PSI - Issue 47

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

548

4

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

Taking into account linearity of the problem one can represent the crack opening u ( x , t ) as follows u ( x , t ) = u ∞ ( x , t ) − u b ( x , t ) , (8) where u ∞ ( x , t ), u b ( x , t ) are the crack opening caused by the external load σ 0 and bond stresses Σ ( x , t ), respectively. By incorporating formulae (5) and (8) one can obtain a system of integral-di ff erential equations relative to bonds stress Σ ( x , t ). Introduce the new variable, s = x /ℓ , and di ff erentiate relation (8) with accounting relation (5) to obtain Here c 0 is the relative bond compliance and the right side of this relation is the given function of the coordinate. The derivatives in relation (9) are defined as follows: the derivative of the crack opening under the action of homogeneous external loading u ′ ∞ ( s ) is determined by the well-knows solution presented in England (1965); the derivative of the crack opening caused by bonds stress action u ′ b ( s , t ) can be obtained starting from the representation for the derivatives of the opening crack under the action of arbitrary static loads on the crack faces. Follow to Goldstein and Perelmuter (1999) and Goldstein and Perelmuter (2012) one can obtain the system of two singular integral di ff erential equations (SIDE) relative to bonds stresses σ yy ( x , t ) and σ xy ( x , t ). The system of SIDE is solved by the time-steps scheme as in Goldstein and Perelmuter (2012). The bond compliances at each time step depend on the density of bonds according to relation (7). The numerical solution of SIDE gives us bonds stress distributions over bridged zone versus time of crack healing. The stress intensity factor for the bridged interfacial crack in 2D problem can be written due to the linearity of the problem in the following complex form as in Goldstein and Perelmuter (1999) K I + iK II = ( K ext I − K int I ) + i ( K ext II − K int II ) , (10) where K ext I , II and K int I , II are the SIFs caused by the external loads and bonds stresses. SIFs in (10) for interface bridged crack are calculated on the basis of the stresses distribution σ yy ( x , t ) and σ xy ( x , t ) ahead a crack tip. If the bonds properties dependent on time, then these SIFs also are time-dependent. Follow to Goldstein and Perelmuter (1999) we obtain K I ( t ) + iK II ( t ) = √ πℓ    σ 0 (1 + 2 i β ) − 2cosh( πβ ) π 1  1 − d ( t ) /ℓ ( q y ( ξ, t ) + i ξ q x ( ξ, t ))  1 − ξ 2 d ξ    , K ( t ) =  K 2 I ( t ) + K 2 II ( t ) (11) where K ( t ) is SIFs modulus (MSIF) and q x , y ( x , t ) are known after the SIDE solution amplitudes of bonds stresses defining as follows q y ( ξ, t ) − iq x ( ξ, t ) = ( σ yy ( ξ, t ) − i σ xy ( ξ, t ))  1 − ξ 1 + ξ  i β , β = ln α 2 π , α = µ 1 + µ 2 κ 1 µ 2 + µ 1 κ 2 , (12) In relation (12) κ 1 , 2 = 3 − 4 ν 1 , 2 or κ 1 , 2 = (3 − ν 1 , 2  / (1 + ν 1 , 2 ) for plane strain and plane stress, respectively, ν 1 , 2 and µ 1 , 2 are Poisson’s ratios and the shear modulus of jointed materials 1 and 2. The decreasing of the stress intensity factors modulus is used as the measure of the healing e ff ect. The time-step algorithm of numerical modeling of cracks self-healing is organized as follow. At the computation start the future healed zone size is defined (crack bridged zone). At every time instant t i = ( i − 1) ∆ t , i ≥ 1 ( ∆ t is time step) the following computations are performed: • the SIDE solution to obtain bonds stresses and crack opening u x , y over the crack bridged zone (see (5)); if i = 1 the solution of the SIDE is performed for an infinitely small bonds sti ff ness (very big compliance which corresponds to a crack without bonds); • SIFs and its module (11) are computed on the SIDE solution basis; • functions τ h ( x ,σ ) and Z ( x , t ,σ ) are determined, it is assumed that Z ( x , t ,σ ) = 0 and σ xy , yy ( x , 0) = 0 if i = 1; • bonds density n h ( x , t ) is computed; at the initial time instant ( i = 1) is assigned n h ( x , 0) = 0; • bonds compliance c ( x , t ) ( see relations (5), (7)) is computed; c 0 ∂ ∂ s φ ( s , t )  σ yy ( s , t ) − i σ xy ( s , t )  + u ′ b ( s , t ) E b = u ′ ∞ ( s ) E b , c 0 = H ℓ , (9)