PSI - Issue 79

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

381

Fig. 2. Components of the bridged interface crack opening u x , y and bonds traction q x , y at the bridged zone trailing edge, d is the bridged zone length.

Fig. 1. Crack with two bridged zones of length d at the materials interface under the tension loading at infinity.

Further in the paper we will use the bridged crack model with the following assumptions: 1) the fracture process is localized at the crack bridged zone and at the crack tip; 2) materials ahead of the crack tips are assumed to be linearly elastic and deformation of these materials ahead of crack tips occurs together with the infinitely thin adhesion layer without loss of its continuity; 3) the total stress intensity factor due to the action of the external loading and the bridging traction is not zero.

3. Nonlocal criterion of bridged crack growth

We will consider a straight crack of length 2 ℓ at an interface of two dissimilar materials with bridged zone of size d (Fig. 1) and present the following relations used in the nonlocal fracture criterion frameworks (Goldstein and Perelmuter, 1999): 1) the deformation energy release rate associated with the modulus of the bridged crack stress intensity factors G tip ( d , ℓ ) = k 1 + 1 µ 1 + k 2 + 1 µ 2 K 2 I + K 2 II 16 cosh 2 ( πβ ) , β = ℓ n α 2 π , α = µ 2 κ 1 + µ 1 µ 1 κ 2 + µ 2 , (4)

2) the rate of the energy absorbtion by bonds at the crack bridged zone

u ( ℓ − d ) 0

∂ u y ( x ) ∂ℓ

q x ( u ) dx − G b + G m , G b =

ℓ ℓ − d

∂ u x ( x ) ∂ℓ

q y ( u ) +

G bond ( d , ℓ ) =

σ ( u ) du

(5)

In relations (4)-(5) the following notations is used: κ 1 , 2 = 3 − 4 ν 1 , 2 in the case of plane strain or κ 1 , 2 = (3 − ν 1 , 2 / (1 + ν 1 , 2 ) in the case of plane stress state; K I = K I , ext − K I , b K II = K II , ext − K II , b , (6) where K I , ext , K II , ext and K I , b , K II , b are the SIF caused by the external loads and bonds stresses, respectively, G m is the energy parameter that determines the crack resistance in a small zone ahead the crack tip and the quantity G b is the density of the deformation energy released upon breaking bonds at the bridged zone trailing edge. For a homogeneous material or an adhesive layer between di ff erent materials, we assume that the bond deformation laws at the crack bridged zone and on its continuation crack ahead are the same, i.e.,

δ cr 0

G m = G b =

σ ( u ) du ,