Issue 74

D. L. Zaidan et alii, Fracture and Structural Integrity, 74 (2025) 42-54; DOI: 10.3221/IGF-ESIS.74.04

minor and the major width of the specimen’s cross-section, y is a vertical distance and dy is a differential of vertical distance. And c is the semi-height and h is the height of the specimen’s cross-section. y y is the elastic-plastic border. Residual stress basics Every step of the residual stress part of the analytical model is explicitly presented and commented on. The value of the bending moment M( ρ i) imposed by the forming tool is inside the range M y < M( ρ i) < M p . The maximum elastic moment M y and the plastic moment M p equations are presented.

2

y

1       2 b

 

  

c



2

2 y dy

2   c

2 M y   

 

2  



S

radius

radius

(01)

y

y

0

3

3

c

0

   

    

  

  

  

 

y

c



2

2 y dy

2      y 

2 M y S E       



radius

(02)

p

y

t

0

3

i 

0

0     ut S S

c

c

c

y

0

i 











p 





radius

radius

c

(03)

E

y

1

2

t

a 



u 

y

u

y

Where S y0 is the material yielding resistance and S ut0 is the material's ultimate resistance, both, as received, E t is the plastic tangent modulus, i  , y  , and p  are the radius of curvature of the specimen, respectively, at loading, at the beginning of yielding, and the full cross-section yielding. The ε a is the strain at point a of Fig 1.a, ε y is the yielding strain, and ε u is the ultimate strain. The elastic-plastic border y y equation is presented:



.   y i

    

y

(04)

y

p

i

y

  i M  in function of the imposed radius of curvature of the specimen:

The equation of the bending moment

    

    

   

   

  

  

  

 

y

y

y

c

  





2

2 y dy   

2

2 y dy

y

2  





2  

 

2      y 



M

y

S

radius

y S E 

radius

i

y

y

t

0

3

0

3

.   y i

i 

y

0

y

  i p M M M    y

(05)

The stress  and strain  evolution of the specimen point a (shown in Fig. 1.a) is presented in Fig. 2.a, whose values are shown in Eqn. (06), Castro and Meggiolaro [6]. 0   0 0 0  

1 0 y S  

1 y   



1

y y       c

  

  

y

2 

S E   





2 







y

t

y

y

0

i 



 



1

y  

  c            0 y S E t y sb i  y   

y

  



c          

3 



(06)

y

3

44