Accuracy of Disk Method to Predict Roll Residual Stress by Measuring the Sliced Disk Stress

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1 ISIJ Intenational, Vol. 57 (2017), ISIJ Intenational, No. 8 Vol. 57 (2017), No. 8, pp Accuacy of Disk Method to Pedict Roll Residual Stess by Measuing the Sliced Disk Stess Nao-Aki NODA,* Kejun HU, Yoshikazu SANO, Yusuke HOSOKAWA and Xu WANG Depatment of Mechanical Engineeing, Kyushu Institute of Technology, 1-1 Sensui-cho, Tobata-ku, Kitakyushu-shi, Fukuoka, Japan. (Received on Novembe 4, 2016; accepted on Apil 13, 2017) Bimetallic olls ae widely used in steel olling industies because of the excellent hadness, wea esistance and high tempeatue popeties. Contolling the esidual stess distibution is necessay since the compessive esidual stess at the suface may impove fatigue life though the tensile esidual stess at the cente may educe the stength. Theefoe, it is necessay to measue the esidual stess distibution fom the suface to cente coectly to ensue the oll quality. The disk method has been widely used in pedicting the oll esidual stess by measuing the stess of the thin sliced disk fom the oll. In this study, theefoe, the elation between the oiginal oll esidual stess and the sliced disk esidual stess is investigated fo the single mateial oll and bimetallic oll on the basis of themo-elastic-plastic FEM analysis. The effect of the quenching time is discussed as well as the effect of the sliced disk thickness on the esidual stess. KEY WORDS: esidual stess; FEM simulation; olling; disk method; heat teatment. 1. Intoduction Wok olls ae widely used in the oughing stands of hot stip mill to educe steel thickness. Duing hot olling pocess, themal stesses ae caused by a cyclic sequence of heating and cooling ove the oll suface. Due to hot stip contact and wate cooling, themal cacks named fiecacks initiate at the oll suface. 1 4) If sevee themal tensile stess is added unde some olling toubles, themal cack stats popagating. Theefoe, a suitable compessive stess is necessay fo peventing themal cack extension. 5) Howeve, a tensile esidual stess always appeas at the oll cente to balance the suface compessive esidual stess. Unde the combined action of themal stess and esidual stess, anothe fom of oll factue is known as themal bael beakage as shown in Fig. 1. This themal beakage oiginates nea the oll cente and beaks out to the bael suface. 6 8) In ou pevious studies, 9,10) theefoe, diffeent quenching methods wee discussed though FEM simulation to poduce suitable suface compessive esidual stesses and educe the cente tensile esidual stess. In eal wok olls, howeve, the existence of suitable esidual stess distibution should be confimed expeimentally. Ove the yeas, diffeent measuing methods have been developed fo wok olls in ode to confim the oll esidual stess distibution. Those methods ae classified into destuctive o non-destuctive ones. Destuctive mechanical methods include deep hole-dilling method, ing coe method, disk method and Sachs boing method. 11,12) Non- * Coesponding autho: noda@mech.kyutech.ac.jp DOI: Fig. 1. An oveview of the oll beakage. destuctive methods include X-ay diffaction method and Bakhausen magnetic method. 13,14) Howeve, X-ay diffaction method and Bakhausen magnetic method ae suitable only fo measuing the suface egions and unsuitable fo the inteio egions of lage olls, as well as hole-dilling method and ing-coe method. Deep hole-dilling method and Sachs boing method can be used fo measuing the esidual stess fom the cente to suface fo vey lage olls although deep hole-dilling method needs special facilities and Sachs boing method is extemely time consuming. Theefoe, the disk method has been developed to pedict 1433

2 ISIJ Intenational, Vol. 57 (2017), No. 8 the oll esidual stess nea the oll cente, as a convenient method because of the convenience only by measuing the stess of the disk cut out fom the oll ) Howeve, attention should be paid fo the elation between the oll stess and the sliced disk stess. Since detailed studies ae not available, in this pape, the accuacy of disk method will be discussed on the basis of FEM simulation. In the fist place, the themo-elastic analysis and themo-elastic-plastic analysis will be consideed to veify the elation between the cylinde stess and the sliced disk stess with single mateial. Next, the themo-elastic-plastic analysis is pefomed to investigate the elation between the bimetallic oll esidual stesses and the sliced disk esidual stesses unde diffeent quenching time. 2. Disk Method and FEM Modeling 2.1. Outline of the Disk Method To evaluate the esidual stess of the cylinde, a thin disk is sliced fom the oiginal cylinde aound the middle potion as shown in Fig. 2. In the fist step, a disk with a thickness of about 30 mm was cut out fom the cylinde. Duing the disk-slicing pocess, cicumfeential and axial stains at the cylinde suface wee ecoded with the aid of stain gauges. Since the axial stess σ Disk z on the sliced disk is completely eleased, the emaining esidual stesses in the sliced disk ae in plane stess. Then, the sliced disk stesses σ Disk and σ Disk θ will be obtained by using X-ay diffaction method o some othe ways including ing slicing and cack compliance method. Finally, the cylinde stess Cylinde σ z will be estimated by using the sliced disk stesses σ Disk and σ Disk θ. Fig. 2. Schematic diagam of the disk method Fundamental Equations Useful fo Calculating Themo-elastic Stesses in Cicula Cylindes and Disks To calculate themo-elastic stesses of the cylinde and disk, the following equations ae available. 19) When a disk is subjected to the tempeatue distibution T(), the themal stesses σ Disk and σ Disk θ ae given by Eqs. (1) (2). On Cylinde Cylinde the othe hand, the cylinde stesses σ z σ and σ Cylinde θ ae given by Eqs. (3) (4) (5). σ σ Disk = b αe 1 T d T d () ()... (1) 0 σ Disk 1 b 1 θ = αe T() + T() d + T() d... (2) Cylinde α E 1 b 1 T d T d ν ν σ Di = 1 = 1 sk () () (3) α σ θcylinde E 1 b 1 = + ν T() d T() d T() (4) 1 = 1 ν σ Disk θ σ α E 2 b = T d T σθ σ 1 ν = + nde () () (5) Cylinde Cylinde Cyli z Fom the above equations, the following elation between the disk stess and the cylinde stess unde the same tempeatue distibution can be found as Eq. (6): Cylinde 1 σ z ν σ Disk σ Disk = + θ ( )... (6) 1 Whee, b is the cylinde o disk adius, T() is the tempeatue distibution, E is the Young s modulus, α is the themal expansion coefficient and ν is the Poisson s atio FEM Analysis Assume bimetallic olls with diamete of 600 mm, body length of mm and shell thickness of 75 mm, which consist of the high speed steel (HSS) as the shell mateial and the ductile casting ion (DCI) as the coe mateial. Table 1 shows the chemical compositions of HSS and DCI fo the common HSS bimetallic olls, and Table 2 shows the mateial popeties of HSS and DCI at oom tempeatue. Figue 3 shows the FEM model and bounday conditions fo the single mateial oll and the bimetallic oll. Hee, MSC.Mac 2012 softwae is used to cay out FEM analysis. Table 1. Chemical composition of high speed steel and ductile casting ion fo high speed steel oll/mass%. Composition C Si Mn P S Ni C Mo Co V W Mg HSS 1 3 <2 <1.5 <5 2 7 <10 < <20 <10 DCI <0.1 <

3 ISIJ Intenational, Vol. 57 (2017), No. 8 Table 2. Mateial popeties of high speed steel and ductile casting ion at oom tempeatue. Popety HSS DCI 0.2% poof stess [MPa] (1 282) *1 415 Young s modulus [GPa] Poisson s atio Density [kg/m 3 ] Themal expansion coefficient [K 1 ] Themal conductivity [W/(m K)] Specific heat [J/(kg K)] *1 Tensile stength of the shell mateial is indicated as the 0.2% poof stess because the defomation at beak is small cutting pocess is pefomed by using the deactivate element setting in the softwae MSC.Mac The element initial status of the sliced disk is set to activate and the est elements ae set to deactivate to simulate the disk cutting opeation fom the cylinde. Fist, the disk method is consideed fo the cylinde unde the tempeatue distibution T() by using the DCI mateial popeties of Young s modulus, themal expansion and Poisson s atio as shown in Figs. 4(a), 4(b) and 4(c). Then, the disk method is consideed fo the single mateial oll and the bimetallic oll unde the diffeent quenching time. Befoe cutting the disk, the oll esidual stess can be obtained though the quenching pocess simulation. Hee, a lage amount of mateial popeties wee expeimentally measued unde vaious tempeatues and utilized as the input data of the quenching simulation. As shown in Figs. 4(a), 4(b), 4(c), 4(d), 4(e) and 4(f), those mateial popeties include Young s modulus, themal expansion coefficient, Poisson s atio, stess-stain cuves, themal conductivity and specific heat. In Fig. 4(b), duing the quenching pocess, the pealite tansfomation occus in the coe mateial DCI and bainite tansfomation occus in the shell mateial HSS. Volume expansions of coe and shell accompany the phase tansfomations. As shown in Fig. 4(b), themal expansion coefficient changes in 250 C 350 C fo HSS and 700 C 720 C fo DCI and ae used as input data to expess the volume expansions of phase tansfomations. Fig. 3. FEM model and bounday conditions. A 4-node linea axisymmetic quad element with the mesh size of 5 5 mm is adopted fo the tansient-static simulation. The displacement bounday conditions and themal isolation conditions ae applied to z = 0 in Fig. 3 due to the symmety. In this study, the tempeatue distibution T() imposed to the cylinde fo the themo-elastic stess analysis and tempeatue is imposed to the oll suface fo the themo-elastic-plastic esidual stess analysis duing quenching. In this pape, the disk with the thickness of 30 mm is cut out fom the oll aound the cental section z = 0. The disk 3. Themal Stess and Residual Stess duing Quenching fo Single Mateial Roll 3.1. Themo-elastic Stess fo Cylinde and Disk In the fist place, the themo-elastic analysis is pefomed fo the cylinde and cicula disk. Assume the cylinde cente tempeatue Tc = T(0) = 200 C, which often appeas afte the standad quenching. Assume the cylinde suface tempeatue Ts = T(300) = 800 C, which may poduce the suface stess σ z (300) 600 MPa often appeaing at the suface afte the standad quenching. 9,10) Hee, assume that all mateial data of DCI ae depending on tempeatue distibution T() as shown in Figs. 4(a), 4(b) and 4(c) fo Young s modulus E, themal expansion coefficient α and Poisson s atio ν. Then, the themo-elastic analysis is pefomed fo the cylinde and the sliced disk. Figue 5 shows the stess distibution fo the cylinde stesses σ Cylinde i ( i = z, θ, ) as the solid lines in compaison with the sliced disk stesses σi Disk ( i = θ, ) as the dashed lines at z = 0. The sliced disk stess ( σ Disk + σ Disk ) calculated fom σi Disk ( i = θ, ) in Eq. (6) is also indicated as the dashed line. It is confimed Cylinde that the cylinde stess σ z coincides with the sliced disk stess ( σ Disk + σ Disk ) as shown in Fig. 5. In Cylinde othe wods, the elation σ z = ( σ Disk + σ Disk ) in Eq. (6) can be used fo the themo-elastic stess of the cylinde and sliced disk even when the mateial popeties ae depending on the tempeatue T() as shown in Figs. 4(a), 4(b) and 4(c) Residual Stess Geneation Mechanism duing Quenching fo Single Mateial Roll As shown in the above discussion, fo the themo-elastic analysis, it is found that the cylinde stess can be evaluated 1435

4 ISIJ Intenational, Vol. 57 (2017), No. 8 Fig. 4. Mateial popeties dependent on tempeatue fo high speed steel and ductile casting ion. by the sliced disk stess. Howeve, the esidual stess of the eal oll can be geneated duing the quenching though themo-elastic-plastic behavio of the oll mateial affected by the tempeatue gadient and phase tansfomation. In this section, theefoe, the esidual stess geneation mechanism will be explained duing quenching pocess. In ou pevious studies, 9,10) the geneation mechanism of esidual stess fo bimetallic oll has been discussed in detail. Hee, fo single mateial oll, the fundamental mechanism of esidual stess will be discussed without consideing the phase tansfomation. Figue 6 shows the histoies of (a) tempeatue Ts, Tc, (b) stess σ z and (c) Young s modulus E, and (d) defomation state fo the single mateial oll duing quenching pocess. Since FEM elastic-plastic analysis equies Young s modulus even unde high tempeatue, 0.05% stain is focused on the stess-stain cuve. Then, the Young s modulus is defined as the gadient of the line connecting the 0.05% stain point and the oigin point. Figue 6(c) shows the Young s modulus E defined in this way duing quenching pocess, which vaies depending on the tempeatue. The quenching pocess is divided into Region Ⅰ (1), Region Ⅱ (2) and Region Ⅲ (3 5) classified by the dominant elastic o plastic state at the suface and cente. In Region Ⅰ, the yield stength of shell and coe is vey low due to high tempeatue, the stess apidly inceases and 1436

5 ISIJ Intenational, Vol. 57 (2017), No. 8 Fig. 5. Themo-elastic stesses of the cylinde and sliced disk assuming E = E(T), α = α(t), ν = ν(t) ae depending on tempeatue T() as shown in Figs. 4(a), 4(b) and 4(c) (Tc = T(0) = 200 C, Ts = T(300) = 800 C). exceeds the yield stess. Theefoe, the lage plastic defomation occus at both oll suface and oll cente (see Fig. 6(d)1). In Region Ⅱ, since the suface becomes elastic due to suface cooling, the suface Young s modulus inceases with deceasing tempeatue although the cente still keeps high tempeatue and plastic state (see Fig. 6(d)2). In Region Ⅲ, since both suface and the cente become elastic (see Fig. 6(d)3 5), both Young s modulus inceases as the cooling continues. In Region Ⅰ, at the beginning of cooling, the suface tempeatue dops faste than the cente tempeatue, leading to the tempeatue gadient in the -diection (see Fig. 6(d)1). Aftewads, the oll suface shinks elative to the cente in the axial diection and esults in tensile stess. In ode to balance the stesses in the oll inteio, the compessive stess appeas in the oll cente. With inceasing the tempeatue gadient, the stess at the oll suface and cente incease togethe continuously. In Region Ⅱ, due to continuous cooling, the oll suface tuns to be elastic with inceasing the Young s modulus. Meanwhile, the oll cente is still plastic keeping high tempeatue (see Fig. 6(d)2). In this peiod, the themal Fig. 6. Residual stess geneation mechanism fo the single mateial oll. 1437

6 ISIJ Intenational, Vol. 57 (2017), No. 8 contaction at the suface is esticted due to the appeaing of elastic state. Howeve, the themal contaction ate at the cente is faste than that one at the suface, causing the themal stain diffeences deceasing. Finally, both the suface and cente stesses each peak values. In Region Ⅲ, themal stain diffeences deceasing due to the cente s themal contaction ate is lage than suface, and both the suface and cente stesses stat deceasing (Fig. 6(d)3). As cooling continues, the suface themal contaction appoximately equals to the cente themal contaction, then the stesses state ae intechanged (see Fig. 6(d)4). Since the cente contaction is lage than that at the suface, the tensile stess inceases at the oll cente (see Fig. 6(d)5). Since Young s modulus inceases in egion Ⅲ (see Fig. 6(c)), the compessive stesses at the oll suface inceases as well as the tensile stess at the oll cente. Finally, the compessive at the suface and tensile stess at the cente ae geneated as shown in Fig. 6(b) Residual Stess Simulation duing Quenching fo Single Mateial Roll The themo-elastic-plastic analysis is pefomed fo the single mateial oll befoe and afte cutting out the cicula disk fom the oll consideing phase tansfomation. The esidual stess is contolled by the heat teatment condition. In ode to investigate the effect of heat teatment on the esidual stess, diffeent quenching time is consideed. As shown in Fig. 7, seveal tempeatue changes ae consideed at the oll suface fom C to 100 C. Hee, the diffeent quenching time = 0.5, 1 7 h coesponds to the eal oll quenching time. Afte quenching pocess, the oll is kept at 100 C until the unifom oll tempeatue is obtained. Hee, the mateial popeties of DCI ae used as shown in Figs. 4(a) 4(f). As shown in the above esults in Fig. 5, fo the themo-elastic analysis, it is found that the atio σ Cylinde z /[( σ Disk + σ Disk )] = 1. Due to the themoelastic-plastic behavio, fo the esidual stess, the atio σ Cylinde z /[( σ Disk + σ Disk )] 1. Theefoe, the ange of the atio σ Cylinde z /[( σ + σ Disk )/( 1 ν )] will be discussed. Disk θ Fig. 7. Quenching time of the oll suface. Figue 8 shows the stess atio σ Cylinde z / [( σ Disk + σ Disk )] of the single mateial oll unde the diffeent quenching time. As shown in Fig. 8(a), it is seen that the atio vaies in the ange of at =0 100 mm nea the oll cente. Hee, this atio is consideed since σ Cylinde z /[( σ Disk + σ Disk )] 1 due to the themoelastic-plastic behavio. At the oll cente, the stess atio inceases with inceasing quenching time t = h. It can be seen that most of the atio σ Cylinde z /[( σ Disk + σ Disk )] at = 100 mm deceases with inceasing the quenching time (o deceasing quenching speed). This is because the plastic stain becomes smalle with inceasing quenching time (o deceasing quenching speed). Fo the single mateial oll, it is found that the atio σ Cylinde z /[( σ Disk + σ Disk )] = fo vaying the quenching time as shown in Fig. 8(a). The disk method can be used fo pedicting the oll esidual stess by consideing the above amounts of accuacy. To claify the effect of sliced disk thickness, FEM analysis is also pefomed to the disk thickness 90 mm and compaed to the esults of disk thickness 30 mm. When the disk thickness is 90 mm shown in Fig. 8(b), the stess atio σ Cylinde z /[( σ Disk + σ Disk )] vaies in the ange of at = mm nea the oll cente and becomes smalle by 6% compaed with the esults of the disk thickness 30 mm. It is found that the effect of disk thickness is small fo the ange mm. The disk thickness esults 30 mm and 90 mm in this study ae useful fo consideing anothe disk thickness. 4. Residual Stess Simulation duing Quenching fo Bimetallic Roll 4.1. Residual Stess Simulation fo Bimetallic Roll Figue 9 shows the esidual stess distibution σ z Roll of the bimetallic oll unde the diffeent quenching time in Fig. 7. It is seen that both the cente tensile stess and the suface compessive stess inceases with deceasing the quenching time. This is because the maximum tempeatue diffeence between the suface and cente inceases with deceasing the quenching time. Figue 10 shows the equivalent plastic stain distibution ε eq of the oll unde the diffeent quenching time in Fig. 7. With deceasing the quenching time, the plastic stain ε eq becomes lage because the tempeatue diffeence between the suface and cente becomes lage. It is also seen that the esidual stesses in Fig. 9 ae closely elated to the plastic stain in Fig. 10, and both stesses at the cente and suface ae inceasing with inceasing the plastic stain Relation between Bimetallic Roll Stess and Sliced Disk Stess As shown in Figs. 5, 8, the elation between the oll stess σ Cylinde z and the sliced disk stess ( σ Disk + σ Disk ) fo the single mateial oll has been discussed. In this section, Roll the elationship between the oll stess σ z and the sliced disk stess ( σ Disk + σ Disk ) fo the bimetallic oll will be discussed. Disk Figue 11 shows the esidual stess distibutions σ and σ Disk θ of the sliced disk fom the bimetallic oll. Figue 12 shows the stess atio σ Roll z / 1438

7 ISIJ Intenational, Vol. 57 (2017), No. 8 Fig. 8. Stess atio σ Cylinde z /[ ( σ Disk + σ Disk ) ] nea the oll cente of the single mateial oll unde the diffeent quenching time. Fig. 9. Roll Residual stess distibutions σ z of the bimetallic oll unde the diffeent quenching time. Fig. 10. Plastic stain ε eq of the bimetallic oll unde the diffeent quenching time. 1439

8 ISIJ Intenational, Vol. 57 (2017), No. 8 Fig. 11. Residual stess distibutions σ Disk, σ θ Disk of the sliced disk unde the diffeent quenching time. Fig. 12. Stess atio σ Roll z /[( σ Disk + σ Disk )] nea the oll cente of the bimetallic oll unde the diffeent quenching time. [( σ Disk + σ Disk )] of the bimetallic oll unde the diffeent quenching time. As shown in Fig. 12, the atio is lage than 1 fo the most cases and vaies in the ange at = mm nea the oll cente. In othe Roll wods, the eal oll stess σ z is usually lage than the sliced disk stess ( σ Disk + σ Disk ) as shown in Fig. 12. On the othe hand, fo the single mateial oll, the atio σ Cylinde z /[( σ Disk + σ Disk ) = was obtained by vaying the quenching time as shown in Fig. 8(a). This study focused on estimating the most impotant esidual stess σ z by the disk method. In fact, since the actual stess σ z is geneated though the plastic defomation pocess, estimating σ z is vey difficult on the basis of the simple elastic theoy. Howeve, since thee is no othe means, the disk method can be used in a pactical way to estimate the oll stess oughly. Since the disk method has been used fo decades, the pesent discussion may be useful fo oll industies. 1440

9 ISIJ Intenational, Vol. 57 (2017), No Conclusions In eal wok olls, it is necessay to confim suitable esidual stess distibutions. In this pape, theefoe, the accuacy of disk method was investigated on the basis of FEM simulation. The elation was discussed between the bimetallic oll esidual stess and the sliced disk esidual stess by vaying quenching time. The conclusions can be summaized in the following way. (1) Fo the single mateial olls it is confimed that the themo-elastic stess can be pedicted exactly fom the sliced disk fom the elation σ Cylinde z /[( σ Disk + σ Disk )] as shown in Fig. 5. (2) Fo the single mateial olls the themo-elasticplastic esidual stess can be pedicted by the disk method by consideing the atio σ Cylinde z /[( σ Disk + σ Disk )] = nea the oll cente as shown in Fig. 8. It is confimed that the eo is insensitive to the disk thickness. (3) Fo the bimetallic oll the esidual stess can be pedicted by the disk method consideing the atio σ Cylinde z / [( σ Disk + σ Disk )] = nea the oll cente as shown in Fig. 12. (4) The disk method has been widely used fo measuing the oll esidual stess. Howeve, the accuacy of this method has not been claified yet. In this pape, theefoe, the accuacy was discussed by vaying the quenching time as shown in Fig. 8 fo the single mateial oll and shown in Fig. 12 fo the bimetallic oll. The disk method can be used fo pedicting the oll esidual stess by consideing the above amounts of accuacy. REFERENCES 1) C. F. Onisa and D. C. J. Faugia: Int. J. Mate. Fom., 1 (2008), ) A. Péez, R. L. Coal, R. Fuentes and R. Colás: J. Mate. Pocess. Technol., 153 (2004), ) D. Benasciutti: J. Stain Anal., 47 (2012), ) D. F. Chang: J. Mate. Pocess. Technol., 94 (1999), 45. 5) Y. Sano, T. Hattoi and M. Haga: ISIJ Int., 32 (1992), ) Y. Sano and K. Kimua: Tetsu-to-Hagané, 73 (1987), ) K. H. Schode: A Basic Undestanding of the Mechanics of Rolling Mill Rolls, ESW-Handbook, Eisenwek Sulzau-Wefen, Tenneck, (2003), 71. 8) The Euopean Foundy Association: Roll Failues Manual, Hot Mill Cast Wok Rolls, CAFÉ, Roll Section, Düsseldof, (2002), 19. 9) N.-A. Noda, K. Hu, Y. Sano, K. Ono and Y. Hosokawa: Steel Res. Int., 87 (2016), ) N.-A. Noda, K. Hu, Y. Sano, K. Ono and Y. Hosokawa: Steel Res. Int., (2016), DOI: /sin ) E. Kingston and D. J. Smith: Ionmaking Steelmaking, 32 (2005), ) X. Zhang, X. Song, L. Zhu and M. V. Li: 5th Int. Conf. on Themal Pocess Modeling and Compute Simulation, ASM Intenational, Olando, Floida, (2014), 6. 13) L. Y. Liu, J. F. Yuan and S. Y. Zhu: Mod. Cast Ion, 1 (1997), 21, (in Chinese). 14) J. Pacyna, A. Kokosza and A. S. Wojtas: The e-jounal of Nondestuctive Testing & Ultasonics, 4 (1999), v04n08/wojtas/wojtas.htm, (accessed ). 15) Y. Higashida, T. Kikuma, T. Kawanami and K. Kimua: Tetsu-to- Hagané, 72 (1986), ) M. Hinnemann, P. J. Mauk, V. Goyany, C. Zybill and R. Baun: Key Eng. Mate., 622 (2014), ) W. Cheng and I. Finnie: Residual Stess Measuement and the Slitting Method, Spinge US, New Yok, (2007), ) Y. Jimbo: J. Adv. Sci., 3 (1991), ) S. P. Timoshenko and J. N. Goodie: Theoy of Elasticity, McGaw-Hill Book Company, New Yok, (1951),