3. 1. Modulus of Elasticity, E
(Young’s Modulus)
s = E e
s
Linear-elastic
E
e
F
F
simple
tension
test
Units:
E: [GPa] ebooks.edhole.com
4. Slope of stress strain plot (which is
proportional to the elastic modulus)
depends on bond strength of metal
Adapted from Fig. 6.7,
Callister 7e.
E=
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5. 2. Poisson's ratio, n
Units:
n: dimensionless
eT
eL
n
F
F
simple
tension
test
“n” is the ratio of
transverse contraction strain
to longitudinal extension
strain in the direction of
stretching force.
Either transverse strain or
longitudional strain is
negative, ν is positive
eT n = -
eL
eT : Transverse Strain
eL : Longitudional Strain
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6. Virtually all common materials undergo a
transverse contraction when stretched in one
direction and a transverse expansion when
compressed.
In an isotropic material the allowable (theoretical)
range of Poisson's ratio is from -1.0 to +0.5,
based on the theory of elasticity.
metals: n ~ 0.33
ceramics: n ~ 0.25
polymers: n ~ 0.40
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7. 3. Shear Modulus, G
t
G
g
t = G g
M
simple
torsion
test
M
Units:
G: [GPa] ebooks.edhole.com
8. 4. Bulk Modulus, K
savg = K
DV
Vo savg
DV
K
Vo
P
P
P
Initial Volume = V0
Volume Change = DV
Units:
K: [GPa]
σavg is the average of
three stresses applied
along three principal
directions.
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9. Elastic Constants
s = E e
t = G g
savg = K
DV
Vo
Stresses Strains
Normal
Shear
Volumetric
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10. Example:
Uniaxial Loading of a Prismatic Specimen
9.9 cm
Before After
10 cm
10 cm
10 cm
10.4 cm
9.9 cm
Determine
E and n
P=1000 kgf
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12. For an isotropic material the stress-strain
relations are as follows:
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13. RELATION B/W K & E
Consider a cube with a unit volume
σ
1
1
1
σ
D
C
A B
σ causes an elongation in the direction
CD and contraction in the directions AB
& BC.
The new dimensions of the cube is :
• CD direction is 1+ε
• BC direction is 1-νε
• AB direction is 1-νε
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14. V0 = 1
Final volume Vf of the cube is now:
(1+ε) (1-νε) (1-νε) = (1+ε) (1-2νε+μ2ε2)
= 1 - 2νε + μ2ε2 + ε-2νε2 + μ2ε3
= 1 + ε - 2νε - 2νε2 + μ2ε2 + μ2ε3
ε is small, ε2 & ε3 are smaller and can be neglected.
Vf = 1+ ε - 2νε → ΔV = Vf - V0 = ε (1-2ν)
If equal tensile stresses are applied to each
of the other two pairs of faces of the cube
than the total change in volume will be :
ebooks.ΔedVh=ole3.cεom(1-2ν)
16. Moreover the relation
between G and E is :
G =
E
2 (1+ν)
The relation between
G, E and K is :
1 1
E
=
1
+
9K 3G
K =
E
3 (1-2ν)
The relation between
K and E is :
Therefore, out of the four elastic
constants only two of them are
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17. For very soft materials such as pastes, gels,
putties, K is very large
Note that as K → ∞ → ν → 0.5 & E ≈ 3G
If K is very large → ΔV/V0 ≈ 0 *No volume
change
For materials like metals, fibers & certain
plastics K must be considered.
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18. Modulus of Elasticity :
• High in covalent compounds such as diamond
• Lower in metallic and ionic crystals
• Lowest in molecular amorphous solids such
as plastics and rubber.
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