Microscopic Description and Model Assumptions

The governing equation for heat conduction with temperature-dependent heat conductivities can be written as  

Membrane-like microstructures are generally several micrometers thick, while the lateral dimensions of the structures and the surrounding package are on the order of a few hundred micrometers. If the layered thin-film structure would be directly transferred to a 3-d geometry model, an enormous number of finite elements would be created, as the smallest structure size determines the mesh density. Averaging the structural information and properties over the different layers in the cross section of the membrane is a good method to avoid such problems. The membrane is, therefore, initially treated as a quasi-two-dimensional object. 

To account for the much larger lateral extension of the membrane in comparison to its thickness, some approximations are made. The variation of the temperature field in the membrane along the z-axis is assumed to be small, i.e., the temperature is nearly constant. For further considerations, the temperature field within the membrane is 

The dynamic Eq. (3.2) is integrated along the z-axis within the interval of the topmost membrane layer at z = 0 and the local membrane thickness z  

The second term in Eq. (3.6) is divided into two parts by using a 2-dimensional gradient  

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