Volume averaged stress in a damascene structure

The volume averaged shear stress components in both the film material and the oxide material are identically zero due to the S5rmmetry of the periodic structure. Therefore, the task is to estimate the volume averaged normal stress components in the coordinate directions. This task is 

Three additional equations are needed to establish the stress estimate. The condition 

This approximation is expected to be most effective for values of h larger than b and p — b. 

The components of averaged stress in the film material can be determined by solving the set of linear equations defined by (3.123), (3.124) and (3.125). It is readily confirmed from these equations that, when the volume fraction of either of the phases approaches zero, the estimate of average stress in the remaining phase approaches the appropriate uniform stress limit. For example, when Eo E, Uo and Qo — Of, the stress estimate has the properties that = (cr ) — - as)AT and (cr ) 0. 

The substrate curvature implied by the average stress estimates from (3.123) (3.125) is also readily obtained. The effect of membrane force resul- 

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