PSI - Issue 68

Oleh Yashiy et al. / Procedia Structural Integrity 68 (2025) 126–131 O. Yashiy et al. / Structural Integrity Procedia 00 (2025) 000–000

129

4

object with a tangible shape and thickness. In essence, the representation of the inhomogeneity as a simple curve does not adequately capture the complex stress distribution that occurs in real materials. Therefore, it is necessary to adopt a different approach for modeling the fracture of the material surrounding the tips of the thread-like inhomogeneity. In particular, it is important to understand that the stress intensity zone near the inclusion tip can only absorb a limited amount of elastic deformation energy before reaching a critical threshold. This threshold represents the maximum energy that can be accommodated elastically within the material without leading to irreversible changes. Once this critical level of elastic deformation energy is exceeded, the material undergoes irreversible processes, which can result in either brittle or quasi-brittle failure. Brittle failure is characterized by sudden fracture without significant plastic deformation, while quasi-brittle failure involves some degree of energy dissipation through micro-cracking before complete fracture occurs. To quantify this failure mechanism, this criterion can be formulated using the following mathematical relation. This relation integrates the strain energy density over a spherical volume centered at the inclusion tip: By analyzing the strain energy density in this manner, we can obtain a clearer understanding of how the energy absorbed by the stress intensity zone influences the failure process. This mathematical approach allows for a more accurate prediction of failure in materials with thread-like inhomogeneities and can help in designing more resilient materials by providing insights into the limits of elastic deformation and the onset of fracture. 5. Numerical examples Pasternak et al. (2022) presented the thermoelastic solutions for a linear thread-like inhomogeneity placed along the section of the axis, which is expressed as a truncated Legendre series: ( ) ! ! ( ) ( ) !" ! " #! $ ! " ! ! . (7)

(

) ! ! ! !

! !"

!

" = = ! ! "

( )

( ) !

,

(8)

! # $

" " % & $

( ) ! " # !

where are Legendre polynomials of the first kind. In view of Eq. (8), the internal fracture criterion for the inclusion (5) can be rewritten as ! ! ! " = ! , and

!

!

$

$

" &

!

!

( ) % ! &! ! ! "

( ) & ' ! &! ! % ! !

'

'

.

(9)

=

#$

#

#

% "#

% "# = #

!

"

"

$

$

According to Olver et al. (2010),

! " ! ! +

"

(

)

( ) !

( ) # $ # $ + ! ! ! ! ( )

,

(10)

# $

=

!

!

!

!

!

"$

therefore, Eq. (9) can be rewritten as

#

! & !" #

!

(

)

% " + & = "

( # $ ! $

) % + +

( ) % $ % $ + $ $ ! ! ( )

.

(11)

"#

%

%

%

"

"

! % "

In the same way, the inclusion debonding criterion (6) can be rewritten using Eqs. (8) and (10) as