Poisson's ratio, n
• Poisson's ratio, n:
n
x
z

y
z
(Isotropic materials only)
metals: n ~ 0.33
ceramics: n ~ 0.25
polymers: n ~ 0.40
Units:
E: [GPa] or [psi]
n: dimensionless
–n > 0.50 density increases
–n < 0.50 density decreases
(voids form)
Derivative Relationship

1
 x   x   ( y   z )
E

Other Elastic Properties
• Elastic Shear
modulus, G:
t
M
G
t=Gg
• Elastic Bulk
modulus, K:
DV
P = -K
Vo
g
M
P
P
K
DV P
Vo
• Special relations for isotropic materials:
E
G
2(1  n)
simple
torsion
test
E
K
3(1  2n)
P
pressure
test: Init.
vol =Vo.
Vol chg.
= DV
Plastic (Permanent) Deformation
(at lower temperatures, i.e. T < Tmelt/3)
• Simple tension test:
Elastic+Plastic
at larger stress
engineering stress, s
Elastic
initially
permanent (plastic)
after load is removed
ep
engineering strain, e
plastic strain
Adapted from Fig. 6.10 (a),
Callister 7e.
Most metals – elasticity only continues until strains of about 0.005
Yield Strength, y
• Stress at which noticeable plastic deformation has
occurred.
when ep = 0.002
tensile stress, 
y
y = yield strength
Note: for 2 inch sample
e = 0.002 = Dz/z
 Dz = 0.004 in
engineering strain, e
p = 0.002
Adapted from Fig. 6.10 (a),
Callister 7e.
Tensile Strength, Tult
• Maximum stress on engineering stress-strain curve.
Adapted from Fig. 6.11,
Callister 7e.
Tult
F = fracture or
ultimate
strength
engineering
stress
y
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
aligned and about to break.
Ductility
• Plastic tensile strain at failure:
Lf  Lo
x 100
%EL 
Lo
smaller %EL
Engineering
tensile
stress, s
larger %EL
Lo
Ao
Af
Adapted from Fig. 6.13,
Callister 7e.
Engineering tensile strain, e
• Another ductility measure:
%RA =
Ao - Af
x 100
Ao
Lf
“Toughness”
• Ability to absorb energy before fracturing
• Approximated by the area under the stress-strain curve.
Engineering
tensile
stress, s
small toughness (ceramics)
large toughness (metals)
very small toughness
(unreinforced polymers)
Adapted from Fig. 6.13,
Callister 7e.
Engineering tensile strain,
Brittle fracture: elastic energy
Ductile fracture: elastic + plastic energy
Note: Not fracture toughness, not notch toughness
e
Effect of Temperature on “Toughness”
Figure 6.14 from text: Engineering stress-strain for iron at three temperatures
Resilience, Ur
• Ability of a material to store energy
– Energy stored best in elastic region
Ur 
y
0
d
If we assume a linear
stress-strain curve this
simplifies to
1
Ur @ y  y
2
Adapted from Fig. 6.15,
Callister 7e.
Elastic Strain Recovery
Adapted from Fig. 6.17,
Callister 7e.
True Stress & Strain
S.A. changes when sample stretched
• True stress
• True Strain
  F Ai
  ln  i  o 
  s1  e
  ln 1  e 
Adapted from Fig. 6.16,
Callister 7e.
Hardening
• An increase in y due to plastic deformation.

large hardening
y
1
y
small hardening
0

• Curve fit to the stress-strain response (flow curve):
K is stress at  = 1
 
 K 
“true” stress (F/A)
n
Strain hardening coefficient:
n = 0.15 (some steels)
to n = 0.5 (some coppers)
“true” strain: ln(L/Lo)
Strain hardening coefficient: slope of log-log plot
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
– Mean
– Standard Deviation
 xn
x
n
2
n



x

x
i

s
 n 1 


where n is the number of data points
1
2