1.18 Lame Task
19
a
b
Fig. 1.8 To derivation of equilibrium equations in the Lame tasks
By projecting all forces acting on the element mm 1 n 1 n to the direction of
the bisecting line of a small angle dϕ, we will obtain the following equilibrium
condition of the separated element:
σ r rdϕ + σ t drdϕ −
σ r +
dσ r
dr
dr
(r + dr)dϕ = 0.
(1.27)
By neglecting the infinitely small values of higher orders in the equality (1.27), we
will obtain
r
dσ r
dr
= σ t − σ r .
(1.28)
Let us express stresses through strains: For the considered case, when using
expressions (1.25) and (1.24), formulas (1.20) take the following form
u
r
=
1
E ∗ (σ t − ν
∗ σ r ),
du
dr
=
1
E ∗ (σ r − ν
∗ σ t ).
By subtracting the respective parts of these equations, we obtain
u
r
−
du
dr
=
1 + ν ∗
E ∗ (σ t − σ r ).
(1.29)
In a similar way, we find
σ r =
E ∗
1 − ν ∗2
du
dr
+ ν
∗ u
r
.
(1.30)
By substituing (1.30) in the Eq. (1.28) and using formula (1.29), we obtain
r
E ∗
1 − ν ∗2
d 2 u
dr 2 +
ν ∗
r
du
dr
−
ν ∗
r 2 u
=
E ∗
1 + ν ∗
u
r
−
du
dr
19
a
b
Fig. 1.8 To derivation of equilibrium equations in the Lame tasks
By projecting all forces acting on the element mm 1 n 1 n to the direction of
the bisecting line of a small angle dϕ, we will obtain the following equilibrium
condition of the separated element:
σ r rdϕ + σ t drdϕ −
σ r +
dσ r
dr
dr
(r + dr)dϕ = 0.
(1.27)
By neglecting the infinitely small values of higher orders in the equality (1.27), we
will obtain
r
dσ r
dr
= σ t − σ r .
(1.28)
Let us express stresses through strains: For the considered case, when using
expressions (1.25) and (1.24), formulas (1.20) take the following form
u
r
=
1
E ∗ (σ t − ν
∗ σ r ),
du
dr
=
1
E ∗ (σ r − ν
∗ σ t ).
By subtracting the respective parts of these equations, we obtain
u
r
−
du
dr
=
1 + ν ∗
E ∗ (σ t − σ r ).
(1.29)
In a similar way, we find
σ r =
E ∗
1 − ν ∗2
du
dr
+ ν
∗ u
r
.
(1.30)
By substituing (1.30) in the Eq. (1.28) and using formula (1.29), we obtain
r
E ∗
1 − ν ∗2
d 2 u
dr 2 +
ν ∗
r
du
dr
−
ν ∗
r 2 u
=
E ∗
1 + ν ∗
u
r
−
du
dr