PSI - Issue 28

Mikhail Perelmuter et al. / Procedia Structural Integrity 28 (2020) 2320–2327

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M.N. Perelmuter / Structural Integrity Procedia 00 (2020) 000–000

Fig. 2. Normal u y and shear u x crack opening at the bridged zone edge.

Fig. 1. Crack with two bridged zones at the materials interface.

homogeneous materials (Weitsman (1986); Rose (1987); Budiansky et al. (1988); Willis (1993); Chen et al. (2011)) and recently the bridged crack model have been extended and developed for interfacial cracks, see Goldstein and Perelmuter (1999); Perelmuter (2007, 2013). New directions in the application of the bridged crack model arise in the analysis of cracks self-healing process, which can be considered as the process of restoring bonds between the fracture faces (bridged zone formation), Perelmuter (2020). It is necessary to use the special fracture criterion to analyse bridged cracks growth which enables to consider the bridged zone and the whole crack length changing during cracks development. The nonlocal fracture criterion for the analysis of quasistatic growth of interfacial bridged cracks has been initially proposed in Goldstein and Perelmuter (1999) and was further extended and analyzed in Perelmuter (2007, 2015). This criterion is based on two fracture conditions: a) the energy condition for the crack tip advancing; b) the kinematic condition is used to determine the crack bridged zone trailing edge advancing. In the frames of this criterion it is possible to consider independently the crack bridged zone length changing and the crack tip advancing. Let’s consider a straight interfacial bridged crack of length 2 , with two bridged zones of size d , see Fig. 1. To describe mathematically the interaction between the crack faces, we assume that there exist bonds with nonlinear deformation law between the faces of the crack in the bridging zone as in Goldstein and Perelmuter (1999). The tractions in the bonds between the interfacial crack faces are the result of the external loading action. These tractions have the normal q y and tangential q x components even for the uniaxial tension case. The crack faces are loaded by the normal and tangential stresses which are numerically equal these tractions. The bridged cracks quasistatic growth criterion is based on the following two conditions: 1) the necessary energy condition - this is the equality of the deformation energy release rate (the energy flux at the crack tip, G tip ( d , )) and the rate of deformation energy absorbed by bonds in the crack bridged zone ( G bond ( d , )); where Π is the total potential energy of the elastic body, U is the deformation energy of bonds in the crack bridged zone, b is the body thickness, G Ic can be regarded as the material intrinsic toughness; 2) the su ffi cient condition - this is the equality of the crack opening at the trailing edge of the bridge zone u ( − d ) and the bond limit stretching δ cr : u ( − d ) = [ u 2 x ( − d ) + u 2 y ( − d )] 1 / 2 = δ cr , (2) where u x , y ( − d ) are the components of the crack opening at the trailing edge of the bridged zone, see Fig. 1. Conditions (1)-(2) represent the nonlocal fracture criterion for quasistatic growth of bridged cracks. Fulfilment of the G tip ( d , ) = G bond ( d , ) , G tip ( d , ) = − ∂ Π ∂ , G bond ( d , ) = ∂ U b ∂ + G Ic , (1) 2. Bridged cracks growth criterion