PSI - Issue 68

Yu.G. Matvienko et al. / Procedia Structural Integrity 68 (2025) 641–645 Matvienko, Pokrovskii / Structural Integrity Procedia 00 (2025) 000–000

642

2

also included in the fracture criterion. Despite the achieved progress in the area under consideration, the problem of fracture toughness assessment of structural elements, taking into account in-plane and out-of-plane constraints along the crack front, is not completely solved yet. Thus, it is necessary to introduce some additional parameters into the basic equations and fracture criteria of fracture mechanics to characterize in-plane and out-of-plane constraints in the vicinity of the crack front. It should be noted that in-plane constraint is controlled by type of loading and geometric parameters of specimens, while out-of-plane constraint is associated with specimen thickness. Currently, there is no brittle fracture criterion that would allow assessing the fracture toughness taking into account in-plane and out-of-plane constraint effects simultaneously. So, in addition to the stress intensity factor, such a criterion should include the Т хx - and Т zz - stresses which describe in-plane and out-of-plane constraint effects in the transverse ( Т xx ) and longitudinal ( Т zz ) directions along the crack front in three-dimensional solids. The aim of this paper is to propose brittle fracture criteria capable of describing in-plane and out-of-plane constraints in the vicinity of the crack front. To confirm the adequacy of developed fracture criteria, it is necessary to carry out verification.

Nomenclature K I

stress intensity factor fracture toughness

K Ic

r c size of the fracture process zone T xx , T zz components of nonsingular T-stress σ Y stress yield σ 0 local strength of the material

2. The fracture criterion based on two-dimensional crack-tip constraint For a linearly elastic isotropic solid, three-dimensional crack tip stress field in the vicinity of the crack front under mode I loading conditions, taking into account the nonsingular terms ( Т хх and Т zz stresses), is given by Nakamura and Parks (1992) in the following form

"

" #$% & %'( %'( ) ! ! ! " # +

$

$

! ! !

" & )

# ' *

#$% & %'( %'( %

)

%

$ =

+

$ =

!

!

& )

' *

!

""

#

*

* *

*

* *

*

*

&

&

(

(

(1)

*

" %'( #$% #$% ) * * * ! ! !

$

$

µ

!

#$%

) % % ' % = , +µ

+

!" + =

$ =

+

!

!

#

##

##

#

!!

*

*

*

&

&

(

(

Where are the stress state components; K I is the mode I stress intensity factor (SIF); Т хх and Т zz are Т -stresses in the plane of the crack directed in the transverse and longitudinal directions along the crack front, respectively; ε z is the strain along the crack front; Е is the Young modulus; μ is the Poisson’s ratio; and r and Θ are polar coordinates. According to expressions (1), the stresses in the crack plane (at Θ = 0) are given by the following equations ! ! ! ! " # !" ! ! ! "

"

!

!

!

µ

(2)

#

#

#

$ %

"

"

! ! = +

! =

! ! = +

"# " =

!

!

""

#

$

$$

"

"

"

#

#

#

#

#

#

To write the criterion equation, the generally accepted approach is employed, according to which the maximum tangential stresses in the fracture process zone are equal to the critical value. Obviously, the tangential stresses are stresses σ у . The local strength of the material is used as the critical stress. Then, according to (2), the criterion equation has the following form