230
16 Solution of the Simplest Problems for the Strain Theory of Plasticity
of tangential stresses for the case when there are elastic (c r b) and plastic
(a r c) zones in the vessel wall at the same time.
By analyzing the given solution, we can note the following properties:
1. In the solution to the problem set by formulas (16.36) and (16.27), there is no
Poisson coefficient ν. Consequently, the compressibility (non-compressibility)
of a material does not affect the stress distribution in the sphere. When finding
strains or displacements, this coefficient would appear in calculation formulas
and its effect on the magnitude of displacements is insignificant.
2. Residual stresses after complete unloading can be calculated using the formulas:
˜
σ ϕ = σ ϕ − σ
y
ϕ ,
˜
σ r = σ r − σ
y
r ,
(16.40)
where the elastic stresses σ
y
ϕ and σ
y
r are defined by formulas (16.27), and instead
of σ ϕ and σ r , it is required to substitute the solution (16.36) at c r b
or solution (16.31), (16.28) at a r c. Formulas (16.40) are true for the
condition that for complete unloading ˜
σ i < σ s . The epure of tangential residual
stresses is shown in Fig. 16.6b.
3. As seen from the epure of residual stresses (Fig. 16.6b), after complete unloading,
there are compression stresses from the inner surface of the sphere. In the case of
repeated loading with the same pressure p exerted while loading, the intensity of
stresses at r = a will be less than the yield point σ s and the vessel remains elastic.
This effect is frequently used by process engineers in machine engineering where
it is known as autofrettage.
16.4 Symmetric Strain of a Cylindrical Tube
Let us define stresses and strains of a thick-wall cylindrical tube loaded by internal
(p a ) and external (p b ) pressures and by the axial elongating force (P ). As in the
Lame problem, we will designate the inner radius of the tube as a and the external
radius as b. Furthermore, let us suggest for simplicity that the tube material is noncompressible.
Let us introduce a cylindrical system of coordinates r, ϕ, z whose axis z
coincides with the tube axis. Due to symmetry, normal stresses σ r , σ ϕ and σ z are
principal. The equilibrium equation will be
dσ r
dr
+
σ r − σ ϕ
r
= 0.
(16.41)
As in the Lame problem, the Cauchy equations for the components of strain and
radial displacement u look as follows :
16 Solution of the Simplest Problems for the Strain Theory of Plasticity
of tangential stresses for the case when there are elastic (c r b) and plastic
(a r c) zones in the vessel wall at the same time.
By analyzing the given solution, we can note the following properties:
1. In the solution to the problem set by formulas (16.36) and (16.27), there is no
Poisson coefficient ν. Consequently, the compressibility (non-compressibility)
of a material does not affect the stress distribution in the sphere. When finding
strains or displacements, this coefficient would appear in calculation formulas
and its effect on the magnitude of displacements is insignificant.
2. Residual stresses after complete unloading can be calculated using the formulas:
˜
σ ϕ = σ ϕ − σ
y
ϕ ,
˜
σ r = σ r − σ
y
r ,
(16.40)
where the elastic stresses σ
y
ϕ and σ
y
r are defined by formulas (16.27), and instead
of σ ϕ and σ r , it is required to substitute the solution (16.36) at c r b
or solution (16.31), (16.28) at a r c. Formulas (16.40) are true for the
condition that for complete unloading ˜
σ i < σ s . The epure of tangential residual
stresses is shown in Fig. 16.6b.
3. As seen from the epure of residual stresses (Fig. 16.6b), after complete unloading,
there are compression stresses from the inner surface of the sphere. In the case of
repeated loading with the same pressure p exerted while loading, the intensity of
stresses at r = a will be less than the yield point σ s and the vessel remains elastic.
This effect is frequently used by process engineers in machine engineering where
it is known as autofrettage.
16.4 Symmetric Strain of a Cylindrical Tube
Let us define stresses and strains of a thick-wall cylindrical tube loaded by internal
(p a ) and external (p b ) pressures and by the axial elongating force (P ). As in the
Lame problem, we will designate the inner radius of the tube as a and the external
radius as b. Furthermore, let us suggest for simplicity that the tube material is noncompressible.
Let us introduce a cylindrical system of coordinates r, ϕ, z whose axis z
coincides with the tube axis. Due to symmetry, normal stresses σ r , σ ϕ and σ z are
principal. The equilibrium equation will be
dσ r
dr
+
σ r − σ ϕ
r
= 0.
(16.41)
As in the Lame problem, the Cauchy equations for the components of strain and
radial displacement u look as follows :