PSI - Issue 47

Mikhail Perelmuter / Procedia Structural Integrity 47 (2023) 545–551 M. Perelmuter / Structural Integrity Procedia 00 (2023) 000–000

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3

Fig. 1. Normal u y and shear u x crack opening at the bridged zone edge.

Fig. 2. Crack with two bridged zones at the materials interface.

This relation will be used in the next section to model the crack bridged zone formation by means of bonds compliance variation over time.

3. Bridged crack model and crack healing

We will use the interface bridged crack model to describe the crack faces interaction in the zones starting from the crack tips. It is assumed that these zones formed in the crack healing process. Let us consider the planar elasticity problem on a crack ( | x | ≤ ℓ ) at the interface of two dissimilar joint half-planes, see Fig. 2. Assume that the uniform loads σ 0 normal to the material interface are acted at infinity. The crack surface interaction is supposed to be existing in the bridged zones, ℓ − d ≤ | x | ≤ ℓ . The size of the interaction zone d = d ( t ) depends on time due to the possibility bonds restoring. As a simple mathematical model of the crack surfaces interaction we will assume that the linearly elastic bonds act through the crack bridged zones. Denote by Σ ( x , t ) the bond stresses, occurring under the external loads action, also depend on time Σ ( x , t ) = σ yy ( x , t ) − i σ xy ( x , t ) , i 2 = − 1; σ ( x , t ) = σ 2 yy ( x , t ) + σ 2 xy ( x , t ) (4) where σ yy , xy ( x , t ) are the normal and shear components of the stress in the crack bridged zone, respectively, σ ( x , t ) is the stress vector modulus (bonds tension). where u y , x ( x , t ) are the projections of the crack opening on the coordinate axes (Fig. 1), c ( x , t ,σ ) is bonds compliance, H is a linear parameter related to the thickness of the intermediate layer adjacent to the interface (related to the thickness of the formed bridged zone), E b is thee ff ective elasticity modulus of the bond and φ ( x , t ,σ ) is dimensionless function defining the compliance variation over the bridging zone and time. The increasing in the bond density over time leads to increasing bonds sti ff ness and to decreasing bonds compliance in the crack bridged zone. Let k s denote the rigidity of a single bond formed between crack faces. Then the e ff ective rigidity and compliance of bonds per unit area of the crack bridged zone are (see (3)) k ( x , t ,σ ) = k s n ( x , t ) = k b Z ( x , t ,σ ) , c ( x , t ,σ ) = k − 1 ( x , t ,σ ) (6) where k b = k s n 0 is the final bond rigidity per unit area of the crack bridged zone. It follows from (6) that the bond compliance over the crack bridged zone is the decreasing function over time and (see (5)) The crack opening, u ( x , t ) inside of the bridged zone ℓ − d ≤ | x | < ℓ is determined as follows u ( x , t ) = u y ( x , t ) − iu x ( x , t ) = c ( x , t ,σ ) ( σ yy ( x , t ) − i σ xy ( x , t )) , c ( x , t ,σ ) = φ ( x , t ,σ ) H E b (5)

E b H

1 ( x , t ,σ ) , k

φ ( x , t ,σ ) = Z −

(7)

b =