# Chapter 3 Plasticity

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29 Οκτ 2013 (πριν από 4 χρόνια και 8 μήνες)

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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
(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
=
σ
o

+

n
).

Idealized Uniaxial Stress
-
Strain Curves

Ludwik
-
Hollomon equation

Plasticity

Voce equation

Johnson
-
Cook equation

Schematic representation of the
change‏ in‏Poisson’s‏ratio‏as‏the‏
deformation regime changes from
elastic to plastic.

True Stress
-

True‏Strain‏Curve‏and‏Poisson’s‏atio

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

/

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

(
A
0
/
A
),
minus strain at necking,
ε
u
. (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 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.

Strain Rate Effects

(a) Compression specimen
between parallel platens.

(b) Length
inhomogeneity

in

specimen.

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. ICMA
6
, 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
Pd
77.5
CU
6
Si
16.5
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

Ni
82.4
Cr
7
Fe
3
Si
4.5
B
3.1
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
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
A. S. Argon, and S. Yip,
Phil. Trans.
Roy. Soc.
Lond
. A
329
(
1989
)
613
.)

Dislocations

Viscosity of soda

lime

silica glass and of

metallic glasses (Au

Si

Ge
, Pd

Cu

Si, Pd

Si, C
0
P)
as a function of normalized temperature. (Adapted
from J. F. Shackelford,
Introduction to Materials
Science for Engineers
,
4
th 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.

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
,
2
nd ed.
(Englewood Cliffs, NJ: Prentice
-
Hall,
1976
), and G. Murphy,
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 Al
2
O
3

(a) Simple model for solid with cracks. (b) Elliptical flaw in elastic

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
-
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
thickness direction,
x
3
. (Values of
R =
σ
2
/
σ
1
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
-
dependence of percent excess boundary hardening in zinc containing Al, Au, or
Cu (
3
-
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

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”‏formed‏in‏a‏deep
-
drawn cup
due to in
-
plane anisotropy. (Courtesy
of Alcoa, Inc.)

Earing in Deep Drawing

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:
ε
1
,
meridional

strain;
ε
2
,
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

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
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