PSI - Issue 7

Robert Goldstein et al. / Procedia Structural Integrity 7 (2017) 222–228 R.Goldstein, M. Perelmuter / Structural Integrity Procedia 00 (2017) 000–000

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Fig. 2. Normal and shear crack opening at the bridged zone edge.

Fig. 1. Plate with bridged interfacial crack of length 2 under tension loading with two bridged zone of length d .

where n 0 = n ( x , 0) is the initial bond density and τ b = τ b ( σ b , x ) is the rupture time of the molecular bond located in the crack bridged zone at the distance − x from its tip (the bond durability). Expression (1) for bonds durability at the crack bridged zone should be rewritten in the form convenient for the following computations of the bond density and new time-dependent functions τ b ( σ b , x ) = τ 0 exp U M − A M ( σ b , x ) RT (4) where U M = U 0 N A and A M ( σ b , x ) = A ( σ b , x ) N A are the activation energy and the work of intermolecular bond deformation per one mole intermolecular bonds, R = kN A is the gas constant, and N A is the Avogadro number. The time variation of the bond density leads to a variation of the bond sti ff ness in the crack bridged zone. Let’s denote by k s ( x ) the sti ff ness of a single molecular bond. Then the e ff ective sti ff ness of bonds per unit area of the crack bridged zone is k ( x , t , σ b ) = k s ( x ) n ( x , t ) = k b ( x ) exp − t τ b ( σ b , x ) (5) where k b ( x ) = k s ( x ) n 0 is the initial bond sti ff ness per unit area of the crack bridged zone. The bond deformation work (per unit width of the body) at the crack bridged zone of size dx is given by the expression (Goldstein and Perelmuter, 2012) dU ( x , t ) =    u x ( x , t ) 0 q x ( u x ) du x + u y ( x , t ) 0 q y ( u y ) du y    dx (6) where u x , y are the opening components in the crack bridged zone (see Fig. 2) at the point with coordinate x and q x , y are the bonds tractions. The quasi-linear spring-like relations between the crack opening components in the crack bridged zone and bonds tractions (the bond deformation law) can be written as follows u x , y ( x , t ) = c ( x , t , σ b ) q x , y ( x , t ) , c ( x , t , σ b ) = 1 / k ( x , t , σ b ) , σ b = q 2 x + q 2 y (7) where c ( x , t , σ b ) is bonds compliance, σ b is the traction vector modulus.