Chapter 3
The Second Primary Problem of
Elasticity Theory
3.1 Definition of Stresses Through Deformations
Let us re-write the first of formulas (1.11) as
ε x =
∂u
∂x
=
1
E
σ x (1 + ν) − ν(σ x + σ y + σ z )
.
Then using formulas (1.15) and (1.16), we find
∂u
∂x
=
1
E
σ x (1 + ν) −
νE
1 − 2ν
.
(3.1)
Formula (3.1) can be written as follows
σ x = 2G
∂u
∂x
+ λλ,
(3.2)
where G is expressed through the Jung modulus E and the Poisson coefficient ν
using formula (1.10), while λ designates
λ =
νE
(1 + ν)(1 − 2ν)
.
(3.3)
The constant values λ and G are referred to as [1] Lame coefficients.
For other components of stress, we can obtain:
σ y = 2G
∂v
∂y
+ λλ, σ z = 2G
∂w
∂z
+ λλ,
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
V. Molotnikov, A. Molotnikova, Theory of Elasticity and Plasticity,
https://doi.org/10.1007/978-3-030-66622-4_3
29
The Second Primary Problem of
Elasticity Theory
3.1 Definition of Stresses Through Deformations
Let us re-write the first of formulas (1.11) as
ε x =
∂u
∂x
=
1
E
σ x (1 + ν) − ν(σ x + σ y + σ z )
.
Then using formulas (1.15) and (1.16), we find
∂u
∂x
=
1
E
σ x (1 + ν) −
νE
1 − 2ν
.
(3.1)
Formula (3.1) can be written as follows
σ x = 2G
∂u
∂x
+ λλ,
(3.2)
where G is expressed through the Jung modulus E and the Poisson coefficient ν
using formula (1.10), while λ designates
λ =
νE
(1 + ν)(1 − 2ν)
.
(3.3)
The constant values λ and G are referred to as [1] Lame coefficients.
For other components of stress, we can obtain:
σ y = 2G
∂v
∂y
+ λλ, σ z = 2G
∂w
∂z
+ λλ,
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
V. Molotnikov, A. Molotnikova, Theory of Elasticity and Plasticity,
https://doi.org/10.1007/978-3-030-66622-4_3
29