PSI - Issue 68

M. Perelmuter et al. / Procedia Structural Integrity 68 (2025) 219–224 Author name / Structural Integrity Procedia 00 (2025) 000–000

221

3

! !

elements are used. For any point at the interface of joint sub-regions or at the interfacial crack bridged zone

nodal

! ! =

unknown functions (

for axisymmetric problem) need to be defined. The multi-domain BIE formulation

unknown functions ( from !

! !

! !

provides

equations, and the auxiliary equations are generated for the remaining

the continuity condition and bridged zones, see (2), or !

from the equilibrium condition across the joined parts of subregions for cases without

!

from the equilibrium conditions and from the bond deformation law (3) for bridged zones of interfacial cracks). The displacements and tractions are regarded as the nodal unknowns on jointed parts of sub-regions whereas the displacements of the upper and lower crack surfaces are regarded as the nodal unknowns on interfacial crack bridged zones and the additional tractions are eliminated by substitution of the bond deformation law (3) into BIE (1). !

with bridged zone of width , are components of the crack opening and bonds tractions. ! !

Fig. 1. Interface crack of radius

! ! ! ! " ! " # $

Fig. 2. SIF module vs crack bridged zone relative width.

3. Computational results The algorithms of numerical solution BIE system (1) for structures with bridged interfacial cracks and the SIFs computation were implemented as the computer code for axisymmetric elasticity problems solution. The quarter-point boundary elements with tractions modification (Martinez and Dominguez, 1984) were used. The SIF module and components for modes are calculated according to the following relations (Perelmuter (2013)) ! ! ! ! !!

! ! !! "

!

!

! $ % & ! =

#

! # !

!

&

$ & =

% $ & =

%

+

!

"

!

"

#

!

##

"

where is the length of boundary element jointed to the crack tip, !

are the modified tractions computed at the

! ! " #

crack tip.

! ! " ! " ! " =

Table 1. Nondimensional SIF for the circular interface crack,

! " # µ µ

! " ! " "

! " !! " "

Murakami (1987)

BIE

Murakami (1987)

BIE

1 4

1.0000 0.9979 0.9960 0.9948 0.9941

1.0200 1.0015 0.9849 0.9751 0.9693

0.0000 0.1315 0.1809 0.2052 0.2186

0.0000 0.1367 0.1855 0.2085 0.2209

10 25

100

!

To verify the implemented BIE solution algorithm the problem for circular interface crack of radius

between two

! !

half-spaces under external tension region divided into two subregions (

was considered. The computation was performed for the finite cylindrical

! !"# $ $ =

"#

!"# $

),

is the external radius of the region. In Table 1 numerical

results are shown with comparison to the exact solution, Murakami (1987),

are shear moduli of materials. The

!"# µ

numerical results have an error lower than 3%. The SIF modulus vs relative bonds stiffness was also computed for finite cylindrical region divided into two