Chapter 3 Plasticity

Common tests used to determine the monotonic strength of materials. (a) Uniaxial tensile test. (b) Upsetting test. (c) Three-point bend test. (d) Plane-strain tensile test. (e) Plane-strain compression (Ford) test. (f) Torsion test. (g) Biaxial test.

Tests for Mechanical Strength of Materials

A servohydraulic universal testing machine linked to a computer. (Courtesy of MTS Systems Corp.)

Mechanical Testing: Servohydraulic Machine

Stress–strain curves for AISI 1040 steel subjected to different heat treatments; curves obtained from tensile tests.

Stress-Strain Curves of a Steel after Different Heat Treatments

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).

Idealized Uniaxial Stress-Strain Curves

Ludwik-Hollomon equation

Plasticity

Voce equation

Johnson-Cook equation

Schematic representation of the change in Poisson’s ratio as the deformation regime changes from elastic to plastic.

True Stress - True Strain Curve and Poisson’s atio

True- and engineering-stress– vs. true -and engineering -strain curves for AISI 4140 hot-rolled steel. R. A. is reduction in area.

Stress-Strain Curves

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

Engineering Stress - Engineering Strain Curves

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.)

Work hardening vs. Strain

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.)

Correction Factor for Necking

Deformation due to Wire Drawing

Stress–strain curves for Fe–0.003% C alloy wire, deformed to increasingstrains by drawing; each curve is started at the strain corresponding to the priorwire-drawing reduction. (Courtesy of H. J. Rack.)

(a) Effect of strain rateon the stress–strain curves for AISI 1040 steel. (b) Strain-rate changes during tensile test. Four strain rates are shown.

Strain Rate Effects

(a) Compression specimen between parallel platens.

(b) Length inhomogeneity inspecimen.

Plastic Deformation in Compressive Testing

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

Stress-Strain Curve for Compression

(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) b) Variation in pressure on surface of cylindrical specimen being compressed.

Finite Element Method

Bauschinger 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. Vöhringer, and E. Macherauch, Proc. ICMA6, Vol. 1 (New York: Pergamon, 1982), p. 255.)

Bauschinger Effect

Schematic of the different types of stress–strain curves in a polymer.

Effect of strain rate and temperature on stress–strain curves.

Plastic Deformation of Polymers

Schematic of necking and drawing in a semicrystalline polymer.

Necking and Drawing in Polymers

(a) Neck propagation in a sheet of linear polyethylene.

(b) Schematic of neck formation and propagation in a specimen,.

Neck Propagation in Polyethylene

Metallic Glasses

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

Stress-Strain Curve of a Metallic Glass

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.)

Shear Steps in a Metallic Glass

(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.)

Dislocations

Viscosity of soda–lime–silica glass and ofmetallic glasses (Au–Si–Ge, Pd–Cu–Si, Pd–Si, C0P) as a function of normalized temperature. (Adapted from J. F. Shackelford, Introduction to Materials Science for Engineers, 4th ed. (Englewood Cliffs, NJ: Prentice Hall, 1991), p. 331, and F. Spaepenand D. Turnbull in Metallic Glasses, ASM.)

Viscosity of Glasses

Viscosity of three glasses as a function of temperature. 1 P=0.1 Pa · s.

Viscosity of Glasses

Rankine, Tresca, and von Mises Criteria

Maximum-Stress Criterion

Maximum-Shear-Stress Criterion

Maximum-Distortion-Energy Criterion

(a) Rankine, von Mises, and Tresca criteria.

(b) Comparison of failure criteria with experimental results. (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.)

Comparison of Rankine, von Mises, and Tresca Criteria

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

Displacement of the Yield Locus due to Plastic Deformation

Tensile and Compressive Curves for Al2O3

(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.

Failure Criteria for Brittle Materials

Mohr-Coulomb failure criterion

Griffith Failure Criterion

McClintock-Walsh Crtierion

Failure Criteria for Brittle Material

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

von Mises Criterion for a Polymer

a b

Shear yielding and crazing for an amorphous polymer under biaxial stress. The thicker line(delineates the failure envelope when crazing occurs in tension.(After S. S. Sternstein and L. Ongchin, Am. Chem. Soc., Div. Of Polymer Chem., Polymer Preprints, 10 (1969), 1117.)

Shear Yielding and Crazing for Amorphous Polymer

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

Failure Envelope for a Fiber Reinforced Composite

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

Plane-Stress Yield Loci for Sheets with Planar Isotropy

Comparison of the impression sizes produced by various hardness tests on a 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.)

Impressions Produced in Hardness Tests

Impression caused by spherical indenter on metal plate in a Brinell hardness test.

Brinell Impression

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, (NewYork: McGraw-Hill, 1941), p. 149.)

Rockwell Hardness Tester

Scales for Rockwell Hardness Tester

Vickers Hardness Test

Relationships Between Yield Stress and Hardness

(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 .)

Hardness Profile near a Grain Boundary

Details of the Knoop indenter, together with its impression.

Knoop Indenter

Nanoindenter apparatus

An impression made by means of Berkovich indenter in a copper sample. (From X. Deng, M. Koopman, N. Chawla, and K.K. Chawla, Acta Mater., 52 (2004) 4291.) (a) An atomic force micrograph, showing 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.

Topographic Features of the Berkovich Indentation

Load vs. Indenter Displacement

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.

Simple Formability Tests for Sheets

“Ears” formed in a deep-drawn cup due to in-plane anisotropy. (Courtesy

of Alcoa, Inc.)

Earing in Deep Drawing

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.)

Fibering

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

Punch-Stretch Test

Schematic of sheet deformed by punch stretching. (a) Representation of strain distribution: ε1, meridional strain; ε2, circumferential strain; h, cup height.

b) Geometry of deformed sheet.

Punch-Stretch Test

Construction of a forming-limit curve (or Keeler–Goodwin diagram).

(Courtesy of S. S. Hecker.)

Forming-Limit Curve

Different strain patterns in stamped part. (Adapted from W. Brazier, Closed Loop, 15, No. 1 (1986) 3.)

Different Strain Patterns in Stamped Part

Stress vs. Strain Rate for Slow-Twitch and Fast Twitch Muscles

Stress–strain response for some biological materials.

Stress-Strain Cures of Some Biological Materials

Mechanical Properties of Biological Materials

Stress–strain response for elastin; it is the ligamentum nuchae of cattle (Adapted from Y.C. Fung and S. S. Sobin, J. Biomech. Eng., 1103 (1981) 121. Also in Y. C. Fung, Biomechanics: Mechanica l Properties of Living Tissues(NewYork: Springer, 1993) p. 244.)

Stress-Strain Response of Elastin

Tensile and compressive stress–strain curves for cortical bone in longitudinal and transverse directions. (Adapted from G. L. Lucas, F. W. Cooke, and E. A. Friis, A Primer on Biomechanics (New York: Springer, 1999).)

Stress-Strain Response of Cortical Bone

Strain-rate dependence of tensile response of cortical bone. (Adapted from J. H. McElhaney, J. Appl. Physiology, 21(1966) 1231.)

Effect of Strain Rate on Tensile Stress-Strain Curve of Cortical Bone