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

127

2

The studies on fracture criteria have primarily focused on two-dimensional problems involving cracks and rigid line inclusions. These investigations typically examine the behavior of cracks and inclusions within planar geometries, using simplifying assumptions such as plane stress or plane strain. The fracture criteria are linked to stress intensity factors, which quantify the magnitude of the square-root stress singularity at the defect tip. While 2D models provide valuable insights into fracture mechanisms, they often overlook the complexities of three-dimensional structures. Berezhnitsky and Kundrat (2000) studied fracture mechanisms caused by debonding and rupture of inclusions. Gdoutos (1981, 2020) introduced the idea of using Sih’s stationary energy density criterion (Sih, 1991) for studying the limiting state of bodies with rigid thin or sharp-edged inclusions. Dal Corso et al. (2008), Giarola et al. (2022), and Misseroni et al. (2014) provided extensive theoretical and experimental studies on thin rigid inclusions in prestressed and ductile materials. Kundrat (2024) examined the delamination of rigid line inclusions under cyclic loading. Nevertheless, fracture problems involving thread-like inhomogeneities differ significantly, as they are inherently three-dimensional. Pasternak et al. (2022) introduced a line model for thread-like inhomogeneities in thermoelastic materials, which captures the 3D nature of these problems. The proposed model replaces the inhomogeneity with a spatial curve along which heat sources and body forces are distributed with specified intensities, simulating the effect of the inhomogeneity. This approach significantly reduces the number of degrees of freedom in the system while maintaining high accuracy and simplifying the resulting equations. This paper focuses on analyzing fracture criteria in materials containing such defects, as the model can be applied to fiber-reinforced composites (notably fiber-reinforced concrete) and textile composites, which are commonly used in engineering applications. The introduced model enhances the ability to predict fracture initiation and progression more accurately by providing a more realistic representation of 3D stress states and interactions. 2. Fracture of the inhomogeneity In modeling thread-like inhomogeneities, Pasternak et al. (2022) derived a 1D line model for heat-conductive, thermoelastic thread-like inclusions embedded in a 3D thermoelastic medium. This mathematical model simplifies the complex 3D interaction by reducing the influence of the inhomogeneity to a set of influence functions, which are distributed along a spatial curve , representing the midline of the thread-like inclusion with volume . By replacing the 3D inclusion with this spatial curve, the model captures the essential thermal and mechanical effects exerted by the inhomogeneity on the surrounding material, significantly reducing the computational complexity while preserving the accuracy of the solution. The inhomogeneity influence functions are defined as ! ! !

!

!

(

)

( ) !

( ) # # $ $ #

" %

+ ! " !

(1)

! "

"

=

#

for heat conduction and

!

!

(

)

( ) !

( ) # " # # $

" $

+ ! " !

(2)

"

#

=

"

!

!

, ! ! ! ( ) ( ) ! ! = ! " # "

( )

( ) ! "

( ) ! "

for thermoelasticity, where vectors of the curve !

;

and

are unit normal and binormal

$ "

"#$

$%&

!

+

!

!

at the point

, respectively;

is the radius of a cross-section of the thread-like is the heat flux at the surface of a medium, which is

!

inhomogeneity; is a polar angle in the plane normal to ; ! !

! "

in the contact with the inhomogeneity; is a corresponding traction vector. Pasternak et al. (2022) also introduced mathematical models for heat conduction and thermoelasticity of a thread like inhomogeneity, which is modeled as a spatial curve . The fracture criterion for these inhomogeneity models is linked to the internal fracture of the inclusion. This internal fracture occurs when the thermal and mechanical stresses within the inclusion exceed a specific critical value . ! " !

!" !