PSI - Issue 15

Mikhail Perelmuter / Procedia Structural Integrity 15 (2019) 60–66

62

M.N. Perelmuter / Structural Integrity Procedia 00 (2019) 000–000

3

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 (2)-(3) one can obtain a system of integral-di ff erential equations relative to bonds stress Σ ( x , t ). Introduce the new variable, s = x / , and di ff erentiate relation (3) with accounting relation (2) to obtain

∂ ∂ s ϕ ( s , t ) σ yy ( s , t ) − i σ xy ( s , t ) + u b ( s , t ) E b = u

H

∞ ( s ) E b , c 0 =

(4)

c 0

,

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 (4) 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) 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 ) in the form

1 1 − d /

E b 2 π c 0

µ 2

dq j ( s , t ) ds

k 1 + 1 µ 1

k 2 + 1

T i j ( s )

, (5)

W i j ( s ) q j ( s , t ) + ε

G i j ( s , ξ ) q j ( ξ, t ) d ξ = Z i ( s ); i , j = 1 , 2; ε =

+

+

where q j ( s , t ) are unknown function depending on bond stresses as follows σ yy ( s , t ) − i σ xy ( s , t ) = ( q y ( s , t ) − iq x ( s , t )) √ 1 − s 1 − s 1 + s − i β , β = ln α 2 π , α = (6) In Eqs. (5) and (6) 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 details of SIDE (5) derivative and the explicit relations for kernels T i j , W i j , G i j , Z i (which depend on coordinates, properties of materials, function ϕ ( s , t ) and its derivatives with respect to s ) are presented in Goldstein and Perelmuter (1999). At every time step for numerical solution of SIDE we use a collocation scheme with piecewise quadratic approxi mation of bonds stresses proposed in Goldstein and Perelmuter (2012). 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) µ 1 + µ 2 κ 1 µ 2 + µ 1 κ 2 ,

int II ) , K = K 2

ext I − K

int I ) + i ( K

ext II − K

2 II ,

K I + iK II = ( K

I + K

(7)

where K ext int I , II are the SIFs caused by the external loads and bonds stresses, K is SIFs modulus (MSIF). SIFs in (7) for interface crack are calculated on the basis of the external loads and the distributions of the bond stresses σ yy ( x , t ) and σ xy ( x , t ) over the crack bridged zone, according to definition given in Rice (1988). If the bonds properties dependent on time, then these SIFs also are time-dependent. I , II and K

3. Kinetic of bridged zone formation - model of crack healing

The restoration of bonds inside of cracks (cracks self-healing) can be considered on the basis of fluctuation kinetic model. When modelling self-healing of bonds, we assume that the increase in the density of bonds between the crack faces over time n h ( x , t ) is governed by a first-order kinetic equation, see Khawam and Flanagan (2006) dn h ( x , t ) dt = n 0 − n h ( x , t ) τ h ( x , σ ) (8) Here n 0 is the maximum of bonds density between crack faces, τ h is the characteristic time of bonds healing ( 1 /τ h is the healing rate constant) defines by the Arrhenius type relation τ h ( x , σ ) = ψ ( x , σ ) A ( T ) , ψ ( x , σ ) = u 2 x ( x , t ) + u 2 y ( x , t ) H , A ( T ) = τ 0 exp U h RT , (9)