CONTINUUM MECHANICS for ENGINEERS

σ xx p G
=− +
σ yy p G
=− +
σ xy G x ∂
=
2 2
⎡⎛
∂x
⎞ ⎛ ∂x
⎞ ⎤
⎢⎜
⎝ ∂Y
⎟ + ⎜
⎠ ⎝ ∂Y
⎟ ⎥
⎣
⎢
⎠
⎦
⎥
2 2
⎡⎛
∂y
⎞ ⎛ ∂y
⎞ ⎤
⎢⎜
⎝ ∂X
⎟ + ⎜
⎠ ⎝ ∂Y
⎟ ⎥
⎣
⎢
⎠
⎦
⎥
⎡ ∂y
∂x
∂y
⎤
⎢ + ⎥
⎣∂X
∂ Y ∂Y
∂Y
⎦
(8.4.8a)
(8.4-8b)
(8.4-8c)
where again it is noted that strict continuum notation is not used for clarity
in this particular solution. The incompressibility condition, J = det [F iA] = 1
may be written as
∂x
∂y
∂x
∂y
− =
∂X
∂Y ∂Y
∂X
1
(8.4-9)
The inverse of the deformation gradient can be directly calculated from
Eq 8.4-7 to be
F
− 1
iA =
⎡∂X
⎢
⎢
∂x
⎢ ∂Y
⎢ ∂x
⎢
∂Z
⎢
⎣⎢
∂x
∂X
∂y
∂Y
∂y
∂Z
∂y
∂X
⎤ ∂y
x
⎥
−
∂z
⎥
∂Y
Y
∂Y
⎥ y x
=
∂z
⎥ X X
∂Z
⎥
⎥
∂z
⎦⎥
∂
∂
− ∂
⎡
⎤
⎢
0⎥
⎢
⎥
⎢ ∂
0
⎥
⎢ ∂ ∂ ⎥
⎢
⎥
⎢ 0 0 1⎥
⎣⎢
⎦⎥
(8.4.10)
Thus, in the case of incompressible material undergoing a plane strain
motion the following relationships must hold:
∂
=
∂
∂ ∂
∂ ∂ =−∂
∂
=−
∂ ∂
∂
x Y x X y Y
; ; ;
X y Y y X ∂x
(8.4-11)
An alternate form of the stress components may be written by factoring out
( )
the quantity Π= σ + σ for future convenience
σ xx
xx yy /2
∂y
∂X
=
∂ Y ∂x
⎡⎛
∂X
⎞ ⎛ ∂Y
⎞ ⎛ ∂X
⎞ ⎛ ∂Y
⎞ ⎤
= Π + G ⎢⎜
⎝ ∂y
⎟ + ⎜
⎠ ⎝ ∂y
⎟ − ⎜
⎠ ⎝ ∂x
⎟ − ⎜
⎠ ⎝ ∂x
⎟ ⎥
⎣
⎢
⎠
⎦
⎥
1
2 2 2 2
2
(8.4-12a)