1 - Erich Schmid Institute

E
E Mechanics of Residually Stressed Coated Systems
α1 · ∆T − F
∆l1 − ɛ1F · l1 = ∆l2 + ɛF 2 · l2
α1 · ∆T · l + σ1
l1 = α2 · ∆T · l + σ2
α1 · ∆T · l − F
α1 · ∆T − F
A1E1
A1E1
E1
A1E1
l2
E2
l1 = α2 · ∆T · l + F
l (1 + ɛ1) = α2 · ∆T + F
l (1 + α1 · ∆T ) = α2 · ∆T + F
1 + α1∆T
∆T (α1 − α2) = F ·
+ 1 + α2∆T
A1E1
A2E2
l2
A2E2
A2E2
(E.19)
(E.20)
(E.21)
l (1 + ɛ2) (E.22)
l (1 + α2 · ∆T ) (E.23)
A2E2
(E.24)
The force F can be calculated from the temperature difference, the dimensions (thicknesses
t1 and t2 and width b) of the specimen and the material parameters:
F =
∆T (α1 − α2)
1+α1∆T
bt1E1
+ 1+α2∆T
bt2E2
(E.25)
E.5 Determination of the Position of the Neutral Axis
The neutral axis passes through the cross-sectional area where the resulting stress from
the bending moment is zero (Fig.E.4). When the coating is much thinner than the
substrate and the Young’s moduli are of the same order of magnitude, the neutral axis
is located in the substrate.
Figure E.4: Location of the neutral axis (dashed line) and the coordinate system.
To calculate the position of the neutral axis, we assume that ∆T = 0K and the system
is bent by an external bending moment Mcurv. At the neutral axis the force FMcurv and
therefore the stress σMcurv due to the bending moment Mcurv is zero:
E–6
σMcurvdA = 0 (E.26)