. In a state of plane strain relative to the ( x, y ) plane, the displacement component w = 0 and the displacement components ( u, v ) are functions of ( x, y ) only. Hence, the components of rotation...

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In a state of plane strain relative to the (x, y) plane, the displacement component
w
=

0 and the displacement components (u, v) are functions of (x, y) only. Hence, the

components of rotation
ωx
=
ωy
= 0 and
ω
=
ωz. For zero body forces (set
V
= 0),

we note that the equations of equilibrium are satisfied by. Show that

σx
+
σy
= 2(λ
+
G)e

where
e
is the volumetric strain or dilatation and where
λ, G
are the Lam´e constants.

Hence, show that in terms of dilatation and rotation the equations of equilibrium are

Thus, show that
e
and
ω
are plane harmonic functions.