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Applied Plasticity pp 51–136Cite as

Problems in Plane Stress

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Abstract

In many problems of practical interest, it is a reasonable approximation to disregard the elastic component of strain in the theoretical analysis even when the body is only partially plastic. In effect, we are then dealing with a hypothetical material which is rigid when stressed below the elastic limit, the modulus of elasticity being considered as infinitely large. If the plastically stressed material has the freedom to flow in some direction, the distribution of stress in the deforming zone of the assumed rigid/plastic body would approximate that in an elastic/plastic body, except in a transition region near the elastic/plastic interface where the deformation is restricted to elastic order of magnitude. The assumption of rigid/plastic material is generally adequate not only for the analysis of technological forming processes, where the plastic part of the strain dominates over the elastic part, but also for the estimation of the yield point load when the rate of work-hardening is sufficiently small (Section 1.2). In the present chapter, we shall be concerned with problems in plane stress involving rigid/plastic bodies which are loaded beyond the range of contained plastic deformation.

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References

  • Alexander, J.M. (1960), An Appraisal of the Theory of Deep Drawing, Metall. Rev., 5, 349.

    Google Scholar 

  • Alexander, J.M., Brewer, R.C., and Rowe, G.W. (1987), Manufacturing Technology, Vols. 1 and 2, Ellis Horwood, Chichester, UK.

    Google Scholar 

  • Alexander, J.M. and Ford, H. (1954), On Expanding a Hole from Zero Radius in a Thin Infinite Plate, Proc. Roy. Soc. London Ser. A 226, 543.

    Article  MathSciNet  MATH  Google Scholar 

  • Alexander, J.M. and Gunasekera, J.S. (1992), Strength of Materials, Volume 2, Ellis Horwood, Chichester, UK.

    Google Scholar 

  • Bishop, J.F.W. (1953), On the Complete Solution to Problems of Deformation of a Plastic-Rigid Material, J. Mech. Phys. Solids, 2, 43.

    Article  MathSciNet  Google Scholar 

  • Blazynski, T.Z. (1989), Plasticity and Modern Metal Forming Technology (ed.), Elsevier Applied Sciences, Amsterdam.

    Google Scholar 

  • Chakrabarty, J. (1970), A Theory of Stretch Forming Over Hemispherical Punch Heads, Int. J. Mech. Sci., 12, 315.

    Article  Google Scholar 

  • Chakrabarty, J. (1971), Elastic-Plastic Expansion of a Circular Hole in an Infinite Plate of Variable Thickness, Int. J. Mech. Sci., 13, 439.

    Article  MATH  Google Scholar 

  • Chakrabarty, J. (1987), Theory of Plasticity, McGraw-Hill, New York.

    Google Scholar 

  • Chakrabarty, J. and Alexander, J.M. (1970), Hydrostatic Bulging of Circular Diaphragms, J. Strain Anal, 5, 155.

    Article  Google Scholar 

  • Chakrabarty, J. and Mellor, P.B. (1968), A New Approach for Predicting the Limiting Drawing Ratio, La Metallurgia Italiana, 8, 791.

    Google Scholar 

  • Chen, F.K. and Liu, J.H. (1997), Analysis of an Equivalent Model for the Finite Element Simulations of a Stamping Process, Int. J. Mach. Tools Manuf., 37, 409.

    Article  Google Scholar 

  • Chern, J. and Nemat-Nasser, S. (1969), The Elargement of a Hole in a Rigid-Workhardening Disk of Non-Uniform Initial Thickness, J. Mech. Phys. Solids, 17, 271.

    Article  MATH  Google Scholar 

  • Chung, S.Y and Swift, H.W. (1951), Cup Drawing from a Flat Blank, Proc. Instn. Mech. Engrs., 165, 199.

    Article  Google Scholar 

  • El-Sebaie, M.G. and Mellor, R.B. (1973), Plastic Instability When Deep Drawing Into a High Pressure Medium, Int. J. Mech. Sci., 15, 485.

    Article  Google Scholar 

  • Ewing, D.J.F and Richards, C.E. (1973), The Yield Point Loads of Singly Notched Pin-Loaded Tensile Strips, J. Mech. Phys. Solids, 21, 27.

    Google Scholar 

  • Ewing, D.J.F. and Spurr, C.E. (1974), The Yield Point Loads of Symmetrically Notched Metal Strips, J Mech. Phys. Solids, 22, 37.

    Article  Google Scholar 

  • Fogg, B. (1968), Theoretical Analysis for the Redrawing of Cylindrical Cups Through Conical Dies, J. Mech. Engng. Sci., 10, 141.

    Article  Google Scholar 

  • Fukui, S. and Hansson, A. (1970), Analytical Study of Wall Ironing Considering Work-Hardening, Ann. of CIRP, 18, 593.

    Google Scholar 

  • Gaydon, F.A. (1952), An Analysis of the Plastic Bending of a Thin Strip in its Plane, J. Mech. Phys. Solids, 1, 103.

    Article  Google Scholar 

  • Gaydon, F.A. (1954), On the Yield Point Loading of a Square Plate with Concentric Circular Hole, J. Mech. Phys. Solids, 2, 170.

    Article  Google Scholar 

  • Gaydon, F.A. and McCrum, A.W. (1954), A Theoretical Investigation of the Yield Point Loading of a Square Plate with a Central Circular Hole., J. Mech. Phys. Solids, 2, 156.

    Article  Google Scholar 

  • Geckeler, J.W. (1928), Plastic Folding of the Walls of Hollow Cylinders and Some Other Folding Phenomena in Bowls and Sheets, Zeit angew. Math. Mech., 8, 341.

    Article  MATH  Google Scholar 

  • Green, A.R (1954a), A Theory of Plastic Yielding due to the Bending of Cantilevers and Fixed-Ended Beams—Part I, J. Mech. Phys. Solids, 3, 1.

    Article  MathSciNet  Google Scholar 

  • Green, A.P. (1954b), A Theory of Plastic Yielding due to Bending of Cantilevers and Fixed-Ended Beams—Part II, J. Mech. Phys. Solids, 3, 143.

    Article  Google Scholar 

  • Hassani, H.A. and Neale, K.W (1991), On the Analysis of Sheet Metal Wrinkling, Int. J. Mech. Sci, 33, 13.

    Article  MATH  Google Scholar 

  • Hill, R. (1949), Plastic Distortion of Non-Uniform Sheets, Phil. Mag. (Ser. 7), 40, 971.

    MATH  Google Scholar 

  • Hill, R. (1950a), The Mathematical Theory of Plasticity, Chap. XI, Clarendon Press, Oxford, UK.

    MATH  Google Scholar 

  • Hill, R (1950b), A Theory of Plastic Bulging of a Metal Diaphragm by Lateral Pressure, Phil. Mag. (Ser. 7), 41, 1133.

    Google Scholar 

  • Hill, R. (1952), On Discontinuous Stress States with Special References to Localized Necking in Thin Sheets, J. Mech. Phys. Solids, 1, 19.

    Article  MathSciNet  Google Scholar 

  • Hill, R. (1953), A New Method for Determining the Yield Criterion and Plastic Potential of Ductile Metals, J. Mech. Phys. Solids, 1, 271.

    Article  Google Scholar 

  • Hill, R. and Storakers, B. (1980), Plasticity and Creep of Pressurized Membranes, J. Mech. Phys. Solids, 26.

    Google Scholar 

  • Hillier, M.J. (1966), Instability Strains in Plane Sheet Under Biaxial Stresses, J. Appl. Mech., 23, 256.

    Google Scholar 

  • Hodge, P.G. (1959), Plastic Analysis of Structures, Chap. 12, McGraw-Hill, New York.

    MATH  Google Scholar 

  • Hodge, P.G. and Perrone, N. (1957), Yield Loads of Slabs with Reinforced Cutouts, J. Appl. Mech., 24, 85.

    MathSciNet  MATH  Google Scholar 

  • Hodge, P.G. and Sankaranarayanan, S. (1958), On Finite Expansion of a Hole in a Thin Infinite Plate, Quart. Appl. Math, 16, 73.

    MathSciNet  MATH  Google Scholar 

  • Hundy, B.B. and Green, A.P. (1954), A Determination of Plastic Stress-Strain Relations, J. Mech. Phys. Solids, 3, 16.

    Article  Google Scholar 

  • Ilahi, M.F., Parmar, A., and Mellor, P.B. (1981), Hydrostatic Bulging of a Circular Aluminium Diaphragm, Int. J. Mech. Sci., 23, 221.

    Article  MATH  Google Scholar 

  • Johnson, W., Chitkara, N.R., and Ranshi, A.S. (1974), Plane Stress Yielding of Cantilevers in Bending due to Combined Shear and Axial Load, J. Strain Anal, 9, 67.

    Article  Google Scholar 

  • Johnson, W. and Mellor, P.B. (1983), Engineering Plasticity, Chap. 11, Ellis Horwood, Chichester, UK.

    MATH  Google Scholar 

  • Kaftanoglu, B. and Alexander J.M. (1970), On Quasi-Static Axisymmetric Stretch Forming, Int. J. Mech. Sci., 12, 1065.

    Article  Google Scholar 

  • Kasuga, Y. and Tsutsumi, S. (1965), Pressure Lubricated Deep Drawing, Bull. Japan Soc. Mech. Engrs., 8, 120.

    Article  Google Scholar 

  • Keeler, S.P. and Backofen, W.A. (1963), Plastic Instability and Fracture in Sheet Stretched Over Rigid Punches, Trans. Amer. Soc. Metals, 56, 25.

    Google Scholar 

  • Kim, J.H. and Kobayashi, S. (1978), Analysis of Stretching of Sheet Metals with Hemispherical Punch, Int. J. Mech. Tool Des. Res., 19, 209.

    Article  Google Scholar 

  • Kim, Y.J. and Yang, D.Y. (1985), A Rigid/Plastic Finite Element Formulation Considering Effect of Geometry Change and its Application to Hydrostatic Bulging, Int. J. Mech. Sci., 27, 453.

    Article  MATH  Google Scholar 

  • Kobayashi, S., Oh, S.I., and Altan, T. (1989), Metal Forming and the Finite Element Method, Oxford University Press, New York.

    Google Scholar 

  • Lee, W.B. and Chan, K.C. (1991), Criteria for the Prediction of Shear Band Angles in F.C.C. Metals, Acta Metall, 39, 411.

    Article  Google Scholar 

  • Lianis, G. and Ford, H. (1957), An Experimental Investigation of the Yield Criterion and the Stress-Strain Law, J. Mech. Phys. Solids, 5, 215.

    Article  Google Scholar 

  • Lo, S.W., Hsu, T.C., and Wilson, W.R.D. (1993), An Analysis of the Hemispherical Punch Hydroforming Processes, 7. Mater. Process. Technol., 37, 225.

    Article  Google Scholar 

  • Mellor, P.B. (1954), Stretch Forming Under Fluid Pressure, J. Mech. Phys, Solids, 5, 41.

    Article  Google Scholar 

  • Nadai, A. (1950), Theory of Flow and Fracture of Solids, Vol. I, p. 275, McGraw-Hill, New York.

    Google Scholar 

  • Neal, B.G. (1961), Effect of Shear Force on the Fully Plastic Moment of an I-Beam, J. Mech. Engng. Sci., 3, 258.

    Article  Google Scholar 

  • Nemat-Nasser, S. (1968), Finite Expansion of a Hole in a Rigid/Work-Hardening Disk of Initially Nonuniform Thickness, J. Mech. Phys. Solids, 16, 195.

    Article  Google Scholar 

  • Nine, H.D. (1978), Drawbead Forces in Sheet Metal Forming, in Mechanics of Sheet Metal Forming (eds., D.P Koistinen and N.M. Wang), Plenum Press, New York, p. 179.

    Chapter  Google Scholar 

  • Nine, H.D. (1982), The Applicability of Coulomb’s Friction Law to Drawbeads in Sheet Metal Forming, J. Appl. Metalworking, 2, 200.

    Article  Google Scholar 

  • Nordgren, R.P. and Naghdi, P.M. (1963), Finite Twisting and Expansion of a Hole in a Rigid /Plastic Plate, J. Appl. Mech., 30, 605.

    Article  MathSciNet  MATH  Google Scholar 

  • Prager, W. (1953), On the Use of Singular Yield Conditions and Associated Flow Rules, J. Appl. Mech., 20, 317.

    MathSciNet  MATH  Google Scholar 

  • Ranshi, A.S., Chitkara, N.R., and Johnson, W (1974), Plane Stress Plastic Collapse Loads for Tapered Cantilevers and Haunched Beams, Int. J. Mech. Sci., 16, 867.

    Article  Google Scholar 

  • Ranshi, A.S., Chitkara, N.R., and Johnson, W. (1976), Plastic Yielding of I-Beams under Shear, and Shear and Axial Load, Int. J. Mech. Sci,, 18, 375.

    Article  Google Scholar 

  • Rogers, T.G. (1967), Finite Expansion and Subsequent Unloading of Hole in Elastic-Plastic Plate of Initially Varying Thickness, Quart. J. Mech. Appl Math., 20, 137.

    Article  MATH  Google Scholar 

  • Ross, E.W and Prager, W. (1954), On the Theory of the Bulge Test, Quart. Appl Math., 12, 86.

    MathSciNet  MATH  Google Scholar 

  • Sanchez, L.R. and Weinmann, K.J. (1996), An Analytical and Experimental Study of the Flow of Sheet Metal Between Circular Drawbeads, J. Engng. Indust., Trans. ASME, 118, 45.

    Article  Google Scholar 

  • Senior, B.W. (1956), Flange Wrinkling in Deep Drawing Operations, J. Mech. Phys. Solids, 4, 235.

    Article  Google Scholar 

  • Shepherd, W.M. and Gaydon, F.A. (1957), Plastic Bending of a Ring Sector by End Couples, J. Mech. Phys. Solids, 5, 296.

    Article  MathSciNet  MATH  Google Scholar 

  • Sokolovsky, V.V. (1969), Theory of Plasticity, 3rd Edition, Moscow.

    Google Scholar 

  • Storen, S. and Rice, J. (1975), Localized Necking in Thin Sheets, J. Mech. Phys. Solids, 23, 421.

    Article  Google Scholar 

  • Storakers, B. (1966), Finite Plastic Deformation of a Circular Membrane Under Hydrostatic Pressure, Int. J. Mech. Sci., 8, 619.

    Article  Google Scholar 

  • Swift, H.W. (1952), Plastic Instability Under Plane Stress, J. Mech. Phys. Solids, 1, 1.

    Article  Google Scholar 

  • Taylor, G.I. (1948), The Formation and Enlargement of a Circular Hole in a Thin Plastic Sheet, Quart. J. Mech. Appl. Math., 1, 101.

    Article  Google Scholar 

  • Thomson, T.R. (1975), Influence of Material Properties in the Forming of Square Shells, J. Australian Inst. Metals, 20, 106.

    Google Scholar 

  • Tomita, Y. and Shindo, W. (1988), Onset and Growth of Wrinkles in Thin Square Plates Subjected to Diagonal Tension, Int. J. Mech. Sci., 30, 921.

    Article  MATH  Google Scholar 

  • Triantafyllidis, N., Maker, B., and Samanta, S.K. (1986), An Analysis of Drawbeads in Sheet Metal Forming, Part I: Problem Formulation, J. Engng. Mater. Technol., Trans. ASME, 108, 321.

    Article  Google Scholar 

  • Tvergaard, V. (1981), Influence of Voids on Shear Band Instabilities Under Plane Strain Conditions, Int. J. Fracture, 17, 389.

    Article  Google Scholar 

  • Wallace, J.F (1960), Improvements in Punches for Cylindrical Deep Drawing, Sheet Metal Industries, 37, 901.

    Google Scholar 

  • Wang, N.M. (1970), Large Plastic Deformation of a Circular Sheet Caused by Punch Stretching, J. Appl. Mech., 37, 431.

    Article  Google Scholar 

  • Wang, N.M. (1982), A Mathematical Model of Drawbead Forces in Sheet Metal Forming, J. Appl. Metalworking, 2, 193.

    Article  Google Scholar 

  • Wang, N.M. and Budiansky, B. (1978), Analysis of Sheet Stamping by a Finite Element Method, J. Appl. Mech., 45, 73.

    Article  MATH  Google Scholar 

  • Wang, N.M. and Shammamy, M.R. (1969), On the Plastic Bulging of Metal Diaphragms Under Hydrostatic Pressure, J. Mech. Phys. Solids, 17, 43.

    Article  Google Scholar 

  • Wang, X. and Lee, L.H.N. (1989), Wrinkling of an Unevenly Stretched Sheet Metal, J. Engng. Mater. Technol., Trans. ASME, 111, 235.

    Article  Google Scholar 

  • Weil, N. A. and Newmark, N.M. (1955), Large Plastic Deformation of Circular Membranes, J. Appl. Mech., 22, 533.

    MATH  Google Scholar 

  • Weiss, H.J., Prager, W., and Hodge, P.G. (1952), Limit Design of a Full Reinforcement for a Circular Cutout in a Uniform Slab, J. Appl. Mech., 19, 307.

    Google Scholar 

  • Wifi, S.A. (1976), An Incremental Complete Solution of the Stretch Forming and Deep Drawing of a Circular Blank Using a Hemispherical Punch, Int. J. Mech. Sci., 18, 23.

    Article  MATH  Google Scholar 

  • Wilson, W.R.D. and Hector, L.G. (1991), Hydrodynamic Lubrication in Axisymmetric Stretch Forming, Part I: Theoretical Analysis, Part II: Experimental Investigation, J. Tribology Trans. ASME, 113, 659 and 667.

    Article  Google Scholar 

  • Woo, D.M. (1964), The Analysis of Axisymmetric Forming of Sheet Metal and the Hydrostatic Bulging Processes, Int. J. Mech. Sci., 6, 303.

    Article  Google Scholar 

  • Woo, D.M. (1968), On the Complete Solution of the Deep Drawing Problem, Int. J. Mech. Sci., 10, 83.

    Article  Google Scholar 

  • Yossifon, S. and Tirosh, J. (1985), Rupture Instability in Hydrodynamic Deep Drawing Process, Int. J. Mech. Sci., 27, 559.

    Article  Google Scholar 

  • Yu, T.X. and Johnson, W. (1982), The Buckling of an Annular Plate in Relation to the Deep Drawing Process, Int. J. Mech. Sci., 24, 175.

    Article  MATH  Google Scholar 

  • Yu, T.X. and Zhang, L.C. (1996), Plastic Bending Theory and Applications, World Scientific, Singapore.

    Book  Google Scholar 

  • Zhang, L.C. and Yu, T.X. (1988), Plastic Wrinkling of an Annular Plate Under Inner Tension, Int. J. Solids Struct., 24, 497.

    Article  MATH  Google Scholar 

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  1. Department of Mechanical Engineering, National Taiwan University, Taipei, 10617, Taiwan, R.O.C.

    J. Chakrabarty (Visiting Professor)

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  1. J. Chakrabarty
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Chakrabarty, J. (2000). Problems in Plane Stress. In: Applied Plasticity. Mechanical Engineering Series. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-3268-9_2

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  • DOI: https://doi.org/10.1007/978-1-4757-3268-9_2

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