4.2 Linear Elasticity—Generalized Hooke’s Law
105
ε x =
λ + G
G(3λ + 2G)
σ x −
λ
2G(3λ + 2G)
σ y + σ z
ε y =
λ + G
G(3λ + 2G)
σ y −
λ
2G(3λ + 2G)
(σ z + σ x )
ε z =
λ + G
G(3λ + 2G)
σ z −
λ
2G(3λ + 2G)
σ x + σ y
(4.23)
γ xy =
τ xy
G
γ yz =
τ yz
G
γ zx =
τ zx
G
Now consider a simple tensile test
Therefore,
ε x =
σ x
E
=
λ + G
G(3λ + 2G)
= σ x
or
1
E
=
λ + G
G(3λ + 2G)
or E =
G(3λ + 2G)
(λ + G)
(4.24)
where E = Modulus of Elasticity
Also,
ε y = −vε x = −v
σ x
E
ε ¯
z = −vε x = −v
σ x
E
where ν = Poisson’s ratio
For σ y = σ z = 0, we get from Eq. (4.23)
−
λ
2G(3λ + 2G)
σ x = −
ν
E
σ x
Therefore,
ν
E
=
λ
2G(3λ + 2G)
(4.25)