CONTINUUM MECHANICS for ENGINEERS

under superposed rigid body motion. A differential element of area in the
current configuration may be written as
Under the superposed rigid body motion the differential area element is
written as
(5.10-26)
where reference quantities do not change, da + is the infinitesimal area, and
n +
k is the unit normal vector to the area, all in the superposed rigid body
motion state. This expression will be further reduced in several steps.
First, the Jacobian can be shown to transform according to J + = J (see
Problem 5.30). Next, the quantity ∂ ∂ is the inverse of the deformation
gradient as represented in the superposed rigid body rotated frame. Application
of the chain rule yields
+
XKxk (5.10-27)
The last term of this equation is evaluated by referring to Eq 5.10-8b and
solving for x k by pre-multiplying by Q ik. Differentiation of the result shows
that the last term of Eq 5.10-27 is Q ik. Thus,
or
da J X
x dA K = k
∂
∂
da da n J X
x dA
+ + + + ∂ K
= =
k k
+
∂
∂X
∂x
∂X
∂x
(5.10-28)
Substitution of the results from the preceding paragraph into Eq 5.10.26
results in
(5.10-29)
Squaring the second and fourth terms of the above equation and equating
them leads to (da + ) 2 = (da) 2 , and since area is always a positive number
k
= ∂X
∂x
K
k
∂x
∂x
K K j
+ +
k j k
K
+
k
( ) =
= ∂X
∂x
Q K
+
kK jK kj
−1
−1
F F Q
+ + +
da da n Q da Q da n
= = = ( )
k k kj j kj j
j
+
da =
da
kj
K
(5.10-30)