PSI - Issue 47

Mikhail Perelmuter / Procedia Structural Integrity 47 (2023) 545–551

549

M. Perelmuter / Structural Integrity Procedia 00 (2023) 000–000

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Fig. 4. Bonds compliance along crack bridged (healing) zone, d /ℓ = 0 . 5, c 0 = 0 . 1.

Fig. 3. Bonds density along crack bridged (healing) zone, d /ℓ = 0 . 5, c 0 = 0 . 1.

The criterion of a crack self-healing process ending (bridged zone formation) is the condition of computations termination in the following form:

ℓ ℓ − d ( t )

1 d ( t )

¯ N ( t i ) ≥ N cr , ¯ N ( t i ) =

n h ( x , t i ) dx

(13)

where ¯ N ( t i ) is the average bonds density along crack bridged zone at the time instant t i and N cr is the limit value of regenerated bonds density.

4. Computation modelling results

This section presents the computational modeling results of crack healing, demonstrating the proposed model and numerical algorithm. A crack at the interface between di ff erent materials was considered, Fig. 2. It was assumed that the adhesion layer between material has self-healing ability (see Tomic´ et al. (2022)) and at the initial time instant (when crack surfaces are free of constraints) some healing process is activated inside a crack, bonds between the crack surfaces are built and formation of the crack bridged zone is started. The computation results were obtained for the following problem parameters: the healed crack size is 2 ℓ = 10 − 3 m , the activation energy of the healing process is U h = 100 kJ / mole , the characteristic time is τ 0 = 10 − 10 s (see relation (2)), the final bonds density is n 0 = 10 18 m − 2 , temperature of self-healing process is T = 310 K , the external tension loading is σ 0 = 10 MPa , the limit value of healed bonds density is N cr = 0 . 95 n 0 . Time of self-healing t is given below as the dimensionless parameter t / t m , where t m = A ( T ) / N st (see relation (2)) and N st = 499 is maximal number of solution time-step. To demonstrate the numerical algorithm we assume that the healing process inside the crack is activated in the zone occupied half of the crack (bridged zone length is d = 0 . 5 ℓ ). In Fig. 3 presented the relative bonds density variation along the crack bridged zone for several time instants during the crack healing process. At the initial healing stage the bonds density variation is rather nonuniform ( i = 30 and i = 60) and at the final healing stage the bond density at the bridged zone edge is n ( ℓ − d , t h ) ≈ 0 . 75 n 0 ( t h is the healing time) and the average bonds density along the whole bridged zone N cr = 0 . 95 n 0 . In numerical computations we use bonds compliance (see relations (5)-(6)). During the healing steps the bonds compliance gradually decreases (see Fig. 4) and approaching to uniform distribution over bridged zone with the average value c / c 0 ≈ 1 . 05, whereas for undamaged material (bridged zone) in our sample c / c 0 = 1.