1 - Erich Schmid Institute

A.4 Calculation Procedure and Results
The actual stress distributions σi−1 and σi−1,i are depicted in Fig.A.7(f). The total
stress distribution in the actual straightened system σi is calculated by adding σi−1 and
σi−1,i. The resulting stress profile as well as the moment that compensates for σi and
straightens the system Mi,s = −Mi are depicted schematically for i = 2 in Fig.A.7(g).
The calculation procedure is repeated until all sublayers are superimposed and the cantilever
has attained its original thickness. The resulting stress profile in the straightened
original cantilever is shown in Fig.A.8.
Figure A.8: Stress profile in the straightened cantilever as a function of the cantilever thickness.
A.4.4 Determination of the Stress Distribution in the Initial System
The stress profiles in the initial system and the straightened cantilever are different,
but they can be converted into each other. The stress profile in the initial system is
calculated from the stress distribution and the thickness of the straightened cantilever
and the substrate thickness of the initial system.
First, the straightened cantilever is considered (Fig.A.9(a)). Then the eight sublayers
and the substrate are separated (Fig.A.9(b)), and the bending moment that straightens
the system M8,s is substituted by nine external forces F [i], with 0 < i < 8, calculated from
the thicknesses t [i] and the stresses in the single sublayers of the straightened cantilever
σ [i,sc] (Eq. (A.9):
F [i] = σ [i],sct [i]
(A.9)
These balanced forces serve two purposes. They ensure that the lengths of the separated
parts are the same as in the straightened cantilever lsc, and induce the same stress
distribution as the moment M8,s.
Now the external forces are removed and the sublayers as well as the substrate extend
or contract until they are stress free (Fig.A.9(c)). The lengths of the separated parts
l [i],stressfree depend on the length of the straightened cantilever, the biaxial Young’s
moduli and the magnitude of the removed external forces or stresses, respectively (Eq.
(A.10)):
A–11
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