Abstract
This paper extends the ideal flow theory, which is well known for isotropic rigid perfectly plastic materials, to quite general orthotropic materials which comply with the principle of maximum plastic dissipation. The new theory is restricted to steady planar flow. The original ideal flow theory is widely used as the basis for inverse methods for the preliminary design of metal forming processes driven by minimum plastic work. The new theory extends this area of application to orthotropic materials. Moreover, another design criterion based on the Cockroft–Latham ductile fracture criterion is incorporated in the theory. To this end, the extended Bernoulli’s theorem relating pressure and velocity along any streamline during the steady planar flow of rigid perfectly plastic solids when the streamline is coincident everywhere with a principal stress trajectory is used. In particular, this theorem and the concept of ideal flow combine to evaluate the integral involved in the ductile fracture criterion. The final result is a simple relation between process parameters and the constitutive parameter involved in the ductile fracture criterion. The simplicity of this relation makes it suitable for quick design of metal forming processes.
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References
Chung K, Richmond O (1994) The mechanics of ideal forming. ASME J Appl Mech 61:176–181
Hill R (1967) Ideal forming operations for perfectly plastic solids. J Mech Phys Solids 15:223–227
Richmond O, Alexandrov S (2002) The theory of general and ideal plastic deformations of Tresca solids. Acta Mech 158:33–42
Weinberger HF (1997) On the nonexistence of certain ideal forming operations. J Mech Phys Solids 45:1275–1280
Chung K, Alexandrov S (2007) Ideal flow in plasticity. Appl Mech Rev 60:316–335
Richmond O, Devenpeck ML (1962) A die profile for maximum efficiency in strip drawing. In: Rosenberg RM (ed) Proceedings of 4th US National Congress Applied Mechanics, vol 2, ASME, New York, pp 1053–1057
Richmond O, Morrison HL (1967) Streamlined wire drawing dies of minimum length. J Mech Phys Solids 15:195–203
Richmond O (1968) Theory of streamlined dies for drawing and extrusion. In: Rimrott FPJ, Schwaighofer J (eds) Mechanics of the solid state. University of Toronto Press, Toronto, pp 154–167
Weinberger HF (2003) Necessary conditions for the optimality of an extrusion die for a rigid-plastic material. Meccanica 38:547–554
Cawthorn CJ, Loukaides EG, Allwood JM (2014) Comparison of analytical models for sheet rolling. Proc Eng 81:2451–2456
Orowan E (1943) The calculation of roll pressure in hot and cold flat rolling. Proc Inst Mech Eng 150:140–167
Zhang G-L, Wang Z-W, Zhang S-H, Cheng M, Song H-W (2013) A fast optimization approach for multipass wire drawing processes based on the analytical model. Proc IMechE Part B J Eng Manufact 227:1023–1031
Hill R (1985) On the kinematics of steady plane flows in elastoplastic media. In: Reid SR (ed) Metal forming and impact mechanics. Pergamon Press, Oxford, pp 3–17
Osakada K, Nakano J, Mori K (1982) Finite element method for rigid-plastic analysis of metal forming-formulation for finite deformation. Int J Mech Sci 24:459–468
Huh H, Lee CH, Yang WH (1999) A general algorithm for plastic flow simulation by finite element limit analysis. Int J Solids Struct 36:1193–1207
Rees DWA (2006) Basic engineering plasticity. Elsevier, Amsterdam
Chung K, Richmond O (1992) Ideal forming—I. Homogeneous deformation with minimum plastic work. Int J Mech Sci 34:575–591
Chung K, Richmond O (1992) Ideal forming—II. Sheet forming with optimum deformation. Int J Mech Sci 34:617–633
Collins IF, Meguid SA (1977) On the influence of hardening and anisotropy on the plane-strain compression of thin metal strip. ASME J Appl Mech 44:271–278
Hosford WF, Caddell RM (1993) Metal forming: mechanics and metallurgy. Prentice Hall, Englewood Cliffs
Atkins AG (1996) Fracture in forming. J Mater Process Technol 56:609–618
Shabara MA, El-Domiaty AA, Kandil MA (1996) Validity assessment of ductile fracture criteria in cold forming. J Mater Eng Perform 5:478–488
Hambli R, Reszka M (2002) Fracture criteria identification using an inverse technique method and blanking experiment. Int J Mech Sci 44:1349–1361
Cockroft MD, Latham DJ (1968) Ductility and the workability of metals. J Inst Metals 96:33–39
Oh SI, Chen CC, Kobayashi S (1979) Ductile fracture in axisymmetric extrusion and drawing. Part 2: workability in extrusion and drawing. ASME J Eng Ind 101:36–44
Clift SE, Hartley P, Sturgess CEN, Rowe GW (1990) Fracture prediction in plastic deformation processes. Int J Mech Sci 32:1–17
Ko D-C, Kim B-M, Choi J-C (1996) Prediction of surface-fracture initiation in the axisymmetric extrusion and simple upsetting of an aluminum alloy. J Mater Process Technol 62:166–174
Jain M, Allin J, Lloyd DJ (1999) Fracture limit prediction using ductile fracture criteria for forming of an automotive aluminum sheet. Int J Mech Sci 41:1273–1288
Domanti ATJ, Horrobin DJ, Bridgwater J (2002) An investigation of fracture criteria for predicting surface fracture in paste extrusion. Int J Mech Sci 44:1381–1410
Landre J, Pertence A, Cetlin PR, Rodrigues JMC, Martins PAF (2003) On the utilisation of ductile fracture criteria in cold forging. Finite Elem Anal Des 39:175–186
Komori K (2003) Effect of ductile fracture criteria on chevron crack formation and evolution in drawing. Int J Mech Sci 45:141–160
Duan X, Velay X, Sheppard T (2004) Application of finite element method in the hot extrusion of aluminium alloys. Mater Sci Eng 369A:66–75
Figueiredo RB, Cetlin PR, Langdon TG (2009) The evolution of damage in perfect-plastic and strain hardening materials processed by equal-channel angular pressing. Mater Sci Eng 518A:124–131
Richmond O, Alexandrov S (2000) Extension of Bernoulli’s theorem on steady flows of inviscid fluids to steady flows of plastic solids. Comp Ren l’Acad Sci Ser IIb 328:835–840
Rice JR (1973) Plane strain slip line theory for anisotropic rigid/plastic materials. J Mech Phys Solids 21:63–74
Hill R (1950) The mathematical theory of plasticity. Clarendon Press, Oxford
Romano G, Barretta R, Diaco M (2014) The geometry of nonlinear elasticity. Acta Mech 225:3199–3235
Romano G, Barretta R, Diaco M (2014) Geometric continuum mechanics. Meccanica 49:111–133
Malvern LE (1969) Introduction to the mechanics of a continuous medium. Prentice-Hall, Englewood Cliffs
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The research described in this paper has been supported by the Grants RFBR-15-58-53075 and NSH-1275.2014.1.
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Alexandrov, S., Mustafa, Y. & Lyamina, E. Steady planar ideal flow of anisotropic materials. Meccanica 51, 2235–2241 (2016). https://doi.org/10.1007/s11012-016-0362-x
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DOI: https://doi.org/10.1007/s11012-016-0362-x