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

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

  • 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♯ =