CC BY
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Sztangret M., Pietrzyk M.
APPLICATIONS OF PLANE STRAIN COMPRESSION TESTS FOR IDENTIFICATION OF FLOW STRESS MODELS OF MATERIALS AND FOR PHYSICAL SIMULATION OF METAL FORMING PROCESSES
Abstract. Possibility of application of the plane strain compression test (PSC) to the determination of the flow stress was evaluated. It was shown that distributions of strains and stresses are very nonuniform, that makes interpretation of results difficult. Inverse analysis eliminates effects of inhomogeneities but in the case of the PSC it involves high computing costs. To improve the efficiency of the analysis PSC was simplified to 2D model. Application of the PSC test to physical simulations of multi pass rolling is discussed in the paper, as well.
Keywords: plane strain compression, flow stress, identification, physical simulation.
1. Introduction
The design of new industrial forming technologies is frequently based on numerical and physical simulations [1].
This approach allows to consider a variety of technological variants and to find the best solution in relatively short time and with low costs. In the case of metal forming processes the flow stress model used in simulation has an essential influence on the accuracy of simulations. Problem of selection and identification of the flow stress model has been discussed in numerous publications, see for example [2-4]. Flow stress models usually contain coefficients, which have to be determined on the basis of plasto-metric tests. Due to low costs the axisymmetric compression is the most commonly used plastometric test [4,5].
Plane strain compression (PSC) is the test, which have one important advantage: the state of strains in this test is similar to that occurring in flat rolling processes. On the other hand, due to very inhomogeneous strains, stresses and temperatures in this test interpretation of results of the plane strain compression is very difficult, what prevent its wide applications. Discussion of capabilities and limitations of the PSC test is the main objective of this paper.
2. Plane strain compression
2.1. General idea
Plane strain compression (PSC) is one of the plasto-metric tests, which is used for determination of the flow stress. In his test a cuboid sample is compressed between two flat dies, see Fig. 1a. This test permits large plastic deformation and the state of strains is similar to that, which occurs in the flat rolling process (Fig. 1b). The plane strain state is obtained due to two factors. The low width of the sample (b) - to width of the die (w) ratio prevents flow of the material in the width direction. It is similar to the flat rolling, where low length of contact (L) - to width of the strip (b) ratio fosters elongation and prevents spread. Influence of the so- called rigid ends is another factor, which constrains spread and involves plane strain state.
Rigid ends are the parts of the sample beyond the area under the die. These parts are not compressed, therefore, they do not have tendency to spread. Moreover, when the samples are heated by resistance heating (eg. on Gleeble 3800) these parts are in lower temperature than the area under the dies and their resistance to deformation is higher. Due to all these discussed facts PSC is frequently used as physical simulation of the flat rolling process.
Fig. 1. Schematic illustration of the PCS test (a) and flat rolling process (b)
Plane state of strains, which is not reachable in other plastometric tests, has for years inspired the scientists to various applications of the PSC tests. Identification of the flow stress model is one of such applications and investigation of the micro structure evolution is another example. Among several research laboratories involved in investigations based on the PSC tests a team led by professor Sellars at the University of Sheffield should be mentioned. During 50-ies and 60-ies of the last century this test was commonly used there for investigation of materials and fundamental works on microstructure evolution [7] and on flow stress models [8] were a result of this research.
It should be emphasized, however, that various disturbances make interpretation of results of PSC tests very difficult. These tests are characterized by large inhomogeneity of deformation (Fig. 2a), which is caused by complex shape of the deformation zone (Fig. 1a) and by the effect of friction. Beyond this, heat generated due to plastic work and friction, as well as heat transfer to the tools and to the surrounding, cause strong inhomogeneity of the temperature in the sample (Fig. 2b).
Fig. 2. Distribution of strains (a) and temperatures (b) in the PSC test
Further analysis of the PSC tests has shown that the flow stress determined from this test as ratio of the force (F) with respect to the contact surface (S) is sensitive to the parameters of the test. It is in contradiction with the theory of plasticity, which states that for isotropic material the flow stress is a property of this material independent of the type of the test, which was used for determination of this stress [9]. The flow stress calculated as the F/S ratio in the PSC for aluminium samples with various initial heights is shown in Fig. 3.
Plots in Fig. 3a represent an influence of the height of the sample on forces and calculated flow stresses. Careful analysis of these plots shows that at the beginning of deformation, when samples are higher, larger stresses are observed for higher samples. It is due to the effect of the rigid ends. When reduction of the height proceeds, the samples become lower and the effect of friction increases. In consequence, larger stresses are observed for lower samples.
Average force plotted in Fig. 3 b is calculated as an integral of the force with respect to the strain and repre-
sents deformation work [10]. It is seen that minimum of this work exists for certain height of the sample. Left to this point an increase of the work is caused by the effect of the friction, which contributes more for lower samples. Right to this point an increase is caused by the effect of the rigid ends, which is larger when the ratio of the height of these ends with respect to the width of the sample increases.
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Fig. 3. Stress (FIS) as a function of strain (a) and integral of force with respect to strain (b) - various height of the sample
Character of the material flow in the PSC tests is sensitive to the height of the sample, as well [10]. For low samples (Fig. 4a) a side of this sample after deformation has the shape of the letter D. For high samples (Fig. 4b) the material flows easier under the die and, in consequence, side of this sample after deformation has the shape of the Letter B.
a
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b
b
Fig. 4. View of the sample after PSC, initial height 2.5 mm (a) and 15 mm (b)
3.2. Sensitivity of the PSC results with respect to the parameters of the test
The scientists working with the PSC tests have realized inaccuracies and problems connected with interpretation of results of this test, which are caused by inhomogeneity of deformation and temperature. A lot of effort was made to development of methods of correction of the PSC test results [7,8,11-16]. In the present work finite element analysis of strains and stresses in the PSC test was performed and selected results are presented in Fig.5.
It is seen in Fig. 5 that the character of the distribution of strains and stresses changes drastically depending on the dimensions of the sample and the die. The idea of the shape coefficient A = w/h, which was introduced in [17], was used in the present work to characterize the dimensions of the deformation zone. Values of this coefficient were 2, 1 and 0.5 respectively in Figs 5a, 5b and 5c. Decrease of the coefficient A leads to more uniform distribution of strains and stresses in the deformation zone. The effect of friction increases with A decreasing. Deformation cross characteristic for compression in flat dies, for example for free forging, is well seen for high coefficient A (Figs 5a and b). Plastic deformation does not penetrate through the whole thickness of the sample in this case. Performed analysis shows that deformation scheme in the PSC test is very sensitive to the dimensions of the sample and the die, which makes direct interpretation of results very difficult. Thus, inverse analysis was applied in the present work to solve this problem.
Fig. 5. Distribution of effective strains (left) and effective stress (right) in the PSC test, initial sample height - to - width of the anvil ratio equals 10/5 mm (a), 10/10 mm (b), 5/10 mm (c)
a
b
3. Inverse analysis
Inverse analysis was used to determine the real flow stress corrected against the effect of friction, rigid ends and deformation heating in the PSC tests. The algorithm descibed in [4] was applied. The flow stress values corresponding to the subsequent strains were determined using optimization techniques. The quadratic norm of the error between measured and calculated compression loads was used as the objective function, see details in [4].
Inverse procedure requires several simulations of the plane strain compression test. Due to high computing costs, 3D inverse analysis is practically impossible. As far as compression of axisymmetrical samples is simulated using 2D mesh, plane strain compression requires 3D mesh, which has to be very fine in the areas of contact with the edge of the die, see Fig. 8. In consequence, the costs of the PSC simulations are about hundred times higher comparing to the axisymmetrical ones.
This correction allowed to use 2D model in simulation of the PSC tests. Analysis of the effectiveness of the inverse solution has shown that even simplified 2D simulation for one PSC test, using FE mesh with 220 elements, requires about 5-10 min. Assuming that at least three temperatures and three strain rates are necessary to determine relation of the flow stress on these parameters, nine simulations of the test are needed for one evaluation of the objective function. In consequence, full inverse analysis would still require very long computing times. In order to make the analysis more efficient, the analysis was performed in two steps [4]. In the first step the stress-strain function was determined in a tabular form for each test separately. This function can be considered a property of material for isothermal, constant strain rate conditions. The functions introduced in the finite element program together with correction for variations of temperature and strain rate gives perfect agreement between measured and predicted forces. The correction is given by the following equation:
Q (1
R
1
Y
T T
(1)
Fig. 8. Finite element mesh in the 3D simulation of the PSC test
These long computing times of the objective function in the PSC tests prevented wide applications of this test for the inverse analysis and evaluation of the flow stress of materials. Assessment of possibility of decreasing computing times necessary for the PSC test simulations and making the inverse analysis more effective is one of
the objectives of the present work. To r
reach this goal, simulations were sim- G = 43 as” exp
plified to the 2D domain at the cross |_
section of the sample. A correction proposed in [19] was used. This correction accounts for the slight increase of the contact surface due to spread under the die. This spread influences the current width of the sample, according to the formula:
0,18 '
where ap is flow stress, ab is isothermal constant strain rate value of the flow stress, determined from the inverse analysis, R is gas constant, Q is activation energy, ¿„ , Tn are nominal values of strain rate and temperature
for a considered test, e, T is current local value of strain rate and temperature in the finite element node (or in the Gauss integration point).
The results given in a tabular form, which were obtained in the first step of the analysis, were approximated using equation, which describes flow stress as a function of strain, strain rate and temperature. The equation proposed in [18] was used:
exP (~qe)+[1 -exp(~qs)] asat exp( ^RrT
r
(2)
C =
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1 + C - C h
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bf f -1 bo
r f ' V h0 J 0,18 5
(1)
1 -
where b0 is initial width, h0 is initial height, bf is final width, hf is final height, b is current width, b is current height, C is coefficient.
where e is effective strain, £ is effective strain rate, T is temperature in K.
The coefficients in equation (2) obtained from the approximation were used as a starting point for the second stage of the inverse analysis. In this stage coefficients in equation (2) were optimization variables. The starting point was very close to the minimum and only few steps of the simplex method were needed to find this minimum. In consequence, a noticeable decrease of the computing costs was obtained
Inverse analysis was performed for the steel containing 0.075%C, 1.375%Mn, 0.25%Si, 0.3%Ni, 0.15%Cu, 0.15%Cr. The samples, which measured 20x25x35 mm, were compressed in the dies with the width of 16 mm at temperatures 800, 850, 900, 950, 1000, 1050 and 1100oC wand at strain rates 0.1, 1 and 10 s-1. Three temperatures (800, 950, and 1100°C) were used in the identification phase and remaining four temperatures were used for the validation of the model. All tests were performed on the
Gleeble 3800 simulator in the Institute for Ferrous Metallurgy in Gliwice, Poland. Selected results of force measurements and flow stress determination are shown in Fig. 9.
Coefficients in equation (2) obtained from the second step of the inverse analysis are given in Table. The final value of the objective function in the inverse analysis is given in the last column of this Table. This value is a measure of the accuracy of the inverse solution.
Coefficients in equation (2) calculated using inverse analysis for the investigated steel
A ß n M asat ßsat q *
8.363 3723.3 0.469 0.099 0.0642 7815.1 1.086 0.129
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100
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850°C 1000-C
900°C 1050°C
950°C 1100°C
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good agreement was obtained in a wide range of temperatures and strain rate. It can be concluded that inverse analysis allows proper interpretation of results of the plane strain compression test and flow stress insensitive to the inhomogeneities in the test can be determined.
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Fig. 10. Selected examples of comparison of forces measured and calculated using FE program with equation (2), with coefficients in Table 1 as the flow stress model
4. Physical simulation of hot rolling
Plane strain compression test has been for years used for physical simulation of rolling processes. Temperature and strain history can be easily reproduced in this test. Samples can be quenched after each deformation stage and micro structure can be investigated. However, problems with the interpretation of results of such physical simulation are similar to those discussed above for the single stage compression. Deformation and temperatures are inhomogeneous and, what is even more important, the shape coefficient A changes during the test (A = 0.667; 0.82; 1.17; 1.55 respectively for stages 1, 2, 3 and 4), what involves changes of the inhomogeneity of deformation. Thus, the general principle followed by the scientists was to investigate the material in the centre of the sample, assuming that strains and temperatures at that location are close to the nominal values in the test calculated as:
( h ^ V hi J
(3)
Fig. 9. Selected examples of recorded forces (a) and flow stress determined using inverse analysis of the PSC tests (b)
Comparison of measured forces and those calculated using FE program with equation (2), with coefficients in Table 1 as the flow stress model is shown in Fig. 10. Very
where hi is initial height, hf is final height.
It seams that FE simulation of the multi stage PSC test should help interpretation of the results.The 4 stage compression test performed on the Gleeble 3800 in the temperatures covering austenitic and ferritic range were investigated. The samples, which measured 15x20x35 mm,
a
0
b
0
0
1
were compressed in the dies with the width of 10 mm. Schematic illustration of the time-temperature-deformation history is given in Fig. 11. Subsequent deformations were carried out at temperatures T1 = 1000°C, T2 = 900°C, T3 = 820°C and T4 = 66680°C. Description of this test and results of measurements are given in [20].
Several samples were deformed in the experiment.
The process was interrupted at various stages and samples were quenched to observe microstructure. All pic- O
tures of microstructure at the centre of the samples are ^
shown in the publication [20]. Four of these pictures g
showing microstructures right after deformation are re- +£
peated in Fig. 12. In the present work FE simulations of &
the multi stage PSC test were performed, and distribu- S-
tion of strains were evaluated. Results of the 2D finite §
element simulations of the 4th experimental stage PSC test are shown in Fig. 13.
It is seen in Fig. 13 that different distributions of strains are observed for different stages of deformation.
The largest inhomogeneity of deformation is observed in the 2nd stage. The most uniform deformation is in the last stage. It seams, however, that problem of strain level in the centre of the sample is even more serious. At this location the microstructure was analysed and it was referred to the nominal homogeneous strain in the compression. It is seen that local strains differ
significantly from the homogeneous one (figure 14). It can be concluded that interpretation of the physical simulation based on the PSC tests should be combined with the FE simulations of these tests.
time, s
Fig. 11. Schematic illustration of the time-temperature-deformation history in the investigated multi-stage PSC test [20]
8
7
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0 1 2 3 4 5 6 7 8 9 10 11 12 13
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8
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6 7 8 9 x, mm
10 11 12 13 14 15
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7 8 9 10 11 12 13 14 15 16 x, mm
Fig. 13. Distribution of strains at subsequent stages of the multi-stage PSC test Conclusions
• PSC test has several advantages, connected mainly with capability to reproduce state of strains, which is characteristic for rolling. It is shown in the paper how multi-stage compression allows to investigate microstructure evolution in multi pass rolling.
• Due to inhomogeneity of deformation and temperature interpretation of results of the PSC tests is difficult. It prevents wider application of this test.
• Finite element simulation combined with optimization methods allows realistic interpretation of results of the PSC test. The application of this solution is twofold. The first is inverse analysis and identification of the flow stress model. The second is interpretation of results of physical simulations of rolling processes.
• FE simulations of the PSC tests combined with optimization techniques require very long computing times. Low efficiency of this approach is its important drawback. Methods of improvement of this efficiency should be searched in the future. Application of the metamodel, which proved to be efficient for the axisymmetric tests [21], seems to be a promising solution and it will be an objective of the future research.
Acknowledgements
Financial assistance of the NCN, project N N508
629 740, is acknowledged.
0.8 0.7 0.6 c 0.5 2 0.4 10 0.3 0.2 0.1
□ homogeneous
□ FE
äm
12 3 4
stage
Fig. 14. Comparison of the homogeneous strains with strains in the centre of the sample calculated by the FE code and
References
Sztangret L, Milenin A., Sztangret M., Walczyk W., Pietrzyk M., Kusiak J., Computer aided design of the best TR forging technology for crank shafts, Computer Methods in Materials Science, 11, 2011, 237-242.
Grosman F., Application of the flow stress function in programmes for computer simulation of plastic working processes, J. Mat. Proc. Techn., 64, 1997, 169-180.
Bariani, P., Dal Negro, T., Bruschi, S., 2004, Testing and modelling of material response to deformation in bulk metal forming, Annals of the CIRP, 53: 573-594.
Szeliga D., Gawqd J., Pietrzyk M., Inverse analysis for identification of rheological and friction models in metal forming, Comp. Meth. Appl. Mech. Eng., 195, 2006, 6778-6798.
Hadasik E., Kuziak R., Kawalla R., Adamczyk M., Pietrzyk M., Rheological model for simulation of hot rolling of new generation steel strips for auto-
0
motive industry, Steel Research Int., 77, 2006, 927-933.
6. Kuc D., Pietrzyk M., Rheological model of the austenitic steel subjected to hot deformation, accounting for the microstructure evolution, Hutnik-Wiadomosci Hutnicze, 76, 2009, 607-610.
7. Sellars C.M., Physical metallurgy of hot working, w: Hot working and forming processes, (ed.), Sellars C.M., Davies G.J., The Metals Soc., London, 1979, 3-15.
8. Davenport S.B., Silk N.J., Sparks C.N., Sellars C.M., Development of constitutive equations for the modelling of hot rolling, Mat. Sci. Techn., 16, 1999, 1-8.
9. Szeliga D., Matuszyk P., Kuziak R., Pietrzyk M., Identification of rheologi-cal parameters on the basis of various types of plastometric tests, J. Mat. Proc. Techn., 125-126, 2002, 150-154.
10. Sztangret M., Pietrzyk M., Proba sciskania probek ptaskich jako metoda symulacji fizycznej procesow przerobki plastycznej oraz narz^dzie do identyfikacji modeli reologicznych materiatow, Hutnik-Wiadomosci Hutnicze, 79, 2012, (in press)
11. Kowalski B., Method of interpretation of results of plane strain compression tests for evaluation of rheological parameters of materials, Praca doktorska, AGH, Krakow, 2003.
12. Pietrzyk M., Lenard J.G., Dalton G.M., A study of the plane strain compression test, Ann. CIRP, 42, 1993, 331-334.
13. Pietrzyk M., Tibbals J.E., Application of the finite element technique to the interpretation of the plane strain compression test for aluminum, Mat.
COMPLAS 4, ed., Owen D.R.J., Onate E., Hinton E., Pineridge Press, Barcelona, 1995, 889-900.
14. Silk N.J., Ban Der Winden M.R., Interpretation of hot plane strain compression testing of aluminum specimens, Mat. Sci. Techn., 15, 1999, 295-300.
15. Kowalski B., Sellars C.M., Pietrzyk M., Development of a computer code for the interpretation of results of hot plane strain compression tests, ISIJ Int., 40, 2000, 1230-1236.
16. Kowalski B., Lacey A.J., Sellars C.M., Correction of plane strain compression data for the effects of inhomogeneous deformation, Mat. Sci. Techn., 19, 2003, 1564-1570.
17. Backofen W.A., Deformation Processing, Addison-Wesley 1972.
18. Gavrus A., Massoni E., Chenot J.L., An inverse analysis using a finite element model for identification of rheological parameters, J. Mat. Proc. Techn., 60, 1996, 447-454.
19. Kowalski B., Wajda W., Pietrzyk M., Sellars C.M., Sensitivity of constitutive equations on strain and strain rate inhomogeneity determined for FE modeling of plane strain compression tests, Proc. 4th Conf. ESAFORM, ed., Habraken A.-M., Liege, 2001, 561-564.
20. Pietrzyk M., Kuziak R., Validation of a model of plastic deformation of C-Mn steels in the two-phase temperature region, Steel Research International, 80, 2009, 767-778.
21. Sztangret L., Szeliga D., Kusiak J., Pietrzyk M., Identyfikacja modelu materiatu w prawie konstytutywnym w oparciu o rozwi^zanie odwrotne z metamodelem, Mechanik, 84, 2011, 32-36 (in Polish).
Pesin A.M.
SCIENTIFIC SCHOOL OF ASYMMETRIC ROLLING IN MAGNITOGORSK
Abstract. The article presents some results of studies of Scientific School of Asymmetric Rolling in Magnitogorsk. This paper presents a classification and practical application of the asymmetric rolling processes. Further development of the asymmetric rolling process is their use as a severe plastic deformation method for ultra-fine structures of the metal.
Keywords: scientific School, classification, asymmetric rolling, shear deformation, severe plastic deformation, parts of large bodies of revolution, finite element method modeling.
Over 30 years Metal Forming Department at the Higher Professional Institution «Magnitogorsk State Technical University» has been developing the scientific direction of both current and new technologies of asymmetric plate rolling. This direction is headed by V. Sal-ganik and dissertations for Ph. D degree on this theme were written by A. Pesin (scientific supervisor M. Polyakov), V. Rudakov, V. Lunev, I. Vier, G. Kunitsyn (scientific supervisor V. Salganik), K. Kuranov, D. Chikishev (scientific supervisor A. Pesin), M. Chernyakovsky (scientific supervisors V. Salganik, A. Pesin). A. Pesin, G. Kunitsyn (scientific advisor V. Salganik) and V. Sal-ganik have written doctorate dissertations.
Since the foundation of Magnitogorsk scholar collaborations with V. Vydrin, L. Ageev (Chelyabinsk school), V. Potapkin, V. Fedorinov, A. Satonin (Kramatorsk school), S. Kotsar, V. Tretiakov, J. Mukhin (Lipetsk school), V. Polukhin, A. Pimenov, V. Skorokhodov, A. Traino, B. Kucheriaev have been established.
Nowadays several scientific projects have been carried out together with K. Dyja, A. Kawalek from Technical University in Czestochowa (Poland), K. Mori from Technical University in Toyohashi (Japan), V. Fedorinov, A. Satonin from State Engineering University in Donbass (Ukrain).
Metal Forming Department at the Higher Professional Institution «Nosov Magnitogorsk State Technical University» has solved the following challenging issues: 1) the processes of asymmetric rolling have been classified; 2)
statics, geometry and kinetics of vertically asymmetric deformation site have been described; 3) special cases of vertically asymmetric rolling have been investigated; 4) a new integrated process of vertically asymmetric rolling and plastic bend of plate mill has been developed; 5) metal cross-section in horizontal asymmetric rolling has been investigated; 6) new technical schemes in horizontally and vertically asymmetric rolling have been found.
These days the research is focused on the following directions: 1) shape control of the front end of the strip in plate rolling; 2) ultrafine grain structure in asymmetric intensive plastic deformation.
1. The classification process of asymmetric rolling
Due to different criteria, there are various approaches to the cases, which take place in asymmetric rolling. The scientists from Magnitogorsk scholar have suggested the classification containing 3 hierarchal levels [1, 2]. The upper level regards the causes of asymmetry occurred deliberately or evoked by any disturbances. The next level considers asymmetry in space in relation to either horizontal or vertical plane or to both of them. Finally, the lower level includes factors that can define asymmetry (geometric, frictional, elastic, kinematic, etc.) Asymmetry of different kinds often occurs simultaneously.
These days the technical approaches to purposeful asymmetry in the vertical plane are widely spread and
