Cambridge-Yield-Lecture

loutsyrianMechanics

Oct 30, 2013 (3 years and 10 months ago)

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Chapter 3:

Plasticity

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.

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 at 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
+
K
ε
n
).

Uniaxial Stress
-
Strain Curve

Ludwik
-
Hollomon equation

Plasticity

Voce equation

Johnson
-
Cook equation

Schematic

representation of the
change in

Poisson’s‏ratio‏as‏the‏
deformation

regime changes from
elastic to

plastic.

True Stress and True Strain Curve

True
-

and

engineering
-
stress

strain curves

for AISI
4140
hot
-
rolled steel. R.

A. is reduction in
area.

Stress
-
Strain Curve

Engineering
-

(or nominal
-
) stress

strain curves (a)
without and (b) with a yield

point.

Engineering Stress and Engineering Strain

Tensile specimen being tested; arrows show onset
of necking.

Tensile tests

Log
d
σ
/d
ε
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
.)


Necking

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
.

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) Variation in pressure on
surface of cylindrical specimen being

compressed.

Finite Element Method

The 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.

V
¨
ohringer, and E. Macherauch,

Proc. ICMA
6
, Vol.
1
(New York:

Pergamon,
1982
), p.
255
.)


Bauschunger 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.

Glassy Polymers

(a) Neck propagation

in a sheet of linear
polyethylene.

(b) Neck formation and

propagation in a
specimen, shown in
schematic fashion.

Neck Propagation in Polyethylene

Metallic Glasses

Compression

stress

strain curves for

Pd
77.5
CU
6
Si
16.5
.
(Adapted with

permission from C. A.
Pampillo and H. S.
Chen,
Mater. Sci. Eng.,
13
(
1974
)
181
.)


Plastic Deformation of Glasses

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

(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
.
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.

Shakelford,
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.)
1
P
=
0.1
Pa

s.

Viscosity of three

glasses as a function of

temperature.
1
P
=
0.1
Pa

s
.

Viscosity of Glass

Rankine, Tresca, and von Mises

Maximum
-
stress Criterion

Maximum
-
Shear
-
Stress Criterion

Maximum
-
Distortion
-
Energy Criterion

(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
,
2
nd ed.

(Englewood Cliffs, NJ:

Prentice
-
Hall,
1976
), and G.

Murphy,
Advanced. Mechanics of

Materials
(New York: McGraw
-
Hill,

1964
), p.
83
.)


Comparison of the Rankine, von Mises, and Tresca

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

Tensile and Compressive Strength of

Al
2
O
3

Mohr
-
Coulomb failure criterion

Griffith Failure Criterion

McClintock
-
Walsh Crtierion

(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

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 Ellipse

Envelopes defining

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

(
1969
),
1117
.)


Shear Yielding and Crazing for Amorphous Polymer

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
.)


Failure Envelope

Plane
-
stress yield loci

for sheets with planar isotropy or

textures that are rotationally

symmetric about the thickness

direction,
x
3
. (Values of
R
indicate

the degree of anisotropy
=

σ
2
/
σ
1
.)


Plane
-
Stress Yield Loci for Sheets with Planar Isotropy

Hardness for Steel

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
.
)


Hardness Tests

Impression caused by

spherical indenter on metal plat
e.

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
, (New

York: McGraw
-
Hill,
1941
), p.
149
.)


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 Distance Profile

Some of the details of

the Knoop indenter, together with

its impression.

Knoop Indenter

A schematic of a

nanoindenter apparatu
s.

Nanoindenter apparatus

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.

Topographic Feature of the Berkovich Indentation

A schematic

representation of load vs. indenter

displacement.

Load vs. Indentation 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”‏formed‏in

deep
-
drawn cups due to in
-
plane

anisotropy. (Courtesy of Alcoa,

Inc.)


Plastic Anisotropy

Effect‏of‏“fibering”‏on‏ formability.‏ The‏ bending‏operation‏is‏often‏ an‏integral

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 in‏the‏metal‏ as‏it‏was‏made‏become‏elongated‏into‏“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:
ε
1
, meridional strains;

ε
2
, circumferential strains;
h
, cup

height. (b) Geomety 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

fore a number of biological

materials.

Strength 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

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
.)


Strain Rate Response of Cortical Bone