Chapter 3: Plasticity

# Chapter 3: Plasticity

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## Chapter 3: Plasticity

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1. Chapter 3: Plasticity

2. Tests for Mechanical Strength of Materials Common tests used to determine the monotonic strength of metals. (a) Uniaxial tensile test. (b) Upsetting test. (c) Three-point bending test. (d) Plane-strain tensile test. (e) Plane-strain compression (Ford) test. (f)‏ Torsion test. (g) Biaxial test.

3. Mechanical Testing: Servohydraulic Machine A servohydraulic universal testing machine linked to a computer. (Courtesy of MTS Systems Corp.)‏

4. Stress-Strain Curves at Different Heat Treatments Stress–strain curves for AISI 1040 steel subjected to different heat treatments; curves obtained from tensile tests.

5. Uniaxial Stress-Strain Curve Idealized shapes of uniaxial stress–strain curve. (a)‏ Perfectly plastic. (b) Ideal elastoplastic. (c) Ideal elastoplastic with linear work-hardening. (d)‏ Parabolic work-hardening (σ = σo + Kεn).

6. Plasticity Ludwik-Hollomon equation Voce equation Johnson-Cook equation

7. True Stress and True Strain Curve Schematic representation of the change in Poisson’s ratio as the deformation regime changes from elastic to plastic.

8. Stress-Strain Curve True- and engineering-stress–strain curves for AISI 4140 hot-rolled steel. R. A. is reduction in area.

9. Engineering Stress and Engineering Strain Engineering- (or nominal-) stress–strain curves (a) without and (b) with a yield point.

10. Tensile tests Tensile specimen being tested; arrows show onset of necking.

11. Work hardening vs. Strain Log dσ/dε versus log ε for stainless steel AISI 302. (Adapted with permission from A. S. de S. e Silva and S. N. Monteiro, Metalurgia-ABM, 33 (1977) 417.)‏

12. Necking Correction factor for necking as a function of strain in neck, ln(A0/A), minus strain at necking, εu. (Adapted with permission from W. J. McGregor Tegart, Elements of Mechanical Metallurgy (New York: MacMillan, 1964), p. 22.)‏ Stress–strain curves for Fe–0.003% C alloy wire, deformed to increasing strains by drawing; each curve is started at the strain corresponding to the prior wire-drawing reduction. (Courtesy of H. J. Rack)‏

13. Strain Rate Effects (a) Effect of strain rate on the stress–strain curves for AISI 1040 steel. (b) Strain-rate changes during tensile test. Four strain rates are shown: 10−1, 10−2, 10−3, and 10−4 s−1.

14. Plastic Deformation in Compressive Testing (a) Compression specimen between parallel platens. (b) Length inhomogeneity in specimen.

15. Stress-Strain Curve for Compression (a) Stress–strain (engineering and true) curves for 70–30 brass in compression. (b)‏ Change of shape of specimen and barreling.

16. Finite Element Method (a) Distortion of Finite Element Method (FEM) grid after 50% reduction in height h of specimen under sticking-friction conditions. (Reprinted with permission from H. Kudo and S. Matsubara, Metal Forming Plasticity (Berlin: Springer, 1979),p. 395.) (b) Variation in pressure on surface of cylindrical specimen being compressed.

17. Bauschunger Effect Ratio of compressive flow stress (0.2% plastic strain) and tensile flow stress at different levels of plastic strain for different steels. (After B. Scholtes, O. V¨ohringer, and E. Macherauch, Proc. ICMA6, Vol. 1 (New York: Pergamon, 1982), p. 255.)‏ The Bauschinger effect.

18. Plastic Deformation of Polymers Schematic of the different types of stress–strain curves in a polymer. Effect of strain rate and temperature on stress–strain curves.

19. Glassy Polymers Schematic of necking and drawing in a semicrystalline polymer.

20. Neck Propagation in Polyethylene (a) Neck propagation in a sheet of linear polyethylene. (b) Neck formation and propagation in a specimen, shown in schematic fashion.

21. Metallic Glasses

22. Plastic Deformation of Glasses Compression stress–strain curves for Pd77.5CU6Si16.5. (Adapted with permission from C. A. Pampillo and H. S. Chen, Mater. Sci. Eng., 13 (1974) 181.)‏

23. Shear Steps Shear steps terminating inside material after annealing at 250◦C/h, produced by (a) bending and decreased by (b)‏ unbending. Metglas Ni82.4Cr7Fe3Si4.5B3.1 strip. (Courtesy of X. Cao and J. C. M. Li.)‏

24. Dislocations (a) Gilman model of dislocations in crystalline and glassy silica, represented by two-dimensional arrays of polyhedra. (Adapted from J. J. Gilman, J. Appl. Phys. 44 (1973)‏ 675) (b) Argon model of displacement fields of atoms (indicated by magnitude and direction of lines) when assemblage of atoms is subjected to shear strain of 5 × 10−2, in molecular dynamics computation. (Adapted from D. Deng, A. S. Argon, and S. Yip, Phil. Trans. Roy. Soc. Lond. A329 (1989) 613.)‏

25. Viscosity of Glass Viscosity of soda–lime–silica glass and of metallic glasses (Au–Si–Ge, Pd–Cu–Si, Pd–Si, C0P) as a function of normalized temperature. (Adapted from J. F. Shakelford, Introduction to Materials Science for Engineers, 4th ed. (Englewood Cliffs, NJ: Prentice Hall, 1991), p. 331, and F. Spaepen and D. Turnbull in Metallic Glasses, ASM.) 1P=0.1 Pa · s. Viscosity of three glasses as a function of temperature. 1 P=0.1 Pa · s.

26. Rankine, Tresca, and von Mises Maximum-stress Criterion Maximum-Shear-Stress Criterion Maximum-Distortion-Energy Criterion

27. Comparison of the Rankine, von Mises, and Tresca (a) Comparison of the Rankine, von Mises, and Tresca criteria. (b) Comparison of failure criteria with test. (Reprinted with permission from E. P. Popov, Mechanics of Materials, 2nd ed. (Englewood Cliffs, NJ: Prentice-Hall, 1976), and G. Murphy, Advanced. Mechanics of Materials (New York: McGraw-Hill, 1964), p. 83.)‏

28. Displacement of the Yield Locus Displacement of the yield locus as the flow stress of the material due to plastic deformation. (a) Isotropic hardening. (b) Kinematic hardening.

29. Tensile and Compressive Strength ofAl2O3

30. Mohr-Coulomb failure criterion Griffith Failure Criterion McClintock-Walsh Crtierion

31. Failure Criteria for Brittle Material (a) Simple model for solid with cracks. (b) Elliptical flaw in elastic solid subjected to compression loading. (c) Biaxial fracture criterion for brittle materials initiated from flaws without (Griffith)‏ and with (McClintock and Walsh) crack friction.

32. von Mises Ellipse Translation of von Mises ellipse for a polymer due to the presence of hydrostatic stress. (a) No hydrostatic stress, (b) with hydrostatic stress.

33. Shear Yielding and Crazing for Amorphous Polymer Envelopes defining shear yielding and crazing for an amorphous polymer under biaxial stress. (After S. S. Sternstein and L. Ongchin, Am. Chem. Soc., Div. of Polymer Chem., Polymer Preprints, 10 (1969), 1117.)‏

34. Failure Envelope Failure envelope for unidirectional E-glass/epoxy composite under biaxial loading at different levels of shear stress. (After I. M. Daniel and O. Ishai, Engineering Mechancis of Composite Materials (New York: Oxford University Press, 1994), p. 121.)‏

35. Plane-Stress Yield Loci for Sheets with Planar Isotropy Plane-stress yield loci for sheets with planar isotropy or textures that are rotationally symmetric about the thickness direction, x3. (Values of R indicate the degree of anisotropy = σ2/σ1.)‏

36. Hardness for Steel

37. Hardness Tests Comparison of the impression sizes produced by various hardness tests on material of 750 HV. BHN = Brinell hardness number, HRC = Rockwell hardness number on C scale, HRN = Rockwell hardness number on N scale, VPN = Vickers hardness number. (Adapted with permission from E. R. Petty, in Techniques of Metals Research, Vol. 5, Pt. 2, R. F. Bunshah, ed. (New York: Wiley-Interscience, 1971), p. 174.)‏

38. Impression Impression caused by spherical indenter on metal plate.

39. Rockwell Hardness Tester Procedure in using Rockwell hardness tester. (Reprinted with permission from H. E. Davis, G. E. Troxel, and C. T. Wiscocil, The Testing and Inspection of Engineering Materials, (New York: McGraw-Hill, 1941), p. 149.)‏

40. Vickers Hardness Test Relationships Between Yield Stress and Hardness

41. Hardness Distance Profile (a) Hardness–distance profiles near a grain boundary in zinc with 100-atom ppm of Al and zinc with 100-atom ppm of Au (1-gf load). (b) Solute concentration dependence of percent excess boundary hardening in zinc containing Al, Au, or Cu (3-gf load). (Adapted with permission from K. T. Aust, R. E. Hanemann, P. Niessen, and J. H. Westbrook, Acta Met., 16 (1968)‏ 291.)‏

42. Knoop Indenter Some of the details of the Knoop indenter, together with its impression.

43. Nanoindenter apparatus A schematic of a nanoindenter apparatus.

44. Topographic Feature of the Berkovich Indentation An impression made by means of Berkovich indenter in a copper sample. (From Deng, Koopman, Chawla, and Chawla, Acta Mater., 52 (2004) 4291.) (a)‏ An atomic force micrograph, which shows very nicely the topographic features of the indentation on the sample surface. The scale is the same along the three axes. (b) Berkovich indentation as seen in an SEM.

45. Load vs. Indentation Displacement A schematic representation of load vs. indenter displacement.

46. Simple Formability Tests for Sheets Simple formability tests for sheets. (a) Simple bending test. (b) Free-bending test. (c)‏ Olsen cup test. (d) Swift cup test. (e) Fukui conical cup test.

47. Plastic Anisotropy “Ears” formed in deep-drawn cups due to in-plane anisotropy. (Courtesy of Alcoa, Inc.)‏

48. Fibering Effect of “fibering” on formability. The bending operation is often an integral part of sheet-metal forming, particularly in making flanges so that the part can be attached to another part. During bending, the fibers of the sheet on the outer side of the bend are under tension, and the inner-side ones are under compression. Impurities introduced in the metal as it was made become elongated into “stringers” when the metal is rolled into sheet form. During bending, the stringers can cause the sheet to fail by cracking if they are oriented perpendicular to the direction of bending (top). If they are oriented in the direction of the bend (bottom), the ductility of the metal remains normal. (Adapted with permission from S. S. Hecker and A. K. Ghosh, Sci. Am., Nov. (1976), p. 100.)‏

49. Punch-Stretch Test Sheet specimen subjected to punch–stretch test until necking; necking can be seen by the clear line. (Courtesy of S. S. Hecker)‏