Combined Hill-Taylor Theory: Theoretical, Experimental and Finite Element Study

Document Type : Research Paper


Department of Materials Science and Engineering, School of Engineering, Shiraz University, Shiraz, Iran


In this study, by combining crystal plasticity notions developed by Taylor and the mathematical expression of Hill’s yield criterion for anisotropic materials, a model is introduced to describe the flow behavior of grains in a grain aggregate. In this model, Hill’s yield criterion coefficients are calculated in terms of Taylor factors for different straining conditions for each grain. The convexity of the proposed model is proved by sign determination of the eigenvalues of the associated Hessian matrix. It is found that the experimental load-displacement curves of specimens showing the size effect are enveloped by the bounds obtained from simulations using the proposed model, which to some extent verifies the applicability of the developed model. Using the developed model, the microforging of miniature rods consisting of 50 and 200 grains in their cross-section are simulated. In agreement with the literature, the results showed that due to the difference in the mechanical behavior of grains, the distribution of strain abruptly changes from one grain to another. Moreover, it is shown that as the number of grains in the cross-section of the specimen increases, the plastic equivalent strain tends toward that predicted by the classical plasticity theories, proving the applicability of the proposed model. Finally, the results suggest that the successful production of microparts by forming processes requires raw materials in microforming to be the products of the severe plastic deformation techniques, where the microstructure is scaled down to the nanometer.


  1. Schmid E, Boas W. Plasticity of crystals : with special reference to metals: Chapman & Hall; 1968.
  2. Sachs G. Plasticity problems in metals. Transactions of the Faraday Society. 1928;24:84.
  3. Cox HL, Sopwith DG. The effect of orientation on stresses in single crystals and of random orientation on strength of polycrystalline aggregates. Proceedings of the Physical Society. 1937;49(2):134-51.
  4. Taylor GI. Plastic Strain in Metals. The Journal of the Institute of Metals. 1938; 62, 307–324
  5. Taylor GI. Analysis of plastic strain in a cubic crystal. Stephen Timoshenko 60th Anniversary Volume. 1938:218-24.
  6. Taylor GI, Elam CF. Bakerian Lecture: The distortion of an aluminium crystal during a tensile test. Proceedings of the Royal Society of London Series A, Containing Papers of a Mathematical and Physical Character. 1923;102(719):643-67.
  7. Bishop JFW, Hill R. XLVI. A theory of the plastic distortion of a polycrystalline aggregate under combined stresses. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 1951;42(327):414-27.
  8. Ashby MF. The deformation of plastically non-homogeneous materials. The Philosophical Magazine: A Journal of Theoretical Experimental and Applied Physics. 1970;21(170):399-424.
  9. Fleck NA, Muller GM, Ashby MF, Hutchinson JW. Strain gradient plasticity: Theory and experiment. Acta Metallurgica et Materialia. 1994;42(2):475-87.
  10. Nye JF. Some geometrical relations in dislocated crystals. Acta Metallurgica. 1953;1(2):153-62.
  11. Ran JQ, Fu MW, Chan WL. The influence of size effect on the ductile fracture in micro-scaled plastic deformation. International Journal of Plasticity. 2013;41:65-81.
  12. Li WT, Fu MW, Shi SQ. Study of deformation and ductile fracture behaviors in micro-scale deformation using a combined surface layer and grain boundary strengthening model. International Journal of Mechanical Sciences. 2017;131-132:924-37.
  13. Li J, Romero I, Segurado J. Development of a thermo-mechanically coupled crystal plasticity modeling framework: Application to polycrystalline homogenization. International Journal of Plasticity. 2019;119:313-30.
  14. Li WT, Li H, Fu MW. Interactive effect of stress state and grain size on fracture behaviours of copper in micro-scaled plastic deformation. International Journal of Plasticity. 2018;114:126-43.
  15. Henning M, Vehoff H. Statistical size effects based on grain size and texture in thin sheets. Materials Science and Engineering: A. 2007;452-453:602-13.
  16. Chen G, Zhang L, Bezold A, Broeckmann C, Weichert D. Statistical investigation on influence of grain size on effective strengths of particulate reinforced metal matrix composites. Computer Methods in Applied Mechanics and Engineering. 2019;352:691-707.
  17. Chan WL, Fu MW, Lu J, Liu JG. Modeling of grain size effect on micro deformation behavior in micro-forming of pure copper. Materials Science and Engineering: A. 2010;527(24-25):6638-48.
  18. Cortellino F, Rouse JP, Cacciapuoti B, Sun W, Hyde TH. Experimental and Numerical Analysis of Initial Plasticity in P91 Steel Small Punch Creep Samples. Exp Mech. 2017;57(8):1193-212.
  19. Hollang L, Hieckmann E, Brunner D, Holste C, Skrotzki W. Scaling effects in the plasticity of nickel. Materials Science and Engineering: A. 2006;424(1-2):138-53.
  20. Hua F, Liu D, Li Y, He Y, Dunstan DJ. On energetic and dissipative gradient effects within higher-order strain gradient plasticity: Size effect, passivation effect, and Bauschinger effect. International Journal of Plasticity. 2021;141:102994.
  21. Jiang J, Britton TB, Wilkinson AJ. The orientation and strain dependence of dislocation structure evolution in monotonically deformed polycrystalline copper. International Journal of Plasticity. 2015;69:102-17.
  22. Kraft O, Gruber PA, Mönig R, Weygand D. Plasticity in Confined Dimensions. Annual Review of Materials Research. 2010;40(1):293-317.
  23. Lu X, Zhang X, Shi M, Roters F, Kang G, Raabe D. Dislocation mechanism based size-dependent crystal plasticity modeling and simulation of gradient nano-grained copper. International Journal of Plasticity. 2019;113:52-73.
  24. Ravaji B, Joshi SP. A crystal plasticity investigation of grain size-texture interaction in magnesium alloys. Acta Materialia. 2021;208:116743.
  25. Zhang B, Dodaran MS, Shao S, Choi J, Park S, Meng WJ. Understanding of plasticity size-effect governed mechanical response and incomplete die filling in a microscale double-punch molding configuration. International journal of mechanical sciences. 2020;172:105406.
  26. Yang LH, Zou GP, Qu J. Yield Criterion and Crack Tip Plastic Zone of Nickel-Based Single Crystal. Key Engineering Materials. 2012;525-526:341-4.
  27. Zhang K, Holmedal B, Mánik T, Saai A. Assessment of advanced Taylor models, the Taylor factor and yield-surface exponent for FCC metals. International Journal of Plasticity. 2019;114:144-60.
  28. Kim HL, Park SH. Loading Direction Dependence of Yield-Point Phenomenon and Bauschinger Effect in API X70 Steel Sheet. Metals and Materials International. 2019;26(1):14-24.
  29. Zhang S, Liu Y, Xu T, Sun M, Zhang Q, Wan Y. Effect of Texture and Microstructure on Tensile Behaviors in the Polycrystalline Pure Niobium. Metals and Materials International. 2021;27(10):4023-34.
  30. Bagherpour E, Qods F, Ebrahimi R, Miyamoto H. Microstructure and Texture Inhomogeneity after Large Non-Monotonic Simple Shear Strains: Achievements of Tensile Properties. Metals. 2018;8(8):583.
  31. A theory of the yielding and plastic flow of anisotropic metals. Proceedings of the Royal Society of London Series A Mathematical and Physical Sciences. 1948;193(1033):281-97.
  32. Johnson W, Mellor PB. Engineering plasticity (pp. 76–78). 1983. England: Ellis Horwood Ltd.
  33. Ebrahimi R, Rezvani A, Bagherpour E. Circular simple shear extrusion as an alternative for simple shear extrusion technique for producing bulk nanostructured materials. Procedia Manufacturing. 2018;15:1502-8.
  34. Rezvani A, Ebrahimi R. Investigation on the Deformation Behavior and Strain Distribution of Commercially Pure Aluminum after Circular Simple Shear Extrusion. Journal of Ultrafine Grained and Nanostructured Materials. 2019 Jun 1;52(1):32-42.
  35. Rezvani A, Bagherpour E, Ebrahimi R. Circular Simple Shear Extrusion as an Alternative to Simple Shear Extrusion Technique. Iranian Journal of Science and Technology, Transactions of Mechanical Engineering. 2018;44(1):193-201.
  36. Bagherpour E, Qods F, Ebrahimi R, Miyamoto H. Microstructure quantification of ultrafine grained pure copper fabricated by simple shear extrusion (SSE) technique. Materials Science and Engineering: A. 2016;674:221-31.
  37. Hosford WF. Mechanical Behavior of Materials: Cambridge University Press; 2009.
  38. Bagherpour E, Qods F, Ebrahimi R, Miyamoto H. Microstructure evolution of pure copper during a single pass of simple shear extrusion (SSE): role of shear reversal. Materials Science and Engineering: A. 2016;666:324-38.
  39. Bishop JFW, Hill R. CXXVIII. A theoretical derivation of the plastic properties of a polycrystalline face-centred metal. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 1951;42(334):1298-307.
  40. Tong W. Algebraic Convexity Conditions for Gotoh's Nonquadratic Yield Function. Journal of Applied Mechanics. 2018;85(7).
  41. Tong W. On the Certification of Positive and Convex Gotoh’s Fourth-Order Yield Function. Journal of Physics: Conference Series. 2018;1063:012093.
  42. Uppaluri R, Helm D. A convex fourth order yield function for orthotropic metal plasticity. European Journal of Mechanics - A/Solids. 2021;87:104196.
  43. Gotoh M. A theory of plastic anisotropy based on a yield function of fourth order (plane stress state)—I. International Journal of Mechanical Sciences. 1977;19(9):505-12.
  44. Pankaj, Arif M, Kaushik SK. Convexity studies of two anisotropic yield criteria in principal stress space. Engineering Computations. 1999;16(2):215-29.
  45. Pardis N, Chen C, Ebrahimi R, Toth LS, Gu CF, Beausir B, et al. Microstructure, texture and mechanical properties of cyclic expansion–extrusion deformed pure copper. Materials Science and Engineering: A. 2015;628:423-32.
  46. Raabe D, Sachtleber M, Zhao Z, Roters F, Zaefferer S. Micromechanical and macromechanical effects in grain scale polycrystal plasticity experimentation and simulation. Acta Materialia. 2001;49(17):3433-41.
  47. Knezevic M, Drach B, Ardeljan M, Beyerlein IJ. Three dimensional predictions of grain scale plasticity and grain boundaries using crystal plasticity finite element models. Computer Methods in Applied Mechanics and Engineering. 2014;277:239-59.
  48. Vidyasagar A, Tutcuoglu AD, Kochmann DM. Deformation patterning in finite-strain crystal plasticity by spectral homogenization with application to magnesium. Computer Methods in Applied Mechanics and Engineering. 2018;335:584-609.
  49. Khajezade A, HABIBI PM, Mirzadeh H, MONTAZERI PM. Grain refinement efficiency of multi-axial incremental forging and shearing: A Crystal Plasticity Analysis.
  50. Torabi M, Eivani AR, Jafarian H, Salehi MT. Re-strengthening in AA6063 alloy during equal channel angular pressing. Journal of Ultrafine Grained and Nanostructured Materials. 2017 Dec 1;50(2):90-7.
  51. Van Houtte P, Wagner F. Development of Textures by Slip and Twinning. Preferred Orientation in Deformed Metal and Rocks: Elsevier; 1985. p. 233-58.
  52. W. F. Hosford. The mechanics of crystals and textured polycrystals. Oxford University Press, New York–Oxford 1993. ISBN 9780195077445.
  53. Zadeh L. Optimality and non-scalar-valued performance criteria. IEEE Transactions on Automatic Control. 1963;8(1):59-60.
  54. Goicoechea A, Duckstein L, Fogel MM. Multiobjective programing in watershed management: A study of the Charleston Watershed. Water Resources Research. 1976;12(6):1085-92.


Volume 54, Issue 2
December 2021
Pages 228-243
  • Receive Date: 13 July 2021
  • Revise Date: 11 September 2021
  • Accept Date: 11 September 2021
  • First Publish Date: 01 December 2021