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. 3. Distribution of the relative bonds density along the crack bridged zone.

Fig. 4. Relative compliance variation along the crack bridged zone.

• A ( σ b , x ), work per one intermolecular bond (10) is defined on the basis of the bond deformation work (6) and parameters N m and λ . The condition of computations termination is the criterion of a crack nucleation (full bonds destruction on a small part of the crack bridged zone). This criterion can be used in the following forms: ¯ N ( t i ) ≤ N cr (13) where ¯ N ( t i ) is the average density of bonds at the time step t i and N cr is the limit value of bonds density on a small part of the weakened region of size ∆ adjoining to the crack bridged zone edge. The modelling further growth of the initial nucleated defect is performed by decreasing the bridged zone size according to condition (13). 3. Analysis of computational results The computations were carried out for a combination of materials typical for microelectronic devices (Khanna, 2011): one of the materials is metal (copper-based alloy) with elasticity modulus E 1 = 130 GPa , and the other one is an epoxy-based polymer with elasticity modulus E 2 = 2 GPa , the Poisson ratios of the materials are ν 1 = 0 . 3 and ν 2 = 0 . 35 , respectively. The elasticity modulus of the bonds assigned as E B = E 2 . The size of the weakened bonds region located along the material interface is set to be equal to 2 = 10 − 5 m . It is assumed that the bonds are formed by chains of polymer molecules with the the molecule monomer chain of the size a m = 10 − 9 m . The weakened bonds regions is modelled as a rectilinear bridged crack located on the interface between two half-planes made of di ff erent materials and the stress analysis is performed by the singular integral-di ff erential equations method (Goldstein and Perelmuter, 1999). The bond deformation law in the computation process was assumed to be linearly elastic with compliance constant along the weakened bond region at the initial time instant (estimates of bonds mechanical parameters in the crack bridged zone can be found in (Goldstein and Perelmuter, 1999, 2009). The kinetic dependencies were calculated for the following parameters: the energy of activation of the molecular bond breakage is U M = 150 kJ / mole , τ 0 = 10 − 12 s (Zhurkov, 1965; Regel et al., 1974), the initial bond density is n 0 = 10 18 m − 2 (Washiyama et al., 1994), and µ = 1. The limit value of the bond density in the initial defect formation region was chosen as N cr = 0 . 1. The external load was assumed to be constant in time. Let’s consider, at first, the case of the initial weakened bonds region, that is d = and will assume that ∆ = 0 . 1 . If the external loading σ 0 = 20 MPa and the relative compliance c 0 = 0 . 1 then condition (13) is attained for t f = 6 . 34 · 10 8 c ( t f is time of crack nucleation) after 318 time steps with the step duration ∆ t = 2 · 10 6 c . Bonds destruction is rather slow in the crack bridged zone the most part of time before the crack formation, and acceleration of this process is started at the last steps of the crack nucleation, see Fig. 3, where the distribution of the relative bonds density along the crack bridged zone is shown. Bonds destruction over time is modelled in the time-step algorithm