CONTINUUM MECHANICS for ENGINEERS

which is the dot product of the vector between positions x i and y i. Since the
material points X A and Y A were arbitrarily chosen, quantities x i and y i are
independent. Subsequent differentiation of Eq 5.10-5 with respect to x i then
y i results in
Since this must hold for all pairs of material points, it is possible to set
(5.10-6)
(5.10-7)
Use of Eq 5.10-7 in Eq 5.10-6 shows that the matrix Qip(t) is orthogonal.
Furthermore, since the special case of the superposed motion as a null
+
motion, that is ( x , t) = x , then matrix Qip(t) must be proper orthogonal
having Qip(t)Qiq(t) = δpq and det(Qip) = +1.
To come up with a particular form for the superposed motion, Eq 5.10-7
may be spatially integrated to obtain
χi j i
or
Vector a i may be written in the alternative form
yielding
where
+ +
∂ ˜ χ ( , ) ∂ ˜
i xj t χi
( yj, t)
∂x
p
(5.10-8a)
(5.10-8b)
(5.10-9)
(5.10-10)
(5.10-11)
A similar development of the superposed motion can be obtained by
assuming two Cartesian reference frames Ox 1x 2x 3 and O + x 1 + x2 + x 3 + which are
separated by vector c i(t) and rotated by an admissible coordinate transformation
defined by Q im (Malvern, 1969). Rather than integrating differential
Eq 5.10-7, the superposed motion can be written as
∂y
∂ ( ) = ∂ + +
˜ χ , ˜
i xj t χi
( yj, t)
∂x
p
∂y
q
p
= δ
pq
= Q () t
˜ +( x , t)= a()+ t Q () t x
χ i j i im m
+
x = a + Q x
i i im m
+ +
a()= t c ( t )− Q () t c () t
i i im m
[ ]
+ +
x = c + Q x −c
i i im m m
Q Q = Q Q = δ , and det(
Q )= 1
im in mi ni mn ij
ip