PSI - Issue 81

Borys Shelestovskyi et al. / Procedia Structural Integrity 81 (2026) 162–169

167

  

exp      

  

2 

2



( )

2 = − − 

.

f

−

1

(21)

2

2

4

4

p

p



For the average integral thickness difference of the main stresses 0  looks like:

( ) 

   

  

1

ch h

−

0  

*

2

G

2

0 



( ) (

) ( exp

)

( ) 

( ) J r d 2

2 2

2 2 − + p r

1 2 − − +

.

(22)

F r

p r

f

=

0 

 

  

( ) sh h h   +

1

2

1

hp

−



3. Results and Discussion A numerical analysis was carried out using experimentally determined parameters of the residual plastic strain field * 0 1.2%,  = 2 0.4 p = , 2 3  = , 5 3  = and a Poisson ratio 0.3  = . Figure 1 demonstrates the distributions of plastic strains along the radial coordinate.

(a)

(b)

Fig. 1. Distribution of (a) plastic radial strains and (b) plastic hoop strains along the radial coordinate for different values of parameter  at * 0 1.2%  = , 2 0.4 p = .

Figures 2 – 6 demonstrate the calculated distributions of residual stresses along the radial coordinate of the plate and through its thickness for different plate thicknesses and parameters of the plastic strain field. The dashed lines in Figures 3 and 4 show the dependence of radial and hoop stresses calculated using the formulas corresponding to the solution of the thin plate problem under the assumption of zero normal zz  and shear stresses rz  , that is, according to the plane stress state (PSS) formulation (Nedoseka, 2008). The graphs are plotted for dimensionless stresses normalized by * 0 G  (indicated in the Figures 2 – 6 by a “~” symbol above the corresponding parameter).

(a)

(b)

Fig. 2. Dependence of residual (a) radial stresses and (b) circumferential stresses on the plate surface versus the radial coordinate for different plate thicknesses h at 2 3  = . The dashed curve demonstrates the corresponding dependence for the plane stress state.

The results shown in Figure 2 testify that the residual stresses caused by welding processes, for which the plastic strain field can be described by expressions (9), computed on the basis of the three-dimensional and two-dimensional plate theories, are in good agreement for plates of about 2 mm thickness. The qualitative nature of the variation of residual stresses along the radial