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16 Solution of the Simplest Problems for the Strain Theory of Plasticity
Let us define the constant C 1 in two cases: (a) the plastic zone covers the entire
thickness of the spherical vessel wall, and (b) the plastic zone has space only in
the case of a r = c < b.
(a) The entire sphere material is covered by yield. There are two boundary
conditions:
r = a : σ s = −p; r = b : σ r = 0.
(16.32)
By using the condition (16.32) from the solution (16.31), we have
−p lim = 2σ s ln a + C 1 ,
0 = 2σ s ln b + C 1 ,
(16.33)
where p lim is the limit pressure when the sphere strains are unlimitedly
increased.
From the system (16.33), we find
C 1 = − [2σ s ln a + p lim ] ,
p lim = 2σ s ln
b
a
.
(16.34)
Then we will find as follows for stresses in the limit state:
σ lim
r
= σ lim
ϕ − σ s ,
σ lim
ϕ
= 2σ s ln
r
a − p lim .
(16.35)
(b) Only a part of the sphere is covered by yield. Let us designate the pressure
intensity of the elastic zone on the plastic zone at the common boundary
as q.
For the elastic zone, we can immediately write the problem solution. To
do it, let us use the solution (16.27):
σ r = q 0
1 −
b 3
r 3
,
σ ϕ = q 0
1 +
1
2
b 3
r 3
,
(16.36)