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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4

The bond compliance in the crack bridged zone can be represented as

exp

t τ b ( σ b , x ) ,

ϕ ( x ) E B

H

c ( x , t , σ b ) = c 0

c 0 =

(8)

where ϕ ( x ) is dimensionless function, E B is the e ff ective Young modulus of bonds, H is a linear scale proportional to the bonding zone thickness and c 0 is named as the relative bond compliance, see (Goldstein and Perelmuter, 1999). Relationship (6) defines the full deformation work of any type bonds (polymer chains, fibers, etc.) between the crack surfaces. The work per one intermolecular bond, which is used in Eqs. (1)-(2), is estimated here under the assumption that the adhesion junction is formed by chains of polymer molecules with the monomer link size of a m . It is assumed that the bond elongation at the bridged zone is within of the linear elasticity limits and this elongation is much less than its initial length. Therefore, the crack opening in the bridged zone at the distance − x from the crack tip is approximately equal to the length of the loaded segment of the chain molecule bridged the crack surfaces. In this case, the number N m of monomer links between the crack surfaces on an interval of length dx is N m = λ dn , λ = u 2 x ( x , t ) + u 2 y ( x , t ) a m (9) where λ is the number of monomer links of the polymer chain forming a bond between the crack surfaces and the number of molecular bonds on the interval dx is dn = n ( x , t ) dx . The work per one intermolecular bond is equal to To calculate distributions over the crack bridged zone the kinetics functions introduced above, one needs to know the stresses and displacement distributions along the bridged zone. For a rectilinear crack located on the interface between two half-planes made of di ff erent materials (see Fig. 1) these functions can be obtained as the solution of the singular integral-di ff erential equations (SIDE) (Goldstein and Perelmuter, 1999, 2012). In cases of curvilinear cracks and finite sized structures the boundary integral equation (BIE) method can be used (Perelmuter, 2013). Having the solution for the tractions over bridged zone we can calculate the stress intensity factors (SIF) for bridged crack on the material interface as follow (Goldstein and Perelmuter, 1999) K I + iK II = ( K ext I + K int I ) + i ( K ext II + K int II ) , i 2 = − 1 (11) where K ext I , II and K int I , II are the SIF caused by the external loads and bond tractions. Accounting for relation (11) we can obtain (see details in Goldstein and Perelmuter (1999)) for a rectilinear bridged crack A ( σ b , x ) = dU ( x , t ) λ dn (10) where K is modulus of the SIFs and bielastic parameter β depends on the Poisson ratios and shear moduli of the joint materials. The equations for the stress state determination are solved numerically by time-step scheme. We assume that the initial time instant is zero ( t 0 = 0) and we define the time step ∆ t depending on bonds durability without external loads (see (1) for σ = 0). On each time step t i = ( i − 1) ∆ t , i ≥ 1 the following computations for any point at the crack bridged zone are performed: • n ( x , t ), the bonds density (3) is computed, at the initial time instant ( i = 1) is assigned n ( x , 0) = n 0 ; • τ b ( σ b , x ), the bonds durability (4) are determined, for i = 1, it is assumed that bonds in the bridged zone are not loaded, q x , y ( x , 0) = 0 and the bonds durability is constant over the bridged zone; • c ( x , t , σ b ), the bonds compliance (8) is computed, for i = 1 is used the initial bonds compliance; • q x , y , tractions and crack opening u x , y over the crack bridged zone (see (7)) are computed as the solution of SIDE or BIE, for i = 1 it is the solution of the problem with the initial bonds compliance (no kinetics); K I ( t ) + iK II ( t ) = √ π (2 ) i β    σ o (1 + 2 i β ) − 2 cosh( πβ ) π 1 1 − d / q y ( s , t ) + itq x ( s , t ) √ 1 − s 2 ds    , K ( t ) = K 2 I + K 2 II (12)