1 - Erich Schmid Institute

A
A A Direct Method of Determining Complex Depth Profiles of Residual Stresses
Figure A.5: System (a): Balanced external forces are applied on the separated coating and substrate to
produce a certain stress distribution. To obtain the ”straightened system” (α), substrate
and coating are joined, the external forces stay applied. System (b) is initially stress free.
External forces are applied to bend system (β). The superposition of the stresses and forces
of (α) and (β) leads to the ”relaxed system” (γ).
A.4.2 Calculation of the Curvature in Section A as a Function of Cantilever
Thickness
First the curvature of section B κb which acts as a curved indicator that amplifies the
curvature of section A, is determined. It is calculated from the lengths of the sections
A and B (lA and lB) and the deflection of the original cantilever δoriginal by solving Eq.
(A.2) numerically for κB:
δoriginal = 1
κb
[1 − cos ((lA + lB) κB)] (A.2)
Now the curvature of section A κA is determined as a function of the actual cantilever
thickness t. It is calculated from the actual deflection δ (t) (Fig.A.4), the lengths of the
sections A and B and the determined κB. To obtain the curvature of section A, Eq.
(A.3) is solved numerically for κA. Because the curvature of section A depends on the
cantilever thickness, Eq. (A.3) must be solved for each step of the gradual thickness
reduction:
δ = 1
[1 − cos (lAκA)] + sin lAκA +
κA
lBκB
2 lBκB
sin
2 κB 2
(A.3)
For the present example, the resulting curvature of section A as a function of the actual
cantilever thickness is depicted in Fig.A.6.
A–6