elastic constants

The two elastic constants are usually expressed as the Young's modulus E and the Poisson's ratio n. However, the alternative elastic constants K (bulk modulus) and/or G (shear modulus) can also be used. For isotropic materials, G and K can be found from E and n by a set of equations, and vice-versa.

1. ELASTIC CONSTANTS IN
ISOTROPIC MATERIALS
1.Elasticity Modulus (E)
2. Poisson’s Ratio (n)
3. Shear Modulus (G)
4. Bulk Modulus (K)
1

2. 1. Modulus of Elasticity, E
(Young’s Modulus)
2

3. 3

4. 4Hooke’s law :- “within elastic limit, the stress is
proportional to the strain.”
Stress-strain curve:-

5. 5

7. Shear Modulus, G 7

8. Bulk modulus(K)

9. Example:
Uniaxial Loading of a Prismatic Specimen
After Before
10 cm
10 cm
10 cm
10.4 cm
9.9 cm
9.9 cm
Determine
E and n
P=1000 kgf
9

10. 10
cm
10
cm
Δl/2=0.2
cm
Δd/2=0.05
cm
1000 kgf
P=1000 kgf
P=1000kgf → σ=
10*10
1000
= 10kgf/cm
2
E=
σ
ε
=
10
0.04
= 250 kgf/cm
2
ε
long
=
Δl
l
0
= =0.04
0.4
10
ε
lat
=
Δd
d
0
= = -0.01
-0.1
10
ν = -
-0.01
0.04
= 0.25
10

11. RELATION B/W K & E
Consider a cube with a unit volume
σ
1
1
1
σ
D
C
B
A
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
11

12. V
0
= 1
Final volume V
f
of the cube is now:
(1+ε) (1-νε) (1-νε) = (1+ε) (1-2νε+μ
2
ε
2
)
= 1 - 2νε + μ
2
ε
2
+ ε-2νε
2
+ μ
2
ε
3
= 1 + ε - 2νε - 2νε
2
+ μ
2
ε
2
+ μ
2
ε
3

ε is small, ε
2
& ε
3
are smaller and can be neglected.
 V
f
= 1+ ε - 2νε → ΔV = V
f
- V
0
= 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 :
ΔV = 3ε (1-2ν)
12

13. σ
σ
σ
σ
σ
σ
Ξ + +
K =
(σ+σ+σ)/3 σ
=
3ε (1-2ν)
=
3ε (1-2ν)
E
3 (1-2ν)
K =
E
3 (1-2ν)
SΔV = 3ε (1-2ν) =ε (1-2ν)ε (1-2ν) ε (1-2ν)+ +
=
s
avg
DV/V
0
13

14.  Moreover the relation between G
and E is :
G =
E
2 (1+ν)
 The relation between The relation between
G, E and K is :G, E and K is :
E
1 1
=
1
+
9K 3G
K =
E
3 (1-2ν)
The relation between The relation between
K and E is :K and E is :
Therefore, out of the four elastic constants
only two of them are independent.
14

15.  For very soft materials such as pastes, gels, For very soft materials such as pastes, gels,
putties, K is very largeputties, K is very large
Note that as K Note that as K → ∞ → → ∞ → νν →→ 0.5 & E ≈ 3G 0.5 & E ≈ 3G
If K is very large → If K is very large → ΔΔV/VV/V
00 ≈ 0 ≈ 0 **No volume No volume
changechange
For materials like metals, fibers & certain For materials like metals, fibers & certain
plastics K must be considered.plastics K must be considered.
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16. 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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17. Elastic Constants of Some
Materials
E(psi)x10E(psi)x10
6 6
(GPa)(GPa)
G(psi)x10G(psi)x10
66

(GPa)(GPa)
νν (-)(-)
Cast IronCast Iron
16 16 110110 7.4 7.4 5050 0.170.17
SteelSteel
30 30 205205 11.8 11.8 8080 0.260.26
AluminumAluminum
10 10 7070 3.6 3.6 2525 0.330.33
ConcreteConcrete
1.5-5.5 1.5-5.5 10-4010-400.62-2.30 0.62-2.30 4-154-150.20.2
WoodWood Long 1.81 Long 1.81 1212
Tang 0.10 Tang 0.10 0.70.7
0.11 0.11 0.70.7
0.03 0.03 0.20.2
??
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18. Thank You
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