Phase angle of the stress concentration field

The stress analysis under consideration is such that both K and Ku must vary linearly with o-m — (7, and both must vanish when (7m — 7a = 0. To render the expressions for stress intensity factors in terms of system parameters dimensionally consistent, each stress intensity factor must be proportional to ((7m — The proportionality factor in each case must 

Suppose that the proportionality factors for K and K i are denoted by Pi and pii, respectively, so that K = pi a — aa,) /Kf and similarly for Ku- The energy release rate for advance of the delamination front is given by (4.34). If this result is substituted for Q in (4.30), it is evident that the proportionality factors must satisfy 

The value of oj Di) can be determined only through solution of the relevant 

Of main interest here is the phase angle if) of the stress state within the stress concentration field at the edge of the delamination zone, as defined in (4.28). From the foregoing discussion, it is evident that the phase depends only on the nondimensional ratio of material parameter D (for T)2 = 0) as defined in (4.29) according to 

In the foregoing discussion, it has been tacitly assumed that (Tm — Ta 0. If this is not the case, the phase angle lies in a different quadrant of the A i, iFii—plane in (4.50) for example. 

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For the time being, suppose that > (7 > 0 which corresponds to a film with residual tensile stress. For this case, it was noted in Section 4.4 that the phase angle of the stress concentration field at the edge of a delamination is approximately -if = 52.1° and that Kj > 0. Because the phase of a delamination crack with this t3cpe of loading is constant, the interfacial fracture energy will be written simply as F, as before, thus suppressing any dependence of this quantity on -if under these special conditions. 

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