# Chapter 6

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```					             Chapter 6:
Mechanical Properties
• Stress and strain: What are they and why are
• Elastic behavior: When loads are small, how much
deformation occurs? What materials deform least?
• Plastic behavior: At what point does permanent
deformation occur? What materials are most
resistant to permanent deformation?
• Toughness and ductility: What are they and how
do we measure them?

Chapter 6 - 1
Elastic Deformation

bonds
stretch

initial
d
F
F             Linear-
elastic
Elastic means reversible!                           Non-Linear-
elastic
d
Chapter 6 - 2
Plastic Deformation (Metals)
bonds
stretch                            planes
& planes                           still
shear                              sheared

dplastic
delastic + plastic

F
F
Plastic means permanent!      linear                 linear
elastic                elastic
d
dplastic
Chapter 6 - 3
Engineering Stress
• Tensile stress, s:         • Shear stress, t:
Ft                  Ft          F

Area, A                     Area, A               Fs

Fs
Ft
Fs              Ft
Ft  lb f  N           t=         F
s=    = 2 or                Ao
Ao  in    m2
original area
 Stress has units:
N/m2 or lbf/in2
Chapter 6 - 4
Common States of Stress
• Simple tension: cable
F                   F
A o = cross sectional
F
s=    s                 s
Ao
Ski lift (photo courtesy
• Torsion (a form of shear): drive shaft          P.M. Anderson)

M           Fs   Ao
Ac
Fs
t =
Ao
M
2R                         Note: t = M/AcR here.
Chapter 6 - 5
OTHER COMMON STRESS STATES (1)
• Simple compression:

Ao

Canyon Bridge, Los Alamos, NM
(photo courtesy P.M. Anderson)

F
s=
Note: compressive
Balanced Rock, Arches                    structure member
National Park
(photo courtesy P.M. Anderson)
Ao   (s < 0 here).

Chapter 6 - 6
OTHER COMMON STRESS STATES (2)
• Bi-axial tension:           • Hydrostatic compression:

Pressurized tank                     Fish under water   (photo courtesy
(photo courtesy                                         P.M. Anderson)
P.M. Anderson)
sq > 0

sz > 0        sh< 0

Chapter 6 - 7
Engineering Strain
• Tensile strain:                                           • Lateral strain:
d/2
-dL
e = d                                                      eL =
Lo                                          Lo               wo
wo

dL /2
• Shear strain:
q
x                          g = x/y = tan q

y                90º - q
Strain is always
90º                                                     dimensionless.
Adapted from Fig. 6.1 (a) and (c), Callister 7e.          Chapter 6 - 8
Stress-Strain Testing
• Typical tensile test                                                 • Typical tensile
machine                                                                specimen

extensometer                                   specimen                                   Fig. 6.2,
Callister 7e.

gauge
length

Adapted from Fig. 6.3, Callister 7e. (Fig. 6.3 is taken from H.W.
Hayden, W.G. Moffatt, and J. Wulff, The Structure and Properties of
Materials, Vol. III, Mechanical Behavior, p. 2, John Wiley and Sons,
New York, 1965.)                                                                   Chapter 6 - 9
Linear Elastic Properties
• Modulus of Elasticity, E:
(also known as Young's modulus)

• Hooke's Law:
s=Ee             s                        F
E

e
Linear-
elastic               F
simple
tension
test

Chapter 6 - 10
Poisson's ratio, n
eL
• Poisson's ratio, n:

eL
n=- e
e

metals: n ~ 0.33                     -n
ceramics: n ~ 0.25
polymers: n ~ 0.40

Units:                 –n > 0.50 density increases
E: [GPa] or [psi]
–n < 0.50 density decreases
n: dimensionless                  (voids form)

Chapter 6 - 11
Mechanical Properties
• Slope of stress strain plot (which is
proportional to the elastic modulus) depends
on bond strength of metal

Callister 7e.

Chapter 6 - 12
Other Elastic Properties
t                     M
• Elastic Shear
modulus, G:
G                     simple
g                 torsion
t=Gg                                              test

M
• Elastic Bulk               P                     P
modulus, K:
V                         V P                  P
P = -K                            Vo
Vo                K                            pressure
test: Init.
vol =Vo.
• Special relations for isotropic materials:              Vol chg.
= V
E                 E
G=                K=
2(1 + n)          3(1 - 2n)
Chapter 6 - 13
Young’s Moduli: Comparison
Graphite
Metals                           Composites
Ceramics Polymers
Alloys                             /fibers
Semicond
1200
1000                   Diamond
800
600
Si carbide
400   Tungsten        Al oxide                  Carbon fibers only
Molybdenum      Si nitride
E(GPa)    200
Steel, Ni
Tantalum         <111>
CFRE(|| fibers)*
Platinum        Si crystal
Cu alloys        <100>                    Aramid fibers only
100   Zinc, Ti
80   Silver, Gold
Glass -soda               AFRE(|| fibers)*     Based on data in Table B2,
Aluminum                                  Glass fibers only
60
40
Magnesium,
Tin                                       GFRE(|| fibers)*     Callister 7e.
Concrete                                       Composite data based on
109 Pa     20                                             GFRE*
CFRE*
reinforced epoxy with 60 vol%
of aligned
Graphite                  GFRE( fibers)*
10                                                                  carbon (CFRE),
8                                             CFRE( fibers) *
6                                             AFRE( fibers) *
aramid (AFRE), or
Polyester                        glass (GFRE)
4                                   PET
PS                            fibers.
PC         Epoxy only
2
PP
1                                  HDPE
0.8
0.6                                             Wood(     grain)
PTFE
0.4

0.2                                  LDPE                                             Chapter 6 - 14
Useful Linear Elastic Relationships
• Simple tension:                 • Simple torsion:
2ML o
d = FL o d = -n Fw o                 a=
L                                    4
EA o        EA o                        pr o G
F                                   M = moment
d/2                           a = angle of twist
Ao
Lo                            Lo
wo

2ro
dL /2
contribute to deflection.
• Larger elastic moduli minimize elastic deflection.
Chapter 6 - 15
Plastic (Permanent) Deformation
(at lower temperatures, i.e. T < Tmelt/3)

• Simple tension test:
Elastic+Plastic
engineering stress, s              at larger stress

Elastic
initially
permanent (plastic)

ep          engineering strain, e

plastic strain    Adapted from Fig. 6.10 (a),
Callister 7e.

Chapter 6 - 16
Yield Strength, sy
• Stress at which noticeable plastic deformation has
occurred.
when ep = 0.002
tensile stress, s
sy = yield strength
sy

Note: for 2 inch sample
e = 0.002 = z/z
 z = 0.004 in

engineering strain, e
ep = 0.002        Adapted from Fig. 6.10 (a),
Callister 7e.
Chapter 6 - 17
Yield Strength : Comparison
Graphite/
Metals/                                                                                                        Composites/
Ceramics/                                                                       Polymers
Alloys                                                                                                           fibers
Semicond
2000
Steel (4140) qt

1000
Yield strength, sy (MPa)

Ti (5Al-2.5Sn) a

in ceramic matrix and epoxy matrix composites, since
700    W (pure)

since in tension, fracture usually occurs before yield.

in tension, fracture usually occurs before yield.
600    Cu (71500) cw
500    Mo (pure)
400    Steel (4140) a
Steel (1020) cd
300
Room T values
Hard to measure ,

Hard to measure,
Al (6061) ag
200    Steel (1020) hr                   ¨
Ti (pure) a
Ta (pure)
Cu (71500) hr                                                                                                                                                                                 Based on data in Table B4,
Callister 7e.
100                                                                                                                                                                                                  a = annealed
dry
70                                                                                                       PC
hr = hot rolled
60    Al (6061) a                                                                                         Nylon 6,6                                                                                 ag = aged
50                                                                                                       PET
cd = cold drawn
40                                                                                                       PVC humid
cw = cold worked
PP
30                                                                                                       HDPE                                                                                       qt = quenched & tempered

20

LDPE
Tin (pure)                                                                                                                                                                                                  Chapter 6 - 18
10
Tensile Strength, TS
• Maximum stress on engineering stress-strain curve.
Callister 7e.
TS
F = fracture or
sy
ultimate
engineering

strength
stress

Typical response of a metal
Neck – acts
as stress
concentrator
strain
engineering strain
• Metals: occurs when noticeable necking starts.
• Polymers: occurs when polymer backbone chains are
Chapter 6 - 19
Tensile Strength : Comparison
Graphite/
Metals/                                   Composites/
Ceramics/    Polymers
Alloys                                      fibers
Semicond
5000                                                   C fibers
Aramid fib
3000                                               E-glass fib
Tensile strength, TS (MPa)

2000   Steel (4140) qt
AFRE(|| fiber)
1000   W (pure)         Diamond                    GFRE(|| fiber)
Ti (5Al-2.5Sn)aa
Steel (4140)                                CFRE(|| fiber)
Cu (71500) cw     Si nitride
Cu (71500) hr    Al oxide
Steel (1020)
300              ag
Al (6061) a
Ti (pure)
200    Ta (pure)                                                     Room Temp. values
Al (6061) a
100                     Si crystal                 wood(|| fiber)    Based on data in Table B4,
<100>       Nylon 6,6
Glass-soda   PC PET                         Callister 7e.
40                                   PVC          GFRE( fiber)      a = annealed
Concrete     PP
30                                                CFRE( fiber)      hr = hot rolled
AFRE( fiber)
HDPE                        ag = aged
20                      Graphite
LDPE                           cd = cold drawn
cw = cold worked
10                                                                  qt = quenched & tempered
AFRE, GFRE, & CFRE =
aramid, glass, & carbon
fiber-reinforced epoxy
wood (   fiber)
composites, with 60 vol%
fibers.
1                                                                                Chapter 6 - 20
Ductility
Lf - Lo
• Plastic tensile strain at failure:          %EL =         x 100
Lo
smaller %EL
Engineering
tensile
stress, s               larger %EL                   Ao
Lo          Af          Lf
Callister 7e.

Engineering tensile strain, e

• Another ductility measure:                      Ao - Af
%RA =           x 100
Ao

Chapter 6 - 21
Toughness
• Energy to break a unit volume of material
• Approximate by the area under the stress-strain
curve.
Engineering               small toughness (ceramics)
tensile                                      large toughness (metals)
stress, s
Adapted from Fig. 6.13,                                  very small toughness
Callister 7e.                                            (unreinforced polymers)

Engineering tensile strain,     e

Brittle fracture: elastic energy
Ductile fracture: elastic + plastic energy
Chapter 6 - 22
Resilience, Ur
• Ability of a material to store energy
– Energy stored best in elastic region

ey
Ur =         sde
0
If we assume a linear
stress-strain curve this
simplifies to

1
Ur @ sy e y
2
Callister 7e.
Chapter 6 - 23
Elastic Strain Recovery

Callister 7e.

Chapter 6 - 24
Hardness
• Resistance to permanently indenting the surface.
• Large hardness means:
--resistance to plastic deformation or cracking in
compression.
--better wear properties.
apply known force               measure size
e.g.,                                                of indent after

Smaller indents
D                          d          mean larger
hardness.

most       brasses         easy to machine              cutting    nitrided
plastics   Al alloys       steels          file hard     tools     steels     diamond

increasing hardness
Chapter 6 - 25
Hardness: Measurement
• Rockwell
– No major sample damage
– Each scale runs to 130 but only useful in range
20-100.
– Major load 60 (A), 100 (B) & 150 (C) kg
• A = diamond, B = 1/16 in. ball, C = diamond

• HB = Brinell Hardness
– TS (psia) = 500 x HB
– TS (MPa) = 3.45 x HB

Chapter 6 - 26
Hardness: Measurement
Table 6.5

Chapter 6 - 27
True Stress & Strain
Note: S.A. changes when sample stretched

• True stress   sT = F Ai           sT = s1 + e
• True Strain   eT = ln i  o    eT = ln1 + e

Callister 7e.

Chapter 6 - 28
Hardening
• An increase in sy due to plastic deformation.
s
large hardening
sy
1
sy                                small hardening
0

e
• Curve fit to the stress-strain response:
hardening exponent:
sT = K eT  n       n = 0.15 (some steels)
to n = 0.5 (some coppers)
“true” stress (F/A)          “true” strain: ln(L/Lo)
Chapter 6 - 29
Variability in Material Properties
• Elastic modulus is material property
• Critical properties depend largely on sample flaws
(defects, etc.). Large sample to sample variability.
• Statistics
n
 xn
– Mean                            x=
n
1
n           2
x i - x  
2

– Standard Deviation           s=
 n -1 
             
where n is the number of data points
Chapter 6 - 30
Design or Safety Factors
• Design uncertainties mean we do not push the limit.
• Factor of safety, N              Often N is
sy         between
sworking =              1.2 and 4
N
• Example: Calculate a diameter, d, to ensure that yield does
not occur in the 1045 carbon steel rod below. Use a
factor of safety of 5.
d
sy
sworking =                1045 plain
carbon steel:
N           sy = 310 MPa                     Lo
220 ,000 N        5               TS = 565 MPa

p d2 / 4
F = 220,000N
d = 0.067 m = 6.7 cm
Chapter 6 - 31
Summary
• Stress and strain: These are size-independent
measures of load and displacement, respectively.
• Elastic behavior: This reversible behavior often
shows a linear relation between stress and strain.
To minimize deformation, select a material with a
large elastic modulus (E or G).
• Plastic behavior: This permanent deformation
behavior occurs when the tensile (or compressive)
uniaxial stress reaches sy.
• Toughness: The energy needed to break a unit
volume of material.
• Ductility: The plastic strain at failure.

Chapter 6 - 32
ANNOUNCEMENTS

Core Problems:

Self-help Problems:

Chapter 6 - 33

```
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