Transcript Chapter 3 Plasticity - University of California, San Diego

Chapter 3
Plasticity
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.
Mechanical Testing: Servohydraulic Machine
A servohydraulic universal testing
machine linked to a computer.
(Courtesy of MTS Systems Corp.)
Stress-Strain Curves of a Steel after Different Heat Treatments
Stress–strain curves for AISI
1040 steel subjected to
different heat treatments;
curves obtained from tensile
tests.
Idealized Uniaxial Stress-Strain Curves
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).
Plasticity
Ludwik-Hollomon equation
Voce equation
Johnson-Cook equation
True Stress - True Strain Curve and Poisson’s atio
Schematic representation of the
change in Poisson’s ratio as the
deformation regime changes from
elastic to plastic.
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.
Engineering Stress - Engineering Strain Curves
Engineering- (or nominal-) stress–strain curves (a) without
the yield point and (b) with a yield point.
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.)
Correction Factor for 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.)
Deformation due to Wire Drawing
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.)
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.
Plastic Deformation in Compressive Testing
(a) Compression specimen
between parallel platens.
(b) Length inhomogeneity in
specimen.
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.
Finite Element Method
(a)
(b)
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.
Bauschinger Effect
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.)
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.
Necking and Drawing in Polymers
Schematic of necking and drawing in a semicrystalline
polymer.
Neck Propagation in Polyethylene
(a) Neck propagation in a sheet of
linear polyethylene.
(b) Schematic of neck formation and
propagation in a specimen,.
Metallic Glasses
Stress-Strain Curve of a Metallic Glass
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.)
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.)
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.)
Viscosity of Glasses
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. 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.)
Viscosity of Glasses
Viscosity of three glasses as a function
of temperature. 1 P=0.1 Pa · s.
Rankine, Tresca, and von Mises Criteria
Maximum-Stress Criterion
Maximum-Shear-Stress Criterion
Maximum-Distortion-Energy Criterion
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.)
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.
Tensile and Compressive Curves for Al2O3
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.
Failure Criteria for Brittle Material
Mohr-Coulomb failure criterion
Griffith Failure Criterion
McClintock-Walsh Crtierion
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.
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.)
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.)
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 =
σ2/σ1 indicate the degree of anisotropy.)
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.)
Brinell Impression
Impression caused by spherical indenter on metal plate in a
Brinell hardness test.
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.)
Scales for Rockwell Hardness Tester
Vickers Hardness Test
Relationships Between Yield Stress and Hardness
Hardness Profile near a Grain Boundary
(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.)
Knoop Indenter
Details of the Knoop indenter, together
with its impression.
Nanoindenter apparatus
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.
Load vs. Indenter Displacement
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.
Earing in Deep Drawing
“Ears” formed in a deep-drawn cup
due to in-plane anisotropy. (Courtesy
of Alcoa, Inc.)
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.)
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.)
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.
Forming-Limit Curve
Construction of a forming-limit curve
(or Keeler–Goodwin diagram).
(Courtesy of S. S. Hecker.)
Different Strain Patterns in Stamped Part
Different strain patterns in stamped part. (Adapted from W.
Brazier, Closed Loop, 15, No. 1 (1986) 3.)
Stress vs. Strain Rate for Slow-Twitch
and Fast Twitch Muscles
Stress-Strain Cures of Some Biological Materials
Stress–strain response for some
biological materials.
Mechanical Properties of Biological Materials
Stress-Strain Response of Elastin
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 Cortical Bone
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).)
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.)