1 - Erich Schmid Institute

E.4 Calculation of the Normal Stresses in the Substrate and the Coating
Figure E.3: Axial forces or the corresponding bending moments, respectively, establish the equilibrium
of the system, compensate for the difference in length and straighten the system.
E.4 Calculation of the Normal Stresses in the Substrate and
the Coating
The force F = −F1 = F2 compensates for the difference in length of the layer and the
substrate. Calculation of the strains ɛ1 and ɛ2 owing to thermal expansion ∆T :
ɛ1 = ∆l1
l = α1∆T (E.9)
ɛ2 = ∆l2
l = α2∆T (E.10)
Calculation of the strains ɛ1F and ɛ2F owing to to the applied forces F1 and F2:
ɛ1F = σ1
E1
ɛ2F = σ2
E2
σ1 = F1
A1
σ2 = F2
A2
A1 = bt1
A2 = bt2
= ∆l1F
l + ∆l1
= ∆l2F
l + ∆l2
= − F
A1
= F
A2
= − F
t1b
= F
t2b
(E.11)
(E.12)
(E.13)
(E.14)
(E.15)
(E.16)
Calculation of the lengths l1 and l2 of the coating and the substrate subjected to ∆T :
l1 = l + ∆l1 = l + ɛ1 · l = l (1 + ɛ1) (E.17)
l2 = l + ∆l2 = l + ɛ2 · l = l (1 + ɛ2) (E.18)
Now the difference in length is compensated by applying F1 and F2 which leads to the
strains ɛ1F and ɛ2F .
E–5
E