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Foundations of tensor algebra and analysis(composed by Dr.-Ing.
Olaf Kintzel, September 2007, reviewed June 2011.)Web-site:
http://www.kintzel.net
1 Tensor algebra
Indices:α, β, γ, δ, . . . ∈ {1, 2}i, j, k, l, m, . . . ∈ {1, 2,
3}
Kronecker delta:
δij = δij = δij = δ
ji
{
1 for i = j0 for i 6= j
.
Einstein summation convention:If an index appears twice within a
tensor component relationin co-variant and contra-variant position,
we have to sum withrespect to these indices. This index is called
summation index(or dummy index) and is different from a free index
which ap-pears only once. Three-fold or four-fold appearing indices
arenot allowed.
Representation of tensors of first order (vectors):
AAA♭ = AiGGGi (co-variant component) ,
AAA♯ = AiGGGi (contra-variant component) .
Remark: In what follows, all tensors are represented with
respect to thereference placement using the material basis vectors
GGGi and GGG
i. A simi-lar representation were imaginable with respect to the
current placement (gggiandgggi) or e.g. the intermediate
placement(ĜGGi and ĜGG
i).
Addition of vectors:AAA♭ ±BBB♭ = AiGGG
i ± BjGGGj = (Ai ± Bi)GGG
i .
Dual basis:
The basis vectors GGGi and GGGi are orthogonal to each
other:
GGGi ·GGGj = δji = GGG
j ·GGGi ,whereby (·) is called scalar product.
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Dyadic product of two vectors:
AAA♭ ⊗BBB♭ = (AiGGGi) ⊗ (BjGGG
j) = AiBjGGGi ⊗GGGj = AijGGG
i ⊗GGGj
with Aij = AiBj .
Metric tensor components:
Gij = Gji = GGGi ·GGGj = GGGj ·GGGi, Gij = Gji = GGGi ·GGGj =
GGGj ·GGGi,
GijGjk = GkjGji = δik .
Using the metric tensor components, the contra-variant
(co-variant) basisvector can be transformed into a co-variant
(contra-variant) basis vector :
GGGi = (GGGi ·GGGj)GGGj = GijGGG
j, GGGi = (GGGi ·GGGj)GGGj = G
ijGGGj .
Metric (Identity) tensors of the reference placement:
G = GijGGGi ⊗GGGj ,
G−1 = GijGGGi ⊗GGGj ,
I = GGGi ⊗GGGi,
I∗ = GGGi ⊗GGGi.
Raising and lowering of indices:
Using the metric tensor components, the contra-variant
(co-variant) compo-nent can be transformed into a co-variant
(contra-variant) component :
Ai = GijAj, Ai = GijAj .
In an absolute notation we can write these expressions as:
AAA♭ = GAAA♯ (Lowering of index),
AAA♯ = G−1AAA♭ (Raising of index).
Dot product of two vectors:
AAA♭ ·BBB♭ = AiBjGij = AiB
i = AjBj = AiBjGij .
Symbolic distinction of the dot product into scalar and vector
product:
< AAA♭,BBB♭ >X∗= AiBj < GGGi,GGGj >X∗= AiBjG
ij , (Vector product)
< AAA♯,BBB♯ >X = AiBj < GGGi,GGGj >X = A
iBjGij , (Vector product)
AAA♭ ·BBB♯ = AAA♯ ·BBB♭. (Scalar product)
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Transformation between scalar and vector product:
< AAA♭,BBB♭ >X∗= G−1AAA♭ ·BBB♭ = AAA♭ ·G−1BBB♭,
< AAA♯,BBB♯ >X = GAAA♯ ·BBB♯ = AAA♯ ·GBBB♯.
Representation of tensors of second order:
A♭ = AijGGG
i ⊗GGGj (co-co-variant)A
♯ = AijGGGi ⊗GGGj (contra-contra-variant)
A\ = Ai. jGGGi ⊗GGG
j (contra-co-variant)
A/ = Ai
j. GGG
i ⊗GGGj (co-contra-variant)
Remark: A symbolic distinction of component variance (position
of in-dices) is redundant if the component decomposition is already
clear.
The dual of a second-order tensor:
The dual is formed by exchanging the order of basis vectors
within the dyadicproduct.
Example: A∗ = (Ai. jGGGi ⊗GGGj)∗ = Ai. jGGG
j ⊗GGGi
(A♭)∗ = AijGGGj ⊗GGGi,
(A♯)∗ = AijGGGj ⊗GGGi,
(A\)∗ = Ai. jGGGj ⊗GGGi,
(A/)∗ = Aij. GGGj ⊗GGG
i.
The transpose of a second-order tensor:
(A\)T = G−1(A\)∗G,(A/)T = G(A/)∗G−1.
Remark: The transpose is identical to the dual after raising and
loweringof indices. Therefore, the component variance is the same
as before. Atranspose for co-co-variant or contra-contra-variant
tensors has no use, butcould be defined by:
(A♭)T = G−1(A♭)∗G−1,(A♯)T = G(A♯)∗G.
Here, the component variance is different than before.
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Inverse:
The inverse of a second-order tensor is defined by:
A\(A−1)\ = (A−1)\A\ = I,
A/(A−1)/ = (A−1)/A/ = I∗,
A♭(A−1)♯ = I∗, (A−1)♯A♭ = I,
A♯(A−1)♭ = I, (A−1)♭A♯ = I∗.
(Skew-)Symmetry of a second-order tensor:
For the definition of symmetry properties it is useful to
consider tensors withequal component variance. Therefore, the
definition of a (skew-)symmetricpart of a tensor is given as
follows:
(A♭)sym =12(A
♭ + (A♭)∗), (A♭)skw =12 (A
♭ − (A♭)∗),
(A♯)sym =12(A
♯ + (A♯)∗), (A♯)skw =12 (A
♯ − (A♯)∗)
(A\)sym =12(A
\ + (A\)T ), (A\)skw =12(A
\ − (A\)T )
(A/)sym =12(A
/ + (A/)T ), (A/)skw =12(A
/ − (A/)T ).
Trace of a second-order tensor of order n:
tr(A♭)n = (A♭G−1)n : I,tr(A♯)n = (A♯G)n : I∗,tr(A\)n = (A\)n :
I∗,tr(A/)n = (A/)n : I.
Deviatoric and spherical part of a second-order tensor:
A♭dev = A
♭ − 13 tr(A♭)G, A♭sph =
13 tr(A
♭)G,
A♯dev = A
♯ − 13 tr(A♯)G−1, A♯sph =
13 tr(A
♯)G−1,
A\dev = A
\ − 13 tr(A\) I, A
\sph =
13 tr(A
\) I,
A/dev = A
/ − 13 tr(A/) I∗, A
/sph =
13 tr(A
/) I∗.
Addition of tensors of second order:
A♭ ± B♭ = (Aij ± Bij) GGG
i ⊗GGGj,A
♯ ± B♯ = (Aij ± Bij) GGGi ⊗GGGj ,
A\ ± B\ = (Ai. j ± B
i. j)GGGi ⊗GGG
j ,
A/ ± B/ = (Ai
j. ± Bi
j. )GGG
i ⊗GGGj.
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Simple contraction of tensors of first and second order:
E.g.:
A♭AAA♯ = (AijGGG
i ⊗GGGj) · (AkGGGk) = AijAjGGGi,
AAA♭A♯ = (AkGGGk) · (AijGGGi ⊗GGGj) = A
kjAkGGGj ,
A♭B
♯ = (AijGGGi ⊗GGGj) · (BklGGGk ⊗GGGl) = AikB
klGGGi ⊗GGGl.
Double contraction of tensors of second-order:
A♭ : B♯ = AijB
ij,
A♯ : B♭ = AijBij ,
A\ : B/ = Ai. jBi
j. ,
A/ : B\ = Ai
j.Bi. j .
Representation of tensors of fourth order:
E♯♯ = EijklGGGi ⊗GGGj ⊗GGGk ⊗GGGl,
E♭♭ = EijklGGG
i ⊗GGGj ⊗GGGk ⊗GGGl.
Remark: Depending on the position of indices there exist 14
additionalcomponent decompositions for a fourth-order tensor.
Tensor products for a tensor of fourth order:
E♭♭ijkl = (A
♭ ⊗ B♭)ijkl = AijBkl,
E♭♭ijkl = (A
♭ × B♭)ijkl = AilBjk,
E♭♭ijkl = (A
♭2× B♭)ijkl = AikBjl.
Simple contractions of tensors of second and fourth order:
E♭\A
♯ = Eijk. mA
mlGGGi ⊗GGGj ⊗GGGk ⊗GGGl,
A\E
♯♯ = Ai.mEmjklGGGi ⊗GGGj ⊗GGGk ⊗GGGl.
Double contractions of tensors of fourth order:
E♭♭ : D♯♭ = EijmnD
mn.. klGGG
i ⊗GGGj ⊗GGGk ⊗GGGl,
E♭♭
q aD♯♯ = EimnjD
mkln GGGi ⊗GGGk ⊗GGGl ⊗GGGj,
E♭♭
a qD♯♯ = EmjknD
imnl GGGi ⊗GGGj ⊗GGGk ⊗GGGl.
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Double contractions of tensors of second and fourth order:
A♯ : E♭♭ = AmnEmnijGGG
i ⊗GGGj , E♭♭ : A♯ = EijmnAmnGGGi ⊗GGGj ,
A♯q aE
♭♭ = AmnEmijnGGGi ⊗GGGj, E♭♭ q aA♯ = EimnjA
mnGGGi ⊗GGGj ,
A♯a qE
♭♭ = AmnEimnjGGGi ⊗GGGj, E♭♭ a qA♯ = EmijnA
mnGGGi ⊗GGGj .
For second-order tensors the distinction of inner and outer
bases is mean-ingless:
A♭ : B♯ = A♭ q aB♯ = A♭ a qB♯.
Transposition operations for fourth-order tensors:
Eijkl → Ejikl =⇒ E♭♭ → (E♭♭)dl,
Eijkl → Eijlk =⇒ E♭♭ → (E♭♭)dr,
Eijkl → Ejilk =⇒ E♭♭ → (E♭♭)d,
Eijkl → Eklij =⇒ E♭♭ → (E♭♭)D.
Eijkl → Eikjl =⇒ E♭♭ → (E♭♭)ti,
Eijkl → Eljki =⇒ E♭♭ → (E♭♭)to,
Eijkl → Elkji =⇒ E♭♭ → (E♭♭)t,
Eijkl → Ejilk =⇒ E♭♭ → (E♭♭)T .
Symmetry-properties of a fourth-order tensor:
A tensor E fulfills minor symmetry if:E = Eti = Eto or E = Edl =
Edr.
A tensor E fulfills major symmetry if:E = ET or E = ED.
A tensor E is supersymmetric if:E = Eti = Eto = ET or E = Edl =
Edr = ED.
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In particular, considering the tensors E = G q aC q aH and F = K
: D : L, thefollowing applies:
ET = HT q aCT q aGT , FD = LD : DD : KD ,
Eti = G q aC q aHti , Fdr = K : D : Ldr ,
Eto = Gto q aC q aH , Fdl = Kdl : D : L .
2 Tensor analysis
Tensor differentiation in absolute notation:
∂f
∂X♭=
∂f
∂XijGGGi ⊗GGGj ,
∂f
∂X♯=
∂f
∂XijGGGi ⊗GGGj ,
∂f
∂X\=
∂f
∂Xi. jGGGi ⊗GGGj ,
∂f
∂X/=
∂f
∂Xij.GGGi ⊗GGG
j,
∂F♭
∂X♭=
∂Fij
∂XklGGGi ⊗GGGk ⊗GGGl ⊗GGG
j,
∂F♭
∂X♯=
∂Fij
∂XklGGGi ⊗GGGk ⊗GGGl ⊗GGGj,
∂F♭
∂X\=
∂Fij
∂Xk. lGGGi ⊗GGGk ⊗GGGl ⊗GGG
j,
∂F♭
∂X/=
∂Fij
∂Xkl.GGGi ⊗GGGk ⊗GGG
l ⊗GGGj,
∂f F♭
∂X♯= f
∂F♭
∂X♯+ F♭ ×
∂f
∂X♯,
∂A♯B♭
∂X♭=
∂A♯
∂X♭B
♭ + A♯∂B♭
∂X♭,
∂A♭ : B♯
∂X♯=
∂A♭
∂X♯a qB
♯ + A♭ q a∂B♯
∂X♯.
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Special rules:
Eikl.. j =
∂Aij∂Akl
= δki δlj ⇒ E
/\ = ∂A♭
∂A♭= I∗ ⊗ I,
Eikl.. j =
∂Aij∂Alk
= δliδkj ⇒ E
/\ = ∂A♭
∂(A♭)∗= I∗ 2× I = (I∗ ⊗ I)ti,
Eikl.. j =
∂Aji∂Akl
= δkj δli ⇒ E
/\ = ∂(A♭)∗
∂A♭= I∗ 2× I = (I∗ ⊗ I)to,
Eikl.. j =
∂Aji∂Alk
= δljδki ⇒ E
/\ = ∂(A♭)∗
∂(A♭)∗= I∗ ⊗ I = (I∗ ⊗ I)t.
Ei. klj = ∂A
ij
∂Akl= δikδ
jl ⇒ E
\/ = ∂A♯
∂A♯= I ⊗ I∗,
Ei. klj = ∂A
ij
∂Alk= δilδ
jk ⇒ E
\/ = ∂A♯
∂(A♯)∗= I 2× I∗ = (I ⊗ I∗)ti,
Ei. klj = ∂A
ji
∂Akl= δjkδ
il ⇒ E
\/ = ∂(A♯)∗
∂A♯= I 2× I∗ = (I ⊗ I∗)to,
Ei. klj = ∂A
ji
∂Alk= δjl δ
ik ⇒ E
\/ = ∂(A♯)∗
∂(A♯)∗= I ⊗ I∗ = (I ⊗ I∗)t.
Ei. kl.j =
∂Ai. j∂Ak. l
= δikδlj ⇒ E
\\ = ∂A\
∂A\= I⊗ I,
Ei.l.kj =
∂Ai. j∂Ak. l
= δikδlj ⇒ E
♯♭ = ∂A\
∂(A\)∗= I 2× I = (I ⊗ I)ti,
Ejkli.. =
∂Ai. j∂Ak. l
= δikδlj ⇒ E
♭♯ = ∂(A\)∗
∂A\= I∗ 2× I∗ = (I ⊗ I)to,
Ejl.k
i. =
∂Ai. j∂Ak. l
= δikδlj ⇒ E
// = ∂(A\)∗
∂(A\)∗= I∗ ⊗ I∗ = (I ⊗ I)t.
Eik. l
j. =
∂Aij.∂Akl.
= δki δjl ⇒ E
// = ∂A/
∂A/= I∗ ⊗ I∗,
Eilk.j. =
∂Aij.∂Akl.
= δki δjl ⇒ E
♭♯ = ∂A/
∂(A/)∗= I∗ 2× I∗ = (I∗ ⊗ I∗)ti,
Ejk.. li =∂Aij.∂Akl.
= δki δjl ⇒ E
♯♭ = ∂(A/)∗
∂A/= I 2× I = (I∗ ⊗ I∗)to,
Ej. lk. i =
∂Aij.∂Akl.
= δki δjl ⇒ E
\\ = ∂(A/)∗
∂(A/)∗= I ⊗ I = (I∗ ⊗ I∗)t.
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Differentiation with respect to a (skew-)symmetrical tensor:
∂F♭
∂X♭
∣
∣
∣
X♭=(X♭)∗=
∂F♭
∂X♭q a
1
2(∂X♭
∂X♭+
∂X♭
∂(X♭)∗)
=∂F♭
∂X♭q a
1
2(I∗ ⊗ I + I∗ 2× I) =
∂F♭
∂X♭q aS
/\,
∂F♭
∂X♯
∣
∣
∣
X♯=(X♯)∗=
∂F♭
∂X♯q a
1
2(∂X♯
∂X♯+
∂X♯
∂(X♯)∗)
=∂F♭
∂X♯q a
1
2(I ⊗ I∗ + I 2× I∗) =
∂F♭
∂X♯q aS
\/,
∂F♭
∂X\
∣
∣
∣
X\=(X\)T=
∂F♭
∂X\q a
1
2(∂X\
∂X\+
∂X\
∂(X\)T)
=∂F♭
∂X\q a
1
2(I ⊗ I + G−1 2× G) =
∂F♭
∂X\q aS
\\,
∂F♭
∂X/
∣
∣
∣
X/=(X/)T=
∂F♭
∂X/q a
1
2(∂X/
∂X/+
∂X/
∂(X/)T)
=∂F♭
∂X/q a
1
2(I∗ ⊗ I∗ + G 2× G−1) =
∂F♭
∂X/q aS
//.
∂F♭
∂X♭
∣
∣
∣
X♭=−(X♭)∗=
∂F♭
∂X♭q a
1
2(∂X♭
∂X♭−
∂X♭
∂(X♭)∗)
=∂F♭
∂X♭q a
1
2(I∗ ⊗ I − I∗ 2× I) =
∂F♭
∂X♭q aA
/\,
∂F♭
∂X♯
∣
∣
∣
X♯=−(X♯)∗=
∂F♭
∂X♯q a
1
2(∂X♯
∂X♯−
∂X♯
∂(X♯)∗)
=∂F♭
∂X♯q a
1
2(I ⊗ I∗ − I 2× I∗) =
∂F♭
∂X♯q aA
\/,
∂F♭
∂X\
∣
∣
∣
X\=−(X\)T=
∂F♭
∂X\q a
1
2(∂X\
∂X\−
∂X\
∂(X\)T)
=∂F♭
∂X\q a
1
2(I ⊗ I − G−1 2× G) =
∂F♭
∂X\q aA
\\,
∂F♭
∂X/
∣
∣
∣
X/=−(X/)T=
∂F♭
∂X/q a
1
2(∂X/
∂X/−
∂X/
∂(X/)T)
=∂F♭
∂X/q a
1
2(I∗ ⊗ I∗ − G 2× G−1) =
∂F♭
∂X/q aA
//.
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Differentiation of the inverse:
∂(X−1)♯
∂X♭= −(X−1)♯ ⊗ (X−1)♯,
∂(X−1)♭
∂X♯= −(X−1)♭ ⊗ (X−1)♭,
∂(X−1)\
∂X\= −(X−1)\ ⊗ (X−1)\,
∂(X−1)/
∂X/= −(X−1)/ ⊗ (X−1)/.
Differentiation with respect to the inverse:
∂X♭
∂(X−1)♯= −X♭ ⊗ X♭,
∂X♯
∂(X−1)♭= −X♯ ⊗ X♯,
∂X\
∂(X−1)\= −X\ ⊗ X\,
∂X/
∂(X−1)/= −X/ ⊗ X/.
Using the chain rule and product rule of differential
calculus:
Using the above rules, we may state :
∆(A♭B♯(X♭)) = ∂A♭B
♯
∂X♭q a∆X♭ = (A♭B♯),
X♭q a∆X♭
= (A♭,X♭
B♯ + A♭B♯,
X♭) q a∆X♭.
Using the contraction rule ( q a) and the representation of a
fourth-order dif-ferential expression in the proposed form, the
product rule of differentialcalculus is fulfilled for a simple
contraction of second-order tensors. Further-more, the chain rule
can be used. Note that for the classical representationof a
differential expression in the form:
∂A♭
∂B♭=
∂Aij∂Bkl
GGGi ⊗GGGj ⊗GGGk ⊗GGGl = A♭;
B♭,
which was often used in the past, the product rule cannot be
applied. Thus:
∆(A♭B♯(X♭)) = ∂A♭B
♯
∂X♭: ∆X♭ = (A♭B♯);
X♭ : ∆X♭
6= (A♭;X♭
B♯ + A♭B♯;
X♭) : ∆X♭.
10
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Note that for traces of second-order tensors we have to use the
contractionrule ( a q) :
∂tr(A♭B♯)∂X♭
= (A♭,X♭
B♯ + A♭B♯,
X♭) a qI.
The contraction rules ( q a) and ( a q) can be used to represent
the same con-traction in absolute notation in different form:
E♭♯
q aI = I a qE♭♯,
or
∂tr(A♭B♯)∂X♭
= I q a(A♭,X♭
B♯ + A♭B♯,
X♭).
It is of importance to consider the contraction of tensors which
is alreadyexisting in a given tensor equation in correct form by
using either ( q a) or( a q)!
Transformation between the new and old convention:
To transform between A♭,B♯ and A♭;B♯ we can use the following
basis rear-
rangement operations:
(A⊗ B)L = A × B , (A× B)L = A 2× B∗ , (A 2× B)L = A⊗ B∗ ,(A⊗ B)R
= A 2× B∗ , (A× B)R = A⊗ B , (A 2× B)R = A× B∗ .
such that:
A♭,B♯ = (A
♭;B♯ )L and A♭;B♯ = (A
♭,B♯ )R.
Also, applying (·)R and (·)L, the sequence of tensors is changed
in a doublecontraction:
E : C = EL q aC = C a qEL , E q aC = C a qE = ER : C ,
(D : E)L = DL q aEL = EL a qDL , (D q aE)R = (E a qD)R = DR : ER
.
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3 Differential geometrical relationships
Material time derivative of a tensor of first and second
order:
ȦAA♯ =˙
AiGGGi + AiĠGGi,
Ȧ♯ = ȦijGGGi ⊗GGGj + A
ijĠGGi ⊗GGGj + AijGGGi ⊗ ĠGGj .
Push-forward and pull-back-relationships:
F�(AAA♭) = F−∗AAA♭, F�(aaa♭) = F∗aaa♭,
F�(AAA♯) = FAAA♯, F�(aaa♯) = F−1aaa♯,
F�(A♭) = F−∗A♭F−1, F�(a♭) = F∗a♭F,
F�(A♯) = FA♯F∗, F�(a♯) = F−1a♯F−∗,
F�(A\) = FA\F−1, F�(a\) = F−1a\F,
F�(A/) = F−∗A/F∗, F�(a/) = F∗a/F−∗,
F�(C♭♭) = (F−∗ ⊗ F−1) q aC♭♭ q a(F−1 ⊗ F−∗),
F�(c ♭♭) = (F∗ ⊗ F) q ac ♭♭ q a(F ⊗F∗),
F�(C♯♯) = (F ⊗ F∗) q aC♯♯ q a(F∗ ⊗ F),
F�(c ♯♯) = (F−1 ⊗ F−∗) q ac ♯♯ q a(F−∗ ⊗ F−1).
Remark: Note that F must be an invertible second-order tensor
mappingvectors onto vectors i.e. F\.
Covariance of tensor functions:
A scalar-valued (second-order-valued, fourth-order-valued)
tensor functionis covariant if the following equations are
satisfied:
F (B,B,B) = F (A�(B),A�(B),B) ,
= F (A�(B),A�(B),B) ,
A�(F(B,B,B)) = F(A�(B),A�(B),B) ,
A�(F(B,B,B)) = F(A�(B),A�(B),B) ,
A�(F(B,B,B)) = F(A�(B),A�(B),B) ,
A�(F(B,B,B)) = F(A�(B),A�(B),B) .
Remark:Covariance of a tensor function is satisfied if it is
constructed us-ing the representation theorem for isotropic tensor
functions. Any tensorialinvariant (usually some linear combinations
of traces) has to be computed in
12
-
mixed-variant form considering certain metric tensors. Normally,
we dis-tinguish the principle of material and spatial covariance
depending on whichmanifold the transformed tensors belong to.
However, we can speak simplyof the principle of covariance whereby
the general case is considered whereany tensor function may be
composed of tensors belonging to different man-ifolds.
The Lie-derivative (for convective coordinates (gggi =
FGGGi):
Definition: LF(·) = F�(˙
F�(·)).
LF(aaa♭) = F�(
˙F�(aaa♭)) = F−∗(
˙F∗aaa♭)
= F−∗(Ḟ∗aaa♭ + F∗ȧaa♭)
= ȧaa♭ + l∗aaa♭ = ȧi gggi,
LF(aaa♯) = F�(
˙F�(aaa♯)) = F(
˙F−1aaa♯)
= F(Ḟ−1aaa♯ + F−1ȧaa♯)
= ȧaa♯ − laaa♯ = ȧi gggi.
LF(a♭) = F�(
˙F�(a♭)) = F−∗(
˙F∗a♭F)F−1
= F−∗(Ḟ∗a♭F + F∗ȧ♭F + F∗a♭Ḟ)F−1
= ȧ♭ + l∗a♭ + a♭l = ȧij gggi ⊗ gggj,
LF(a♯) = F�(
˙F�(a♯)) = F(
˙F−1a♯F−∗)F∗
= F(Ḟ−1a♯F−∗ + F−1ȧ♯F−∗ + F−1a♯Ḟ−∗)F∗
= ȧ♯ − la♯ − a♯l∗ = ȧij gggi ⊗ gggj,
LF(a\) = F�(
˙F�(a\)) = F(
˙F−1a\F)F−1
= F(Ḟ−1a\F + F−1ȧ\F + F−1a\Ḟ)F−1
= ȧ\ − la\ + a\l = ȧi.j gggi ⊗ gggj,
LF(a/) = F�(
˙F�(a/)) = F−∗(
˙F∗a/F−∗)F∗
= F−∗(Ḟ∗a/F−∗ + F∗ȧ/F−∗ + F∗a/Ḟ−∗)F∗
= ȧ/ + l∗a/ − a/l∗ = ȧij. ggg
i ⊗ gggj.
13
-
A time derivative is not unique:
We are able to use any time derivative (material, Lie,
co-rotated) for thetime differentiation of covariant tensor
functions. Thus, the choice of a timederivative is not unique and
can only be motivated on the grounds of specialconstitutive
assumptions. Thus, we find:
˙F (a♭,b♯) =
∂F (a♭,b♯)
∂a♭: ȧ♭ +
∂F (a♭,b♯)
∂b♯: ḃ♯
=∂F (c\
�
(a♭), c\�
(b♯))
∂c\�(a♭)
:˙
c\�(a♭) +
∂F (c\�
(a♭), c\�
(b♯))
∂c\�(b♯)
:˙
c\�(b♯)
= c\�
(
∂F (a♭,b♯)
∂a♭
)
:˙
c\�(a♭) + c\
�
(
∂F (a♭,b♯)
∂b♯
)
:˙
c\�(b♯)
=∂F (a♭,b♯)
∂a♭: c\�(
˙c\
�(a♭)) +
∂F (a♭,b♯)
∂b♯: c\�(
˙c\
�(b♯))
=∂F (a♭,b♯)
∂a♭: L
c\(a♭) +
∂F (a♭,b♯)
∂b♯: L
c\(b♯) = L
c\(F ) .
or
Lc\1
(F(c\2
�
(a♭), c\2
�
(b♯))) =∂F(c
\2
�
(a♭), c\2
�
(b♯))
∂c\2
�
(a♭)q aL
c\1
(c\2
�
(a♭)) + · · ·
=∂F(c
\2
�
(a♭), c\2
�
(b♯))
c\2�(∂c
\2
�
(a♭))q ac
\2�(Lc\
1
(c\2
�
(a♭))) + · · ·
=∂c
\2
�
(F(a♭,b♯))
∂a♭q aL
c\2c\1
(a♭) + · · ·
=c\2
�
(∂F(a♭,b♯))
∂a♭q aL
c\2c\1
(a♭) + · · ·
= c\2
�
(Lc\2c\1
(F(a♭,b♯))) .
Remark: Note in particular that∂(·)(c\
�
(a♭), c\�
(b♯))
∂c\�(a♭)
= c\�
(
∂(·)(a♭,b♯)
∂a♭
)
.
14
-
The following couplings hold for a scalar-valued function
f(a♭j,b♯i , c
\k,d
/l ):
−m1∑
j=1(a♭)
∗j
∂F∂(a♭)j
+m2∑
i=1
∂F∂(b♯)i
(b♯)∗i +
m3∑
k=1
(
∂F∂(c\)k
(c\)∗k − (c
\)∗k
∂F∂(c\)k
)
= 0,
j ∈ [1, . . . ,m1], i ∈ [1, . . . ,m2], k ∈ [1, . . . ,m3],
and
−m1∑
j=1
∂F∂(a♭)j
(a♭)∗j +
m2∑
i=1(b♯)
∗i
∂F∂(b♯)i
+m4∑
l=1
(
(d/)∗l
∂F∂(d/)l
− ∂F∂(d/)l
(d/)∗l
)
= 0,
j ∈ [1, . . . ,m1], i ∈ [1, . . . ,m2], l ∈ [1, . . . ,m4].
Remark: A valid function F would be e.g.
F = tr((d/)2a♭b♯a♭b♯) tr((c\)2b♯a♭b♯a♭).
For a second-order-valued tensor function only very restricted
cases are con-sidered. E.g. we consider the functions:
F\(a♭j,b
♯i) and F
/(a♭j,b♯i).
Then the following couplings hold:
−m1∑
j=1
∂F\
∂(a♭)jq a(i∗ ⊗ (a♭)j) +
m2∑
i=1
∂F\
∂(b♯)iq a((b♯)i ⊗ i
∗) = 0,
−m1∑
j=1
∂F/
∂(a♭)jq a((a♭)j ⊗ i) +
m2∑
i=1
∂F/
∂(b♯)iq a(i⊗ (b♯)i) = 0,
j ∈ [1, . . . ,m1], i ∈ [1, . . . ,m2].
15
-
For the second-order derivative of the above introduced
scalar-valued func-tion similar couplings hold! Thus, we find:
m1∑
j=1
m2∑
i=1
(
A2Ti
q a∂2F
∂(b♯)i∂(a♭)jq aA1j + A1
Tj
q a∂2F
∂(a♭)j∂(b♯)iq aA2i
)
+m1∑
j=1
m3∑
k=1
(
A3Tk
q a∂2F
∂(c\)k∂(a♭)jq aA1j + A1
Tj
q a∂2F
∂(a♭)j∂(c\)kq aA3k
)
+m2∑
i=1
m3∑
k=1
(
A3Tk
q a∂2F
∂(c\)k∂(b♯)iq aA2i + A2
Ti
q a∂2F
∂(b♯)i∂(c\)kq aA3k
)
+m1∑
i,k=1A1
Ti
q a∂2F
∂(a♭)i∂(a♭)kq aA1k +
m2∑
j,l=1A2
Tj
q a∂2F
∂(b♯)j∂(b♯)lq aA2l
+m3∑
i,k=1A3
Ti
q a∂2F
∂(c\)i∂(c\)kq aA3k + G = 0,
with
A1j = −((a♭)j ⊗ i), A2i = (i⊗ (b
♯)i), A3k = −((c\)k ⊗ i) + (i⊗ (c
\)k),
G =∑m1
j=1 i∗
2× (a♭)∗j
∂F∂(a♭)j
+∑m1
j=1 (a♭)
∗j
∂F∂(a♭)j
2× i∗ +∑m3
k=1 i∗
2× (c\)∗k
∂F∂(c\)k
+∑m3
k=1 (c\)
∗
k∂F
∂(c\)k2× i∗ −
∑m3k=1 (c
\)∗
k 2× ∂F
∂(c\)k−∑m3
k=1∂F
∂(c\)k2× (c\)
∗
k,
and:
m1∑
j=1
m2∑
i=1
(
A2Ti
q a∂2F
∂(b♯)i∂(a♭)jq aA1j + A1
Tj
q a∂2F
∂(a♭)j∂(b♯)iq aA2i
)
+m1∑
j=1
m4∑
l=1
(
A4Tl
q a∂2F
∂(d/)l∂(a♭)jq aA1j + A1
Tj
q a∂2F
∂(a♭)j∂(d/)lq aA4l
)
+m2∑
i=1
m4∑
l=1
(
A4Tl
q a∂2F
∂(d/)l∂(b♯)iq aA2i + A2
Ti
q a∂2F
∂(b♯)i∂(d/)lq aA4l
)
+m1∑
i,k=1A1
Ti
q a∂2F
∂(a♭)i∂(a♭)kq aA1k +
m2∑
j,l=1A2
Tj
q a∂2F
∂(b♯)j∂(b♯)lq aA2l
+m4∑
j,l=1A4
Tj
q a∂2W
∂(d/)j∂(d/)lq aA4l + G = 0,
with
A1j = −(i∗⊗(a♭)j), A2i = ((b
♯)i⊗ i∗), A4l = −(i
∗⊗(d/)l)+((d/)l⊗ i
∗),
G =∑m1
j=1 i 2×∂F
∂(a♭)j(a♭)
∗
j +∑m1
j=1∂F
∂(a♭)j(a♭)
∗
j 2× i +
∑m4l=1
∂F∂(d/)l
(d/)∗
l 2× i
+∑m4
l=1 i 2×∂F
∂(d/)l(d/)
∗l −
∑m4l=1
∂F∂(d/)l
2× (d/)∗l −
∑m4l=1 (d
/)∗l 2× ∂F
∂(d/)l,
j ∈ [1, . . . ,m1], i ∈ [1, . . . ,m2], k ∈ [1, . . . ,m3], l ∈
[1, . . . ,m4].
16
-
The Lie-variation:
δFf(X) = F�(δ(F�(f(X)))) = F�(
d
dǫ
(
F�(f(X + ǫ δX))
)
∣
∣
∣
ǫ=0)
including the typical variation as special case:
δf(X) =d
dǫf(X + ǫ δX)
∣
∣
∣
ǫ=0=
∂f
∂X
∣
∣
∣
δX=0q aδX.
Analogously, the variation of a second-order tensor function can
be found:
δA(X) =d
dǫA(X + δX)
∣
∣
∣
ǫ=0=
∂A
∂X
∣
∣
∣
δX=0q aδX,
or its general Lie-variation:
δFA(X) = F�(d
dǫ
(
F�(A(X + δX))
)
∣
∣
∣
ǫ=0).
The linearization of a second-order tensor function can be
computed as:
Lin(A(X,∆X)) = A(X)+∆A(X,∆X) = A(X)+∂A
∂X
∣
∣
∣
∆X=0q a∆X.
The gradient and divergence of a first and second-order
tensor:
Grad(AAA♯) =∂AAA♯
∂XXX=
∂AAA♯
∂θi⊗GGGi,
Grad(A♯) =∂A♯
∂XXX=
∂A♯
∂θi⊗GGGi,
Div(AAA♯) = Grad(AAA♯) : I∗ =∂Ai
∂θi,
Div(A♯) = Grad(A♯) : I∗ =∂Aik
∂θkGGGi.
In particular, we have:
Div(A♯AAA♭) = Grad(A♯AAA♭) : I∗ = AAA♭Div((A♯)∗)+(A♯)∗ :
Grad(AAA♭),
Div(AAA♭A♯) = Grad(AAA♭A♯) : I∗ = AAA♭Div(A♯) + A♯ :
Grad(AAA♭).
17
-
4 Application of the tensor differentiation rules
Here, we use an invariant representation of tensors. In fact,
these tensorscould have arbitrary component variance. Although the
actual kind of com-ponent variance may be important for the
implementation process, the useof invariant relations merits
consideration for their simple expressions (e.g.I∗ is simply
written as I and no symbols (·)♭, (·)♯, (·)\ or (·)/).
Differentiation of F (A) = tr(A3):
F (A),A = I q a((I ⊗ I)A2 + A(I ⊗ I)A + A2(I ⊗ I))
= I q a(I ⊗ A2 + A ⊗A + A2 ⊗ I)= 3 (A2)∗.
Rule: tr(An),A = n (An−1)∗.
Differentiation of F (A) = tr(A3)tr(A2):
F (A),A = 3 (A2)∗ tr(A2) + tr(A3) 2A∗,
F (A),A×A = 6((A2)∗ × A∗ + A∗ × (A2)∗)
+3 tr(A2) (A∗ 2× I + I 2× A∗) + 2 tr(A3) (I 2× I).
Differentiation of F(A) = Atr(A2) + A∗tr(A3):
F(A),A = tr(A2)I ⊗ I + tr(A3) I 2× I + 2A × A∗ + 3A∗ ×
(A2)∗.
Differentiation of F(A) = dev(A):
F(A) = A− 13tr(A) I,
F(A),A = I⊗ I −13 I × I.
for a symmetric tensor we have: F(A),A = S −13 I × I.
Differentiation of F(A) = (dev(A))3:
F(A),A = ((dev(A))2 ⊗ I + dev(A) ⊗ dev(A) + I ⊗ (dev(A))2) q
a(dev(A)),A
= ((dev(A))2 ⊗ I + dev(A) ⊗ dev(A) + I ⊗ (dev(A))2) q a(I ⊗ I −
13 I × I)
= (dev(A))2 ⊗ I + dev(A) ⊗ dev(A) + I ⊗ (dev(A))2 − (dev(A))2 ×
I.
18
-
5 Tables
(A⊗ B)C = A⊗ (BC) C (A⊗ B) = (CA) ⊗ B
(A × B)C = (AC) × B C (A× B) = (CA) × B
(A 2× B)C = A 2× (BC) C (A 2× B) = (CA) 2× B
Table 1: Simple contractions of tensors of second and fourth
order.
(A⊗ B) : C = A (B : C) C : (A ⊗B) = (C : A)B
(A ⊗ B) q aC = ACB C q a(A ⊗ B) = A∗ CB∗
(A⊗ B) a qC = A∗ CB∗ C a q(A ⊗ B) = ACB
(A× B) : C = AC∗ B∗ C : (A ×B) = B∗ C∗ A
(A × B) q aC = (B : C)A C q a(A × B) = (A : C)B
(A× B) a qC = (A : C)B C a q(A × B) = (C : B)A
(A 2× B) : C = ACB∗ C : (A 2× B) = A∗ CB
(A 2× B) q aC = AC∗ B C q a(A 2× B) = BC∗ A
(A 2× B) a qC = BC∗ A C a q(A 2× B) = AC∗ B
Table 2: Double contractions of tensors of second and fourth
order.
19
-
(A ⊗ B) : (C ⊗ D) = (B : C)A⊗ D
(A⊗ B) q a(C ⊗ D) = (AC) ⊗ (DB)
(A ⊗ B) a q(C ⊗D) = (CA) ⊗ (BD)
(A ⊗ B) : (C × D) = A⊗ (D∗ B∗ C) (A× B) : (C ⊗ D) = (AC∗ B∗)
⊗D
(A⊗ B) q a(C × D) = (ACB) × D (A× B) q a(C ⊗ D) = A× (C∗
BD∗)
(A ⊗ B) a q(C ×D) = C × (A∗ DB∗) (A× B) a q(C ⊗ D) = (CAD) ×
B
(A × B) : (C × D) = (AD) 2× (BC)
(A × B) q a(C ×D) = (B : C)A× D
(A × B) a q(C ×D) = (A : D)C × B
(A ⊗ B) : (C 2× D) = A⊗ (C∗ BD) (A 2× B) : (C ⊗ D) = (ACB∗) ⊗
D
(A ⊗ B) q a(C 2× D) = (AC) 2× (DB) (A 2× B) q a(C ⊗ D) = (AD∗)
2× (C∗ B)
(A ⊗ B) a q(C 2× D) = (CB∗) 2× (A∗ D) (A 2× B) a q(C ⊗ D) = (CA)
2× (BD)
(A 2× B) : (C 2× D) = (AC) 2× (BD)
(A 2× B) q a(C 2× D) = (AD∗) ⊗ (C∗ B)
(A 2× B) a q(C 2× D) = (CB∗) ⊗ (A∗ D)
(A × B) : (C 2× D) = (AD) × (BC) (A 2× B) : (C × D) = (AC) ×
(BD)
(A × B) q a(C 2× D) = A × (DB∗ C) (A 2× B) q a(C × D) = (AC∗ B)
× D
(A × B) a q(C 2× D) = (CA∗ D) × B (A 2× B) a q(C × D) = C× (BD∗
A)
Table 3: Double contractions of tensors of fourth order.
20
-
(A ⊗ B)T = A∗ ⊗ B∗ (A ⊗ B)D = B ⊗A
(A× B)T = B × A (A × B)D = B∗ × A∗
(A 2× B)T = B 2× A (A 2× B)D = A∗ 2× B∗
(A ⊗ B)ti = A 2× B (A ⊗ B)dl = A∗ ⊗ B
(A× B)ti = A× B∗ (A × B)dl = B 2× A
(A 2× B)ti = A ⊗ B (A 2× B)dl = B × A
(A ⊗ B)to = B∗ 2× A∗ (A ⊗ B)dr = A⊗ B∗
(A× B)to = A∗ × B (A × B)dr = A 2× B
(A 2× B)to = B∗ ⊗ A∗ (A 2× B)dr = A ×B
(A ⊗ B)t = B∗ ⊗ A∗ (A ⊗ B)d = A∗ ⊗ B∗
(A× B)t = A∗ ×B∗ (A × B)d = B × A
(A 2× B)t = B∗ 2× A∗ (A 2× B)d = B 2× A
Table 4: Transposition operations for tensors of fourth
order.
A♭,
A♭= I∗ ⊗ I
A♭,(A♭)∗ = I
∗2× I
(A♭)∗,A♭
= I∗ 2× I
(A♭)∗,(A♭)∗ = I∗ ⊗ I
A♯,A♯ = I ⊗ I
∗
A♯,(A♯)∗ = I 2× I
∗
(A♯)∗,A♯ = I 2× I∗
(A♯)∗,(A♯)∗ = I ⊗ I∗
A\,
A\= I ⊗ I
A\,(A\)∗ = I 2× I
(A\)∗,A\
= I∗ 2× I∗
(A\)∗,(A\)∗ = I∗ ⊗ I∗
A/,
A/= I∗ ⊗ I∗
A/,(A/)∗ = I
∗2× I∗
(A/)∗,A/
= I 2× I
(A/)∗,(A/)∗ = I ⊗ I
Table 5: Identity tensors of fourth order.
21
-
(E : F)T = Ed : Fd (E : F)D = FD : ED
(E q aF)T = FT q aET (E q aF)D = Fti T q aEto T
(E a qF)T = FT a qET (E a qF)D = Fto T a qEti T
(E : F)dl = Edl : F
(E q aF)ti = E q aFti
(E a qF)ti = Eti a qF
(E : F)dr = E : Fdr
(E q aF)to = Eto q aF
(E a qF)to = E a qFto
(E : F)t = FDd : EDd (E : F)d = Ed : Fd
(E q aF)t = Et q aFt (E q aF)d = FT q aET
(E a qF)t = Et a qFt (E a qF)d = FT a qET
Table 6: Transposition operations applied to contracted
4th-order tensors.
ET ti = Eto T ED dl = Edr D Eti D = ED to = Ed ti = Eto d
ET to = Eti T ED dr = Edl D ED ti = Eto D = Ed to = Eti d
Et T = ET t = ED ED d = Ed D = Et Et dl = Edr t Et dr = Edl
t
Eti to = Eto ti = Et Edl dr = Edr dl = Ed Et D = ED t Et d = Ed
t
ET dl = Edl T = Edr ET dr = Edr T = Edl ET D = ED T ET d = Ed T
= E
Table 7: Connections between various transposition operations
(ET = Ed).
22