1 - Erich Schmid Institute

E
E Mechanics of Residually Stressed Coated Systems
Figure E.2: Introduced sign convention: The end where the x-axis leaves the cut system (in this case
the right end) is defined as positive. The end where the x-axis enters the system is defined
as negative. A negative bending moment on the positive end creates a negative curvature
(bending radius respectively), a positive bending moment on the positive end creates a
positive curvature. A positive bending moment on the negative end creates a negative
curvature, a negative bending moment on the negative end creates a positive curvature
With
ɛ = ∆l
l
(E.3)
the strain as a function of the curvature and the distance from the neutral plane is
calculated:
ɛ = pɛ − p
p
= 2 (ρ − y) π − 2ρπ
2ρπ
= 2ρπ − 2yπ − 2ρπ
2ρπ
= − 2yπ
2ρπ
With Eq. (E.4), Hookes Law (Eq. (E.5)) can be written as
E.3 Force Balance
= −y = −κy (E.4)
ρ
σ = Eɛ (E.5)
σ = −Eκy. (E.6)
Since the system is in a state of equilibrium, both forces F1 and F2 have the same magnitude
but opposite signs (Fig.E.3). Otherwise, the system would drift in one direction.
E–4
F1 + F2 = 0 (E.7)
F2 = −F1 = F (E.8)