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Plane Strain Plastic Deformation

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Engineering Plasticity
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Abstract

In previous chapters, plastic deformation of an element of material on a microscopic scale was considered as though the element were independent of the body of material. For many metal forming processes such as forging, drawing and rolling the geometrical configuration of the tooling, frictional conditions at the workpiece—tooling interfaces and the constraint of neigh­bouring material will result in the plastically deforming region being subjected to a field of stress and strain which may differ from point to point. This field of stress—strain states makes it extremely difficult to obtain a solution to a problem. Only by making certain assumptions is it possible to derive good first approximations to the force, energy and power required to perform a metal forming operation and also provide useful information about the manner in which the workpiece material deforms.

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References

  1. Hill, R., The Mathematical Theory of Plasticity, ch. VI and app. III, Oxford University Press, London (1950)

    Google Scholar 

  2. Johnson, W., Sowerby, R. and Haddow, J. B., Plane Strain Slip Line Fields: Theory and Bibliography, ch. 3, and app. 2, Edward Arnold, London (1970)

    Google Scholar 

  3. Hencky, H. Z., Über einige statisch bestimmte Fälle des Gleichgewichts in plastischen Körpern, Z. angew. Math. Mech., 3, 241 (1923)

    Article  Google Scholar 

  4. Christopherson, D. G., Oxley, P. L. B. and Palmer, W. B., Orthogonal cutting of a work-hardening material, Engineering, Lond., 186, 113 (1958)

    Google Scholar 

  5. Palmer, W. B. and Oxley, P. L. B., The mechanics of orthogonal machining, Proc. Instn mech. Engrs, 173, 623 (1959)

    Article  Google Scholar 

  6. Geiringer, H. Beitrag zum Vollständigen ebenen Plastizitäts-problem, Proc. 3rd Intern. Congr. appl. Mech. 2 185 (1930)

    Google Scholar 

  7. Prager, W., A geometrical discussion of the slip line field in plane plastic flow, Trans. R. Inst. Technol., Stockholm, 65, 27 (1953)

    Google Scholar 

  8. Prager, W. and Hodge, P. G., Theory of Perfectly Plastic Solids, Wiley, New York (1951)

    Google Scholar 

  9. Thomas, T. Y., Plastic Flow and Fracture in Solids, Academic Press, London (1961)

    Google Scholar 

  10. Prager, W., The theory of plasticity: A survey of recent achievements, Proc. Instn mech. Engrs, 169, 1 (1955)

    Article  Google Scholar 

  11. Ford, H., The theory of plasticity in relation to engineering application. J. appl. Math. Phys., 5, 1 (1954)

    Article  Google Scholar 

  12. Alexander, J. M., Deformation modes in metal forming processes, Proc. Conf. Technol. Engng Manuf., Pap. 42, Instn Mech. Engrs (1958)

    Google Scholar 

  13. Bishop, J. F. W., Green, A. P. and Hill, R., A note on the deformable region in a rigid—plastic body, J. Mech. Phys. Solids., 4, 256 (1956)

    Article  Google Scholar 

  14. Green, A. P., Plastic yielding of notched bars due to bending, Q. J. Mech. appl. Math., 6, 223 (1953)

    Article  Google Scholar 

  15. Green, A. P., On the use of hodographs in problems of plane plastic strain, J. Mech. Phys. Solids., 2, 73 (1954)

    Article  Google Scholar 

  16. Ford, H., Advanced Mechanics of Materials, pt IV, ch. 32, p. 518, Longmans Green, London (1963)

    Google Scholar 

  17. Hill, R. Discussion of a paper by E. Siebel, Application to shaping processes of Hencky’s laws of equilibrium, J. Iron Steel Inst. 156 513 (1947)

    Google Scholar 

  18. Hill, R. A theoretical analysis of stresses and strains in extrusion and piercing, J. Iron Steel Inst. 159 177 (1948)

    Google Scholar 

  19. Prandtl, L. Über die Härte Plastischer Körper, Nachr. Ges. Wiss. Göttingen, p. 74 (1920)

    Google Scholar 

  20. Bishop, J. F. W., On the complete solution to problems of deformation of a rigid—plastic material, J. Mech. Phys. Solids, 2, 43 (1953)

    Article  Google Scholar 

  21. Hill, R. On the mechanics of cutting metal with knife-edge tools, J. Mech. Phys. Solids 1 265 (1953)

    Article  Google Scholar 

  22. Hill, R., The Mathematical Theory of Plasticity, ch. IX, p. 254, Oxford University Press, London (1950)

    Google Scholar 

  23. Hill, R. Lee, E. H. and Tupper, S. J., A theory of wedge indentation of ductile materials, Proc. R. Soc., Ser. A 188 273 (1947)

    Article  Google Scholar 

  24. Grunzweig, J., Longman, I. M. and Petch, N. J., Calculations and measurements on wedge-indentation, J. Mech. Phys. Solids, 2, 81 (1954)

    Article  Google Scholar 

  25. Johnson, W. and Kudo, H., The cutting of metal strips between partly rough knife edge tools, Int. J. mech. Sci., 2, 294 (1961)

    Article  Google Scholar 

  26. Johnson, W., The cutting of round wire with knife-edge and flat edge tools, Appl. Sci. Res. (A), 7, 65 (1957)

    Google Scholar 

  27. Mahtab, F. U. and Johnson, W., The cutting of strip with knife-edge and flat-edge face tools, J. Mach. Tool Des. Res., 2, 335 (1962)

    Article  Google Scholar 

  28. Green, A. P., A theoretical investigation of the compression of a ductile material between smooth flat dies, Phil. Mag., 42, 900 (1951)

    Article  Google Scholar 

  29. Collins, I. F., Geometric properties of some slip line fields for compression and extrusion, J. Mech. Phys. Solids 16 137 (1968)

    Article  Google Scholar 

  30. Johnson, W. and McShane, I. E., A note on calculations concerning the plastic compression of thin material between smooth plates under conditions of plane strain, Appl. Sci. Res., A, 9, 169 (1960)

    Article  Google Scholar 

  31. Prandtl, L. On the penetration hardness of plastic materials and the hardness of indenters, Z. angew. Math. Mech., 1 15 (1920)

    Article  Google Scholar 

  32. Hill, R. Lee, E. H. and Tupper, S. J. A method of numerical analysis of plastic flow in plane strain and its application to the compression of a ductile material between rough plates, Trans. Am. Soc. Mech. Engrs, J. Appl. Mech. 18 46 (1951)

    Google Scholar 

  33. Alexander, J. M., The effect of Coulomb friction in the plane strain compression of a plastic—rigid material, J. Mech. Phys. Solids, 3, 233 (1955)

    Article  Google Scholar 

  34. Sachs, G., Spanlose Formung der Metalle, Springer Verlag, Berlin (1931)

    Book  Google Scholar 

  35. Hoffman, O. and Sachs, G., Introduction to the Theory of Plasticity for Engineers, McGraw-Hill, New York (1953)

    Google Scholar 

  36. Sendzimir, M. G., The Sendzimir cold strip mill, J. Metals, N. Y., 8, 1154 (1956)

    Google Scholar 

  37. Underwood, L. R., The Rolling of Metals, John Wiley, New York (1950)

    Google Scholar 

  38. Larke, E. C., The Rolling of Strip, Sheet and Plate, Chapman and Hall, London (1957)

    Google Scholar 

  39. Alexander, J. M. and Brewer, R. C., Manufacturing Properties of Materials, ch. 5, p. 256, Van Nostrand Reinhold, London (1963)

    Google Scholar 

  40. Avitzur, B., Metal Forming: Processes and Analysis, ch. 15, p. 436, McGraw-Hill, New York (1968)

    Google Scholar 

  41. Rowe, G. W., An Introduction to the Principles of Metalworking, ch. 9, p. 194, Edward Arnold, London (1965)

    Google Scholar 

  42. Tarnovskii, I. Y., Pozdeyev, A. A. and Lyashkov, V. B., Deformation of Metals During Rolling (English translation), Pergamon Press, Oxford (1965)

    Google Scholar 

  43. Tselikov, A. I. and Smirnov, V. V., Rolling Mills (English translation), Pergamon Press, Oxford (1965)

    Google Scholar 

  44. von Karman, T., Beitrag zur Theorie des Walzvorganges, Z. angew, Math. Mech., 5, 139 (1925)

    Google Scholar 

  45. Orowan, E., The calculation of roll pressure in hot and cold flat rolling, Proc. Instn mech. Engrs, 150, 140 (1943)

    Article  Google Scholar 

  46. Bland, D. R. and Ford, H., The calculation of roll force and torque in cold strip rolling with tensions, Proc. Instn mech. Engrs 159 144 (1948)

    Article  Google Scholar 

  47. Hitchcock, J., Roll neck bearing, Report of Am. Soc. Mech. Engrs, Special Research Committee on Heavy-duty Anti-friction Bearings (1935)

    Google Scholar 

  48. Stone, M. D., Rolling of thin strips, Iron Steel Engr, 30 (2), 61 (1953)

    Google Scholar 

  49. Hill, R., The Mathematical Theory of Plasticity, ch. VII, p. 188, Oxford University Press, London (1950)

    Google Scholar 

  50. Ford, H., Roll hard materials in thin gauges: Basic considerations, J. Inst. Metals, 88, 193 (1960)

    Google Scholar 

  51. Avitzur, B. Maximum reduction in cold strip rolling, Proc. Instn mech. Engrs 174 865 (1960)

    Article  Google Scholar 

  52. Alexander, J. M. A slip-line field for the hot rolling process, Proc. Instn mech. Engrs 169 1021 (1955)

    Article  Google Scholar 

  53. Alexander, J. M. and Ford, H., Limit analysis of hot rolling, in Progress in Applied Mechanics: The Prager Anniversary Volume, p. 191, Collier-Macmillan, London (1963)

    Google Scholar 

  54. Druyanov, A. B., Kinematic problems in the theory of the plane plastic flow of ideally plastic bodies, in Investigation of Processes of Plastic Deformation of Metals, p. 134, Moscow (1965)

    Google Scholar 

  55. Firbank, T. C. and Lancaster, P. R., A suggested slip line field for cold rolling with slipping friction, Int. J. mech. Sci., 7, 847 (1965)

    Article  Google Scholar 

  56. Johnson, W. and Kudo, H., The use of upper bound solutions for the determination of temperature distributions in fast hot rolling, Int. J. mech. Sci., 1, 175 (1960)

    Article  Google Scholar 

  57. Ford, H. Researches into the deformation of metals by cold rolling, Proc. Instn mech. Engrs, 159 115 (1948)

    Article  Google Scholar 

  58. Watts, A. B. and Ford, H., On the basic yield stress curve for a metal, Proc. Instn mech. Engrs, 169, 1141 (1955)

    Article  Google Scholar 

  59. Hessenberg, W. C. F. and Sims, R. B., The effect of tension on torque and roll force in cold strip rolling, J. Iron Steel Inst., 168, 155 (1951)

    Google Scholar 

  60. Whitton, P. W. and Ford, H. Surface friction and lubrication in cold strip rolling, Proc. Instn mech. Engrs 169 123 (1955)

    Article  Google Scholar 

  61. Wilcox, R. J. and Whitton, P. W., The rolling of thin titanium strip, J. Inst. Metals, 88, 200 (1960)

    Google Scholar 

  62. Bedi, D. S. and Hillier, M. J. Hydrodynamic model for cold strip rolling, Proc. Instn mech. Engrs 182(1) 153 (1967)

    Article  Google Scholar 

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© 1977 R. A. C. Slater

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Slater, R.A.C. (1977). Plane Strain Plastic Deformation. In: Engineering Plasticity. Palgrave, London. https://doi.org/10.1007/978-1-349-02160-4_7

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  • DOI: https://doi.org/10.1007/978-1-349-02160-4_7

  • Publisher Name: Palgrave, London

  • Print ISBN: 978-1-349-02162-8

  • Online ISBN: 978-1-349-02160-4

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