PSI - Issue 79

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

383

The solution of the criterion equations (7) and (8) for linear elastic bonds at a constant compliance along the crack bridged zone is graphically shown in Fig. 3 and Fig. 4 for three values of the relative bond compliances - c 0 = 0 . 1 , 0 . 25 , 0 . 5. On the graphs in Fig. 3 the following notations are used

G tip ( d , ℓ ) G tip (0 , ℓ ) ,

G bond ( d , ℓ ) G tip (0 , ℓ ) ,

¯ G bond =

¯ G tip =

where G tip (0 , ℓ ) is the energy flux into the crack tip in bonds absence. The position of intersection point of graphs functions ¯ G tip and ¯ G bond (points A 1 , 2 , 3 ) corresponds to the relative length of the critical crack bridged zone t cr = d cr /ℓ , it is the solution of equation (7). The critical external load σ cr , corresponding to the obtained value of t cr and to the critical crack opening at the bridged zone trailing edge is determined from the solution of equation (8). This solution can be illustrated in Fig. 4, where the dependencies of the crack opening at the bridged zone trailing edge (at x = ℓ − d ) versus the bridged zone relative length and also the critical line u = δ cr are shown. Each graph in Fig. 4 shows the crack opening u ( ℓ − d ) computed at the critical external load ( σ 0 = σ cr ) obtained from equation (8) for every of the three compliance values. The solution t cr , satisfying both equations (7) and (8) corresponds to the intersection point positions (points A 1 , 2 , 3 ) of the critical line u = δ cr with dependencies of the crack opening, see Fig. 4. The second solution of equation (8) (points B 1 , 2 , 3 in Fig. 4) also existing for the critical load σ cr does not fulfill condition (7) and, therefore, it does not determine the state of limiting equilibrium of the bridged zone and the crack tip.

(a)

(b)

(c)

Fig. 3. The deformation energy release rate G tip ( d , ℓ ) and the rate of the energy absorbtion by bonds G bond ( d , ℓ ) versus d /ℓ (the relative bridged zone length), c 0 = H /ℓ is the relative bond compliance, t cr = d cr /ℓ is the solution equation (7): (a) t cr ≈ 0 . 041; (b) t cr ≈ 0 . 105; (c) t cr ≈ 0 . 225.

(a)

(b)

(c)

Fig. 4. The crack opening at the bridge zone trailing edge x = ℓ − d versus the bridge zone relative length d /ℓ ; the critical external load of the crack quasistatic growth start: (a) σ cr ≈ 25 . 1 MPa ; (b) σ cr ≈ 16 . 1 MPa ; (c) σ cr ≈ 11 . 7 MPa .