108
4 Stress–Strain Relations
Table 4.1 Relation between elastic constants
E
v
K
G
λ
E.v
E
v
E
3(1+2v)
E
2(1+v)
Ev
(1+v)(1−2v)
E.K
E
3K −E
6K
K
3K E
9K −E
3K (3K −E)
9K −E
E.G
E
E−2G
2G
G E
3(3G−E)
G
G(E−2G)
3G−E
E.λ
E
R−E−λ
4λ
R+E+3λ
6
R+E−3λ
4
λ
v.K
3k(1 − 2v)
v
K
3K (1−2v)
2(1+v)
3K v
1+v
v.G
2G(1 + v)
v
2G(1+v)
3(1−2v)
G
2Gv
1−2v
v.λ
λ(1+v)(1−2v)
v
v
λ(1+v)
3v
λ(1−2v)
2v
λ
K.G
9K G
3K +G
3K −2G
6K +2G
K
G
K −
2
3 G
K.λ
9K G(k−λ)
3K −λ
λ
3K −λ
K
3K −λ
2
λ
G.λ
G(3λ+2G)
λ+G
λ
2(λ+G)
3λ+2G
3
G
λ
R =
√
E 2 + 9λ 2 + 2Eλ > 0
Table 4.2 Typical values of elastic moduli for common engineering materials
E(GPa)
v
G(GPa)
λ(GPa)
k(GPa)
α(10 −6/ °C)
Aluminium
68.9
0.34
25.7
54.6
71.8
25.5
Concrete
27.6
0.20
11.5
7.7
15.3
11
Copper
89.6
0.34
33.4
71
93.3
18
Glass
68.9
0.25
27.6
27.6
45.9
8.8
Nylon
28.3
0.40
10.1
4.04
47.2
102
Rubber
0.0019
0.499
0.654 × 10 −3
0.326
0.326
200
Steel
207
0.29
80.2
111
164
13.5
Fig. 4.1 Element subjected to a normal stress