PSI - Issue 79

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

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where δ cr is the critical bond extension at the bridged zone trailing edge. The following two conditions are lied in the base of the nonlocal fracture criterion (Goldstein and Perelmuter, 1999): 1) equality of the deformation energy release rate G tip ( d , ℓ ) and the rate of deformation energy absorbed by bonds at the crack bridged zone G bond ( d , ℓ ) (the necessary energy condition): G tip ( d , ℓ ) = G bond ( d , ℓ ) , (7) 2) equality of the crack opening at the trailing edge of the bridge zone u ( ℓ − d ) and the bond limit stretching δ cr (the su ffi cient condition): u ( ℓ − d ) = [ u 2 x ( ℓ − d ) + u 2 y ( ℓ − d )] 1 / 2 = δ cr , (8) where u x , y ( ℓ − d ) are the crack opening components at the bridged zone trailing edge, see Fig. 1. Conditions (7) and (8) represent the nonlocal fracture criterion for quasistatic growth of bridged cracks. Fulfilment of these conditions corresponds to the limit equilibrium state of the crack tip and the crack bridged zone trailing edge. The parameter δ cr is defined by the bonds properties at the crack bridged zone and can also depend on the crack scale (for example, when the type of bonds changes as the crack grows). From the simultaneous solution of Eqs. (7)-(8) it is possible to determine the bridged zone size d cr and the critical external stress σ 0 = σ cr in the crack limit equilibrium state. The following cases of the equilibrium and quasi-static crack growth can be distinguished using the nonlocal criterion of the bridged crack growth (Perelmuter, 2007): -the crack tip advancing, the crack length increases with simultaneous increasing of the crack bridged zone length without bonds breaking G tip ( d , ℓ ) ≥ G bond ( d , ℓ ) , u ( ℓ − d ) < δ cr (9) - the crack length does not increase, the bridged zone size is decreasing due to bonds breaking at the bridged zone trailing edge G tip ( d , ℓ ) < G bond ( d , ℓ ) , u ( ℓ − d ) ≥ δ cr (10) These two stages of the crack growth (9)-(10) are processes of the crack adaptability to a given level of external loads (it is subcritical cracks growth). The crack tip advancing with the simultaneous breaking of bonds at the bridged zone trailing edge occurs while the following conditions are met G tip ( d , ℓ ) ≥ G bond ( d , ℓ ) , u ( ℓ − d ) ≥ δ cr (11) The quasi-static crack growth corresponds to equal signs in relation (11). For the given bonds deformation curve (the bond stress depends on the crack opening), the nonlocal fracture criterion (7) and (8) application consists of two steps: 1) solving the problem to determine the bonds traction and the crack opening over the crack bridged zone; 2) determining the crack or / and its bridged zone propagation mode depending on conditions (9)-(11). We will assume that the bonds traction and the crack opening over the crack bridged zone are known (numerical solution methods for these problems given in (Goldstein and Perelmuter, 1999; Perelmuter, 2013)) and will consider the second step of the nonlocal criterion application in details. The following material data are used: E 1 = 135 GPa, E 2 = 25 GPa, ν 1 = ν 2 = 0 . 35 (Cu-epoxy polymer junction), E b = E 2 , φ x , y = 1 (linear-elastic bonds). Initial size of the crack half is ℓ = 0 . 5 · 10 − 3 m and the critical crack opening is taken to δ cr = 2 · 10 − 7 m . 4. Computation modelling results