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Chapter VII Change in Observer. Invariance of Material Response
CHAPTER
VII Change in Observer. Invariance of Material Response
20. CHANGE IN OBSERVER Two observers viewing a moving body will generally record different motions for the body; in fact, the two recorded motions will differ by the rigid motion which represents the movement of one observer relative to the other. We now make this idea precise. Let x and x* be motions of fJI. Then x and x* are related by a change in observer if x*(p, t) = q(t) + Q(t) [x(p, r) - 0]
(1)
for every material point p and time t, where q(t) is a point of space and Q(t) is a rotation. That is, writing f(x, t)
= q(t) + Q(t)(x -
0),
(2)
then at each time t the deformation x*(., t) is simply the deformation x(', t) followed by the rigid deformation f~·, t): x*(', t) = f(·, t) x(', t). 0
139
(3)
140
VII.
OBSERVER CHANGE. INVARIANCE OF RESPONSE
o
X(o,'>j
f(o, I)
,,*(,/)
• Figure I
For convenience, we write !!Ai for the region of space occupied by !!A at time t in the motion x*: !!Ai = x*(!!A, t);
then the relation between x and x* is illustrated by Fig. 1. We now determine the manner.in which the various kinematical quantities transform under a change in observer. Letting
F
=
Vx,
F* = \lx*
and differentiating (1) with respect to p, we arrive at F*(p, t) = Q(t)F(p, r),
(4)
which is the transformation law for the deformation gradient. Note that, since det Q = 1, (4) implies det F*
=
det F.
(5)
Next, let F = RU = VR,
F* = R*U* = V*R*
be the polar decompositions ofF and F*. Then (4) implies F* = R*U* = QRU, and, since QR is a rotation, we conclude from the uniqueness of the polar decomposition that R* = QR, Also,
U* =
u.
141
20. CHANGE IN OBSERVER ~
91,
:/~I •
Figure 2
and therefore, since Y
fJI~
f( -, I)
= RUR T , = QYQT.
y*
From these relations it follows that the Cauchy-Green strain tensors C = U 2, B = v-, C* = U*2, B* = y*2 transform according to C*
=
C,
Intuitively it is clear that if x E !!AI and x* E x*
=
!!A~
are related through
f(x, t),
(6)
then x and x* must correspond to the same material point. To verify that this is indeed the case, note that, by (3), x(-, t)
= f(·, t) - 1 x*(-, t), 0
which is easily inverted to give pC t)
=
p*C t) f(·, t), 0
where p and p* are the reference maps corresponding to the motions x and x*, respectively (Fig. 2). Thus p(x, t)
= p*(f(x, t), t),
or equivalently, by (6), p(x, t) = p*(x*, t),
(7)
so that x and x* correspond to the same material point. As a consequence of (1), x*(p, t) = q(t)
+ Q(t)x(p, t) + Q(t) [x(p, t)
- o].
If we take p = p*(x*, t) in the left side and p = p(x, t) in the right side of this relation-a substitution justified by (7) provided x* and x are related by (6)-we conclude that
v*(x*, t) = cj'
+ Q(t)v(x, t) + Q(t)(x
- 0),
(8)
142
VII.
OBSERVER CHANGE. INVARIANCE OF RESPONSE
where
are the spatial descriptions of the velocity in the two motions. Equation (8) is the transformation law for the velocity. Now let L = grad v,
L* = grad v*.
Ifwe take x* = f(x, t) in (8) and differentiate with respect to x using the chain rule and the fact that grad f = Q, we arrive at L*(x*, t)Q(t) = Q(t)L(x, t)
+ Q(t),
so that the velocity gradient transforms according to L*(x*, t)
=
Q(t)L(x, t)Q(t)T
+ Q(t)Q(t)T.
(9)
Since QQT = I, it follows that QQT
+ QQT
=
0,
or equivalently that QQT = _(QQT)T,
and the tensor QQT is skew. Thus the symmetric part of (9) is the following transformation law for the stretching: (10)
D*(x*, t) = Q(t)D(x, t)Q(t?, where
EXERCISES
1.
Show that the spin W transforms according to W*(x*, t)
2.
=
Q(t)W(x, t)Q(t?
+ Q(t)Q(t)T.
A spatial tensor field A is indifferent if, during any change in observer, A transforms according to A*(x*, t) = Q(t)A(x, t)Q(t)T. Assume that A is smooth and indifferent. (a)
Show that and t.
Ais indifferent if and only if A(x,
t) =
P(x, t)1 for all x
21. INVARIANCE UNDER A CHANGE IN OBSERVER
(b)
143
Show that the tensor fields
A= A-
+ AW, A= A + LTA + AL WA
o
(c)
6.
are indifferent. In the case of the stress T [cf. (21.1)], T and Tare called, respectively, the corotational and convected stress rates. Show that o
•
6.
•
W=W, W=W+DW+WD.
(11)
3. Use (9.14) to show that
C:(y*, t)
= Q(T)C.(y, t)Q(r?,
and use this result to show that the Rivlin-Ericksen tensors (9.18) are indifferent.
21. INVARIANCE UNDER A CHANGE IN OBSERVER One of the main axioms of mechanics is the requirement that material response be independent ofthe observer. This axiom is never stated explicitly in elementary books on mechanics, probably because it seems so obvious; nonetheless it is instructive to look at a simple example from that subject. A thin elastic string weighted at the end is spun with constant angular velocity and suffers an elongation of amount D. The axiom above asserts that the force in the string which produces this extension is the same as the force required to produce the elongation {) when the string is held in one place. Indeed, the two motions of the string differ only by a change in observer; that is, an observer spinning with the string sees the string undergoing the same motion as a "still observer" sees when he extends the string by hand. We now give a precise statement of this axiom within our general framework. To do this we must first deduce the manner in which we would expect the stress to transform under a change in observer. Thus let (x, T) and (x*, T*) be dynamical processes with x and x* related through (20.1). IfT and T* are also related through this change in observer, then the corresponding surface force fields should transform as shown in Fig. 3. Thus if
Q
n* = Qn,
=
we would expect that s*(n*)
=
Qs(n).
Q(t)
144
VII.
OBSERVER CHANGE. INVARIANCE OF RESPONSE
Figure 3
But s(n)
=
Tn,
s*(n*) = T*n*,
and hence T*n*
=
QTQT n*.
Since this relation should hold for every unit vector n*, we would expect the following transformation law for the stress tensor: T*(x*, t) = Q(t)T(x, t)Q(t)T.
(1)
Of course, in (1) it is understood that x* = f(x, t) with f given by (20.2). This discussion should motivate the following definition: (x, T) and (x*, T*) are related by a change in observer if there exist C3 functions q:IR-.S,
Q: IR -. Orth '
such that (a) the transformation law (20.1) holds for all p E fJI and t E IR; (b) the transformation law (1) holds for all (x, t) in the trajectory of x. We say that the response of a material body is independent of the observer provided its constitutive class ce has the following property: if a process (x, T) belongs to ce, then so does every dynamical process related to (x, T) by a change in observer. The content of this definition is, of course, the axiom discussed previously. Theorem.
The response ofideal fluids and elastic fluids are independent ofthe
observer.
The proof of this result makes use of the following lemma, which is a direct consequence of (16.2) and (20.5).
145
SELECTED REFERENCES
Lemma. A dynamical process related to an isochoric dynamical process by a change in observer is itself isochoric. Proofofthe Theorem. Let ~ be the constitutive class of an ideal fluid, let (x, T) E~, and let (x*, T*) be related to (x, T) by a change in observer. To show that (x*, T*)E~ we must show that (x*, T*) is isochoric and Eulerian. The former assertion follows from the lemma; to verify the latter note that, since T = - nI, (l) yields
T* = QTQT = _nQQT = -ni. Therefore ideal fluids are independent of the observer. The proof that elastic fluids also have this property proceeds in the same manner and is left as an exercise. D EXERCISES
1. Prove that the response of an elastic fluid is independent of the observer. In the next two exercises we consider the change in observer defined by (20.1). 2. Show that
div.; T*(x*, t) = Q(t) div, T(x, t). 3. Show that the acceleration transforms according to v*(x*, t)
=
Q(t)v(x, t)
+ q(t) + 2Q(t)v(x,
t)
+ Q(t)(x
- 0).
Show further that if the body force transforms according to b*(x*, t)
=
Q(t)b(x, t),
then div T*
+ b* + k
=
p*v*,
where k(x*, t) = p(x, t)[q(t)
+ 2Q(t)v(x, t) + Q(t)(x -
0)].
Because of the additional term k, the equation of motion is not invariant under all changes in observer. The observers for whom this equation transforms "properly" are exactly those with Q and q constant. Such observers are called Galilean and have the property of being accelerationless with respect to the underlying inertial observer. SELECTED REFERENCES
Noll [1, 21Truesdell and Noll [1, §§17-19J.
VII Change in Observer. Invariance of Material Response
20. CHANGE IN OBSERVER Two observers viewing a moving body will generally record different motions for the body; in fact, the two recorded motions will differ by the rigid motion which represents the movement of one observer relative to the other. We now make this idea precise. Let x and x* be motions of fJI. Then x and x* are related by a change in observer if x*(p, t) = q(t) + Q(t) [x(p, r) - 0]
(1)
for every material point p and time t, where q(t) is a point of space and Q(t) is a rotation. That is, writing f(x, t)
= q(t) + Q(t)(x -
0),
(2)
then at each time t the deformation x*(., t) is simply the deformation x(', t) followed by the rigid deformation f~·, t): x*(', t) = f(·, t) x(', t). 0
139
(3)
140
VII.
OBSERVER CHANGE. INVARIANCE OF RESPONSE
o
X(o,'>j
f(o, I)
,,*(,/)
• Figure I
For convenience, we write !!Ai for the region of space occupied by !!A at time t in the motion x*: !!Ai = x*(!!A, t);
then the relation between x and x* is illustrated by Fig. 1. We now determine the manner.in which the various kinematical quantities transform under a change in observer. Letting
F
=
Vx,
F* = \lx*
and differentiating (1) with respect to p, we arrive at F*(p, t) = Q(t)F(p, r),
(4)
which is the transformation law for the deformation gradient. Note that, since det Q = 1, (4) implies det F*
=
det F.
(5)
Next, let F = RU = VR,
F* = R*U* = V*R*
be the polar decompositions ofF and F*. Then (4) implies F* = R*U* = QRU, and, since QR is a rotation, we conclude from the uniqueness of the polar decomposition that R* = QR, Also,
U* =
u.
141
20. CHANGE IN OBSERVER ~
91,
:/~I •
Figure 2
and therefore, since Y
fJI~
f( -, I)
= RUR T , = QYQT.
y*
From these relations it follows that the Cauchy-Green strain tensors C = U 2, B = v-, C* = U*2, B* = y*2 transform according to C*
=
C,
Intuitively it is clear that if x E !!AI and x* E x*
=
!!A~
are related through
f(x, t),
(6)
then x and x* must correspond to the same material point. To verify that this is indeed the case, note that, by (3), x(-, t)
= f(·, t) - 1 x*(-, t), 0
which is easily inverted to give pC t)
=
p*C t) f(·, t), 0
where p and p* are the reference maps corresponding to the motions x and x*, respectively (Fig. 2). Thus p(x, t)
= p*(f(x, t), t),
or equivalently, by (6), p(x, t) = p*(x*, t),
(7)
so that x and x* correspond to the same material point. As a consequence of (1), x*(p, t) = q(t)
+ Q(t)x(p, t) + Q(t) [x(p, t)
- o].
If we take p = p*(x*, t) in the left side and p = p(x, t) in the right side of this relation-a substitution justified by (7) provided x* and x are related by (6)-we conclude that
v*(x*, t) = cj'
+ Q(t)v(x, t) + Q(t)(x
- 0),
(8)
142
VII.
OBSERVER CHANGE. INVARIANCE OF RESPONSE
where
are the spatial descriptions of the velocity in the two motions. Equation (8) is the transformation law for the velocity. Now let L = grad v,
L* = grad v*.
Ifwe take x* = f(x, t) in (8) and differentiate with respect to x using the chain rule and the fact that grad f = Q, we arrive at L*(x*, t)Q(t) = Q(t)L(x, t)
+ Q(t),
so that the velocity gradient transforms according to L*(x*, t)
=
Q(t)L(x, t)Q(t)T
+ Q(t)Q(t)T.
(9)
Since QQT = I, it follows that QQT
+ QQT
=
0,
or equivalently that QQT = _(QQT)T,
and the tensor QQT is skew. Thus the symmetric part of (9) is the following transformation law for the stretching: (10)
D*(x*, t) = Q(t)D(x, t)Q(t?, where
EXERCISES
1.
Show that the spin W transforms according to W*(x*, t)
2.
=
Q(t)W(x, t)Q(t?
+ Q(t)Q(t)T.
A spatial tensor field A is indifferent if, during any change in observer, A transforms according to A*(x*, t) = Q(t)A(x, t)Q(t)T. Assume that A is smooth and indifferent. (a)
Show that and t.
Ais indifferent if and only if A(x,
t) =
P(x, t)1 for all x
21. INVARIANCE UNDER A CHANGE IN OBSERVER
(b)
143
Show that the tensor fields
A= A-
+ AW, A= A + LTA + AL WA
o
(c)
6.
are indifferent. In the case of the stress T [cf. (21.1)], T and Tare called, respectively, the corotational and convected stress rates. Show that o
•
6.
•
W=W, W=W+DW+WD.
(11)
3. Use (9.14) to show that
C:(y*, t)
= Q(T)C.(y, t)Q(r?,
and use this result to show that the Rivlin-Ericksen tensors (9.18) are indifferent.
21. INVARIANCE UNDER A CHANGE IN OBSERVER One of the main axioms of mechanics is the requirement that material response be independent ofthe observer. This axiom is never stated explicitly in elementary books on mechanics, probably because it seems so obvious; nonetheless it is instructive to look at a simple example from that subject. A thin elastic string weighted at the end is spun with constant angular velocity and suffers an elongation of amount D. The axiom above asserts that the force in the string which produces this extension is the same as the force required to produce the elongation {) when the string is held in one place. Indeed, the two motions of the string differ only by a change in observer; that is, an observer spinning with the string sees the string undergoing the same motion as a "still observer" sees when he extends the string by hand. We now give a precise statement of this axiom within our general framework. To do this we must first deduce the manner in which we would expect the stress to transform under a change in observer. Thus let (x, T) and (x*, T*) be dynamical processes with x and x* related through (20.1). IfT and T* are also related through this change in observer, then the corresponding surface force fields should transform as shown in Fig. 3. Thus if
Q
n* = Qn,
=
we would expect that s*(n*)
=
Qs(n).
Q(t)
144
VII.
OBSERVER CHANGE. INVARIANCE OF RESPONSE
Figure 3
But s(n)
=
Tn,
s*(n*) = T*n*,
and hence T*n*
=
QTQT n*.
Since this relation should hold for every unit vector n*, we would expect the following transformation law for the stress tensor: T*(x*, t) = Q(t)T(x, t)Q(t)T.
(1)
Of course, in (1) it is understood that x* = f(x, t) with f given by (20.2). This discussion should motivate the following definition: (x, T) and (x*, T*) are related by a change in observer if there exist C3 functions q:IR-.S,
Q: IR -. Orth '
such that (a) the transformation law (20.1) holds for all p E fJI and t E IR; (b) the transformation law (1) holds for all (x, t) in the trajectory of x. We say that the response of a material body is independent of the observer provided its constitutive class ce has the following property: if a process (x, T) belongs to ce, then so does every dynamical process related to (x, T) by a change in observer. The content of this definition is, of course, the axiom discussed previously. Theorem.
The response ofideal fluids and elastic fluids are independent ofthe
observer.
The proof of this result makes use of the following lemma, which is a direct consequence of (16.2) and (20.5).
145
SELECTED REFERENCES
Lemma. A dynamical process related to an isochoric dynamical process by a change in observer is itself isochoric. Proofofthe Theorem. Let ~ be the constitutive class of an ideal fluid, let (x, T) E~, and let (x*, T*) be related to (x, T) by a change in observer. To show that (x*, T*)E~ we must show that (x*, T*) is isochoric and Eulerian. The former assertion follows from the lemma; to verify the latter note that, since T = - nI, (l) yields
T* = QTQT = _nQQT = -ni. Therefore ideal fluids are independent of the observer. The proof that elastic fluids also have this property proceeds in the same manner and is left as an exercise. D EXERCISES
1. Prove that the response of an elastic fluid is independent of the observer. In the next two exercises we consider the change in observer defined by (20.1). 2. Show that
div.; T*(x*, t) = Q(t) div, T(x, t). 3. Show that the acceleration transforms according to v*(x*, t)
=
Q(t)v(x, t)
+ q(t) + 2Q(t)v(x,
t)
+ Q(t)(x
- 0).
Show further that if the body force transforms according to b*(x*, t)
=
Q(t)b(x, t),
then div T*
+ b* + k
=
p*v*,
where k(x*, t) = p(x, t)[q(t)
+ 2Q(t)v(x, t) + Q(t)(x -
0)].
Because of the additional term k, the equation of motion is not invariant under all changes in observer. The observers for whom this equation transforms "properly" are exactly those with Q and q constant. Such observers are called Galilean and have the property of being accelerationless with respect to the underlying inertial observer. SELECTED REFERENCES
Noll [1, 21Truesdell and Noll [1, §§17-19J.