PSI - Issue 68

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

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2

for bridged interfacial cracks between two semi-infinite plates were solved by Goldstein, Perelmuter (1999) and Perelmuter (2011) by the singular integral-differential equation method (planar problems). In the last two decades a number of papers were devoted to the application of the boundary integral equations (BIE) method to the SIF computation for cracks on bi-material interfaces in finite size structures, see, for example Yuuki and Cho (1889), Raveendra and Banerjee (1991), Hadjesfandiari and Dargush (2011), Gu and Zhang (2020). In all these papers interaction between crack surfaces was not assumed nor considered. Only few papers have been directed to analysis two-dimensional problems for bridged cracks in finite size structures, see Liu et al. (1998), Selvadurai (2010), Perelmuter (2013). The main objectives of this paper are the stress analysis of composite structures with bridged cracks and damaged zones and the SIF computation using the direct boundary integral equations method in axisymmetric formulation. 2. Boundary integral equations formulation for axisymmetric bridged interfacial crack The modeling of bridged interfacial cracks is based on the multi-domain BIE formulation. Within this approach, direct boundary integral equations for elasticity problems are used for each homogeneous sub-region of the structure. The supplementary boundary conditions at the interfacial boundaries and at the bridged zones of cracks are introduced and used to eliminate additional variables on joint boundaries of sub-regions. The direct BIE for axisymmetric elasticity problems without body forces for any sub-region of the structure is given as Brebbia et al. (1984):

( ) ( ) " ! # $% $ & $'('F ' G $'('% ' +,' !" (- ! " # = $ = % & ' ( ) ( ) ( ) ! ( ) ( ) ( ) ( ) ! ! ! ! !" !" " " "

(1)

! !

where is the line that, when rotated about axis symmetry , forms the axisymmetric surface ! !

, the

" # !" ! ! = "

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

are displacements and tractions in the direction at field point , due to unit ! !

kernel functions

( ) ! !" # $ %

ring load in the direction, applied at source point . The expressions of the kernel function can be derived by integration of the fundamental solution for three-dimensional case over an axisymmetric ring for radial and axial loads and the expressions for kernel function are calculated by differentiation respect to and according to stress-displacement relations in cylindrical coordinates. The coefficient in Eq. (1) depends on local geometry of the boundary at a point , for a smooth boundary it is . If the point lies on - axis then must consider limiting relations of the axisymmetric kernels. The displacements continuity and the tractions equilibrium supplementary conditions at the interfacial boundaries without cracks are the following ! ! ( ) ! !" # $ % ! ! ( ) !" # $ ! ( ) !"# !" !" # $ ! = ! !

( )

( )

(2)

! # $ # $ # $ % & % & ' & ' & = = ! ! # $ ! ! ! " # " # ! " ! "

( ) ! ! " # $

!

where and !

are joint sub-regions numbers,

and

are displacements and tractions components at

! " # ! " # $

the boundary point . The relationships between bond tractions and displacements difference of the upper and lower crack surfaces (the crack opening, see Fig. 1) at the crack bridged zone or at the damaged interface zone can be written in the following generalized form (the bonds deformation law) (3) where and are the components of tractions vector and crack opening in the local coordinate system connected with the axial and radial directions at the point , are the stiffness of bonds depending on the distance from the crack tip, is the tractions vector modulus at the current point , is length dimension parameter proportional to the bonding zone thickness, is the effective elasticity modulus of the bond. The boundaries of all sub-regions of the structure are subdivided into quadratic isoparametric elements for numerical solution of the BIE (1). For the interfacial crack displacements and stresses asymptotic modeling the special crack tip ! ( ) ( ) ! ! " " " " " " " " # $ # " $ # $" # $ " # " $ # " $ " ! " # $ % $ % $ % $ % $ % $ % $ % $ % $ % & D E E ) E ) E ) E ) E E E D * D ! " ! " # " " = = $ = = + % % ( ) ! ! " # $ ( ) ! ! " # $ ! ! ! ! ( ) ! ! ! " # ! " ! ! ! ! "