PSI - Issue 79

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

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Fig. 5. Dimensionless critical length of the crack bridged zone vs the relative bonds sti ff ness κ 0 = ℓ/ H for fixed crack length.

Fig. 6. Dimensionless critical external load vs the relative crack length ℓ/ H for fixed parameter H .

We further consider the modes of crack growth with an arbitrary size of the bridged zone 0 < d ≤ ℓ under the external load increasing monotonically. The crack bridge length d can lie in two di ff erent ranges: 1) it belongs to the interval 0 < d < d cr ; or 2) it belongs to the interval d cr < d < ℓ . In the first case, the inequality G tip ( d , ℓ ) > G bond ( d , ℓ ) is satisfied and the crack opening at the trailing edge of the bridged zone does not exceed the critical value ( u < δ cr ), which corresponds to the fulfillment of conditions (9) (see Fig. 3). In this case, an increase of the crack length without breaking the bonds at the trailing edge of the bridged zone is possible. When the critical size of the bridged zone and the critical value of the external load are reached, a transition to a quasistatic fracture mode described by conditions (11) occurs. In the second case the first condition of (10) G tip ( d , ℓ ) < G bond ( d , ℓ ) is satisfied and under monotonic loading condition u = δ cr is reached. The crack tip does not advance and the size of the crack bridged zone is reduced due to breaking bonds at the trailing edge of the bridged zone, d → d cr . The transition to a quasistatic fracture mode occurs, as in the first case, when the critical size of the bridged zone and the critical value of the external load are reached. Next, for three sets of mechanical properties of materials (for all three cases, it was assumed E b = E 2 ) we will consider the dependence of the crack bridged zone length in the limit state according relation (7) versus the relative bonds sti ff ness κ 0 = 1 / c 0 , see Fig. 5. The change in the relative bonds sti ff ness at the fixed crack length is controlled by variation the parameter H in the range 0 . 1 ℓ ≤ H ≤ ℓ . Increasing the relative sti ff ness of jointed materials E 1 / E 2 leads to raise of the critical bridged zone length d cr . In the considered range of relative bonds sti ff ness in the crack bridged zone, the smallest critical size of the bridged zone is observed at E 1 / E 2 = 0 . 2, due to the fact that deformation of a relatively soft materials allows to achieve the critical crack opening with a small size of the crack bridged zone. At relative bond sti ff ness κ 0 = 1 and E 1 / E 2 = 0 . 2, we have t cr = d cr /ℓ = 0 . 082. With a 10-fold increase in bond sti ff ness, the critical size of the bridged zone decreases by approximately 10 times, also. The most significant change in the critical size of the bridged zone is observed for E 1 / E 2 = 5 . 4. In this case at relative bond sti ff ness κ 0 = 1we have t cr = d cr /ℓ = 0 . 59, the bridged zone occupies more than half the crack length. The points A 1 , 2 , 3 on the graph with the relative material sti ff ness E 1 / E 2 = 5 . 4 shown the positions of the bonds compliances and the critical bridged zones corresponding to Fig. 3 and Fig. 4. The dependencies of the critical external stress σ cr , determined from conditions (7) and (8), versus the crack length is shown in Fig. 6. The initial crack length is defined above, the parameter H was assumed constant and equal to this initial crack length H = ℓ . The quasistatic crack growth is considered over the interval H ≤ ℓ ≤ 10 H . Normalization of stress σ cr is performed by the bonds strength σ f = ( E b / H ) δ cr . For the chosen materials properties σ f = 10 MPa . The external critical stress decreases with the crack growth, since at a constant value of parameter H , the relative sti ff ness of bonds increases with increasing crack length while the critical size of the crack bridged zone