PSI - Issue 79

Mikhail Perelmuter et al. / Procedia Structural Integrity 79 (2026) 379–385

380

The bridged crack model is more general and includes the cohesive model as a particular case, (Sun and Jin, 2006; Stang et al., 2007). The bridged zone length can be compared to the whole crack length or occupy full crack space. To modelling growth of cracks with bridged zone at least two separate conditions are required: one for the trailing edge of the bridging zone growth and one for the limit state of the crack tip (Cox and Marshall, 1994; Morozov et al., 1997; Grekov and Morozov, 2006). It should be noted that the criteria used in the cited papers do not take into account the energy expended during bonds deformation. In this paper for modelling cracks growth with bridged zones we use the nonlocal two-parameter criterion proposed in (Goldstein and Perelmuter, 1999; Perelmuter, 2007, 2015). This fracture criterion based on the necessary energy condition of the crack tip limit equilibrium and the su ffi cient condition of the bond limit stretching at the trailing edge of the bridge zone. From these two conditions the equilibrium size of the bridged zone and the critical external stress can be determined and also equilibrium and growth regimes of the bridged zone and the crack tip can be analyzed. This paper is organized as follows. In section (2) the brief description of the bridged crack model presented; section (3) describes the nonlocal fracture criterion for two-dimensional bridged cracks problem, the regimes of bridged cracks growth and small scale bridged zone case are also defined. In section (4) the application of this criterion in the case of a straight interface bridged crack is presented and the regimes of bridged cracks growth are illustrated. In section (5) final comments are given. The main statements of the interface bridged crack model proposed in (Goldstein and Perelmuter, 1999; Perelmuter, 2011) are presented shortly below. Let us consider a straight crack of length 2 ℓ at an interface of two dissimilar materials with shear modulus µ 1 , 2 and Poisson factors ν 1 , 2 , Fig. 1. Assume that the uniform tensile stress σ 0 is applied at infinity transverse to the interface. We suppose that the crack parts started from the crack tips (( ℓ − d ) ≤ | x | ≤ ℓ, y = 0) are occupied by bonds connecting the crack faces. We will call these parts of cracks bridged zones; noted that the size of these zones can be comparable to the whole crack length (0 < d /ℓ ≤ 1). Under external loads (even for normal to a crack line) in bonds at the bridged zone of a straight interface crack traction with normal q y ( x ) and tangential q x ( x ) components (Fig. 2) are arisen and a general form the traction-separation law (bonds deformation law) can be written as u x ( x ) u y ( x ) = c x ( x ,σ ) 0 0 c y ( x ,σ ) q x ( x ) q y ( x ) , (1) where u x , y ( x ) are the crack opening components along the axes coordinates (Fig. 1), c x , y ( x ,σ ) are the quasi-linear bonds compliances, which can vary along the bridged zone and also depend on bonds tension σ ( x ) = q 2 y ( x ) + q 2 x ( x ) and these compliances are defined as c x , y ( x ,σ ) = φ x , y ( x ,σ ) H E b , (2) where φ x , y ( x ,σ ) are dimensionless functions which defines variation of bonds compliance over bridged zone and bonds tension σ ( x ), H is a length dimension parameter proportional to the thickness of the interphase layer between the materials and E b is the bond elasticity modulus. If functions φ x , y ( x ,σ ) are constant, then expression (2) determines the linear elastic law of bonds deformation with the constant bond compliance along the crack bridged zone. The dependence these functions on the bond stress arises when considering the nonlinear laws of bonds deformation. Next, we introduce the relative bond compliance c 0 as 2. Bridged crack model

H ℓ

c 0 =

(3)

which defines the stress state in the crack bridged zone for fixed materials elastic properties. Note that in various models the crack process zone need to introduce, in addition to the bridged zone length, supplemental parameter with dimension of length ( H in this case), which defines bonds compliance (sti ff ness) in the crack bridged zone (Rose, 1987).