Axisymmetric and Related Problems

  • J. Chakrabarty
Part of the Mechanical Engineering Series book series (MES)

Abstract

Many physically important problems of plasticity involve solids of revolution which are loaded symmetrically about their geometrical axes. Typical examples of axially symmetrical distribution of stress and strain are provided by the expansion of circular cylindrical tubes, upsetting of cylindrical blocks, extrusion of cylindrical billets, and the axial drawing of wires and tubes. In the theoretical analysis of such problems, which involve large plastic strains, it is customary to assume the material to be rigid/plastic, for which no deformation can occur until the load attains the yield point value. The rigid/plastic assumption is adequate for the estimation of the yield point load itself, and also for the determination of stresses and strains in the plastically deforming zone during the continued loading. In this chapter, we shall deal with several examples in which the stress distribution is axially symmetrical, together with a few related problems involving three-dimensional states of stress.

Keywords

Yield Criterion Meridian Plane Extrusion Pressure Frictional Stress Hydrostatic Extrusion 
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References

  1. Adie, J.F. and Alexander, J.M. (1967), A Graphical Method of Obtaining Hodographs to Axisymmetric Problems, Int. J. Mech. Sci., 9, 349.MATHGoogle Scholar
  2. Alexander, J.M., Brewer, R.C., and Rowe, G.W. (1987), Manufacturing Technology, Vols. 1 and 2, Ellis Horwood, Chichester, UK.Google Scholar
  3. Alexander, J.M. and Lengyel, B. (1971), Hydrostatic Extrusion, Mills and Boon, London.Google Scholar
  4. Atkins, A.G. and Caddell, R.M. (1968), The Incorporation of Work-Hardening and Redundant Work in Rod Drawing Analysis, Int. J. Mech. Sci., 10, 15.Google Scholar
  5. Avitzur, B. (1968), Metal Forming Processes and Analysis, McGraw-Hill, New York.Google Scholar
  6. Avitzur, B. (1983), Handbook of Metal Forming, Processes, Wiley, New York.Google Scholar
  7. Bishop, J.F.W. (1958), On the Effect of Friction on Compression and Indentation Between Flat Dies, J. Meck Phys. Solids, 6, 132.MathSciNetMATHGoogle Scholar
  8. Blazynski, T.Z. (1971), Optimization of Die Design in the Extrusion of Rod, Int. J. Mech. Sci., 13, 113.Google Scholar
  9. Blazynski, T.Z. (1976), Metal Forming: Tool Profiles and Flow, Macmillan, London.Google Scholar
  10. Blazynski, T.Z. and Cole, I.M. (1960), An Investigation of the Plug Drawing Process, Proc. Inst. Mech. Engrs., 174, 797.Google Scholar
  11. Bridgman, P.W. (1944), Stress Distribution in the Neck of a Tension Specimen, Trans Amer. Soc. Metals, 32, 553.Google Scholar
  12. Bridgman, P.W. (1952), Studies in Large Plastic Flow and Fracture, McGraw-Hill, New York.MATHGoogle Scholar
  13. Caddell, R.M. and Atkins, A.G. (1969), Optimum Die Angles and Maximum Attainable Reductions in Rod Drawing, J. Engng. Indust., Trans. ASME, 91, 664.Google Scholar
  14. Calladine, CR. (1985), Plasticity for Engineers, Ellis Horwood, Chichester, UK.Google Scholar
  15. Chakrabarty, J. (1972), The Necking of Cylindrical Bars Under Lateral Fluid Pressure, Proc. 13th MTDR Conference, Birmingham UK, p. 565.Google Scholar
  16. Chakrabarty, J. (1987), Theory of Plasticity, McGraw-Hill, New York.Google Scholar
  17. Chakrabarty, J. (1993), An Analysis of Axisymmetric Extrusion Including Redundant Shear and Strain-Hardening, in Advances in Engineering Plasticity and its Applications (ed., W.B. Lee), Elsevier, Amsterdam, p. 901.Google Scholar
  18. Chen, C.C. and Kobayashi, S. (1978), Rigid Plastic Finite Element Analysis of Ring Compression, ASME, AMD, 28, 163.Google Scholar
  19. Chen, C.C., Oh, S.I., and Kobayashi, S. (1979), Ductile Fracture in Axisymmetric Extrusion and Drawing, Part I, Deformation Mechanics of Extrusion and Drawing, J. Engng. Indust, Trans. ASME, 101, 23.Google Scholar
  20. Chen, C.T. and Ling, F.F. (1968), Upper Bound Solutions to Axisymmetric Extrusion Problems, Int. J. Mech. Sci., 10, 863.Google Scholar
  21. Cheng, S.Y., Ariaratnam, S.T., and Dubey, R.N. (1971), Axisymmetric Bifurcation in an Elastic/Plastic Cylinder Under Axial Load and Lateral Hydrostatic Pressure, Quart. Appl. Math., 29, 41.MATHGoogle Scholar
  22. Chenot, J.L., Feigeres, L., Lavarenne, B., and Salecon, J. (1978), A Numerical Application of the Slipline Field Method to Extrusion Through Conical Dies, Int. J. Engng. Sci., 16, 263.Google Scholar
  23. Chitkara, N.R. and Bhutta, M.A. (1996), Near Net Shape Forging of Spur Gear Forms. An Analysis and Some Experiments, Int. J. Mech. Sci., 38, 891.Google Scholar
  24. Chung, S.W. and Swift, H.W (1952), A Theory of Tube Sinking, J. Iron Steel Inst., 170, 29.Google Scholar
  25. Collins, I.F. (1968), The Upper Bound Theorem for Rigid/Plastic Solids Generalized to Include Coulomb Friction, J. Mech. Phys. Solids, 17, 323.Google Scholar
  26. Collins, I.F. and Williams, B.K. (1985), Slipline Fields for Axisymmetric Tube Drawing, 7. Mech. Eng. Sci., 27 225. Google Scholar
  27. Danyluk, H.T. and Haddow, J.B. (1969), Elastic/Plastic Flow Through a Converging Conical Channel, Acta Mech., 7, 35.MATHGoogle Scholar
  28. Davidenkov, N.N. and Spiridonova, N.I. (1946), Analysis of Tensile Stress in the Neck of an Elongated Test Specimen, Proc. ASTM., 46, 1147.Google Scholar
  29. Denton, A.A. and Alexander, J.M. (1963), On the Determination of Residual Stresses in Tubes, J. Mech. Engng. Sci., 5, 75.Google Scholar
  30. Desdo, D. (1971), Principal and Slipline Methods of Numerical Analysis in Plane and Axially Summetric Deformations of Rigid/Plastic Media, J. Mech. Phys. Solids, 19, 313.Google Scholar
  31. Duffill, A.W. and Mellor P.B. (1969), A Comparison Between Conventional and Hydrostatic Extrusion Methods of Cold Extrusion Through Conical Dies, Ann. of CIRP, 17, 97.Google Scholar
  32. Dugdale, D.S. (1954), Cone Indentation Experiments, J. Mech. Phys. Solids, 2, 265.Google Scholar
  33. Durban, D. (1979), Axially Symmetric Radial Flow of Rigid-Linear Hardening Material, J. Appl. Mech., 46, 322.MATHGoogle Scholar
  34. Durban, D. and Reck, N.A. (1992), Singular Plastic Fields in Steady Penetration of a Rigid Cone, J. Appl. Mech., 59, 706.MATHGoogle Scholar
  35. Eason, G. and Shield, R.T. (1960), The Plastic Indentation of a Semi-Infinite Solid by a Rough Circular Punch, Z. Angew. Math. Phys., 11, 33.MathSciNetMATHGoogle Scholar
  36. Haar, A. and von Karman, Th. (1909), Nachr. Ges. Wiss. Göttingen, Math. Phys. Kl., p. 204.Google Scholar
  37. Haddow, J.B. (1962), Ideal Die Pressures for Axisymmetric Tube Extrusion, Int. J. Mech. Sci., 4, 447.Google Scholar
  38. Haddow, J.B. and Danyluk, H.T. (1964), Indentation of a Rigid-Plastic Semi-Infinite Medium by a Smooth Rigid Cone Fitted in a Prepared Cavity, J. Mech. Sci., 5, 1.Google Scholar
  39. Haddow, J.B. and Johnson, W. (1962), Bounds for the Load to Compress Plastically a Square Disc Between Rough Dies, Appl. Sci. Res., A10, 476.Google Scholar
  40. Hailing, J. and Mitchell, L.A. (1965), An Upper Bound for Axisymmetric Extrusion, Int. J. Mech. Sci., 1, 277.Google Scholar
  41. Hardy, C, Baronnet, C.N., and Tordion, G.V. (1971), The Elasto-Plastic Indentation of a Half Space by a Rigid Sphere, Int. J. Numer. Methods Engng., 3, 451.Google Scholar
  42. Hawkyard, J.B. and Johnson, W. (1967), An Analysis of the Changes in Geometry of a Short Hollow Cylinder During Axial Compression, Int. J. Mech. Sci., 9, 163.Google Scholar
  43. Hill, R. (1950a), The Mathematical Theory of Plasticity, Clarendon Press, Oxford, UK.MATHGoogle Scholar
  44. Hill, R. (1950b), On the Inhomogeneous Deformation of a Plastic Lamina in a Compression Test, Phil. Mag., Ser. 7, 41, 733.Google Scholar
  45. Hill, R. (1963), A General Method of Analysis for Metal Working Processes, J. Mech. Phys. Solids, 11, 305.Google Scholar
  46. Hill, R. (1967), Ideal Forming for Perfectly Plastic Solids, J. Mech. Phys. Solids, 15, 223.Google Scholar
  47. Hill, R. (1976), Plastic Analysis of Pressurized Cylinders Under Axial Load, Int. J. Mech. Sci., 18, 145.MATHGoogle Scholar
  48. Hill, R., Storakers, B., and Zdunek, A.B. (1989), A Theoretical Study of the Brinell Hardness Test, Proc. Roy. Soc. London Ser. A, 423, 301.MATHGoogle Scholar
  49. Hutchinson, J.W. and Miles, J.P. (1974), Bifurcation Analysis of the Onset of Necking in an Elastic/Plastic Cylinder Under Uniaxial Tension, J. Mech. Phys. Solids, 22, 61.MATHGoogle Scholar
  50. Ishlinsky, A. (1944), Prikl Mat. Mekh. Leningrad, 8, 201.MathSciNetGoogle Scholar
  51. Iwata, K., Osakada, K., and Jujeno, S. (1972), Analysis of Hydrostatic Extrusion by the Finite Element Method, J. Engng. Indust., Trans. ASME, 94, 697.Google Scholar
  52. Johnson, R.W. and Rowe, G.W. (1968), Redundant Work in Drawing Cylindrical Stock, J. Inst. Metals, 96, 97.Google Scholar
  53. Johnson, W (1957), The Pressure for the Cold Extrusion of Lubricated Rod Through Square Dies of Moderate Reduction at Slow Speeds, J. Inst, of Metals, 85, 403.Google Scholar
  54. Johnson, W and Kudo, H. (1962), Mechanics of Metal Extrusion, Manchester University Press, Manchester, UK.Google Scholar
  55. Johnson, W and Mellor, PB. (1983), Engineering Plasticity, Ellis Horwood, Chichester, UK.Google Scholar
  56. Johnson, W, Slater, R.A.C., and Yu, A.S. (1966), The Quasi-Static Compression of Non-Circular Prismatic Blocks Between Very Rough Platens, Int. J. Mech. Sci., 8, 731.Google Scholar
  57. Juneja, B.L. (1973), Forging of Polygonal Discs with Barrelling, Int. J. Mach. Tool Des. Res., 13, 87.Google Scholar
  58. Kim, J.H. and Yang, D.Y. (1985), An Analysis of Upset Forging of Square Blocks Considering the Three-Dimensional Bulging of Sides, Int. J. Mech. Tool Des. Res., 25, 327.Google Scholar
  59. Kim, J.H., Yang, D.Y, and Kim, M.U. (1987), Analysis of Three-Dimensional Upset Forging of Arbitrary Shaped Rectangular Blocks, Int. J. Mach. Tool Manufact., 27, 311.Google Scholar
  60. Kobayashi, S. and Thomsen, E.G. (1965), Upper and Lower Bound Solutions to Axisym-metric Compression and Extrusion Problems, Int. J. Mech. Sci., 7, 127.Google Scholar
  61. Kobayashi, S. Oh, S.I., and Altan, T. (1989), Metal Forming and the Finite Element Method, Oxford University Press, New York.Google Scholar
  62. Kudo, H. (1960), Some Analytical and Experimental Studies of Axisymmetric Cold Forging and Extrusion, Int. J. Mech. Sci., 1, 175.Google Scholar
  63. Kummerling, R. and Lippmann, H. (1975), On Spread in Rolling, Mech. Res. Commun., 2, 113.Google Scholar
  64. Kwasczynska, K. and Mröz, Z. (1967), A Theoretical Analysis of Plastic Compression of Short Circular Cylinders, Arch. Mech. Stos., 19, 787.Google Scholar
  65. Lahoti, G.D. and Kobayashi, S. (1974), On Hill’s General Method of Analysis for Metal Working Professes, Int. J. Mech. Sci., 16, 521.Google Scholar
  66. Lambert, E.R. and Kobayashi, S. (1969), An Approximate Solution for the Mechanics of Axisymmetric Extrusion, Proc. 9th M.T.D.R. Conference, Pergamon Press, Oxford.Google Scholar
  67. Lange, K. (1985), Handbook of Metal Forming, McGraw-Hill, New York.Google Scholar
  68. Lee, C.H. and Kobayashi, S. (1971), Analysis of Axisymmetric Upsetting and Plane Strain Side Pressing of Solid Cylinders by the Finite Element Method, J. Engng. Indust., Trans. ASME, 93, 445.Google Scholar
  69. Lippmann, H. (1962), Principal Line Theory of Axially Symmetrical Plastic Deformation, J. Mech. Phys. Solids, 10, 111.MathSciNetMATHGoogle Scholar
  70. Lippmann, H. (1965), Statics and Dynamics of Axially Symmetric Plastic Row, J. Mech. Phys. Solids, 13, 29.MathSciNetMATHGoogle Scholar
  71. Lippmann, H. and Mahrenholtz, O, (1967), The Mechanics of Plastic Forming of Metals (in German), Springer-Verlag, Berlin.Google Scholar
  72. Lockett, F.J. (1963), Indentation of a Rigid/Plastic Material by a Conical Indenter, J. Mech. Phys. Solids, 11, 345.Google Scholar
  73. MacDonald, A., Kobayashi, S., and Thomsen, E.G. (1960), Some problems of Press Forging Lead and Aluminum, J. Engng. Indust., Trans. ASME, 82, 246.Google Scholar
  74. Maclellan, G.D.S. (1952), Some Friction Factors in Wire Drawing, J. Inst. Metals, 81, 1.Google Scholar
  75. Male, A. T. and Cockroft, M.G. (1964), The Ring Test, J. Inst. Metals, 93, 38.Google Scholar
  76. Male, A.T. and DePierre, V. (1970), The Validity of Mathematical Solutions for Determining Friction from the Ring Compression Test, J. Lubrication Technol., Trans. ASME, 92, 389.Google Scholar
  77. Marshall, E.R. and Shaw, M.C. (1952), The Determination of Row Stress from a Tensile Specimen, Trans. Amer. Soc. Metals, 44, 705.Google Scholar
  78. Meyer, O.E. (1908), Untersuchungen über Härteprüfung und Harte, Z. ver. deutsche Ing., 52, 645.Google Scholar
  79. Moore, G.G. and Wallace, J.F. (1967), Theories and Experiments on Hollow Sinking Through Conical Dies, Proc. Inst. Mech. Engrs., 182, 19.Google Scholar
  80. Mori, K. and Osakada, K. (1982), Simulation of Three-Dimensional Rolling by the Rigid/ Plastic Finite Element Method, in Numerical Methods in Industrial Forming Processes (eds. J.M. Alexander and O.C. Ziekiewicz), Pineridge Press, Swansea, UK, p. 747.Google Scholar
  81. Mröz, Z. (1967), Graphical Solution of Axially Symmetric Problems of Plastic Flow, Z Angew. Math. Mech., 18, 219.Google Scholar
  82. Nagamatsu, A., Murota, T., and Jimma, T. (1970), On the Non-Uniform Deformation of Block in Plane Strain Compression Caused by Friction, Bull. Japan Soc. Mech. Engrs. 13, 1389.Google Scholar
  83. Nagpal, V. (1977), On the Solution of Three-Dimensional Metal Forming Processes, J. Engng. Indust., Trans. ASME, 99, 624.Google Scholar
  84. Needleman, A. (1972), A Numerical Study of Necking in Circular Cylindrical Bars, J. Mech., Phys. Solids, 20, 111.MATHGoogle Scholar
  85. Norbury, A.L. and Samuel, T. (1928), The Recovery and Sinking in or Piling up of Material in the Brinell Test, and the Effects of these Factors on the Correlation of the Brinell Test with Certain Other Hardness Tests, J. Iron. Steel Inst., 117, 673.Google Scholar
  86. Oh, S. I. (1982), Finite Element Analysis of Metal Forming Problems with Arbitrary Shaped Dies, Int. J. Mech. Sci., 17, 293.Google Scholar
  87. Oh, S.I. and Kobayashi, S. (1975), An Approximate Method for a Three-Dimensional Analysis of Rolling, Int. J. Mech. Sci., 17, 293.MATHGoogle Scholar
  88. Osakada, K. and Nimi, Y. (1975), A Study of Radial Flow Fields for Extrusion Through Conical Dies, Int. J. Mech. Sci., 17, 241.Google Scholar
  89. Park, J.J. and Kobayashi, S. (1984), Three-Dimensional Finite Element Analysis of Block Compression, Int. J. Mech. Sci., 26, 165.MATHGoogle Scholar
  90. Pugh, H. LI. D. (1970), Mechanical Behavior of Metals under Pressure, Elsevier, Amster dam.Google Scholar
  91. Richmond, O. (1965), Theory of Streamlined Dies for Drawing and Extrusion, J. Mech. Phys. Solids, 13, 154.Google Scholar
  92. Richmond, O. and Morrison, H.L. (1967), Streamlined Wire Drawing Dies of Minimum Length, J. Mech. Phys. Solids, 15, 195.Google Scholar
  93. Rowe, G.W. (1977), Introduction to the Principles of Industrial Metalworking, Arnold, London.Google Scholar
  94. Sachs, G. and Baldwin, W.M. (1946), Stress Analysis of Tube Sinking, Trans. Amer. Soc. Mech. Engrs., 68, 655.Google Scholar
  95. Sagar, R. and Juneja, B.L. (1979), An Upper Bound Solution for Flat Tool Forging Taking into Account the Bulging of Sides, Int. J. Mech. Tool Des. Res., 19, 253.Google Scholar
  96. Samanta, S.K. (1968), The Application of the Upper Bound Theorem to the Prediction of Indenting and Compressing Leads, Acta Polytech. Scand., Mc-38.Google Scholar
  97. Samanta, S.K. (1971), A New Die Profile with Higher Process Efficiencies, Appl. Sci. Res., 25, 54.Google Scholar
  98. Sha, S.N. and Kobayashi, S. (1977), A Theory on Metal Flow in Axisymmetric Piercing and Extrusion, J. Prod. Engng., 1, 73.Google Scholar
  99. Shield, R.T. (1955a), Plastic Flow in a Converging Conical Channel, J. Mech. Phys. Solids, 3, 246.MathSciNetGoogle Scholar
  100. Shield, R.T. (1955b), On the Plastic Flow of Metals Under Conditions of Axial Symmetry, Proc. Roy. Soc. London Sec. A, 233, 267.MathSciNetMATHGoogle Scholar
  101. Shield, R.T. (1955c), The Plastic Indentation of a Layer by a Flat Punch, Quart. Appl. Math., 13, 27.MathSciNetMATHGoogle Scholar
  102. Shield, R.T. and Drucker, D.C. (1953), The Application of Limit Analysis to Punch Indentation Problems, J. Appl. Mech., Trans. ASME, 20, 453.MATHGoogle Scholar
  103. Siebel, E. (1923), Stahl und Eisen, Düsseldorf, 43, 1295.Google Scholar
  104. Siebel, E. (1947), Der derzeitige stand der Erkenntnisse über die mechanischen Vorgange beim Drahtziehen, Stahl und Eisen, 66, 171.Google Scholar
  105. Sinclair, G.B., Follansbee, P.S., and Johnson, K.L. (1985), Quasi-Static Normal Indentation of an Elastic-Plastic Half Space by a Rigid Sphere, Int. J. Solids Struct., 21, 865.MATHGoogle Scholar
  106. Slater, R.A.C. (1979), Engineering Plasticity: Theory and Application to Metal Forming, Macmillan, London.Google Scholar
  107. Sokolovsky, W.W. (1969), Theory of Plasticity (in Russian), 3rd ed., Nauka, Moscow.Google Scholar
  108. Sortais, H.C. and Kobayashi, S. (1968), An Optimum Die Profile for Axisymmetric Extrusion, Int. J. Mach. Tool Des. Res.,8, 61.Google Scholar
  109. Spencer, A.J.M. (1964), The Approximate Solution of Certain Problems of Axially Symmetric Plastic Flow, J. Mech. Phys. Solids, 12, 231.MATHGoogle Scholar
  110. Storakers, B. and Larson, P. (1994), On Brinell and Boussinesq Indentation of Creeping Solids., J. Mech. Phys. Solids, 42, 307.MATHGoogle Scholar
  111. Swift, H.W. (1949), Stresses and Strains in Tube Drawing, Phil. Mag., Ser. 7, 11, 883.Google Scholar
  112. Szczepinski, W. (1962), The Method of Successive Approximation of Some Strain-Hardening Solutions, Proc. 4th U.S. Nat. Congr. Appl. Mech., p. 131.Google Scholar
  113. Szczepinski, W., Dietrich, L., Drescher, E., and Miastkowski, J. (1966), Plastic Flow of Axially Symmetric Notched Bars Pulled in Tension, Int. J. Solid Struct., 2, 543.Google Scholar
  114. Tabor, D. (1951), The Hardness of Metals, Clarendon Press, Oxford, UK.Google Scholar
  115. Thomsen, E.G., Yang, C.T., and Kobayashi, S. (1965), Mechanics of Plastic Deformation in Metal Processing, Macmillan, New York.Google Scholar
  116. Tirosh, J. (1971), Dead Zone Formation in Plastic Axially Symmetric Converging Flow, J. Mech. Phys. Solids, 19, 237.Google Scholar
  117. Tvergaard, V. and Needlemann A. (1984), Analysis of Cup-Cone Fracture in a Round Tensile Bar, Acta Metall., 32, 157.Google Scholar
  118. Unksov, E.P. (1961), An Engineering Theory of Plasticity, Butterworth, London.Google Scholar
  119. Van Rooyen, G.T. and Backofen, W.A. (1960), A Study of Interface Friction in Plastic Compression, Int. J. Mech. Sci., 8, 1.Google Scholar
  120. Wilcox, R.J. and Whitton, P.W. (1958), The Cold Extrusion of Metals Using Lubrication at Slow Speeds, J. Instn. Metals, 87, 289.Google Scholar
  121. Wistreich, J.G. (1955), Investigation of the Mechanics of Wire Drawing, Proc. Inst. Mech. Engrs., 169, 123.Google Scholar
  122. Wistreich, J.G. (1958), Fundamentals of Wire Drawing, Metall. Rev. 3, 97.Google Scholar
  123. Yang, D.Y. and Han, C.H. (1987), A New Formulation of Generalized Velocity Field for Axisymmetric Forward Extrusion Through Arbitrary Curved Dies, J. Engng. Indust., Trans. ASME, 109, 161.Google Scholar
  124. Yang, D.Y. and Kim, J.H. (1986), An Analysis for Three-Dimensional Upset Forging of Elliptical Disks, Int. J. Mach. Tool Des. Res., 26, 147.Google Scholar
  125. Yang, D.Y. and Lee, C.H. (1978), Analysis of Three-Dimensional Extrusion of Sections Through Curved Dies by Conformai Transformation, Int. J. Mech. Sci., 20, 541.Google Scholar
  126. Yang, D.Y, Lee, CM., and Cho, J.R. (1990), Analysis of Axisymmetric Extrusion by the Method of Weighted Residuals Using Coordinate Transformation, Int. J. Mech. Sci., 32, 101.MATHGoogle Scholar
  127. Zienkiewicz, O.C., Jain, P.C., and Onate, E. (1978), Row of Solids During Forming and Extrusion: Some Aspects of Numerical Solutions, Int. J. Solids Struct., 14, 15.Google Scholar

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© Springer Science+Business Media New York 2000

Authors and Affiliations

  • J. Chakrabarty
    • 1
  1. 1.Department of Mechanical EngineeringNational Taiwan UniversityTaipeiTaiwan, R.O.C.

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