# Foundations of tensor algebra and analysis 1 Tensor Symmetry-properties of a fourth-order tensor: A

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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 = δ ij = δij = δ

j i

{

1 for i = j 0 for i 6= j

.

Einstein summation convention: If an index appears twice within a tensor component relation in co-variant and contra-variant position, we have to sum with respect 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 are not allowed.

Representation of tensors of first order (vectors):

AAA♭ = AiGGG i (co-variant component) ,

AAA♯ = AiGGGi (contra-variant component) .

Remark: In what follows, all tensors are represented with respect to the reference placement using the material basis vectors GGGi and GGG

i. A simi- lar representation were imaginable with respect to the current placement (gggi andgggi) or e.g. the intermediate placement(ĜGGi and ĜGG

i).

Addition of vectors: AAA♭ ±BBB♭ = AiGGG

i ± BjGGG j = (Ai ± Bi)GGG

i .

Dual basis:

The basis vectors GGGi and GGG i are orthogonal to each other:

GGGi ·GGG j = δji = GGG

j ·GGGi , whereby (·) is called scalar product.

1

Dyadic product of two vectors:

AAA♭ ⊗BBB♭ = (AiGGG i) ⊗ (BjGGG

j) = AiBjGGG i ⊗GGGj = AijGGG

i ⊗GGGj

with Aij = AiBj .

Metric tensor components:

Gij = Gji = GGGi ·GGGj = GGGj ·GGGi, G ij = Gji = GGGi ·GGGj = GGGj ·GGGi,

GijGjk = GkjG ji = δik .

Using the metric tensor components, the contra-variant (co-variant) basis vector can be transformed into a co-variant (contra-variant) basis vector :

GGGi = (GGGi ·GGGj)GGG j = GijGGG

j , GGGi = (GGGi ·GGGj)GGGj = G

ijGGGj .

Metric (Identity) tensors of the reference placement:

G = GijGGG i ⊗GGGj ,

G −1 = GijGGGi ⊗GGGj ,

I = GGGi ⊗GGG i ,

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 = GijA j, 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♭ = AiBjG ij = AiB

i = AjBj = A iBjGij .

Symbolic distinction of the dot product into scalar and vector product:

< AAA♭,BBB♭ >X∗= AiBj < GGG i ,GGGj >X∗= AiBjG

ij , (Vector product)

< AAA♯,BBB♯ >X = A iBj < GGGi,GGGj >X = A

iBjGij , (Vector product)

AAA♭ ·BBB♯ = AAA♯ ·BBB♭. (Scalar product)

2

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 dyadic product.

Example: A∗ = (Ai. jGGGi ⊗GGG j)∗ = Ai. jGGG

j ⊗GGGi

(A♭)∗ = AijGGG j ⊗GGGi,

(A♯)∗ = AijGGGj ⊗GGGi,

(A\)∗ = Ai. jGGG j ⊗GGGi,

(A/)∗ = Ai j . 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 lowering of indices. Therefore, the component variance is the same as before. A transpose for co-co-variant or contra-contra-variant tensors has no use, but could be defined by:

(A♭)T = G−1(A♭)∗G−1, (A♯)T = G(A♯)∗G.

Here, the component variance is different than before.

3

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 with equal component variance. Therefore, the definition of a (skew-)symmetric part of a tensor is given as follows:

(A♭)sym = 1 2(A

♭ + (A♭)∗), (A♭)skw = 1 2 (A

♭ − (A♭)∗),

(A♯)sym = 1 2(A

♯ + (A♯)∗), (A♯)skw = 1 2 (A

♯ − (A♯)∗)

(A\)sym = 1 2(A

\ + (A\)T ), (A\)skw = 1 2(A

\ − (A\)T )

(A/)sym = 1 2(A

/ + (A/)T ), (A/)skw = 1 2(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 =

1 3 tr(A

♭)G,

A ♯ dev = A

♯ − 13 tr(A ♯)G−1, A♯sph =

1 3 tr(A

♯)G−1,

A \ dev = A

\ − 13 tr(A \) I, A

\ sph =

1 3 tr(A

\) I,

A / dev = A

/ − 13 tr(A /) I∗, A

/ sph =

1 3 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.

4

Simple contraction of tensors of first and second order:

E.g.:

A ♭AAA♯ = (AijGGG

i ⊗GGGj) · (AkGGGk) = AijA jGGGi,

AAA♭A♯ = (AkGGG k) · (AijGGGi ⊗GGGj) = A

kjAkGGGj ,

A ♭ B

♯ = (AijGGG i ⊗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 additional component 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

♯ = Eij k . mA

mlGGGi ⊗GGGj ⊗GGGk ⊗GGGl,

A \E

♯♯ = Ai.mE mjklGGGi ⊗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 ⊗GGG j ,

E ♭♭

a qD ♯♯ = EmjknD

imnl GGGi ⊗GGG j ⊗GGGk ⊗GGGl.

5

Double contractions of tensors of second and fourth order:

A ♯ : E♭♭ = AmnEmnijGGG

i ⊗GGGj , E♭♭ : A♯ = EijmnA mnGGGi ⊗GGGj ,

A ♯ q aE

♭♭ = AmnEmijnGGG i ⊗GGGj, E♭♭ q aA♯ = EimnjA

mnGGGi ⊗GGGj ,

A ♯ a qE

♭♭ = AmnEimnjGGG i ⊗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.

6

In particular, considering the tensors E = G q aC q aH and F = K : D : L, the following applies:

E T = HT q aCT q aGT , FD = LD : DD : KD ,

E ti = G q aC q aHti , Fdr = K : D : Ldr ,

E to = Gto q aC q aH , Fdl = Kdl : D : L .

2 Tensor analysis

Tensor differentiation in absolute notation:

∂f

∂X♭ =

∂f

∂Xij GGGi ⊗GGGj ,

∂f

∂X♯ =

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