PSI - Issue 81

Ivan Shatskyi et al. / Procedia Structural Integrity 81 (2026) 129–134

132

  

   

4| |

1

m h

x



2       

2       



  

  

2

2

(14)

[ ]( ) u x

Atanh cos 

tan

tan

,







y

(1 ) 

2

B

l

 

and from expression (6) – the contact reaction on crack faces:

| | m  



(15)

( ,0)

.

y N x

1

h





3. Results and Discussion 3.1. Intensity Factors and Limit Load Stress and moment intensity factors characterize the singularity of stress fields near crack tips. Based on solutions (13), (14), accounting for crack face contact, we obtain:

| | m

l 

m

l 



(16)

tan ,

tan .

K

d

K

d

 







N

M

1

1

h

d

d









Crack propagation as a through-thickness object can be evaluated by the thickness-averaged energy flux into its tip (Wynn and Smith, 1969; Shatskyi et al., 2024):

2

K

2  

  

2 N

M

K

  

h

(17)

* 2 ,    

G   

(2 )

h E

 

where E   is the thickness-averaged elastic modulus, From this, we find the critical value of the bending moment:

*    is the average specific surface energy.

1

1

cot . l 





2

|

| 2 2 m h E 

(18)

    



*

*

d d



3.2. Example Calculation for Different Inhomogeneity Profiles Consider a symmetrical three-layer structure with parameters 0.3   . Case 1: Steel-Aluminum-Steel (S-A-S) – configuration with stiffer periphery: outer E Case 2: Aluminum-Steel-Aluminum (A-S-A) – configuration with rigid core: outer E 0.5   and

inner 200 GPa, 70 GPa. E  inner 70 GPa, 200 GPa. E 





Substituting these values into formula (11), we obtain:

0.876,

1.867.









(19)

S-A-S

A-S-A

Figs. 1 – 3 present calculation results according to formulas (16) – (19) for both configurations. Analysis of the results allows the following observations.

As cracks approach each other ( / (2 ) 1 d l  ), intensity factors increase sharply, and the load-bearing capacity decreases catastrophically regardless of the inhomogeneity profile. This is explained by the mutual amplification of stress fields from neighboring cracks. The stress intensity factor N K for the A-S-A configuration is higher than for S-A-S. Conversely, the moment intensity factor M K for the S-A-S configuration is higher than for A-S-A. This is because a smaller value of  means a smaller fraction of energy transferred to the membrane component. The limit load for a plate with a stiffer periphery (S-A-S) is significantly higher compared to a plate with a rigid core (A-S-A). This non-obvious result is explained by the inequality S-A-S A-S-A    (see (19)). As the distance between cracks increases ( / (2 ) d l  ), the results asymptotically approach the case of a single isolated crack.