Chapter 3 Plasticity

1
Chapter 3 Plasticity
2
Tests for Mechanical Strength of Materials
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.
3
Mechanical Testing Servohydraulic Machine
A servohydraulic universal testing machine linked
to a computer. (Courtesy of MTS Systems Corp.)?
4
Stress-Strain Curves of a Steel after Different
Heat Treatments
Stressstrain curves for AISI 1040 steel
subjected to different heat treatments curves
obtained from tensile tests.
5
Idealized Uniaxial Stress-Strain Curves
Idealized shapes of uniaxial stressstrain curve.
(a)? Perfectly plastic. (b) Ideal elastoplastic.
(c) Ideal elastoplastic with linear
work-hardening. (d)? Parabolic work-hardening (s
so Ken).
6
Plasticity
Ludwik-Hollomon equation
Voce equation
Johnson-Cook equation
7
True Stress - True Strain Curve and Poissons atio
Schematic representation of the change in
Poissons ratio as the deformation regime changes
from elastic to plastic.
8
Stress-Strain Curves
True- and engineering-stress vs. true -and
engineering -strain curves for AISI 4140
hot-rolled steel. R. A. is reduction in area.
9
Engineering Stress - Engineering Strain Curves
Engineering- (or nominal-) stressstrain curves
(a) without the yield point and (b) with a yield
point.
10
Work hardening vs. Strain
Log ds/de versus log e for stainless steel AISI
302. (Adapted with permission from A. S. de S. e
Silva and S. N. Monteiro, Metalurgia-ABM, 33
(1977) 417.)?
11
Correction Factor for Necking
Correction factor for necking as a function of
strain in neck, ln (A0/A), minus strain at
necking, eu. (Adapted with permission from W. J.
McGregor Tegart, Elements of Mechanical
Metallurgy (New York MacMillan,1964), p. 22.)?
12
Deformation due to Wire Drawing
Stressstrain curves for Fe0.003 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 stressstrain
curves for AISI 1040 steel. (b) Strain-rate
changes during tensile test. Four strain rates
are shown.
14
Plastic Deformation in Compressive Testing
(a) Compression specimen between parallel
platens. (b) Length inhomogeneity in specimen.
15
Stress-Strain Curve for Compression
(a) Stressstrain (engineering and true) curves
for 7030 brass in compression. (b)? Change of
shape of specimen and barreling.
16
Finite Element Method

1. 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.) (
2. b) Variation in pressure on surface of
   cylindrical specimen being compressed.

17
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.
18
Plastic Deformation of Polymers
Effect of strain rate and temperature on
stressstrain curves.
Schematic of the different types of stressstrain
curves in a polymer.
19
Necking and Drawing in 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) Schematic of neck formation
and propagation in a specimen,.
21
Metallic Glasses
22
Stress-Strain Curve of a Metallic Glass
Compressive stressstrain 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 in 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.)?
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 Glasses
Viscosity of sodalimesilica glass and
of metallic glasses (AuSiGe, PdCuSi, PdSi,
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. Spaepen and D. Turnbull in Metallic
Glasses, ASM.)
26
Viscosity of Glasses
Viscosity of three glasses as a function of
temperature. 1 P0.1 Pa s.
27
Rankine, Tresca, and von Mises Criteria
Maximum-Stress Criterion
Maximum-Shear-Stress Criterion
Maximum-Distortion-Energy Criterion
28
Comparison of Rankine, von Mises, and Tresca
Criteria
(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.)?
29
Displacement of the Yield Locus due to Plastic
Deformation
Displacement of the yield locus as the flow
stress of the material due to plastic
deformation. (a) Isotropic hardening. (b)
Kinematic hardening.
30
Tensile and Compressive Curves for Al2O3
31
Failure Criteria for Brittle Materials
(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
Failure Criteria for Brittle Material
Mohr-Coulomb failure criterion
Griffith Failure Criterion
McClintock-Walsh Crtierion
33
von Mises Criterion for a Polymer
a
b
Translation of von Mises ellipse for a polymer
due to the presence of hydrostatic stress. (a) No
hydrostatic stress, (b) with hydrostatic stress.
34
Shear Yielding and Crazing for Amorphous Polymer
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.)?
35
Failure Envelope for a Fiber Reinforced Composite
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.)?
36
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 s2/s1 indicate the degree of
anisotropy.)?
37
Impressions Produced in Hardness Tests
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.)?
38
Brinell Impression
Impression caused by spherical indenter on metal
plate in a Brinell hardness test.
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, (NewYork
McGraw-Hill, 1941), p. 149.)?
40
Scales for Rockwell Hardness Tester
41
Vickers Hardness Test
Relationships Between Yield Stress and Hardness
42
Hardness Profile near a Grain Boundary
(a) Hardnessdistance 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.)?
43
Knoop Indenter
Details of the Knoop indenter, together with its
impression.
44
Nanoindenter apparatus
45
Topographic Features of the Berkovich Indentation
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.
46
Load vs. Indenter Displacement
47
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.
48
Earing in Deep Drawing
Ears formed in a deep-drawn cup due to in-plane
anisotropy. (Courtesy of Alcoa, Inc.)?
49
Fibering
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.)?
50
Punch-Stretch Test
Sheet specimen subjected to punchstretch test
until necking necking can be seen by the clear
line. (Courtesy of S. S. Hecker.)?
51
Punch-Stretch Test
Schematic of sheet deformed by punch stretching.
(a)? Representation of strain distribution e1,
meridional strain e2, circumferential strain h,
cup height. b) Geometry of deformed sheet.
52
Forming-Limit Curve
Construction of a forming-limit curve (or
KeelerGoodwin diagram). (Courtesy of S. S.
Hecker.)?
53
Different Strain Patterns in Stamped Part
Different strain patterns in stamped part.
(Adapted from W. Brazier, Closed Loop, 15, No. 1
(1986) 3.)?
54
Stress vs. Strain Rate for Slow-Twitch and Fast
Twitch Muscles
55
Stress-Strain Cures of Some Biological Materials
Stressstrain response for some biological
materials.
56
Mechanical Properties of Biological Materials
57
Stress-Strain Response of Elastin
Stressstrain 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.)?
58
Stress-Strain Response of Cortical Bone
Tensile and compressive stressstrain 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).)?
59
Effect of Strain Rate on Tensile Stress-Strain
Curve of Cortical Bone
Strain-rate dependence of tensile response of
cortical bone. (Adapted from J. H. McElhaney, J.
Appl. Physiology, 21(1966) 1231.)?