Chapter 3: Plasticity. 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
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.
universal testing machine linked to
a computer. (Courtesy of MTS
Stress–strain curves for
AISI 1040 steel subjected to
different heat treatments; curves
obtained from tensile tests.
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).
representation of the change in
Poisson’s ratio as the deformation
regime changes from elastic to
for AISI 4140 hot-rolled steel. R.
A. is reduction in area.
Engineering- (or nominal-) stress–strain curves (a) without and (b) with a yield
Tensile specimen being tested; arrows show onset of necking.
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.)
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)
(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.
specimen between parallel platens.
(b) Length inhomogeneity in
(engineering and true) curves for
70–30 brass in compression. (b)
Change of shape of specimen and
(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
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.
Schematic of the
different types of stress–strain
curves in a polymer.
Effect of strain rate
and temperature on stress–strain curves.
Schematic of necking
and drawing in a semicrystalline polymer.
(a) Neck propagation
in a sheet of linear polyethylene.
(b) Neck formation and
propagation in a specimen, shown in schematic fashion.
stress–strain curves for
Pd77.5CU6Si16.5. (Adapted with
permission from C. A. Pampillo and H. S. Chen, Mater. Sci. Eng., 13 (1974) 181.)
terminating inside material after
annealing at 250◦C/h, produced by (a) bending and decreased by (b)
Ni82.4Cr7Fe3Si4.5B3.1 strip. (Courtesy of X. Cao and J. C. M. Li.)
(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.)
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.
(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.)
Displacement of the
yield locus as the flow stress of the
material due to plastic
deformation. (a) Isotropic
hardening. (b) Kinematic
Griffith Failure Criterion
(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.
Translation of von
Mises ellipse for a polymer due to
the presence of hydrostatic stress.
(a) No hydrostatic stress, (b) with
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
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.)
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 =
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.)
Impression caused by
spherical indenter on metal plate.
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.)
Relationships Between Yield Stress and Hardness
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)
Some of the details of
the Knoop indenter, together with
A schematic of a
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.
representation of load vs. indenter
tests for sheets. (a) Simple bending
test. (b) Free-bending test. (c)
Olsen cup test. (d) Swift cup test.
(e) Fukui conical cup test.
“Ears” formed in
deep-drawn cups due to in-plane
anisotropy. (Courtesy of Alcoa,
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.)
subjected to punch–stretch test
until necking; necking can be seen
by the clear line. (Courtesy of S. S.
Schematic of sheet
deformed by punch stretching. (a)
Representation of strain
distribution: ε1, meridional strains;
ε2, circumferential strains; h, cup
height. (b) Geomety of deformed
Construction of a
forming-limit curve (or
(Courtesy of S. S. Hecker.)
patterns in stamped part. (Adapted
from W. Brazier, Closed Loop, 15,
No. 1 (1986) 3.)
fore a number of biological
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
properties of Living Tissues
(NewYork: Springer, 1993) p. 244.)
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,
dependence of tensile response of
cortical bone. (Adapted from J. H.
McElhaney, J. Appl. Physiology,