ABSTRACT
The components of displacement can be represented in a power series in the variable z. If the plate is thin enough, we can obtain a satisfactory approximation by retaining only the lower-order terms:
u x y z u x y z u z
v x y z v x y z
( , , ) ( , , ) ,
( , , ) ( , , )
+ ∂ ∂
+ ∂
v z
w x y z w x y
z ∂
,
( , , ) ( , , ).
(12.1)
That is,
u x y z u x y z x y
v x y z v x y z x y
( , , ) ( , ) ( , ),
( , , ) ( , ) ( , )
= +
= +
β , ( , , ) ( , ).w x y z w x y=
(12.2)
The components of the strain tensor are therefore given by
ε ε β γ γ α βxx xx yy yy xy xyz x z y z y x= +
∂ ∂ = +
∂ ∂ = +
∂ ∂ +
∂ ∂
, ,
=
∂ ∂ + =
∂ ∂ +
,
, .γ β γ αyz zxwy w x
(12.3)
The in-plane strains for z = 0 are
u x
v y
u y
v x
=
∂ ∂ =
∂ ∂ =
∂ ∂ +
∂ ∂, , .
