PSI - Issue 28

Mikhail Perelmuter et al. / Procedia Structural Integrity 28 (2020) 2320–2327 M.N. Perelmuter / Structural Integrity Procedia 00 (2020) 000–000

2322

3

necessary and su ffi cient conditions (1)-(2) corresponds to the limit equilibrium state of the crack tip and the trailing edge of the crack bridged zone. The parameter δ cr is defined by the bonds properties in 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. (1)-(2) it is possible to determine the bridged zone size d cr and the critical external stress σ cr in the crack limit equilibrium state. The deformation energy absorption rate obtained from this solution is the energy characteristic of the adhesive fracture toughness, G cr = G bond ( d cr , ), and the quantity G cr does not remain constant when the crack length changes. Noted, the approximated equation for a small scale bridged zone (obtained by J-integral approach) similar to (1) was used in (Budiansky et al., 1988). The expression for the deformation energy release rate in the case of a crack on the interface of di ff erent materials still holds when there are bonds within the crack bridged zone since the e ff ect of the bonds is expressed in the application of the loads to the crack faces in the bridged zone. Hence, regardless of the bond deformation law, the deformation energy release rate is given by the expression (Salganik, 1963) G tip ( d , ) = κ 1 + 1 µ 1 + κ 2 + 1 µ 2 K 2 B 16 cosh 2 ( πβ ) , β = n α 2 π , α = µ 2 κ 1 + µ 1 µ 1 κ 2 + µ 2 , K B = K 2 I + K 2 II , (3) where 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, ν 1 , 2 and µ 1 , 2 are Poisson’s ratios and the shear moduli of jointed materials 1 ( y > 0) and 2 ( y < 0), see Fig. 1, K B is the modulus of the stress intensity factors (SIF) and it defines on the basis of the SIF K I , II K I + iK II = ( K ext I + K int I ) + i ( K ext II + K int II ) , i 2 = − 1 , (4) where K ext I , II and K int I , II are the SIF caused by the external loads and bonds tractions; note that K int I , II < 0. The deformation energy of the bonds in the crack bridged zone can be defined as follows

u ( x ) 0

− d

σ ( u ) du , u ( x ) = u 2

2 y , σ ( u ) = q 2

2 y ,

U = b

Φ ( u ) dx ,

Φ ( u ) =

x + u

x + q

(5)

where Φ ( u ) is the density of the bonds deformation energy in the crack bridged zone, u x , y ( x ) are the components of the crack opening in the bridged zone, q y ( s ) and q x ( s ) are the normal and tangential components of bonds traction and σ ( u ) is the modulus of the traction vector in bonds. Substituting the first two expressions from (5) for the last equality of (1), we obtain

   u ( x ) 0

σ ( u ) du  

− d

∂ U b ∂

∂ ∂

G bond ( d , ) =

+ G Ic =

 dx + G Ic

(6)

The derivative of the integral in expression (6) will be obtained with the assumption that the change in the bridged zone size can be the result of the bonds rupture at the trailing edge of the bridged zone (when x = − d ) and also the result of the simultaneous crack tip advancing. Thus, the bridged zone length can be changed during crack growth (non self-similar crack growth). The autonomy condition for the crack bridged zone is fulfilled in the limit case of a long crack. Di ff erentiating with respect to the upper and lower limits in Eq. (6), we obtain (Perelmuter, 2007)

u ( − d ) 0

− d

σ ( u ) dx + G Ic − G b , G b =

∂ u ( x ) ∂

σ ( u ) du ,

(7)

G bond ( d , ) =

where in the crack limit equilibrium state u ( − d ) = δ cr (according to condition (2)) and the quantity G b in (7) is the deformation energy density released during bonds rupture in the bridged zone trailing edge x = − d . The relations between bond tractions and displacements di ff erence of the upper and lower crack faces (the crack opening, see Fig. 2) in the crack bridged zone are used in the following generalized form (Goldstein and Perelmuter, 1999) q y ( x ) − iq x ( x ) = c ( x , u )( u y ( x ) − iu x ( x )) (8)