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DIFFERENTIATIONOF ROTATIONAND ROTOTRANSLATIONby
Teodoro Merlini
Scientific ReportDIA-SR 02-16
2002
Politecnico di MilanoDipartimento di Ingegneria Aerospaziale
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2
Abstract The differentiation of the rotation and rototranslation
tensors up to the 3rd-order is performed, onuse in variational
formulations of continuum mechanics. Detailed mathematical
derivations and a widecollection of all the formulae pertaining the
subject are included.
Chapters 2 to 4 refer to the case of rotations. Three
differential rotation vectors characterizing the 1st, 2ndand 3rd
differentiation respectively are identified and the relevant
expressions and corotational differentiationformulae are given.
Formulae for composite rotations are also supplied, and a
specialization follows for thecase of kinematics of a
continuum.
Chapters 5 to 8 refer to the case of rototranslations, namely
orthonormal dual tensors. All the formulae
drawn for the rotations apply as are to rototranslations, and
this enables us to duplicate the results. Moreover,the
differentiation of the rototranslation decomposition as a rotation
followed by a translation is discussed.
Keywords Finite rotations, finite rototranslations, dual
numbers.
Scientific Report DIA-SR 02-16
Author:
Teodoro Merlini, Full Professor,
[email protected] di Ingegneria Aerospaziale,
Politecnico di MilanoVia La Masa 34, I-20158 Milano, Italy
Published by:
Politecnico di Milano, Dipartimento di Ingegneria
Aerospaziale,Campus Bovisa, Via La Masa 34, I-20158 Milano,
Italy
Printed in ItalyNovember 2002 Revised May 2003
mailto:[email protected]:[email protected]
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3
Contents
Contents.............................................................................................................................................................................3
1
Introduction....................................................................................................................................................................5
2 Differentiation of the
rotation.......................................................................................................................................6
2.1 First differentiation of the
rotation...........................................................................................................................................6
2.2 Second differentiation of the
rotation.......................................................................................................................................6Variations
of the 1st differential rotation
vector.........................................................................................................................7
2.3 Third differentiation of the rotation
.........................................................................................................................................7Variations
of the 1st and 2nd differential rotation
vectors.........................................................................................................10
2.4 Differential rotation
vectors.....................................................................................................................................................
10
2.5 Corotational
differentiations...................................................................................................................................................
11
3 Differentiation of rotation
composition......................................................................................................................14
3.1 Differential rotation
vectors.....................................................................................................................................................
14
3.2 Corotational
differentiations...................................................................................................................................................
18
4 Differentiation of rotations in a
continuum...............................................................................................................21
4.1 Differentiation of the
rotation..................................................................................................................................................
214.1.1 Characteristic rotation
differentials..................................................................................................................214.1.2
Corotational
differentiations............................................................................................................................23
4.2 Differentiation of rotation
composition...................................................................................................................................244.2.1
Characteristic rotation
differentials..................................................................................................................244.2.2
Corotational
differentiations............................................................................................................................26
5 Differentiation of the
rototranslation.........................................................................................................................30
5.1 Differential
helices....................................................................................................................................................................30
5.2 Corototranslational
differentiations........................................................................................................................................30
6 Differentiation of rototranslation
composition..........................................................................................................32
6.1 Differential
helices....................................................................................................................................................................32
6.2 Corototranslational
differentiations........................................................................................................................................32
7 Differentiation of the rototranslation
decomposition................................................................................................34
7.1 Differential
helices....................................................................................................................................................................34
7.2 Explicit form of the differential
helices...................................................................................................................................36
8 Differentiation of rototranslations in a
continuum...................................................................................................37
8.1 Differentiation of the
rototranslation......................................................................................................................................378.1.1
Characteristic rototranslation
differentials.......................................................................................................378.1.2
Corototranslational
differentiations..................................................................................................................38
8.2 Differentiation of rototranslation
composition.......................................................................................................................388.2.1
Characteristic rototranslation
differentials.......................................................................................................388.2.2
Corototranslational
differentiations..................................................................................................................39
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8.3 Differentiation of the rototranslation
decomposition.............................................................................................................408.3.1
Characteristic rototranslation
differentials.......................................................................................................408.3.2
Explicit form of the characteristic rototranslation
differentials.......................................................................
41
References.......................................................................................................................................................................
43
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1. Introduction 5
1IntroductionThis Report deals with the differentiation of the
rotation and rototranslation tensors up to the 3rd-order, to be
used in
variational formulations of continuum mechanics, when three
mixed but independent differentiations are involvedavirtual
differentiation inherent in the variational framework, an
incremental differentiation due to the solutionprocedure, and a
spatial differentiation to describe the generalized curvatures. The
Report consists mostly of thedetailed mathematical derivation and
of a wide collection of all the formulae pertaining the subject,
with the aim ofstanding as a reference guide for interested
researchers.The work is divided in two parts, relevant to rotations
(Chapters 2 to 4) and rototranslations (Chapters 5 to 8),
respectively.First, the differentiation of therotation tensor up
to the 3rd-order is performed in Chapter 2. Threedifferential
rotation
vectors characterizing the 1st, 2nd and 3rd differentiation
respectively are identified and the relevant expressions aregiven.
Their variations yield meaningful corotational differentiation
formulae. It is worth noting that the matter dealtwith is intrinsic
to an orthonormal rotation tensor independently of any particular
parameterization of the rotation tensoritself.The same matter is
therefore extended to the composition of rotations in Chapter 3.
Composite rotations and
rototranslations are repeatedly used in formulations of
continuum mechanics. In fact, consider that an orientation
canalways be seen as the rotation from another orientation, so the
knowledge of the relationship among the characteristicdifferential
vectors of subsequent rotations becomes important.
A specialization of the proposed differentiations follows in
next Chapter 4, being useful in formulating the kinematicsof
continua, in particular polar media. The variations so far used
assume now a specific mechanical meaning, beingassociated
respectively with virtual variations, incremental variations and
spatial variations.The second part of the Report deals with
rototranslations. The same considerations as for the rotation apply
as are to
the case of therototranslation tensoran orthonormal dual tensor,
see Merlini and Morandini (2003). This enables usto duplicate the
results, avoiding any further derivation. In particular,
threedifferential helices (dualrotation vectors)are identified in
Chapter 5. Composition of rototranslations is dealt with as well in
Chapter 6, and this becomes helpfulin investigating the
differentiation of the rototranslation decomposition as a rotation
followed by a translation (Chapter7). Again, a specialization for
the kinematics of polar continua follows in Chapter 8.
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2. Differentiation of the rotation6
2Differentiation of the rotationLet be arotationtensor, obeying
to theorthonormality condition
T = I (2-1)
Since we have to characterize three independent variations of
therotation tensor, we consider in sequence a singlevariation , a
double variation , and a triple variation d of the orthonormality
condition itself.
2.1First differentiation of the rotationBy differentiating the
orthonormality condition,
T
T T
T T
( )
( )
= +
= +
= 0
T
it follows that tensor turns out to be skew-symmetric. By
denoting withT the vector that characterizes it, onewrites
T = . (2-2)
So,the differential of the rotation tensor is
characterizedbyvector , called the1st differential rotation
vector,axial of which is actually a skew-symmetric tensorT 1:
T S( ) = 0 (2-3)Tax( ) = (2-4)
When refers to a virtual variation, vector is often referred to
as virtual rotation vector. However, it is worth
noting that no parameterization of the rotation tensor is
implied in the above characterization; rather, the choice of
suchname and symbol is supported by the fact that taking the
rotation tensor between a rotation and a varied rotation
+ very close to it, it results in =T( )+ + I , a tensor matching
exactly the exponential map definitionexp any time is an
infinitesimal vector.
2.2Second differentiation of the rotationBy extracting the
second differential of the orthonormality relation T = I ,
( )
T
T T T T
TT T T T T1 1
2 2
( )
( ) ( )
= + + +
= + + + + +
= 0
T
we can recognize as skew-symmetric the tensor T T12( + + T ) .
By denoting with the vector
that characterizes it, one writes
T T T1
2( )
+ + =
.
1 Notation denote the symmetric part of a tensor.S()
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2. Differentiation of the rotation 7
It is then noted that the second term on the left-hand-side of
this formula is a symmetric tensor. This means that this
equation behaves as aEuclidean decomposition of tensor T , i.e.T
T S( ) = +
)
(2-5)
with a known function of the first variationsT S( and . Notice
that the differentials and in Eq.(2-5) are interchangeable.
By resorting to the 1st differential rotationvector, the
symmetric and skew-symmetric parts can be worked out asfollows,
( )
T S
T T12
12
( )
( )
= +
=
( )T T12 ( )
= T
( )
( )
T T T T12
T T T T12
( ) ( )
( ) ( )
= +
= +
T
T
T12
T12
( )
( )
= +
= +
12
12
( )
( )
=
=
and lead to
(T S 12( ) = + ) (2-6)T
12
12
12
ax( )
( )
=
=
= = +
(2-7)
So, the 2nd differential of the rotation tensor is characterized
by the 1st differential rotation vectors and ,and furthermore by
vector , the axial of , that can be called the2
T nd differential rotation vector and accounts
for the variations of the first characteristic vectors and .
Variations of the 1st differential rotationvector
By inverting Eqs. (2-7), the following expressions for the
variation of the 1st differential rotationvector are obtainedas
function of just the differential rotationvectors:
12
12
= +
= +
(2-8)
2.3Third differentiation of the rotationBy extracting the third
differential of the orthonormality relation T = I ,
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2. Differentiation of the rotation8
( )
T
T T T T T T T
T T T T T T T12
TT T T T T T T1
2
( )
( )
( )
d
d d d d d d d d
d d d d d d d
d d d d d d d
= + + + + + + +
= + + + + + +
+ + + + + + +
= 0
T
we can now recognize tensor +Td T T T T T12( )d d d d d d + + +
+ + T as a
skew-symmetric one. By denoting with d the vector that
characterizes it, one writes
T T T T T T T12( ) dd d d d d d d + + + + + + = .
Again, it is noted that the second term on the left-hand-side of
this formula is a symmetric tensor. So, this equation
behaves as aEuclidean decomposition of tensor Td , i.e.T T S( )
dd d
= +
)
(2-9)
with a known function of the first and second variations dT S(d
, , , , d and d . Noticethat the differentialsd, and in Eq. (2-9)
are interchangeable.
By resorting to the 1st and 2nd differential rotationvectors,
the symmetric and skew-symmetric parts can be workedout as
follows:
(
T S
T T T T T T12
12
12
( )
( )
(d d d d d d
d
d
d d d d d d
= + + + + +
=
d + 12) ( + 12) (d d d+ 12
)
(d
+
+ 12) ( d + d+ 12) ( d + d + ))
( )T T12 ( )d
d d
= T
( )
( )
( )
( )
T T T T T12
T T T T T12
T T T T T12
T T T T T T T T12
T T T T12
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
d d d d
d d d d
d d d d
d d d d d d d d
d d d d
= +
= +
= +
= + + +
=
( )
T
( )
T T T T
T T T T T T T T12
( ) ( )
d d d
d d d d d d d d
+ + + = + + +
T
T
d
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2. Differentiation of the rotation 9
( )
( )
T1 1 12 2 2
T1 1 12 2 2
T1 1 12 2
( ) (
( ) (
( )
d d d d
d d d d d d d d
d d d d d d
d d
= + + + + +
= + + + + +
= + + + +
)
)
( )
(
2
T1
212
( )
(
d d
d d d d d d
d d
d d
+
= + + +
+ +
1
2) ( + 12) (d d + d +
12
)
( d
d + 12) ( d + 12) (d d + + + )
( T1212
)
(
d d d d d d d d
d d
= + + +
+ +
1
2) ( d + 12) (d + +
12
)
(
d
d
+ 12) ( d + 12) (d d d + + + )
( T1212
)
(
d d d d d dd d
= + + +
+ +
1
2) (d d + 12) (d d + d+
12
)
( d
d+
12) (d +
12) ( d d + + + ))
( )( )( )
1 12 4
1 12 4
1 12 4
12
( )
( )
( )
(
d d d d d
d d d d d d
d d d d d d
d
d
d
= + + + +
= + + + +
= + + + +
= +
12
1 12 2
1 12 2
) ( )
( ) (
( ) (
d d d d
d d d d d d
d d d d dd
+ + = + + + +
= + + +
)
)
( )( )
( )
1 12 4
1 12 4
1 1
2 41 12 2
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( )
d d d d d
d d d d
d d d d d
d d d
d
d
= + + +
= + + +
= + + +
= +
( )
d
( )( )
1 12 2
1 12 2
( )
( ) (
( ) (
d d
d d d d d
d d d d dd
+ = + + +
= + +
)
)
In passing, note that
( )1212
( ) ( ) (
( )
d d
d d
d d
d
d
d d d
d d d d
d d
)
+ +
= + + + + +
= + + + + +
= + +
yielding the following identity:
d d dd d d + + = + + (2-10)Finally, it results:
( )T S 12( ) d d d d dd = + + + + + d (2-11)
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2. Differentiation of the rotation10
T
1 12 4
1 12 4
1 12 4
1 1
3 613
ax( )
(2 )
( 2 )
( 2
( ) ( )
d
d d d d
d d d d d
d d d d d
d d d d d
d
d
d
=
= + +
= + +
= + +
= + + + +
)
1 12 2
1 12 2
1 12 2
13
( )
( ) (
( ) (
( ) (
(
d d
d d d d
d d d d d d
d d d d d
d
d
d
d
)
)
)
d
d
+ +
= + +
= + + +
= + +
= + +
1
6
13
) ( )
( )
d d d
d d
d
+ + + +
d
(2-12)
So, the 3rd differential d of the rotation tensor is
characterized by the 1st and 2nd differential rotation vectors d ,
, , d , and d , and furthermore by vector d , the axial of
Td , that can be called the 3rddifferential rotation vectorand
accounts for the variations of the preceding characteristic
vectors.
Variations of the 1st and 2nd differentialrotation vectors
By inverting Eqs. (2-12), the following expressions for the 1st
variation of the 2nd differential rotationvector and forthe 2nd
variation of the 1st differential rotationvector are obtained as
function of just the differential rotationvectors:
( )
( )
1 12 4
S12
1 12 4
S12
1 1
2 4
(2 )
( )
( 2
( )
(
d d d d d
d d
d d d d d d
d d d d
d d d d
d
= + + + +
=
= + + + +
=
= + + +
I
I
)
( )S12
2 )( )
d d
d d d d
+ = I
(2-13)
( )
( )
1 12 2
S1 12 2
1 12 2
S1 12 2
( ) (
( ) ( )
( ) (
( ) ( )
d d d d d d
d d d d d
d d d d d d d
d d d d
d
d
= + + + +
= + + + +
= + + + + +
= + + + +
)
)
( )
1 12 2
S1 12 2
( ) (
( ) ( )
d d d d d d
d d d d d
= + + + +
= + + + +
)
(2-14)
2.4Differential rotation vectorsLets summarize the work
discussed so far. Starting from the orthonormality condition, it is
found that a differential
vector characterizes the first differentiation of the rotation
tensor, and that every subsequent differentiation introduces a
new characteristic differential vector. It is also seen that
theEuclidean decompositionof tensors ,T T andleads to a symmetric
part which is a known function of the characteristic differential
vectors of the lower
differentiations (null for the 1
Td st differentiation). So, it seems a good choice to regard the
axials , and d of
such tensors as the characteristic differential vectors of the
rotation differentiations. They are called the 1st, 2ndand
3rddifferential rotation vectors, respectively, and account for the
lowest variations of the rotation tensor. However, it mustbe noted
that this characterization is not unique: another choice for d
would be, for instance, the first part of Eqs.
(2-12)i.e. 1/3 of the members of identity Eq. (2-10).
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2. Differentiation of the rotation 11
From now on, we assume three possible independent variations
referred to as d, and respectively andpresent the picture relevant
to each single variation and to any combination of them. Thus, the
proposed characterizationleads to the following recursive
expressions for the subsequent variations of the rotation,
T
T
T
T 12
T 12
T 12
T 12
( )
( )
( )
( )
d
d d d
d d d
d d d d d d d
d
d
d
d
=
= =
= + +
= + +
= + +
= + + + + + +
(2-15)
and the characteristic differential vectors allow different
equivalent expressions, to be chosen from Eqs. (2-7) and(2-12), for
instance:
T
T
T
T 12
T 12
T 12
T 1 13 6
13
1
ax( )
ax( )
ax( )
ax( ) ( )
ax( ) ( )
ax( ) ( )
ax( ) ( ) ( )
( )
d
d d
d d
d d d d d d
d d
d
d d
d d
d d
=
=
=
= = +
= = +
= = +
= = + + + +
+ +
=
d1
3 613
( ) ( )
( )d d d d
d d
d d
+ + + +
+ +
d
(2-16)
The variations of the 1st and 2nd differentialrotation vectors
are gathered in the following formulae:
1 12 2
12
1 12 2
d d d d
d d d d d
d
d
12
d
= + = +
= + = +
= + = +
(2-17)
( )
( )
( )
S12
S12
S12
( )
( )
( )
d d
d d d d d
d d d d d
d
=
=
=
I
I
I
(2-18)
( )
( )
( )
S1 12 2
S1 12 2
S1 12 2
( ) (
( ) (
( ) (
d d d d d
d d d d d
d d d d d
d
d
= + + + +
= + + + +
= + + + +
)
)
)
(2-19)
2.5Corotational differentiationsFrom Eqs. (2-17)-(2-19)several
useful relations follow, for instance:
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2. Differentiation of the rotation12
d d d d
d d d
d d
d d d
= =
= =
= =
(2-20)
( )( ) ( )( ) ( )
S12
1 1 1 12 4 2 4
1 1 1 12 4 2 4
S12
( )( ) ( )
( ) ( )
( )
d d
d d d d d d
d d d d d d
d d d d
dd
d
= = + = +
= + = +
=
I d
d
( )( ) ( )( ) ( )
( )
1 1 1 12 4 2 4
1 1 1 12 4 2 4
S12
12
( ) ( )
( ) ( )
( )
d
d d d d d d d
d d d d d d d
d d d d d
d d
d
d
d
= + = +
= + = +
=
= +
I
I
( ) ( )
( ) ( )
1 1 14 2 4
1 1 1 12 4 2 4
( ) ( )
( ) ( )d d d d d
d d d d d d dd
= +
= + = +
(2-21)
S12
1 1 12 4 2
1 1 12 4 2
12
( ) ( )
( ) ( ) (
( ) ( ) (
(
d d d d d
d d d d d d d
d d d d d d
d d d
d
d
d
= + + +
= + = +
= + = +
= +
)
) d
S1 1 12 4 2
1 1 12 4 2
12
) ( )
( ) ( ) (
( ) ( ) (
( )
d d d
d d d d d d d
d d d d d d d
d d d d
d
d
d
)
)
+ + +
= + = +
= + = +
= + + +
S1 1 12 4 2
1 1 12 4 2
( )
( ) ( ) ( )
( ) ( ) (
d
d d d d d d
d d d d d d d
d
d
)d
= + = +
= + = +
(2-22)
Then, the following formulae ofcorotational differentiationof
the differential rotationvectors are obtained:
T T
1 12 2
T T
1 1
2 2
T T
12
( ) ( )
( ) ( )
( ) (
d
d d d d
d d d d
d
d
d d
d
d d
d d
= =
= = = =
= =
= = = = =
= =
12
)d d
d d
d
=
= =
d
(2-23)
T
T
T
( )
( )
( )
d
d d
d d d
d d
=
=
=
d
(2-24)
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2. Differentiation of the rotation 13
TS1
2
TS1
2
T
( )( )
( ) (
( )( )
( ) (
(( )
d d d
d d d d d
d d d d
dd d d d
d d d
dd
dd
= +
= + +
= + +
= + + +
= +
)
) d
S1
2
)( ) (d d d d d
= + + )
(2-25)
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3. Differentiation of rotation composition14
3Differentiation of rotation compositionIn this Chapter a
rotation is considered as being the composition of a first rotation
followed by a second rotation, giving the resultant rotation
= (3-1)Formulae for differential rotation vectors and the
relevant corotational differentiations are studied with reference
tothree possible independent variationsd, and .
3.1Differential rotation vectorsThe definitions of the
differential rotation tensors up to 3rd-order, Eqs. (2-15), apply
as are to the resultant rotation,
and apply of course to each of the subsequent rotations and ,
namely
( )( )( )
( )
T
T
T
T 12
T 12
T 12
T 12
d
d d d
d d d
d d d d d d d
d
d
d
d
=
=
=
= + +
= + +
= + +
= + + + + + +
(3-2)
and
( )
( )
( )
T
T
T
T 12
T 12
T 12
T 12
d
d d d
d d d
d d d d d d
d
d
d
d
=
= =
= + +
= + +
= + +
= + + + + + +
( ) d
(3-3)
Now, we are looking for the relationship among the differential
rotation vectors of the resultant rotation and thedifferential
rotationvectors of the two subsequent rotations and . Working
out:
T
T T T T
T
( )
= +
= +
= +
=
d d d
= +
= +
= +
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3. Differentiation of rotation composition 15
( )
T
T T T T T T T T
T T T1 12 2
1 12 2
( ) ( )
( ) ( ) ( ) ( ) (
= + + +
= + + + + + + +
= + + + + + +
( )
( ) ( )1 12 2
) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
+
= + + + + + + + + +
( ) ( )
( ) ( ) ( )( )
1 12 2
1 1 12 2 2
1 1 12 2 2
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( ) (
= + + + + +
= + + +
= + + + + +
( ) ( ) ( )1 1 1 12 2 2 2
)
( ) ( ) ( ) ( )
= + + + + + + +
( ) ( )1 12 212
( )
( )
= + + + + +
= + +
12
12
12
( )
( )
( )
d d d d d
d d d d d
= + + +
= + + +
= + + +
( )
T
T T
T T T T T T T
T T T T T T T T
12
12
( )
(
d d d d d d d
d
d
d d
d d d d
d d d d
= +
= + + +
+ + + +
= + + + + + +
+ +
( ) ( )
( )
( )
( ) ( )
T T12
T12
T12
T T1 12 2
) ( )
( )
( )
( ) (
d d d d d
d
d
d d d d d d
d
+ + + +
+ + +
+ + +
+ + + + + +
+ +
) ( ) T12 ( )d d d d d d + + + + +
( )121 12 2
12
( ) ( )( ) ( ) ( )( )
( ) ( ) ( ) ( ) (
d d d d d d d
d d d d d d
d d
= + + + + + +
+ + + + + +
+ + +
( )
( )
( )
12
12
12
12
)
( ) ( )( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) (
d d
d d d
d d d
d d d d
+ + +
+ + +
+ + +
+ + + +
(
)
)
( ) ( ) ( ) ( ) ( ) ( )d d d
+ + +
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3. Differentiation of rotation composition16
( )
( )
( )
12
12
1
2
12
( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) (
(
d d
d d d d
d d d d d
d d d d d
d
= +
+ + + + + +
+ + + + + +
+ + + + + +
+ +
)
( )
( )
( )
1 12 2
1 12 2
12
) ( ) ( ) ( ) ( ) ( )
( )( ) ( ) ( ) ( ) ( )
( )( ) ( ) ( ) ( ) ( )
(
d d
d d
d d d d
d
+ + + +
+ + + +
+ + + +
+ +
d
( )12 )( ) ( ) ( ) ( ) ( )d d d + +
( )
( )
( )( )
12
12
14
12
12
( )
( ) ( ) ( )
( ) ( ) (
( ) (
d d
d d d
d d d
d d d d d
d d d
d
= +
+ + +
+ +
+ + + + +
+ + +
+
)
( )
( )
( )
14
12
12
) ( )
( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) (
d d
d d d d
d d d d d
d d
+
+ + + + +
+ + + + + +
+ + + +
d
( )
( )
( )
1 12 2
1 12 2
1 12 2
) ( ) ( )
( )( ) ( ) ( ) ( ) ( )
( )( ) ( ) ( ) ( ) ( )
( )( ) ( ) (
d d
d d
d d d d
d d d
+ +
+ + + +
+ + + +
+ + +
( ) ) ( ) ( )d +
( ) ( )
( )
1 12 2
12
12
( )
( ) ( ) ( )
( ) ( )
d d
d d d d d d
d d d
d
= +
+ + + + + +
+ +
( ) ( )d+ ( ) ( )d + ( )
( )
( )
14
1
4
12
( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
d d d d d
d d d d d
d d
+ + + + +
+ + + + + + +
( )
( )
12
( ) ( )
d
d
( ) ( )d+ ( ) ( )d + ( )
( )
( )
14
14
12
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
(
d d d d
d d d d
d d
+ + + + +
+ + + + +
+ +
d
d
) ( ) ( + + ) ( ) (d d d + + ( )12
) ( )
( d d
+ +
) ( ) ( + + ) ( ) (d + + ( )) ( )d
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3. Differentiation of rotation composition 17
( )
( )
( )
1 12 2
1 12 2
1 12 2
( )( ) ( ) ( ) ( ) ( )
( )( ) ( ) ( ) ( ) ( )
( )( ) ( ) ( ) ( ) ( )
d d
d d d d
d d d
+ + + +
+ + + +
+ + + +
d
( ) ( )
( ) ( )
1 12 2
1 12 2
14
( )
( )
d d
d d d d d d
d d d d d d
d
= + + + + + + +
+ + + + + +
( )d+ ( ) ( ) + ( ) + ( )(
( )
d
d
+
( )d + ( ) )
( ) ( ) ( )( )( ) ( )(
( ) )
14
14
14
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
( )
d d d d d
d d d
d d
d
+ + + + + +
+ + +
+ +
+
( )d + ( ) ( ) ( ) + ( ) + ( )( ( ) ( )
d
d
+
( ) d+ ( ) )
14
( )
( ) ( )d
( ) + ( ) ( ) ( )d+ ( )d + ( )(
( ) ( )d + ( )d+ ( ))( ) ( )(
( ))( ) ( ) ( )( )
14
14
14
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( )
d d d
d d
d d d d
d
+ + + +
+ +
+ + + + +
+
d
( ) + ( ) ( )d + ( ) d+ ( )( ( ) d + ( )d + ( ))
( )
( )
( )
1 12 2
1 12 2
1 12 2
( )( ) ( ) ( ) ( ) ( )
( )( ) ( ) ( ) ( ) ( )
( )( ) ( ) ( ) ( ) ( )
d d
d d d d
d d d d
+ + + +
+ + + +
+ + + +
( ) ( )( ) ( )
1 12 2
1 12 2
14
( )
( ) ( ) ( ) ( )
d d d d d d d d
d d d d d d
d d d d
= + + + + + + +
+ + + + + +
+ + + +
(
)14
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
d d d d
d d d d
d d d d
+ + + +
+ + + +
+ + + +
(
)
( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
d d d d
d d d
+ + + +
+ + + +
d
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3. Differentiation of rotation composition18
( )( ) ( )
( )(
1 12 2
1 12 2
14
( ) ( )
d d d d d d d d
d d d d d d
d
d d
= + + + + + + +
+ + + + + +
+ + +
+ + +
( )( ) )
( )(( )
( ) )
14
d d
d d d d
d
d d d d
d d d d
+
+ + + +
+ + +
+ + + +
+ + + +
(
(
))
1 12 2
S S12
S
12
( ) ( )
( ) ( ) ( ) ( )
( ) ( )
d d d d d d d
d d d d
d d
d d
= + + + + + + +
+ + + + +
+ + +
+ + +
( )
d
( )
( ) ( )
12
1 12 2
d d d d
d d d d d d d
+ + +
= + + + + + +
(
)
1 12 2
S S12
S
( ) ( )
( ) ( ) ( ) ( )
( ) ( )
d d d d d d d d d
d d d d
d d
= + + + + + + +
+ + + + +
+ + +
The following relationship among thedifferential rotation
vectors in the case of rotation composition is therefore found:
12
12
12
1 12 2
( )
( )
( )
( ) (
d d d
d d d d d
d d d d d
d d d d d d d
= +
= +
= +
= + + +
= + + +
= + + +
= + + + + +
(
)
S S12
S
)
( ) ( ) ( ) ( )
( ) ( )
d d
d d d d
d d
+ +
+ + + + +
+ + +
(3-4)
3.2Corotational differentiationsThe formulae of corotational
differentiation, Eqs. (2-23)-(2-25), can be given a form based on
the differentialrotation
vectors of the two subsequent rotations. Working out:
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3. Differentiation of rotation composition 19
( )
T
T
T T T
T T T T
T1 12 2
1 12 2
( )
( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
= +
= +
= +
= +
= + +
( )( )( )
( )
T
T 12
T T T T T12
T T T T
T T T T1 12 2
( )
( )
( ) ( ) ( )
( ) ( )
( ) ( ) (
d
d
d d d
d d d
d d
= + + +
= + + +
= +
+ + +
T T T T
T T T T1 1 1 1 12 2 2 2 2
)
( ) ( ) ( ) ( ) ( ) ( ) (
d d d
d d d d d
= + + + +
) T 1 1 1 1
2 2 2 2
T1 1 1 1 12 2 2 2 2
( ) ( ) ( ) ( ) (
( ) ( ) ( )
( ) (
d d d d d d d
d d d d d
d d
d d
d d
)
= +
+ + +
= +
1 1 1 12 2 2 2
T T T T1 1 1 1 12 2 2 2 2
) ( ) ( )
( ) ( ) ( )
d d d d d
d d d d d
+ + +
+ +
( ) ( )
1 1 1 12 2 2 2
T T1 1 1 12 2 2 2
( ) ( ) ( ) ( )
( ) ( )
d d d d d d d
d d d d d d
d d
= + + + +
+ +
( )
T
T
T T T
T T T T T T T T
T T T
T T T T T T T
( )
( )
( ) ( )
( ) ( ) ( ) ( )
( )
( ) ( )
d
d
d d
d d d d d
d d
d d d
= +
= +
= +
=
+
( ) ( )
T T T T
S T1 12 2
T T T1 12 2
( )
( ) ( ) (
( ) ( )
d d d d d d d d
d d d d d
d
d d
T
)
+ +
= + + + +
+ +
+
( )S12 ( ) ( )d d d d + +
( )
( )
( )
S12
S12
T1 1 12 2 2
( ) ( )
( ) ( )
( ) ( ) ( )
d d d d d
d d d d d
d d d d d d
= + +
+ + +
+ + +
The following formulae for thecorotational differentiation in
the case of rotation composition are found:
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3. Differentiation of rotation composition20
T T1 12 2
1 12 2
T T1 12 2
1 12 2
( ) ( ) ( ) ( )
( ) ( )
( ) ( ) ( ) ( )
( ) ( )
d d d d d
d d d d d
d
= =
+ + + +
= =
+ + + +
d
T T1 12 2
1 12 2
( ) ( ) ( ) ( ) ( ) ( )
d d d d d
d d d d d d
d
= = + + + +
(3-5)
( ) ( )
T
1 1 1 12 2 2 2
T T1 1 1 12 2 2 2
T
( ) ( ) ( )
( ) ( )
( ) ( )
( ) ( ) (
d d d
d d d d
d d d d d d
d d d d
d d d
= + +
+ +
+ +
= +
( ) ( )
1 1 1 12 2 2 2
T T1 1 1 12 2 2 2
T
1 12 2
)
( ) ( )
( ) ( ) ( ) ( ) ( )
(
d d
d d d
d d d d
d d d d d d
+
+ +
+ +
= + +
+
( ) ( )
1 12 2
T T1 1 1 12 2 2 2
) ( )
( ) ( )
d d d
d d d d
+
+ +
(3-6)
( )
( )
( )
T S12
S12
1 12 2
T12
T 12
( ) ( ) ( )
( ) (
( ) ( )
( )
( ) (
d d d d d
d d d d d
d d d
d d d
d d
d
= + +
+ + +
+ +
+
=
( )
)
( )
( )
S
S12
1 12 2
T12
T 12
) ( )
( ) (
( ) ( )
( )
( ) (
d d d d
d d d d
d d d d
d
d d dd
+ +
+ + +
+ +
+
= +
( )
) d
( )
( )
S
S12
1 12 2
T12
) ( )
( ) (
( ) ( )
( )
d d
d d d d d
d d d
d d d
+
+ + +
+ +
+
)
(3-7)
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4. Differentiation of rotations in a continuum 21
4Differentiation of rotations in a continuumThe differentiations
of rotation so far studied apply to a number of fields in
mechanics, according to the meaning of
variations d, and . For instance, in a boundary-value
incremental problem of rigid body mechanics, variations d, and
could be associated with variations in time (velocity), incremental
variations and virtual variations, respectively.
Our interest is focused on elastostatics of polar continua. In
view of that, we draw in this Chapter a specialization ofthe
differentiation of rotations where variation d refers to aspatial
change, variation refers to an incremental change,and variation
refers to avirtual change. As usual, we convert variationd into
afinitespatial change by means of the
notion of gradient, d dx. First, let us introduce a coordinate
system describing the continuum, made ofn families of
coordinate lines j (withj ranging from 1 to n, and in general n
=3 for a three-dimensional continuum), with jg =
jd dx and jg the reciprocal frames of base vectors. Then, the
dyadic expression /() (),j
j = g for the gradient
()d dx enables us to consider n spatial derivatives (), () jj d
d= in place of a 1-order higher tensor, the spatial
gradient2. However, note that suchnspatial derivatives are
actually separate components of a unique spatial gradient, sothey
do not combine into mixed derivatives unless one takes a second
spatial variation.Therefore, in this Section we will deal with the
following set of rotational changes: n+2 simple
1st-orderrotational
changes (namely, n spatial derivatives plus 1 incremental and 1
virtual variations); 2n+1 mixed 2nd-order rotational
changes (namely, n incremental variations of spatial derivatives
plus n virtual variations of spatial derivatives plus 1mixed
incremental-virtual variation); and n doubly-mixed 3rd-order
rotational changes (namely, n doubly-mixedincremental-virtual
variations of spatial derivatives). However, since the spatial
derivatives are actually separatecomponents of a unique spatial
gradient, the set of independent rotational changes reduces to: 3
simple 1st-orderrotational changes, the spatial gradient and the
incremental and virtual variations; 3 mixed 2nd-order rotational
changes,the incremental and virtual variations of the spatial
gradient and the mixed incremental-virtual variation; and 1
doubly-mixed 3rd-order rotational change, the doubly-mixed
incremental-virtual variation of the spatial gradient.
4.1Differentiation of the rotation4.1.1Characteristic rotation
differentials
We replace the 1st differential rotationvector d with the finite
spatial angular velocities aj , and keep the 1st
differential rotationvectors and with the meaning of incremental
rotation vector and virtual rotation vector,respectively. We use
the general notation spatial angular velocity with reference to the
concept of angular strain.Moreover, we replace the 2nd differential
rotation vectors d and d with the incremental spatial angular
velocities a j and thevirtual spatial angular velocities a j ,
respectively, and keep the 2nd differential rotation
vector with the meaning of mixedvirtual-incremental rotation
vector. Finally, we replace the 3rd differential
rotationvectors d with themixed virtual-incremental spatial
angular velocities a j . Then, from Eqs. (2-1) and
(2-15), and rearranging the order, we obtain
T
T
T
T 12( )
=
=
= = + +
I
and
T
T 12
T 12
T 1 12 2
,
, ( )
, ( )
, ( ) (
j a j
j a j a j a j
j a j a j a j
j a j a j a j a j a j a j a j
=
= + +
= + +
= + + + + + +
)
or, basing equivalently on the back-rotated differential
rotational vectors,
2 Short notations /() (),j
j = g and /() (),j
j = g denote the gradient grad =() () and divergence di =v() ()
.
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4. Differentiation of rotations in a continuum22
( )
T
T T
T T
T T T T T T12
( )
( )
( ) ( ) ( ) ( ) ( )
=
=
=
= + +
I
and
( )
( )
( )
T T
T T T T T T12
T T T T T T12
T T T T T T12
T T T T12
, ( )
, ( ) ( ) ( ) ( ) ( )
, ( ) ( ) ( ) ( ) ( )
, ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) (
j a j
j a j a j a j
j a j a j a j
j a j a j a j
a j a
=
= + +
= + +
= + +
+ +
( )T T T T) ( ) ( ) ( ) ( )j a j a j + +
Then, we introduce the spatial angular velocitytensor ja aj= g
and the relevant virtual, incremental andmixed virtual-incremental
tensors ja a j = g , ja a j = g and ja a j = g , and work out as
followssome mixed variations,
T T T T T12
T T T T T12
T T T T T12
T T T T T T12
, ( ) ( ) ( ) ( )
, ( ) ( ) ( ) ( )
, ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
j a j a j a j
j a j a j a j
j a j a j a j
a j a j a j a j
= +
= +
= +
+ + +
( )
The variations of the rotation tensor and of its gradient are
finally given the following expressions3:
( )
T
T T
T T
T T T T T T12
( )
( )
( ) ( ) ( ) ( ) ( )
=
=
=
= + +
I
(4-1)
and
T T/
T T T T T1/ 2T T T T T1
/ 2T T T T T1
/ 2T T T T T T1
2
( )
( ) ( ) ( ) ( )( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
a
a a a
a a a
a a a
a a a
=
= + = +
= +
+ + +
( )a
(4-2)
So, the variations of the rotation tensor are characterized by
the virtual, incremental and mixedvirtual-incrementalrotation
vectors, thefinite spatial angular velocitytensor, and thevirtual,
incremental and mixed virtual-incrementalspatial angular
velocitytensors. Suchcharacteristic rotation differentials are the
following axials, together with therelevant expressions as drawn
from Eqs. (2-16):
( )3 The tensor-cross notation ( ) = jj g refers to a 3rd-order
tensor of skew-symmetric nature, built on three skew-
symmetric tensors, hence characterized by the second-order
tensor ( ) = ( ) jj g built on the respective axials and referred
to as
the relevant axial tensor.
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4. Differentiation of rotations in a continuum 23
T T
T T
T T 12
ax( ) ax( )
ax( ) ax( )
ax( ) ax( ) ( )
= =
= =
= = = +
(4-3)
and
( )
T/
T 1/ /2
T 1/ /2
T 1 1/ /3 6
S1 13 2
1 1/ /3 6
1 13 2
ax( )
ax( ) ( )
ax( ) ( )
ax( ) ( ) ( )
2( )
( ) ( )
a
a a
a a
a a a a a
a
a a a
=
= = +
= = +
= = + + +
+
= + + +
+
I
( )S2( ) a I
(4-4)
The variations of the characteristic rotation differentials are
reported in the following.
1 12 2
12
1 1/ /2 2
a a a a a
a a a
= + = +
= + = +
= =
12 a
a
(4-5)
( )
( )
( )
S1 12 4
S1 12 4
S1
/ 2
( )
( )
( )
a a a
a a a
a a
= + +
= + +
= I
a
a
(4-6)
( ) ( )( ) ( )
( ) ( )
S1 12 2
1 1/ 2 2
1 1/ 2 2
( )a a a a a
a a a
a a a
= + + + +
= +
=
I
I
a
a
(4-7)
4.1.2Corotational differentiationsFrom Eqs. (4-5)-(4-7) follow
thecorotational differentiation formulae of the characteristic
rotation differentials:
T T
1 12 2
T T/ / / /
1 12 2
/
T
12
( ) ( )
( ) ( )
( )
a a
a
a a a
a a a
a a
= =
= = = =
= =
= + = + = + = +
=
= =
a
a
/
T
12
( )a a a
a a
=
= =
(4-8)
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4. Differentiation of rotations in a continuum24
T
T
T/ /
( )
( )
( )
a a
a a
a
=
=
= +
a
a
(4-9)
( )
(
TS1 1
2 2
/T/ 1 1
2 2
/T/
( )( )
( ) ( )
( )( )
( )
(( )
a a a a
aa a a a
a a a
a a a
a
= +
= +
= +
= + + +
=
I
) a
( )1 12 2
)
( )a a
a a a
+
= + + + I
a
(4-10)
4.2Differentiation of rotation compositionWe consider now a
rotation = resultant of the composition of two subsequent rotations
and .
4.2.1Characteristic rotation differentialsEqs. (4-1) and (4-2)
hold for the resultant rotation . Analogously, we can write the
following characterizations of
the variations of subsequent rotations and separately,
( )
T
T T
T T
T T T T T T12
( )
( )
( ) ( ) ( ) ( ) ( )
=
=
=
= + +
I
and
T T/
T T T T T1/ 2
T T T T T1/ 2
T T T T T1/ 2
T T T T T T12
( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
a
a a a
a a a
a a a
a a a
=
= +
= +
= +
+ + +
( )a
and then
( )
T/
T T
T T
T T T T T T12
( )
( )
( ) ( ) ( ) ( ) ( )
=
=
=
= + +
I
and
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4. Differentiation of rotations in a continuum 25
T T/
T T T T T1/ 2
T T T T T1/ 2
T T T T T1/ 2
T12
( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
(
a
a a a
a a a
a a a
a
=
= +
= +
= +
+
T T T T ) ( ) ( ) ( ) ( )a a + + T a
Now, by specializing Eqs. (3-4) and working out,
12
12
12
12
( )
( )
( )
( )
a a aj j j
a a a a aj jj j j
a a a a aj jj j j
a a a a aj jj j j
= +
= +
= +
= + + +
= + + +
= + + +
= + + +
(
)
12
S12
S S
( )
( ) ( )
( ) ( ) ( ) ( )
a a a aj j j j
a aj j
a a a aj j j j
+ + + +
+ +
+ + + + + +
( )
( )
( )
( )
12
12
12
1 12
( )
( )
( )
a a aj j j
a a a a aj jj j j
a a a a aj jj j j
a a a a aj jj j j
= +
= +
= +
= + + +
= + +
= + +
= + + +
( )
( )
( )
2
S12
14
1
4
( ) ( )
( )
( ) ( ) ( ) ( )
( ) ( ) (
a a a aj j j j
a aj j
aj
+
+ +
+ + + + + + +
+ + + + +
I I
( ) ) ( ) a
j + + I I
the following relationship among thecharacteristicrotation
differentials in the case of rotation composition are found:
( )12
= +
= +
= + + +
(4-11)
and
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4. Differentiation of rotations in a continuum26
( )
( )
( ) ( )( )
12
12
1 12 2
S12
( )
( )
( ) ( ) ( )
(
a a a
a a a a a
a a a a a
a a a a a a a a a
a
= +
= + +
= + +
= + + + +
+ +
( )
( )
14
14
)
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
a
a
a
+ + + + + + +
+ + + + + + +
I I
I I
(4-12)
4.2.2Corotational differentiationsThe formulae of corotational
differentiation are now worked out as a specialization of Eqs.
(3-5)-(3-7):
T 1 12 2
T 1 12 2
T 1 12 2
T 1 12 2
( ) ( ) ( )
( ) ( ) ( )
( ), ( ) ( )
( ), ( ) (
j a a a aj jj j
j a a a aj jj j
= + +
= + +
= + +
= +
aj
T 1 12 2
T 1 12 2
)
( ) ( ) ( )
( ) ( ) ( )
aj
a a a a a aj j jj j
a a a a a aj j jj j
j
j
+
= + +
= + +
( ) ( )
T
1 1 1 12 2 2 2
T T1 1 1 12 2 2 2
T
( ) ( ) ( )
( ) ( )
( ) ( )
( ) (
a a a a a aj j j j j j
a a aj jj
a a a aj j jj
a aj j
= + +
+ +
+ +
=
( ) ( )
1 1 1 12 2 2 2
T T1 1 1 12 2 2 2
T
) ( )
( ) ( )
( ) ( )
( ), ( , ) ( ,
a a a aj j j j
a a aj jj
a a a aj j jj
j j a j aj j
+ +
+ +
+ +
= +
( ) ( )
1 1 1 12 2 2 2
T T1 1 1 1
2 2 2 2
)
( ) ( )
( ) (
aj
a a a aj jj j
a a a a a aj j j jj j
)
+
+ +
+ +
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4. Differentiation of rotations in a continuum 27
( )
( )( )
T S12
S12
T1 1 12 2 2
( ) ( ) ( )
( ) ( )
( ) ( ) ( )
(
a a a a a aj j jj j j
a a a a aj jj j j
a a a a aj j jj j
= + +
+ + +
+ + +
( )( )
( )
T S1
2
S12
T1 1 12 2 2
T
), ( ) ( ) ( ) ( )
( ) ( ) ( )
(
j a a a a aj jj j j
a a a a aj jj j j
a a a a a aj j j jj j
= + +
+ + +
+ + +
( )( )
( )
S12
S12
T1 1 12 2 2
), ( ) ( )
( ) ( )
( ) ( ) ( )
j a a a a aj jj j j
a a a a aj jj j j
a a a a a aj j jj j
= + +
+ + +
+ + +
j
T 1 12 2
T 1 12 2
T 1 12 2
T 1 12 2
( ) ( ) ( )
( ) ( ) ( )
( ), ( ) ( )
( ), ( ) (
j a a a a aj jj j
j a a a ajj j
= + +
= + +
= + + + +
= + + +
j
T 1 12 2
T 1 12 2
)
( ) ( ) ( ) (
( ) ( ) ( ) ( )
aj j
a a a a aj j jj j
a a a a aj j jj j
+
= +
= +
) aj
aj
( )( ) ( )
T
1 1 1 1 12 2 2 2 2
1 1 12 2 4
( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
a a a a a aj j j j j j
a a a aj jj j
aj
= +
+ +
+ +
( )( )
T
1 1 1 1 12 2 2 2 2
1 1 12 2 4
( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
aj
a a a a a aj j j j j j
a a a aj jj j
aj
= +
+ +
+ +
( )
( )
T
1 1 1 1 1 12 2 2 2 2 2
14
( ), ( , ) ( , )
( ) ( ) ( ) ( )
( ) ( ) ( )
aj
j j a j a aj j j
a a a a a aj jj j j
aj
= + + + +
+ + + +
+ + +
j
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4. Differentiation of rotations in a continuum28
( )( )( )( )
T S1 12 2
S1 12 2
1 12 2
1
2
( ) ( ) ( )
( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) (
a a a a aj jj j j
a a a ajj j j
a a a aj jj j
= +
+ +
+
( )( )
( )( )
( )( )
T 1 12 2
1 12 2
1 12 2
14
)
( ), ( )
( )
( ) ( ) ( )
(
aj
j a a a ajj j j
a a a jj j j
a a aj jj
a j
= + + +
+ + + +
+ +
+ +
I
I
a
( )
( )( )
( )( )
T 1 12 2
1 12 2
1 12 2
) ( )
( ), ( )
( )
( ) ( ) ( )
aj
j a a a ajj j j
a a a jj j j
a a aj jj
= + + +
+ + + +
+ +
I
I
a
( )14 ( ) ( )a ajj + +
This yields the following formulae for thecorotational
differentiation in the case of rotation composition:
T T1 12 2
1 12 2
T T1 1/ /2 2
1 12 2
( ) ( ) ( ) ( )
( ) ( )
( ) ( ) ( ) ( )
( ) ( )
a a a a
a a a a a
= =
+ + + +
= + = +
+ + + + + +
T T1 12 2
1 12 2
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
a
a a a a a a
a a a a a
= = + +
a
(4-13)
( )( )( )
T
1 1 1 1 12 2 2 2 2
1 12 2
14
T
( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( )
(
a a a a a a
a a a a
a
a
a
= +
+ +
+
+
( )( )( )
1 1 1 1 12 2 2 2 2
1 12 2
14
T/
) ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( )
( ) (
a a a a a
a a a a
a
a
= +
+ +
+
+
=
( )
/ /
1 1 1 1 1 12 2 2 2 2 2
14
) ( )
( ) ( ) ( ) ( )
( ) ( ) ( )
a a a
a a a a a
a
+ + + +
+ + + +
+ + +
a
(4-14)
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4. Differentiation of rotations in a continuum 29
( )( )( )( )
( )( )
T S1 12 2
S1 12 2
1 12 2
12
( ) ( ) ( )
( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
(
a a a a a
a a a a
a a a a
a
= +
+ +
+
( )( )( )( )
( )
T 1 1/ 2 2
1 12 2
1 12 2
14
) ( )
( )
( ) ( ) ( )
( ) ( )
a a a a
a a a
a a a
a a
= + + +
+ + + +
+ +
+ +
I
I
a
( )( )
( )( )
( )
T 1 1/ 2 2
1 12 2
1 12 2
14
( ) ( )
( )
( ) ( ) ( )
( ) ( )
a a a a
a a a
a a a
a
= + + +
+ + + +
+ +
+ +
I
I
a
a
(4-15)
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5. Differentiation of the rototranslation30
5Differentiation of the rototranslationA rototranslation is
defined as a dual tensor that transforms a pole-based dual vector
into another dual vector of the
same magnitude based on the same pole (Merlini and Morandini
2003). The reader is addressed to a paper by Borri,Trainelli and
Bottasso (2000) for a deep theoretical settlement of
rototranslations, and to a paper by Angeles (1998) forthe
application of dual algebra to tensors and vector transforms. A
rototranslation must be an orthonormal dual tensor.
Let be a rototranslationtensor, namely a dual rotation tensor.
To be orthonormal, a dual tensor is required topossess such a
form
= + t , (5-1)with an orthonormal tensor, namely a rotation
tensor, and ta length vector, referred to as translation vector.
Theorthonormality condition reads
T = I (5-2)
As for the case of rotation, we have to characterize three
independent variations of therototranslation tensor
referred to asd, and respectively. Due to the rules of the dual
tensor algebra, such characterization turns out to beidentical to
that of the differential of the rotation tensor. Thus, no
derivation will be attempted here, and just theoutcome from the
case of the rotation will be concisely presented.
5.1Differential helicesThe characterization leads to the
following recursive expressions for the subsequent variations of
the rototranslation,
T
T
T
T 1
2T 1
2T 1
2
T 12
( )( )
( )
( )
d
d d d
d d d
d d d d d d d
d
d
d
d
=
=
=
= + +
= + +
= + +
= + + + + + +
(5-3)
where the characteristic differential vectors are the following
axials called 1st, 2ndand 3rd differential helicestogether with the
relevant expressions:
T
T
T
T 12
T 12
T 12
T 1 13 6
13
1
ax( )
ax( )
ax( )
ax( ) ( )
ax( ) ( )
ax( ) ( )
ax( ) ( ) ( )
( )
d
d d
d d
d d d d d d
d d
d
d d
d d
d d
=
=
=
= = +
= = +
= = +
= = + + + +
+ +
=
d1
3 6
13
( ) ( )
( )d d d d
d d
d d
+ + + +
+ +
d
(5-4)
5.2Corototranslational differentiationsThe formulae
ofcorototranslational differentiationof the differential helices
follow:
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5. Differentiation of the rototranslation 31
T T
1 12 2
T T
1 12 2
T T
12
( ) ( )
( ) ( )
( ) (
d
d d d d
d d d d
d
d
d d
d
d d
d d
= =
= = = =
= == =
= =
=
= =
1
2
)d d
d d
d
=
= =
d
(5-5)
T
T
T
( )
( )
( )
d
d d
d d
d d
=
=
=
d
d
(5-6)
TS1
2
TS1
2
T
( )( )
( ) (
( )( )
( ) (
(( )
d d d
d d d d d
d d d d
dd d d d
d d d
dd
dd
= +
= + +
= + +
= + + +
= +
)
) d
S1
2
)
( ) (d d d d d
= + + )
(5-7)
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6. Differentiation of rototranslation composition32
6Differentiation of rototranslation compositionWe consider now a
rototranslation composition of a first rototranslation followed by
a second rototranslation ,
giving the resultant rototranslation
= (6-1)Formulae for differential helices and the relevant
corotational differentiations are studied with reference to
threepossible independent variationsd, and .
6.1Differential helicesThe relationship among thedifferential
helicesof the resultant and the subsequent rototranslations are the
same as for
the case of the rotation:
12
12
12
1 12 2
( )
( )
( )
( ) (
d d d
d d d d d
d d d d d
d d d d d d d
= +
= +
= +
= + + +
= + + +
= + + +
= + + + + +
(
)
S S12
S
)
( ) ( ) ( ) ( )
( ) ( )
d d
d d d d
d d
+ +
+ + + + +
+ + +
(6-2)
6.2CorototranslationaldifferentiationsThe following formulae
ofcorototranslational differentiationare reported:
T T1 12 2
1 12 2
T T1 12 2
1 12 2
( ) ( ) ( ) ( )
( ) ( )
( ) ( ) ( ) ( )
( ) ( )
d d d d d
d d d d d
d
= =
+ + + +
= =
+ + + +
d
T T1 1
2 2
1 12 2
( ) ( ) ( ) ( )
( ) ( )
d d d d d
d d d d d d
d
= =
+ + + +
(6-3)
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6. Differentiation of rototranslation composition 33
( ) ( )
T
1 1 1 12 2 2 2
T T1 1 1 12 2 2 2
T
( ) ( ) ( )
( ) ( )
( ) ( )
( ) ( ) (
d d d
d d d d
d d d d d d
d d d d
d d d
= + +
+ +
+ +
= +
( ) ( )
1 1 1 12 2 2 2
T T1 1 1 12 2 2 2
T
1 12 2
)
( ) ( )
( ) ( )
( ) ( ) ( )
(
d d
d d d
d d d d
d d d d d d
+
+ +
+ +
= + +
+
( ) ( )
1 12 2
T T1 1 1 12 2 2 2
) ( )
( ) ( )
d d d
d d d d
+
+ +
(6-4)
( )
( )
( )
T S12
S12
1 12 2
T12
T 12
( ) ( ) ( )
( ) (
( ) ( )
( )
( ) (
d d d d d
d d d d d
d d d
d d d
d d
d
= + +
+ + +
+ +
+
=
( )
)
( )
( )
S
S12
1 12 2
T12
T 12
) ( )
( ) (
( ) ( )
( )
( ) (
d d d d
d d d d
d d d d
d
d d dd
+ +
+ + +
+ +
+
= +
( )
) d
( )
( )
S
S1
2
1 12 2
T12
) ( )
( ) ( ( ) ( )
( )
d d
d d d d d
d d d
d d d
+
+ + +
+ +
+
)
(6-5)
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7. Differentiation of the rototranslation decomposition34
7Differentiation of the rototranslation decompositionA
rototranslationtensor = , see Eq. (5-1), can always be given
amultiplicative decomposition+ t
= T (7-1)
into arotation around a pole followed by atranslation= + T I t .
(7-2)
It is worth noting that the rototranslation is actually
thecompositionof two subsequent particular rototranslations,
therotation apurely real rototranslationand thedual translation,
namely the orthonormal translationtensorT=
, built on the translationvector t. This consideration enables
us to take advantage of the issues of the
preceding Chapter about composition of rototranslations.
exp( ) t
7.1Differential helicesThe characterization of first (real)
rototranslation yields the well-knowndifferential rotation vectors
d , , ,d , , d and d as defined in Eqs. (3-2). As far as concerns
the subsequent pure translation T, multiple
differentiation of Eq. (7-2) is immediate due to the rules of
dual algebra, and is conveniently put in the form
T
T
T
T
T
T
T
d
d
d
d
d
d
d
d
=
=
=
=
=
=
=
(7-3)
where the characteristicdifferential translation vectors d , , ,
d , , d and d arepure dual vectors given
by:
d
d
d
d
d
d
dd
=
=
=
=
=
= =
t
t
t
t
t
t
t
(7-4)
With the differentiation of composition of rototranslations at
hand, the differential helices can then be written in termsof the
differential vectors of the subsequent separate rotation and
translation. The following expressions are obtained
forthedecomposed differential helices d , , d , , d and d :
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7. Differentiation of the rototranslation decomposition 35
( )
12
1
212
12
S12
( )
( )( )
( ) (
d d d
d d d d d
d d d d d
d d d d d d d d d
d d d
= +
= +
= +
= + + +
= + + +
= + + +
= + + + + + + +
+ + +
( )S S) ( )d d + +
(7-5)
Notice that, for each characteristic differential, the number of
vectors keeps unchanged namely a real vector
together with a pure dual vector instead of a proper dual vector
, etc.
Inversion of Eqs. (7-5) yields thedifferential translation
vectorshence the variations of vector tas a function ofthe
differential helices and differential rotation vectors. Working
out:
d
d d
=
=
=
( )
( )
( )
12
12
12
12
12
12
( )
( ) (
( )( ) (
( )
( ) ( )
d
d d d d
d d d d d
d
d d d d
d d d d d
= + +
= + +
= + + = + +
= + +
= + +
)
)
( )
( )( )
12
S S S12
12
1
4
( ) ( ) ( )
( ) ( ) ( )
( ) (
d
d d d d d d d d
d d d d d
d d d d d d d
d d
= + + + + + +
+ + + + + +
= + + +
+ + + +
( )( )
( ) ( ) ( )( )
12
S S S12
) ( )
( ) ( ) ( )
d d d
d d d
d d d d d
+ + + + +
+ + + + + + I I
I
( ) ( )( )
( ) ( ) ( )( )
1 12 2
14
S S S12
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
d d d d d d d d d d
d d d d d
d d d d d
d
= + + + + + +
+ + + + + +
+ + + + + +
=
I I I
( ) ( )
( )
1 12 2
S S S
12
12
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) (
d d d d d d d d d
d d d
d d d d d d d
d d d d
+ + + + + +
+ + +
= + + +
+ + +
( )
S S S)
( ) ( ) ( ) ( ) ( ) ( )d d d d
+ + +
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7. Differentiation of the rototranslation decomposition36
The following expressions are then obtained:
( )( )( )
( )
12
12
12
12
( ) ( )( ) ( )
( ) ( )
( ) ( ) (
d d d
d d d d d d
d d d d d d
d d d d d d d d
=
=
=
= + + = + +
= + +
= + + +
)
( )12S S S
( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) (d d d d
d d d d
)
+ + +
+ + +
(7-6)
Given the fact that the differential rotation vectors are just
the primal parts of the differential helices themselves (seebelow),
Eqs. (7-6) constitute the means of extracting the variations of the
translation vectort from the differentialhelices themselves.
7.2Explicit form of the differential helicesBy using the
explicit form of the rototranslation tensor Eq. (5-1) within the
definition of the differential helices, the
explicit form of thedifferential helicescan be drawn as
T T
T T
T T
T T
T
ax( ) ax( ( ) )
ax( ) ax( ( ) )
ax( ) ax( ( ) )
ax( ) ax( ( ) )
ax(
d d d d d
d d d d d
d d d
d d d
= + = =
= + = =
= + = =
= + = =
= + =
t t
t t
t t
t
T T
T T
T T
) ax( ( )
ax( ) ax( ( ) )
ax( ) ax( ( ) )
d d d d d
d d d d d
d d
d d
=
= + = =
= + = =
t t
t t
t t
T
T
T
Tt
T
T
)
d
d
(7-7)
It is seen that the primal parts are just the differential
rotation vectors. In order to obtain an expression for the dual
parts,we resort to the issues of the decomposition of the
rototranslation, Eqs. (7-5).The dual parts of the differential
helices,i.e. the real vectors d , , , d , , d and d , are easily
recognized to be
( )
( )
12
1
212
12
S12
( )
( )( )
( )
d d
d d d
d d d
d d d d d
d d
d
d d
d d
d d d d
=
=
=
= +
= + = +
= + + + + +
+ +
t t
t t
t t
t t t t
t t t t t t t t
t t t t t t t t
I t
( ) (( )S S( ) ( )d dd ) + + + I t I t
(7-8)
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8. Differentiation of rototranslations in a continuum 37
8Differentiation of rototranslations in a continuumAs for the
case of the rotation, we deal in this Section with aspecialization
of the differentiation of rototranslations
where variation d refers to a spatial change, variation refers
to an incremental change, and variation refers to avirtual
change.
8.1Differentiation of the rototranslation8.1.1Characteristic
rototranslation differentialsThe differentiation of
rototranslations mirrors strictly the case of rotations. The
virtual, incremental and mixed
virtual-incremental rotation vectors are replaced by the
corresponding helices (namely dual rotation vectors). Thespatial
angular velocity and the relevant virtual, incremental and mixed
virtual-incremental tensors are replaced bythe corresponding
generalizedspatial velocity dual tensors , , , and . J ust the
outcome borrowed fromthe case of rotation will be presented.The
incremental and virtual variations and the spatial gradients take
the expressions
( )
T
T T
T T
T T T T T T12
( )
( )
( ) ( ) ( ) ( ) ( )
=
=
=
= + +
I
(8-1)
and
T T/
T T T T T1/ 2
T T T T T1/ 2
T T T T T1/ 2
T T T T T T12
( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
=
= +
= +
= +
+ + +
( )
(8-2)
and are characterized by the virtual,incremental and
mixedvirtual-incremental helices, the finite
generalizedspatialvelocitytensor, and the virtual,incremental and
mixedvirtual-incremental spatial velocitytensors, namely
thefollowing axials, together with the relevant expressions:
T T
T T
T T 12
ax( ) ax( )
ax( ) ax( )
ax( ) ax( ) ( )
= =
= =
= = = +
(8-3)
and
( )
T/
T 1/ /2
T 1/ /2
T 1 1/ /3 6
S1 13 2
1 1/ /3 6
S1 1
3 2
ax( )
ax( ) ( )
ax( ) ( )
ax( ) ( ) ( )
2( )
( ) ( )
2( )
=
= = +
= = +
= = + + +
+
= + + +
+
I
( ) I
(8-4)
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8. Differentiation of rototranslations in a continuum38
8.1.2Corototranslational differentiationsThecorototranslational
differentiation formulae are reported:
T T
1 12 2
T T/ / / /
1 12 2
/ /
T T
12
( ) ( )
( ) ( )
( ) ( )
= =
= =
= =
= =
= + = + = + = +
= =
= ==
1
2
=
(8-5)
T
T
T/ /
( )
( )
( )
=
=
= +
(8-6)
( )
(
TS1 1
2 2
/T/ 1 1
2 2
/T
/
( )( )
( ) ( )
( )( )
( )
( )( )
= +
= +
= +
= + + +
= +
=
I
) ( )1 12 2( )
+ + + I
(8-7)
8.2Differentiation of rototranslation compositionWe consider now
a rototranslation = resultant of the composition of two subsequent
rototranslations and.
8.2.1Characteristic rototranslation differentialsThe following
relationship holds among thecharacteristicrototranslation
differentials in the case of rototranslation
composition:
( )12
= +
= +
= + + +
(8-8)
and
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8. Differentiation of rototranslations in a continuum 39
( )
( )
( ) ( )(
