stresses as a function of steel composition and input parameters.

+ ((p Al ∗ p P) ∗ p S))/(Time − p Al)))) + (((((p S + p Ni) + (p Ti + p Cr)) + (p Ti + p Cr)) ∗ (C Pot − p Mn)) + (((((p Cu − TQuench) ∗ p P) + (p Cr + ((p Cu − tp T) ∗ p Ti)))

C 0.5 = −6.60242045283854

+ (p Cr + ((p Pb − TQuench) ∗ p P))) + (((((p Si − 10)/

+ ln((((((((ln((Temp ∗ Time)) + Time) ∗ ((C Pot + p Cu) ∗ ln((Temp − C Pot)))) + ((p Fe ∗ sin(exp(time Quench))) − TQuench))

(C Pot − p Al)) + (Diff T ∗ p S)) ∗ (C Pot − p Al))

+ (((((C Pot + C Pot) ∗ (C Pot − p Cr))/ cos(p Fe))/ sin(ln((Temp − p Fe))))/

+ ((p Ni − p Al) ∗ ((C Pot + p Al) − ((p Mo ∗ 10)

sin(ln(((Temp − Time) − p Fe))))) + (((ln((Temp − p Fe))

+ (p Ni − p Al))))))))) − Diff T)

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∗ ln(((Temp/Time) + C Pot)))/ cos(exp(time Quench))) + (p Fe ∗ sin((Temp − exp(time Quench))))))

sigZZ 0.5

∗ ln((cos(cos(sin(ln((Temp − p Fe))))) + C Pot)))

= −209.5361088879514 + ((((exp(cos(p Ni)) ∗ (exp(((cos(p Ni)

+ (((Temp − exp(time Quench))

∗ ln(Time)) ∗ cos(exp(cos(p Mn))))) ∗ ln(Time)))

∗ (C Pot + cos(ln((Temp − exp(C Pot))))))

∗ (cos(((ln(TQuench) ∗ (cos(p Ni) ∗ ln(TQuench)))

+ ((((((p Ni ∗ p V) ∗ (C Pot + (p Ni ∗ p V)))/ cos(p Fe))/

∗ ln(cos(cos(cos(p Ni)))))) ∗ (cos((cos(exp(cos(p Cr)))

sin(ln((Temp − exp(p Ti))))) − p Fe) + cos(sin(ln(((Temp − Time) − p Fe))))))))

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∗ ((ln(TQuench) ∗ cos(Diff T)) ∗ cos(p Ni)))) ∗ (exp(cos(p Ni)) ∗ ln(Time))))) + (((cos(ln(C Pot)) ∗ (sin(p Ni) ∗ (cos((cos(p Ni) ∗ ln(Time))) ∗ ln(Time)))) − exp(exp((cos(Diff T)

Mart 0.5

∗ cos(exp(cos(p Ni))))))) ∗ (((cos(cos(p Ni)) ∗ (cos(ln(C Pot))

= −0.11323201854021092 + cos(((((p Cr ∗ (cos((p N − cos((p Cr ∗ 0.1))))/

∗ (cos(ln(TQuench)) + ln(Time)))) ∗ ln(Time)) − ((cos(p Ni)

(p Cu + ((p N/0.1) + 10)))) − (((((p N/0.1) − cos((p Cr ∗ p C)))

∗ (exp(cos(cos(p Ni))) + exp(exp(cos(p Cr)))))

− ((C Pot ∗ p Mn) ∗ (p Cr ∗ 0.1))) − cos(((0.1 ∗ (C Pot ∗ Time)) ∗ 0.1)))/

∗ cos(ln(TQuench))))))/ cos(p Ni))

(((p Cu − cos((p Cr ∗ p Cr))) − cos((p C + p Mn))) + (((p Cu − cos(p Mn))

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− cos((p C − 0.1))) + ((p N/0.1) + (p C + (p C + 10)))))))/C Pot) − cos(cos(((p Co/(p Cu − cos(p Mn))) + (cos(((p C/Time) + (p C + (p C + 10)))) − (cos(((p C ∗ Temp) ∗ (p Cr ∗ Diff T)))/(cos((p Mo + 10)) + (p Ni + (p W + 10))))))))))

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