Determination of Coefficient for Transformation Plasticity in Terms

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Determination of Coefficient for Transformation Plasticity in Terms of Three-Point Bending System∗

Kenichi OSHITA∗∗, Shigeru NAGAKI∗∗, Shinichi ASAOKA∗∗∗ and Katsuyuki MOROZUMI∗∗∗∗

To determine the transformation plasticity coefficient, a three-point bending system is employed and the large deflection due to transformation plasticity in bending is analyzed un- der the simplified assumption that transformations occur uniformly across the cross-section of a beam. It is shown that the deflection due to transformation plasticity is similar to that due to elastic deformation, and simple relationships are also derived between the ratio of the maximum deflection to the elastic deflection and material constants (transformation plasticity coefficient K and Young’s modulus E). Experiments are carried out for a slender bar speci- men, which is loaded in a three-point bending system, and the austenized specimen is cooled so that martensite transformation accompanied by transformation plasticity occurs and the profile and maximum value of the deflection of the specimen are measured. The measured profiles of deflection agree very well with the theoretical results. This proves the validity of the proposed method. The transformation plasticity coefficient is also determined by the proposed method.

Key Words: Plasticity, Transformation Plasticity, Constitutive Equation, Three-Point Bend- ing System

many materials. 1. Introduction Recently, Tutumi et al. have clearly shown experi- In the stress and transformation analyses of heat- mentally that the deflection due to transformation plastic- treatment and welding, the importance of accurately tak- ity occurs in bending a slender bar specimen, which was ing the effect of the transformation plasticity into ac- then employed to identify the transformation plasticity co- ffi (10) count is pointed out(1) – (4) and many studies on this com- e cient . Moreover, Inoue et al. proposed a method for ffi position rule have been carried out(5) – (7). For example, determining the transformation plasticity coe cient using (1) the material constant of transformation plasticity, that is, a four-point bending system . It seems that it is easy for the transformation plasticity coefficient, is obtained from a four-point bending system to theoretically treat test re- ffi the temperature-deflection curves obtained under a ten- sults and that the transformation plasticity coe cient ob- sile load(8), (9). However, an experimental method of accu- tained using this system is extremely precise because the rately measuring transformation plasticity coefficient us- distributions of bending moment and stress are uniform. ing a simple device has not been established yet and reli- However, three-point bending tests are simpler and easier able data on this parameter are still limited at present in to perform than four-point bending tests. In the present paper, we propose a method for the easy determination ffi ∗ Received 1st April, 2004 (No. 02-1470). Japanese Orig- of the transformation plasticity coe cient in terms of the inal: Trans. Jpn. Soc. Mech. Eng., Vol.69, No.684, A three-point bending system and experimentally discuss the (2003), pp.1230–1235 (Received 13th December, 2002) validity of this method. ∗∗ Department of Mechanical Systems Engineering, Tokyo University of Agriculture and Technology, 2–24–16 Naka- 2. Analysis cho, Koganei-city, Tokyo 184–8588, Japan. In bending tests, to determine the transformation plas- E-mail: [email protected] ffi ∗∗∗ Pioneer Display Products Co., Ltd., 15–1 Nishinotani, ticity coe cient, a constant load (moment) is applied to Washizu, Hukuroi-city, Shizuoka 437–8511, Japan a heated slender bar specimen, which is cooled. Deflec- ∗∗∗∗ Nihon ESI K.K., 45–18 Oyama-cho, Shibuya-ku, Tokyo tion is then measured at the center of the specimen dur- 107–0062, Japan. E-mail: [email protected] ing cooling. It is assumed that neither restraint nor force

JSME International Journal Series A, Vol. 48, No. 1, 2005 2 axially exist in this specimen, which is simply supported occurs, where the volume fraction of the generated phase during test. In this section, the procedure of this analytical is defined as ξ. Furthermore, Young’s modulus E and the method is discussed and then it is shown that the transfor- coefficient of linear expansion α are equal in each phase, mation plasticity coefficient can be obtained by examining and β and T are the expansion coefficient due to trans- maximum bending deflection experimentally. formation and the temperature, respectively. In the present 2. 1 Basic equation in bending study, since the applied load is small so as not to cause The analysis of the bending of a one-dimensional plastic deflection, it seems that the plastic strain rateε ˙ p is beam is in accordance with the following relation. Under equal to 0. the assumption of Euler-Bernoulli bending theories, strain The constitutive equation on transformation plasticity ε is given by is defined by Greenwood-Johnson’s relation(11) η ε = (1) ε˙tp = 2K (1−ξ)ξσ,˙ (10) R and strain rateε ˙ by where K is the transformation plasticity coefficient. Substituting each strain rate component [Eqs. (7) – R˙ ε˙ = − η, (2) (11)] into the total strain rate [Eq. (6)], we obtain R2 where R and η are the radius of curvature and the distance σ˙ = E{ε˙ −αT˙ −βξ˙ −2K(1−ξ)ξσ˙ }. (11) from a neutral plane, respectively. Under the assumption Since this stress rate should fulfill Eq. (5) at a given cross-    that the slope of a beam is very small ( dy/dx 1), if the section, profile of deflection is defined by y = y(x) at a given time, we can approximate the radius of curvature using E{ε˙ −αT˙ −βξ˙ −2K(1−ξ)ξσ˙ }ηdA= 0. (12) A 2 1 = − d y If each material’s constant is independent of temperature 2 . (3) R dx and the position of the specimen, Eq. (12) becomes On the other hand, considering the equilibrium state of moment in a given cross-section, the bending moment εη˙ dA−α T˙ηdA−β ξη˙ dA A A A becomes − − = = 2K (1 ξ)ξση˙ dA 0. (13) M σηdA, (4) A A where A and M are the cross-sectional area of the beam In the experiment, it is assumed that the specimen is and the bending moment, respectively. Furthermore, when so small that the change in temperature occurs uniformly the applied load is constant, the bending moment rate be- across its cross-section during cooling, the transformation, therefore, occurs uniformly across the cross-section of the comes specimen. Then, the second and third terms of the above M˙ = ση˙ dA= 0. (5) equation become first-order moments around the neutral A plane, which are equal to 0. Meanwhile, if the speci- 2. 2 Constitutive equation of transformation plas- men, which is symmetric to the neutral plane, is cooled ticity uniformly from the upper and lower-surfaces despite the Let us consider the constitutive equation in uniaxial transformation occurring nonuniformly, it can be consid- stress problem in which only bending stress is applied. ered that the rate of change in temperature T˙ becomes an Here, the process in which the specimen is subjected to even function of the distance from the neutral plane η. elastic loading and then cooled under constant loading is Similarly considering that the rate of volume fraction ξ˙ discussed. In general, the total strain rateε ˙ is calculated by also becomes an even function of η, the second and third the sum of elasticity, plasticity, thermal expansion, trans- terms of Eq. (13) become 0. Finally, the equation of equi- formation expansion and transformation plasticity(5). librium for moment can be written as ε˙ = ε˙e +ε˙ p +ε˙T +ε˙ ph +ε˙tp (6) εη˙ dA−2K(1−ξ)ξ˙ σηdA= 0. (14) Here, elastic strain rate, thermal expansion strain rate and A A transformation plastic strain rate are respectively repre- 2. 3 Profile of deflection sented by In this section, the profile of deflection is considered σ˙ as follows: Assuming the deflection y, which is a function ε˙e = (7) E of the position x and the time t from loading (i.e., cooling ε˙T = αT˙ (8) start time), is written as ε˙ ph = βξ.˙ (9) y(x,t) = δ(t)g(x) (15) As an example, it is assumed that a type of trans- (the validity of this assumption is experimentally dis- formation from the austenite phase to the martensite phase cussed later),

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0 ≤ g(x) ≤ 1 (16) For a simply supported beam with a span of l sub- e jected to three-point bending, the applied load W at its δ ≤ δ(t) ≤ δmax, (17) center, a nondimensional function of the deflection g(x), where δe and δ are the maximum elastic deflection and max is expressed as final maximum deflection, respectively, and g(x) can be     W  x3 3 l2  considered a function of the dimensionless deflection in- = −  − 1 − 1 −  g(x)   x x, (26) duced by the maximum deflection. Then we obtain E I 12 6 2 16 d2y d2g(x) where  = δ(t) . (18)  dx2 dx2  0ifx < 0 x =  (27) Using Eq. (3), the radius of curvature can be expressed as  x if x ≥ 0 1 d2g(x) ffi = −δ(t) (19) 2. 4 Transformation plasticity coe cient R dx2 Considering g(l/2) = 1 in Eq. (26), the transformation and in the derivative form as plasticity coefficient K can be expressed by 2 2 d 1 2 dδ d g(x) 48I 1 R˙ = −R = R . (20) K = δ − . (28) dt R dt dx2 Wl3 max E From Eqs. (19) and (20), the integration of the first terms That is, since the shape of the beam and the applied load in Eq. (14) can be rewritten as are already known, the transformation plasticity coeffi- R˙ R˙ cient K can be calculated if only the maximum deflection εη˙ dA= − η2dA= − η2dA 2 2 δ is measured. A A R R A max R˙ dδ d2g(x) Moreover, according to Eq. (25), the ratio of the max- = − I = −I , 2 2 (21) imum deflection due to transformation plasticity δmax to R dt dx e where I is the geometrical moment of inertia. Since the the maximum elastic deflection δ is also derived using integration of the second terms of Eq. (14) is equal to the Young’s modulus E and the transformation plasticity co- ffi bending moment M, the equation of equilibrium for mo- e cient K. δ ment is finally given by max = KE+1 (29) δe dδ d2g(x) I +2K(1−ξ)ξ˙M = 0. (22) This equation is independent of the loading procedures dt dx2 and type of bending system used. For example, if Young’s By integrating the above equation from the cooling start modulus E is 200 GPa and the transformation plasticity = = t 0 to the transformation end t t0, with consideration of coefficient is on the order of about 1 × 10−5 –1× 10−4 in = e = = = = = δ δ and ξ 0att 0, and δ δmax and ξ 1att t0, carbon steel, it is easily determined that the deflection due ff the di erential equation for the nondimensional function to transformation plasticity is about 3 – 21 times as large of deflection is obtained. as the elastic deflection. Then, Eq. (29) is simplified to d2g(x) KM KM = − = − (23) reflect the deflection due to transformation plasticity. 2 − e tp dx (δmax δ )I δ I δtp tp = − e = KE (30) Where, δ δmax δ is the deflection occurring after load- δe ing, and can be considered the deflection due to trans- 2. 5 Other deformation forms formation under the assumption that Young’s modulus and In this section, we show that a relation similar to other parameters are independent of temperature and the Eq. (30) is obtained under the loading conditions exclud- type of structure. By introducing ing bending system, and discuss the difference between e tp  δ −δ δ the deflections under bending system and other systems, E = max = , (24) K K e.g., tensile or torsional system. Eq. (23) can be rewritten as Let us consider the case that the tensile specimen is d2g(x) M cooled under constant loading and then displacement is = − . (25) dx2 EI measured. Under the assumption that the whole speci- This is of the same form as the basic differential equation men transforms uniformly during cooling, a relation sim- of deflection in general strength of materials. ilar to Eq. (30) in bending is approved between the trans- That is, under the above-mentioned assumptions, the formation plasticity strain εtpand the elastic strain εe. profile of the deflection of the beam due to transformation εtp = KE (31) plasticity is similar to that of Hooke’s elastic body ob- εe tained from basic beam theories in the strength of mate- However, in this case, the thermal expansion strain and rials, and can be determined easily even if the distribution transformation expansion strain are superimposed to the of bending moment is known. Thus, the transformation measured total strain, and thus only transformation plas- plasticity coefficient K can also be calculated easily. ticity strain cannot be measured directly. Furthermore,

JSME International Journal Series A, Vol. 48, No. 1, 2005 4 since the displacement generated in the specimen is so suring the displacement of the plate set on the loading pin small under this condition, this procedure is not suitable with a laser-beam-type extensometer. for measuring the transformation plasticity coefficient. The specimen was machined from SCM 440 steel On the other hand, in the case that the round bar of l with a good quenching property into a rectangular cross- length and d diameter is cooled under a constant torsional sectional bar of 15.8 × 240 × 5mm3. Table 1 shows the moment, the relation between the maximum angle of twist chemical composition of the tested material. φmax and the transformation plasticity coefficient K can be The temperature of the specimen was measured with derived as an Alumel-Chromel thermocouple fixed by spot welding Ip in a hole of 2 mm diameter and 1 mm depth made 15 mm K = (φ −φe), (32) 3T max from the center of an upper surface. First, the specimen e ◦ where Ip and φ are the polar moment of inertia of the area was heated up to 900 C in an electric furnace and then and the elastic specific angle of twist by loading, respec- transformed into its austenitic phase by maintaining at this tively. Since the change in volume due to thermal expan- temperature for 30 minutes. Then, it was removed from sion and phase transformation does not influence shearing the electric furnace and set up in the bending system, and deformation under the above conditions, the deformation the three-point bending test was performed under a con- ◦ due to transformation plasticity can be measured directly stant load from about 750 C. The deflection and the tem- in bending tests. Thus, the transformation plasticity coef- perature of the center of the specimen were measured and ficient can be simply estimated in torsional tests. In the recorded at a sampling frequency of 0.1 seconds during case of transformation plasticity occurring, the ratio of the cooling, which was performed in ambient air. maximum specific angle of twist φmax to the specific angle The loads used were 5.86 kg, 4.65 kg, and 2.85 kg, of twist during elastic deformation φe is given by which corresponded to 43.6 MPa, 34.6 MPa and 21.2 MPa φ in maximum bending stress, respectively. It seems that max = 3KG+1 (33) φe these stresses were lower than the yield stress of this ma- ◦ where G is the modulus of shearing elasticity. This equa- terial at 800 C, about 80 MPa, so that plastic deformation tion shows that a large torsional deformation similar to the does not occur during cooling. After the test, the specimen deflection in the bending system occurs. was set up on feed, and the final profile of deflection was measured with a dial indicator. 3. Test Procedures 4. Test Results and Discussion To examine the validity of the method for obtain- ing the transformation plasticity coefficient in terms of The changes in the temperature of the specimen mea- the bending system, which was proposed in chapter 2, sured under a constant bending load during cooling were three-point bending tests were performed for a rectangu- plotted against time in Fig. 3. In this figure, the ordi- lar cross-sectional bar. Figures 1 and 2 show a schematic nate and abscissa describe the temperature of the speci- view and a general view of the experimental equipment, men and time after specimen was removed from the elec- respectively. In the present tests, the distance between two support- ing points is 200 mm, and a load is applied by hanging the weight at the center of the specimen. The loading and sup- porting pins were machined from stainless steel of 10 mm diameter. The maximum deflection is obtained by mea-

Fig. 2 Photograph of experimental equipment

Table 1 Chemical composition of SCM 440 [wt.%]

Fig. 1 Schematic view of experimental equipment

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Fig. 3 Temperature variation with time during cooling Fig. 5 Nondimensional profile of deflection after cooling

Fig. 4 Relationship between temperature and maximum Fig. 6 Relationship between applied load and deflection due to deflection transformation plasticity tric furnace, respectively. In this figure, the curves for the three types of applied stress are almost the same. A pin to the central loading pin, that is, half of specimen temperature-increase induced by the generation of trans- length, on account of the symmetry of loading system. In formation latent heat can be observed at about 330◦C, sug- this figure, •, and
symbols indicate the final profile gesting that the transformation caused here is a marten- of deflection obtained from the experiment under different sitic transformation. In fact, the Vickers hardness of the applied stresses (43.6 MPa, 34.6 MPa and 21.2 MPa). On specimen surface measured after the tests is approximately the other hand, the solid line is a third-order function, hav- 1 300. ing the form of Eq. (26), that shows the deflection in the Figure 4 shows the changes in maximum deflec- three-point bending system. Although elastic deflection tion measured during cooling. The ordinate is deflec- in loading is not included in the final profiles of deflec- tion and the abscissa is temperature. It should be noted tion because the elastic deflection is a third-order func- that the deflection begins to increase gradually from about tion having the form of Eq. (26), the nondimensional pro- 400◦C and increases rapidly at the martensitic trans- file of deflection should almost be expressed by Eq. (26). formation temperature of about 330◦C. Then, the deflec- Actually, the measured profiles of deflection are in quite tion hardly changes after transformation and temperature good agreement with theoretical results, suggesting that decreases, suggesting that the sequence of deflection is the assumptions described previously, e.g., Eq. (15), are due to transformation plasticity according to martensitic appropriate. Moreover, it was experimentally confirmed transformation. It can be seen that the maximum deflec- from this figure that the profile of deflection due to trans- tion δmax increases with an applied stress, and both param- formation plasticity is similar to that due to elasticity. eters have a linear relation. The relationship between the applied load W and the Then the final profiles of the dimensionless deflection maximum deflection δtp due to transformation plasticity formed by the maximum deflection δ¯ were examined as (maximum value of the final profile of deflection) is shown shown in Fig. 5. The axis of abscissa is the distance from in Fig. 6. In this figure, the open circles are test results, and the supporting pin, which is shown from the supporting the solid lines are determined by least-squares regression

JSME International Journal Series A, Vol. 48, No. 1, 2005 6 analysis of the test results. The slope of the solid line is Annual Meeting, Soc. Mat. Sci. Jpn., (in Japanese), 1.52×101 N/mm. From Eq. (28) in section 2.4, the deflec- Vol.5 (2002), pp.555–556. tion due to the transformation plasticity δtp is related to the ( 2 ) Miyao, K., Wang, Z.-G. and Inoue, T., Analy- applied load W according to sis of Temperature, Stress and Metallic Structure in Carburized-Quenched Gear Considering Transforma- 48I W = δtp. (34) tion Plasticity, J. Soc. Mat. Sci. Jpn., (in Japanese), Kl3 Vol.35, No.399 (1986), pp.1352–1357. Supposing that the specimens are transformed into 100% ( 3 ) Morozumi, K., Asaoka, S. and Nagaki, S., Experi- martensite after tests, the transformation plasticity coeffi- mental and Calculational Study of Distortion Mech- cient should be K = 6.51×10−5 MPa−1 in the present study. anism on Welding Process, Proc. 46th Material Re- search Meeting, Soc. Mat. Sci. Jpn., (in Japanese), 5. Conclusion No.9 (2002), pp.73–74. ( 4 ) Yamanaka, S., Sakanoue, T., Yoshii, T. and Inoue, To obtain the transformation plasticity experimen- T., Transformation Plasticity — The Effect on tally, a method of analyzing the deformation due to trans- Metallo-Thermo-Mechanical Simulation of Carburized formation plasticity in terms of bending was proposed. Quenching Process, J. Shanghai Jiatong Univ., E-5-1 Then, it was proved that the profile of deflection is sim- (2000), pp.185–195. ilar to that based on Hooke’s elastic theories when trans- ( 5 ) Inoue, T., Tanaka, K. and Nagaki, S., Solid Mechanics formations occur uniformly across the cross-section of a and Analysis of Phase Transformation, (in Japanese), beam. Thus, the transformation plasticity coefficient can (1995), p.175, Taiga Shuppan. be obtained easily by measuring the maximum deflection ( 6 ) Fischer, F.D., Sun, G.F. and Tanaka, K., Trans- formation-Induced Plasticity (TRIP), Appl. Mech. in a three-point bending system. Furthermore, it was de- Rev., Vol.49, No.6 (1996), pp.317–364. rived that the ratio of the deflection due to transformation ( 7 ) Fischer, F.D., Reisner, G., Werner, E., Tanaka, K., Cail- plasticity to elastic deflection in bending is the product of letaud, G. and Antetter, T., A New View on Trans- Young’s modulus and the transformation plasticity coef- formation Induced Plasticity (TRIP), Int. J. Plast., ficient. In addition, three-point bending tests were per- Vol.17, No.7 (2000), pp.723–748. formed for SCM 440 steel, and then the profile of deflec- ( 8 ) Taleb, L., Cavallo, N. and Waeckel, F., Experimen- tion due to transformation plasticity was measured. After tal Analysis of Transformation Plasticity, Int. J. Plast., the validity of the above assumption was proved, the plas- Vol.17, No.1 (2001), pp.1–20. ( 9 ) Liu, C.C., Yao, K.F. and Xu, X.J., Models for Transfor- ticity coefficient due to martensitic transformation could mation Plasticity in Loaded Steels Subjected to Bainitic be identified on the basis of the test results. and Martensitic Transformation, Mater. Sci. Technol., Acknowledgement Vol.17 (2001), pp.983–988. (10) Tutumi, K., Nakayama, K., Morimoto, T., Yamauchi, We would like to thank Professor T. Inoue of M. and Ohe, K., Development of Constitutive Law of Fukuyama University for suggesting the problem of trans- Phase Transformation (Prediction of Residual Stress of formation plasticity in bending deformation and for stim- TMCP-steel-plate 1), Proc. 141st Spring Meeting, ISIJ, ulating our interest in it. (in Japanese), Vol.13, No.4 (2001), p.1051. (11) Greenwood, G.W. and Johnson, R.H., The Deforma- References tion of Metals under Small Stresses during Phase Transformation, Proc. R. Soc. London, A, No.283 ( 1 ) Ohtsuka, T. and Inoue, T., An Experimental Method of (1965), pp.403–422. Identifying Transformation Plastic Behavior, Proc. 51st

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