PREDICTION OF DISTORTIONS IN THROUGH HARDENING OF CYLINDRICAL STEEL WORKPIECES BY DIMENSIONAL ANALYSIS

C. Şimşir1, T. Lübben1 , F. Hoffmann1, H.-W. Zoch1, M. Wolff2 1 Foundation Institute of Materials Science (IWT) ,Badgasteinestr. 3, 28359, Bremen, Germany, simsir@iwt-bremen.de 2 Centre for Industrial Mathematics, Fachbereich 3, University of Bremen, Postfach 330440D-28334 Bremen, Germany

Abstract : Former investigations on the dimensional changes during transformation-free gas nozzle field quenching of cylindrical workpieces exhibited a typical behavior which can be described by few dimensionless numbers [1-3]. Recently, in the outlook of [3], it was proposed to extend the previous approach to the case with martensitic transformation. In this article, an application of this extension to through hardening process of long steel cylinders is presented. The significance of Biot number is demonstrated by computer simulations performed on commercial FEA software SYSWELD. The simulations were verified experimentally for gas nozzle field quenching of 100Cr6 steel cylinders. Results indicate that Biot number based on the thermal conductivity of austenite may be used as a measure of distortion in combination with other dimensionless numbers.

Keywords : Distortion, through hardening, dimensionless numbers, cylinders, SAE 52100

1. INTRODUCTION Isotropic dimensional changes during through hardening process are unavoidable due to density differences between the initial and final microstructures. However, the major concern of heat treatment industry is the prediction of anisotropic dimensional changes due to plasticity processes. In contrast to transformation-free quenching , for which Wolff et al. [3] have proved that there is a critical Biot number above which plasticity can occur, anisotropic dimensional change is unavoidable in through hardening process, even at very low cooling rates due to transformation-induced plasticity(TRIP) which occurs without a yield limit [4].

Former studies have shown that dimensional changes during transformation free quenching of austenitic stainless can be predicted by dimensional analysis[1-3]. Frerichs and Landek et al. [1-2] have shown that the Biot number is the most important dimensionless number in transformation-free quenching. In this study, the extension of former approach to the case with martensitic transformation [5] is demonstrated by focusing on the Biot number :

(1)

where α, λ are the convective heat transfer coefficient and thermal conductivity, whereas; (V/S) is the volume to surface area ratio. The choice of using (V/S) instead of diameter (D) as in the classical definition allows a better comparison between different geometries[6].

Another important dimensionless number that governs the transient heat conduction problem is the Fourier number defined as ,

(2)

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where ρ, cp are the density and specific heat capacity, respectively; whereas, t is a characteristic time [3]. Fourier number is important in calculation of the critical Biot number as a function of diameter ( Bi*(D) ) above which through hardening can be achieved. However, the calculation of Bi*(D) is left out of the scope of this paper for the sake of brevity.

One can also utilize the Fourier number in comparison of the evolution of dimensional changes. However, it should be emphasized that as long as through hardening is achieved, Fourier number has no influence on the final dimensions due to infinite time requirement of a complete thermal equilibrium.

In the extension of the former approach to through hardening, a problem arise in the decision of thermal properties which will be used in the definition of the Biot and Fourier numbers since there exists a continuously evolving microstructure consisting of two phases in the system. As a first approach, we will intuitively use the mean thermal properties of the austente which are calculated by,

(3)

where θo, θ∞ are the austenitization and ambient temperature. These new definitions will be denoted as Bia and Foa henceforth. This choice of definition may be owed to the fact that the temperature interval in which austenite is the only phase is considerably larger than the transformation range. The consequences of this choice will be justified later in Section 3.2.

2. EXPERIMENTAL AND NUMERICAL INVESTIGATION PROCEDURE The SAE52100 steel specimens used in experiments are machined from a larger billet and each specimen is marked in such a way that exact position of the shaft is known with respect to the initial billet. After machining of the specimens, initial coordinate measurements was performed to determine the geometry before heat treatment. Then, the specimens are heat treated in a gas nozzle field quenching system which consists of a vertical tube furnace and a gas nozzle array of 48 nozzles which is arranged as shown in Fig. 1. Details of the system can be found in [7].

∫ ⋅⋅⋅⋅⋅

=R

02 drr2Δz

R2Δl π

πz

r

dzΔrL2Δd

L

0

⋅⋅= ∫

R

L

∫ ⋅⋅⋅⋅⋅

=R

02 drr2Δz

R2Δl π

πz

r

dzΔrL2Δd

L

0

⋅⋅= ∫

R

L

top αT

bottom αT

shell αS

top αT

bottom αT

shell αS

Fig. 1 : Side and top views of the gas nozzle field quenching system

used in experimental verification of simulations. Fig. 2. : Calculation of mean dimensional

changes and heat transfer boundary conditions.

During heat treatment, specimens are heated up and austenitized at 850°C in a furnace for 10 min under a nitrogen atmosphere, then, immediately taken into the nozzle array. After the hardening process, another coordinate measurement is carried out to determine the final geometry. Experiments for each testing condition were repeated 5 times to ensure the repeatability.

The numerical simulations have been performed on commercial FEA software SYSWELD using material data set published in [8,9]. Broadening the former methodology of [1-2] by considering multiphase material and the phase transformation effects, a temperature independent

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(average) material data set for dimensional analysis was generated (Table 1). The consequences of this averaging process will be discussed in the next section.

Previous work confirmed that an axisymmetric model with heat transfer boundary conditions as shown on the right of Fig. 2 is adequate for the simulation of that gas nozzle field quenching of shafts for prediction of dimensional changes[7]. During parametric analysis, a very conservative automatic time stepping algorithm that ensures proper integration of material laws and convergence up to Bia = 5 was employed in the simulations. The mesh density and element type were determined by a convergence analysis performed on the extreme case (Bia = 5). The results indicated that required accuracy can most efficiently be achieved by using a mesh of 8-node quadrilateral elements (reduced integration) with 20 elements in radial, 500 elements in axial direction. Due to the lack of space, the reader is urged to refer to [10] for the details of simulation. After performing the simulations, mean dimensional changes were calculated according to Fig 2. (left) from the extracted contour displacements. Table 1 : Average material properties and their units and dimensions (M: mass, L: length, T: time, θ: Temperature)

Dimensions Phase Property Unit M L T θ Austenite Martensite

λ Thermal Conductivity [W/mm∙°C] -1 2 -3 -1 0.02012 0.04273 cp Specific Heat Capacity [J/kg∙°C] 0 -2 2 -1 560.7 501.3 ρ Density [kg/m3] 1 -3 0 0 7798 7741 E Elastic Modulus [MPa] 1 1 -2 0 170665 204671 ν Poisson`s Ratio - 0 0 0 0 0.3318 0.3556 K Ramberg-Osgood Coefficient [MPa] 1 1 -2 0 510.9 820.3 n Ramberg-Osgood Exponent - 0 0 0 0 0.1092 0.2794 σo Initial Yield Strength [MPa] 1 1 -2 0 173.3 1278.4 αth Thermal Expansion Coefficient [10-6∙°C-1] 0 0 0 -1 23.8 10.9

ΔHa-m Transformation Enthalpy [J/kg] 0 2 -2 0 -78520 εa-m Transformation Strain [10-6] 0 0 0 0 993

3. RESULTS AND DISCUSSION 3.1 Experimental verification and the effect of averaging of material data The comparison of experimental results with simulations (isotropic and kinematic hardening) is presented in Fig. 3. It can be seen that the simulations with both hardening models are in reasonable agreement with experimental results. This makes an objective decision of hardening model practically impossible based solely on these results. For the sake of brevity, the focus of the rest of this paper will be based on isotropic hardening assumption. It should also be noted that there is an ongoing research at IWT-Bremen on determination of hardening model for 100Cr6 steel based on residual stress measurements and Bauschinger effect tests.

Fig. 4 illustrates the comparison of the results obtained using average(*) and θ-dependent data sets as a function of Bia. In the figure, it can be seen that these two sets are quantitatively comparable up to Bia ≈ 0.25. Bia ≈ 0.25 is a typical maximum for SAE52100 in the specified gas nozzle field quenching system because of the limitation in diameter due to through hardening restriction . On the other hand, it can also be observed that the increasing difference between two sets makes the quantitative evaluation impossible for higher Biot numbers. Nevertheless, it is still possible to use the average data set for qualitative analysis for understanding of quenching since both curves exhibit similar trends.

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In order to understand the basis of the similarity in the region of concern, a closer look on both the thermal and mechanical response of a cylinder for a quenching condition (Bia = 0.145 , Do =10 mm, αo=1200 W/m2∙°C) - which will be henceforth referred as "standard" - is presented in Fig. 5. In Fig. 5(a), it can be clearly seen that the thermal histories of the core and the surface of the cylinder, calculated using θ-dependent and average(*) data sets are almost identical. Similarly, Fig. 5(b) reveals that mechanical response of two datasets is highly similar by illustrating the residual stress distributions calculated at the mid-height of the cylinder. Thus, one can conclude that it is no surprise to have similar distortion behavior between θ-dependent and average data sets in the investigated region since both, the thermal and mechanical analyses, yield similar results in terms of dimensional changes.

Fig.3 : Experimental verification of simulations. Fig.4 : Comparison of θ-dependent and average(*) sets

Fig. 5 : Comparison of (a) cooling curves (b) residual stress distribution of temperature dependent and average data

sets for a quenched cylinder (Bia = 0.145 , Do =10 mm, αo =1200 W/m2∙°C).

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3.2 The significance of Biot number A systematic variation of parameters in Biot number was carried out to demonstrate its significance and to justify the assumption of using λa in its definition. Fig. 6(a) illustrates the relative dimensional changes as a function of Bia after independent variations of α, c·λa and a collective variation of c·λa and c·λm in the standard case; where c is a factor between 1/4 and 4. On the legend of Fig 6(a), it should be noted that the parameter which was varied is expressed in the brackets; while the other parameters were kept at their standard values. Fig. 6(a) clarifies that the assumption of using Bia as a measure for distortion is only valid up to Bia ≈ 1.0 if a tolerance of 1 μm/mm is considered acceptable; whilst, the collective variation of λa and λm results in a very good agreement event at higher Biot numbers. This brings out the problem of definition of more generalized Biot number which takes thermal conductivities of both phases into account. Nevertheless, suggested assumption of using of λa only in definition of Biot number is a simple but still practically applicable to a wide range of quenching problems. The definition of a more generalized Biot number which takes phase transformations into account is a subject of ongoing research by the authors.

In addition to the variation of thermal conductivities of phases, Fig. 6 (b) clearly indicates that simulations with similar Bia by variation of diameter yield similar relative dimensional changes. Minor differences may be owed to numerical reasons such as the differences in FE meshes and different amounts of bainitic transformation.

Fig. 6 : Comparison of relative dimensional changes as a function of Bia calculated from simulations with variation

of a) thermal conductivity (λ) of phases, b) diameter of the cylinder (D).

After independent variations, a collective variation of D and λ was carried out to highlight the fact that Biot number can be used as a measure of relative dimensional changes after quenching process rather than individual parameters. For this purpose, a detailed comparison of two quenching processes is presented in Fig. 7-9. One of the quenching conditions compared on the figures is the "standard case" , while the second case which yields the same Biot number (Bia = 0.145) is obtained by simultaneous doubling of α and λa while keeping the diameter (D) constant (D = 10 mm, α= 2αo, λo= 2λo).

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A comparison of relative dimensional changes as a function of normalized distance at the side and top surface is presented in Fig. 7 (a) and Fig. 7(b), respectively. From the results, it can be concluded that the two different conditions with same Bia produce similar distributions of relative dimensional changes : The differences in diameter changes is not practically measurable whilst the length changes are barely measurable for the given geometry.

The similarity of systems can also be emphasized by comparing the evolution of other physical quantities. For example, from Fig. 8(a), it can easily be observed that the systems bring out similar results also in terms residual stress distributions. In Fig 8(b), it can be further observed that the distribution of total radial strain, which is the sum of elastic, plastic, thermal, transformation and TRIP strains, exhibit a similar behavior.

Fig. 7 : Comparison of relative dimensional changes as function of normalized distance at (a) the side and (b) top of

2 cylinders quenched under different conditions resulting in the same Biot number (Bia = 0.145).

Fig. 8: Comparison of a) residual stress b) total strain distribution at the center line of 2 cylinders quenched under

different conditions resulting in the same Biot number (Bia = 0.145).

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Although it is not major concern of distortion engineering, it can also be demonstrated (Fig. 9) that the systems are also kinematically similar if the evolution of quantities such as temperature and stresses are expressed in terms of a " normalized dimensionless time" (τ) which is defined as:

( )( ( ))

a

a eqm

Fo tFo t

tq q

=»

(4)

where t(θ ≈ θeqm) is the time at which the system practically reached the thermal equilibrium (20.1°C for this case). It should be noted that τ = 1 by definition at the end of quenching process.

Fig. 9: Comparison of evolution a) temperature and b) tangential stress as function of normalized dimensionless time for 2 cylinders quenched under different conditions resulting in the same Biot number (Bia = 0.145).

4. CONCLUSIONS AND OUTLOOK Dimensional analysis is a scarcely used tool in solid mechanics, especially in elastoplasticity; in contrast to its diverse applications in fluid mechanics. Dimensional analysis approach have recently been used by several authors for prediction of the dimensional changes during transformation-free gas nozzle field quenching of cylindrical workpieces[1-3]. This study suggests a possible extension of former approach to the case with martensitic transformation by demonstrating one of most important dimensionless number that affects distortion. In the near future, the same approach will be extended to other dimensionless numbers in combination with Biot number. In the light of the presented results, following conclusions can be drawn :

• Biot number has a prime importance prediction of quenching distortion. • There exist a critical Biot number as a function of diameter (Bi*(D)) above which

through hardening can be achieved and the presented approach is valid. Bi*(D) can be determined by quenching simulations or analytically. However, the analytical derivation of Bi*(D) was left out of the scope of this article.

• Quantitative prediction of relative dimensional changes of shafts by dimensional analysis with a temperature independent data set is possible for small Biot numbers up to Bia ≈ 0.25, which covers the typical range of gas nozzle field quenching of small sized

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specimens. On the other hand, dimensional analysis can still be used qualitatively in understanding the physical phenomena that leads to distortion during quenching.

• The use of mean thermal conductivity of austenite in the definition of Biot number is a reasonable approach up to Bia ≈ 1 if an error of 1 μm/mm is tolerable in relative dimensional changes. However, it is necessary to define a more general Biot number that takes the effect of phase transformation into account for higher Biot numbers. This fact is also included in the outlook of this research.

Aknowledgement : This research was carried out within the scope of the Collaborative Research Center SFB570 “Distortion Engineering” in the subproject A6 at the University of Bremen. The authors would like to acknowledge the Deutsche Forschungsgemeinschaft (DFG) for the financial support.

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[2] Landek,D., Lisjak D., Frerichs, F.: Prediction of unavoidable distortions in transformation- free cooling by a newly developed dimensionless model, Proceedings of 2nd International Congress on Distortion Engineering (IDE) , Bremen, Germany, (2008) 237-246.

[3] Wolff, M., Böhm, M., Frerichs, F.: Dimensional analysis of a model problem in thermoelasto-plasticity for cylindrical bodies under heating and cooling, Z. Angew. Math. Mech. (ZAMM) 88 (10) (2008) 758-775.

[4] Fischer F.D., Sun Q-P., Tanaka K.: Transformation-Induced Plasticity, Applied Mechanics Reviews 49 (1996), 317-364

[5] Wolff, M., Böhm, M., Suhr, B. : Dimensional Analysis of a Model Problem for Cylindrical Steel Work-pieces in the case of Phase Transformations, Proceedings of IFHTSE 2009 Congress, Dubrovnik, Croatia, (2009)

[6] Kobasko, N. I., Morganyuk, V. S., Dobrivecher, V. V. : Control of residual stress formation and steel deformation during rapid heating and cooling, in: Totten, G., Howes, M., Inoue, T. (Eds.) Handbook of residual stress and deformation of steel. ASM Int., Materials Park, Ohio, USA (2002) 312-330.

[7] Frerichs, F., Lübben, T., Fritsching, U. et al. : Simulation of Gas Quenching, Journal de Physique IV 120 (2004) 727-736.

[8] Acht,C., Dalgic,M., Frerichs,F. et al. : Ermittlung der Materialdaten zur Simulation des Durchhärtens von Komponenten aus 100Cr6 - Teil 1, HTM 63 (2008) 234-244.

[9] Acht,C., Dalgic,M., Frerichs,F. et al. : Ermittlung der Materialdaten zur Simulation des Durchhärtens von Komponenten aus 100Cr6 - Teil 2, HTM 64 (2008) 362-371.

[10] Gür, C.H., Pan, J. : Handbook of Thermal Processing of Steels, ISBN 0849350190, CRC Press, (2008), 341-427

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