PSI - Issue 13

Ivan Shatskyi et al. / Procedia Structural Integrity 13 (2018) 1482–1487 Author name / Structural Integrity Procedia 00 (2018) 000–000

1485

4

 

  x

 

 

u x

u x

B

B

n l

s l

y

1

2

lim x l 

,

,

K

l

lim x l 





K

l





N

S

4

1

4

1



1 



2 

2 2

2 2

l

x

l

x



 

  x

  x

  x

y   

B

m l

3

4

lim x l 

.

(8)

,

K

l

lim x l 

0





K B l  



M

H

4

1

4



3 

2 2

2 2

l

x

l

x





By placing the result (7) into presentations (4), we also obtained expressions for forces and moments in the filler:

s

m

1 

2 

3 

0 xy M  .

,

,

,

(9)

1 y N n  

N



M



xy

y

1

1



1 

2 





3

By taking into account results (8), (9), we can resume the distribution of characteristics of stressed state by the width of the plate and inclusion respectively:   1 2 2 1 3 1 3 1 3 2 2 1 1 N M z n l z m l k z K K h h h h                       ,   2 2 1 1 3 1 2 2 1 S H z n l k z K K h h h                    ,   3 1 2 2 1 3 1 3 1 3 2 2 1 1 y y y m n z z z N M h h h h                        ,   2 2 2 1 3 1 2 2 1 xy xy xy s z z N M h h h                    . (10) 3.2. Critical equilibrium Let us examine two mechanisms of fracture: cracking of the plate in places of high concentration of stresses near the peaks of the crack and disruption of integrity of the injection material, which assumes part of the external load. For the first case, we will use the local force criterion of linear fracture mechanics (Panasyuk (1968)):   1 max c z eq K k z   , (11) and for the second we will use the classical theory of the filler strength:     0 max z eq z    . (12) Here k eq , σ eq is equivalent coefficient of the stress intensity and equivalent stress respectively; K 1c is crack resistance of the plate material and 0 [ ]  is admissible stress for the filler material. 3.3. Example. Tension with shear Let us assume that 0, 0 n s   . In this case, by using the   -criterion (Panasyuk (1968)) for mixed mode of brittle fracture of the types tension-shear, we will find:

2





1 sgn 1 8 k  

k k

 

 

 



1

2 1

2

2arctan

.

(13)

,

 



2 3 tan cos 2  

k

k k

  





1

eq

4

k k

2

2 1

With the help of the theory of maximum normal stresses in the inclusion, we will accept

2 4     

2 y

y

xy

.

(14)





eq

2

By placing expressions (10), (13) and (14) into criteria (11) and (12), after transformation, we obtained curve equations, which limit safe areas, where the integrity of the plate and the inclusion are secured respectively: it safe areas, where the integrity of the plate and the inclusion are secured respectively: