CONTINUUM MECHANICS for ENGINEERS

In the first, a continuum body’s response to applied forces or prescribed
motion must be the same as observed from two different reference frames.
The body and the applied forces remain the same; only the observer’s reference
frame changes. In superposing a rigid body motion to the body, the
observer maintains the same reference frame. Here, each material point has
a superposed motion added to it. The forces applied to the body are rotated
with the superposed motion.
Both of these methods produce the same restrictions on constitutive
responses within the context of this book. In this text, the method of superposed
rigid body motion will be presented as the means for enforcing invariance.
The superposition of a rigid motion along with a time shift can be applied
to the basic definition of the motion
x i = χ i(X A, t) (5.10-1)
From this state a superposed rigid body motion is applied, maintaining all
relative distances between material points. Also, since the motion is a function
of time, a time shift is imposed on the motion. After the application of
the superposed motion, the position of the material point becomes x i + at time
t + = t + a where a is a constant. A superscript “+” is used for quantities having
the superposed motion. Some literature uses a superscript “*” to denote this,
but since we use this to represent principal stress quantities, the “+” is used.
The motion in terms of the superposed motion is written as
+ +
x χ X , t
= ( )
i i A
(5.10-2)
Assuming sufficient continuity, the motion can be written in terms of the
current configuration since X A = χ –1 (x i, t). That is to say
˜χ i
+ + +
x χ X , t ˜ χ x , t
= ( )= ( )
i i A i j
(5.10-3)
where is written because the substitution of XA = χ –1 +
(xi, t) results in a
+
different function than χ . i
To represent the relative distance between two particles, a second material
point is selected. In an analogous manner, it is straightforward to determine
+ + +
y χ Y , t ˜ χ y , t
= ( )= ( )
i i A i j
Relative distance between material particles X A and Y A is written as
( ) −
[ ] [ ( )− ( ) ]
( )= ( )− ( )
+ + + +
x − y x y ˜ χ x , t ˜ χ y , t ˜ χ x , t ˜ χ
y , t
i i i i i j i j i j i j
(5.10-4)
(5.10-5)