Modeling the temperature rise in high-pressure torsion

Pedro Henrique R. Pereira a, Roberto B. Figueiredo b,n, Yi Huang c, Paulo R. Cetlin a,Terence G. Langdon c,d

a Department of Mechanical Engineering, Universidade Federal de Minas Gerais, Belo Horizonte 31270-901, MG, Brazilb Department of Materials Engineering and Civil Construction, Universidade Federal de Minas Gerais, Belo Horizonte 31270-901, MG, Brazilc Materials Research Group, Faculty of Engineering and the Environment, University of Southampton, Southampton SO17 1BJ, UKd Departments of Aerospace & Mechanical Engineering and Materials Science, University of Southern California, Los Angeles, CA 90089-1453, USA

a r t i c l e i n f o

Article history:Received 18 October 2013Received in revised form4 November 2013Accepted 7 November 2013Available online 16 November 2013

Keywords:Finite element modelingHigh-pressure torsionSevere plastic deformationTemperature rise

a b s t r a c t

Experiments and finite element modeling were used to estimate the temperature rise during high-pressure torsion. The results show the temperature rise is dependent upon the material strength, therotation rate, the sample radius, the heat capacity and the volume of the anvils. A general relationship isderived which predicts the temperature rise in samples of different geometries processed using differentanvil sizes. A simplified version of the equation is presented for general use.

& 2013 Elsevier B.V. All rights reserved.

1. Introduction

Severe plastic deformation (SPD) techniques [1] have beenextensively used to refine the grain structure in metallicmaterials. The large amounts of plastic deformation appliedto the samples lead to the introduction of internal defects inthe crystalline structure of the materials but also producesignificant heating. Among the various SPD techniques,Equal-Channel Angular Pressing (ECAP) [2] and high-pressuretorsion (HPT) [3] are the most common. It was shown experi-mentally that the temperature rise during ECAP may reach�70 K in a high strength aluminum alloy when pressing at aspeed as high as 18 mm s�1 [4]. Later, Finite Element Modeling(FEM) was used to estimate the temperature rise during ECAPand the results agreed well with the experimental values [5].Other similar experimental results were also reported [6]together with a heat transfer analysis [7]. Nevertheless, it isimportant to note that the temperature rise in ECAP occursabruptly and there is a gradual cooling after crossing theshearing zone. This means in practice that the temperaturerise is not incremental during multiple passes. By contrast, thedeformation during HPT is continuous and therefore thetemperature is expected to continue rising while processingis maintained. Accordingly, it is important to determine thetemperature rise for samples processed to large numbers of

turns in HPT, especially for high strength materials processedat high rotation rates.

The temperature rise, ΔT, during plastic deformation is given bythe following relationship [8]:

ΔT ¼ 0:9C

Zs dε ð1Þ

where C is the heat capacity of the sample (C¼ρCp where ρ is thedensity and Cp is the specific heat capacity), s is the flow stress, ε isthe plastic deformation and the fraction of plastic deformationwork converted into heat is assumed as 0.9. Early reports esti-mated temperature rises of �300 K [9] and �400 K [10] duringHPT processing of an aluminum alloy and a Cu-based metallicglass, respectively. However, these calculations failed to includethe heat loss to the massive HPT anvils. Later, experiments wereconducted to determine the temperature rise in the anvils duringHPT processing of different materials [11] and these values wereused in FEM to estimate the temperature rises in the samples[11,12]. Through these calculations, it was shown that the tem-perature rise is proportional to the sample strength and therotation rate [11,12] and a graphical representation was used toestimate the temperature rise at any time in a typical HPT facility[12]. Nevertheless, this approach failed to incorporate the effect ontemperature rise of either the size of the anvils or the size of theworkpiece. Accordingly, the present calculations were conductedwith the objective of deriving a general relationship to predict thetemperature rise during HPT processing when all variables areincluded.

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Materials Science & Engineering A

0921-5093/$ - see front matter & 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.msea.2013.11.015

n Corresponding author.E-mail address: figueiredo@demc.ufmg.br (R.B. Figueiredo).

Materials Science & Engineering A 593 (2014) 185–188

2. Experimental procedure and modeling

In order to critically evaluate the temperature rise during HPTprocessing, samples of titanium grade 2 with a 5 mm radius and a0.8 mm thickness were processed at rotation rates of 1 rpm and0.2 rpm under a pressure of 6.0 GPa at room temperature using anHPT facility operating under quasi-constrained conditions [12,13].The anvils used in the experiments have diameters of 50 mm andheights of �30 mm. The surface of the anvil has a conical shape atan inclination of �51 to avoid contact between the anvils duringprocessing. Marks were scribed on both surfaces of the samplebefore processing and these marks exhibited perfect alignmentafter HPT processing which shows no slippage had occurred. Thetemperature rise in the anvil was tracked during processing byplacing a K-type (chromel–alumel) thermocouple within thecenter of the anvil at 10 mm from the contact surface of the anvilwith the sample.

Simulations of HPT processing were performed using the FEMDEFORM 2D 10.0 software (Scientific Forming Technologies Corpora-tion, Columbus, OH). The geometries and meshes of the anvils andthe workpiece used in this study are depicted in Fig. 1. Since HPTexhibits axial symmetry, the model was simplified to a two dimen-sional situation considering only the axial and radial directions. Thesimulations considered sticking conditions between the sample andthe anvils on the top and bottom surfaces which means no slippagewas allowed in these surfaces. A coefficient of friction of 0.6 [14] wasconsidered in the contact between the sample and the anvils in thearea of material outflow. The simulations considered workpieceswith different values for the radius, rw, and thickness, hw, anvils withdifferent volumes, V, and different rotation rates, ω. Further details onthe modeling parameters and the boundary conditions were givenearlier [12]. The thermal conductivity and heat capacity of the anvilswere taken as 42Wm�1 K�1 and 3.72 MJ m�3 K�1, respectively,where these values were selected from the library of the software forthe tool steel.

The material of the workpiece was titanium grade 2 with athermal conductivity of 20 W m�1 K�1 and a thermal capacity of2.36 MJ m�3 K�1 [15]. The flow stress was considered constantand taken as the saturation stress, 940 MPa [16]. Additional

simulations were carried out considering the early variation inflow stress in the strain-hardening regime of the material and theresults showed the difference in temperature rise is not significant.This shows that considering the flow stress as constant during HPTdoes not compromise the prediction of temperature rise.The valueof the flow stress was considered as a constant since the HPTprocessing is expected to impose a sufficiently large level ofdeformation that the flow stress saturates in the early stagesleading to only minor variations in the flow stress during the laterstages of processing.

3. Results and discussion

3.1. Validation of the simulation model

Fig. 2 shows the temperature plotted as a function of time forthe experimental HPT processing of titanium at two differentrotation rates together with the simulation values. The tempera-ture in the experiments was recorded within the anvils and thetemperature in the simulations was estimated both in the anvilsand in the workpiece. It is readily apparent that the values of thetemperatures in the simulations agree very well with the valuesobserved in the experiments at both rotation rates, therebyconfirming the general validity of the model. It is also clear thatthe temperature in the workpiece follows the same general trendas in the anvils but with a shift towards higher values. The earliersimulations also showed good agreement with experimental datafor the temperature rise during HPT [12].

3.2. General trend of temperature rise

Further simulations were conducted in order to evaluate theeffect of rotation rate, workpiece radius and thickness and theanvil volume on the temperature evolution and the results werethen analyzed in order to provide a general trend. The overalltrend of the temperature evolution during HPT can be explained interms of heat generation, heat storage and heat dissipation. Thus,in the very early stage of deformation the heat generated by plasticdeformation of the workpiece is mostly stored in the workpieceleading to a rapid increase in temperature. This leads to an initialtemperature rise which is incorporated as ΔT0. Following this earlystage, the temperature gradient between the workpiece and theanvil leads to heat transfer so that the heat generated in the

Fig. 1. Illustration of geometry of anvils and workpiece used in the simulations. Thedistribution of temperature is shown for a specific processing condition.

Fig. 2. Temperature evolution as a function of time using experimental data fortitanium grade 2 processed at rotation rates of 1.0 rpm (0.105 rad s-1) and 0.2 rpm(0.02 rad s-1). The dashed and continuous lines represent, respectively, the tem-perature predicted in the simulation in the anvil, Pt, and in the workpiece, Pmax.

P.H.R. Pereira et al. / Materials Science & Engineering A 593 (2014) 185–188186

workpiece is then stored in both the workpiece and the anvil andthe rate of temperature rise is given by ΔT′0. Thereafter, theincrease in temperature of the anvils leads to heat losses to theenvironment and this causes a continuous decrease in the rate oftemperature rise and, ultimately, a saturation in the temperatureat a final value given by ΔTf. It follows that any description of thetemperature evolution during HPT must incorporate all of theseparameters which are illustrated in Fig. 3 for the evolution oftemperature observed in a simulation of HPT. Also, the solid line inFig. 3 shows the temperature evolution predicted by an equationdetermined empirically.

3.3. The effect of HPT processing variables

Simulations were carried out considering different values ofsample thickness, hw, sample radius, rw, rotation rate, ω, andvolume of the anvils, V, in order to determine their effect on theparameters ΔT0, ΔT′0 and ΔTf. Fig. 4 shows the plots of ΔT0, ΔT′0and ΔTf as a function of three different processing variables. It isobserved that the initial temperature rise, ΔT0, is proportional tothe sample radius, rw, and to the rotation rate, ω, and it is notdependent on the volume of the anvils, V. The initial temperaturerise rate, ΔT′0, is proportional to the rotation rate, to the square ofthe sample radius and inversely proportional to the volume of theanvils. The saturation temperature, ΔTf, is also proportional to therotation rate, to the sample radius raised to a power of 2 but it isinversely proportional to the cube root of the volume of the anvilsat least over the range of anvil volumes evaluated in this analysis.It is important to note these results suggest that the saturationtemperature may become lower than the initial temperature risewhen processing samples with very small radii and/or when usinganvils with very large volumes. However, this is not possible inreal experiments and in practice the processing of samples withvery small radii, or when using massive anvils, will lead to amaximum saturation temperature which is equal to the initialtemperature rise. Additional simulations showed that the flowstress of the sample and rotation rate play a major role and thesample thickness plays only a minor role in the temperature riseparameters.

The exponents determined from the simulation results may beused to establish phenomenological relationships between thetemperature rise parameters and the HPT processing variablesover the range considered in the simulations. The best fit for the various parameters are given by the following relationships:

ΔT0 ¼ K1sωrw ð2Þ

ΔT ′0 ¼ K2sωr2wCV

ð3Þ

ΔT f ¼ K3sωr2wffiffiffiffiffiffiffiCV3

p ð4Þ

where K1, K2 and K3 are constants. The value of K1 is equalto 4.5�10�2, K2 is 10 and K3 is 1.5 when s is in MPa, ω is inrad/s, rw is in mm, C is in MJ/m3 and V is in mm3 so that thecalculated values of ΔT0 and ΔTf have units of K and ΔT′0 has unitsof K/s.

3.4. General phenomenological relationship for the temperature risein HPT

It follows from the solid curve in Fig. 3 that the temperatureevolution of the sample during HPT processing can be describedby the following simple equation:

ΔT ¼ aþb 1�e� ct� � ð5Þ

Fig. 3. Variation of temperature in the workpiece, at Pmax, as a function of time inHPT processing for a titanium workpiece with rw¼5 mm and hw¼0.8 mmprocessed at 1.0 rpm. The parameters used to develop an equation for thetemperature rise during HPT are shown.

Fig. 4. Thermal parameters plotted as a function of three HPT processing variables.

P.H.R. Pereira et al. / Materials Science & Engineering A 593 (2014) 185–188 187

where t is the time in seconds and a, b and c are defined by thefollowing relationships:

a¼ΔT0 ð6Þ

b¼ΔT f �ΔT0 ð7Þ

c¼ ΔT ′0ðΔT f �ΔT0Þ

ð8Þ

It is apparent from inspection in Fig. 3 that there is very goodagreement between Eq. (5) and the temperature evolution as afunction of time during the simulation. Other examples are shownin Fig. 5 for the processing of samples with different radii andthicknesses using different rotation rates and anvils having differ-ent volumes. It is readily apparent that there is an excellentagreement between Eq. (5) and the FEM results such that thedifference between the FEM results and the values predicted bythe general relationship is within o15%.

3.5. Practical implications of the relationship for the temperature rise

In practice, the radius of most samples used in HPT processingis �5 mm and the anvils are generally made of tool steel withvolumes for each anvil of �58,200 mm³. Thus, taking rw¼5�10�3 m and V¼1.164�10�4 m3, it is possible to derive a simpli-fied relationship for the temperature rise, ΔTusual, in a quasi-constrained HPT facility as follows:

ΔTusual ¼ 0:22sω 1þ1:28 1�e�ðt=482Þ� �h i

ð9Þ

Eq. (9) shows that the temperature rise in Kelvin for the earlystage of HPT is given by ΔTusual ¼ 0:22sω and for the saturationcondition it is given by ΔTusual ¼ 0:5sω.

The present simulations were performed for a quasi-constrainedHPT facility but the simulations were repeated also for unconstrainedand constrained HPT where the workpiece is either free to flowoutwards without restriction under the applied pressure or confinedwithin a cavity on the lower anvil so that no outward flow is possible.The results for the temperature rises in these conditions are reason-ably consistent with the results for the quasi-constrained condition.Thus, it is concluded that the relationship for the temperature rise inHPT derived in this report, as shown in Eq. (9), may be used equallyfor HPT processing under quasi-constrained, constrained or uncon-strained conditions and the relationship applies for all materials.

4. Summary and conclusions

1. FEM simulations were used to estimate the temperature rise insamples during HPT processing. The simulations were coupledwith experimental data measuring the temperature rise in atitanium grade 2 alloy processed at different rotation rates.

2. An empirical equation for the temperature rise in HPT wasderived for any sample geometry.

3. A simplified relationship, as given in Eq. (9), may be used toestimate the temperature rise in samples processed using aconventional HPT facility.

Acknowledgements

The authors acknowledge support from the FAPEMIG, CNPqunder Grant no. 483077/2011-9, the CAPES under Grant no. PVE037/2012 and the European Research Council under ERC GrantAgreement no. 267464-SPDMETALS.

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Fig. 5. Variation of temperature at Pmax as a function of time in HPT disks of grade2 titanium with different geometries processed at different rotation rates usinganvils with different volumes.

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