Tensor algebra

Daniel Weschke

December 18, 2018

List of Symbols

\(()\cdot()\), \(():()\)	Scalar product, tensor contraction, inner product
\(()()\), \(()\otimes()\)	Dyadic product, Kronecker product, outer product
\(()\circ()\)	Hadamard product, Schur product, entrywise product

1 Transformation tensor

Given basis \(\{\tensorI{e}_x, \tensorI{e}_y, \tensorI{e}_z\}\).

Wanted basis \(\{\tensorI{e}_x^\prime, \tensorI{e}_y^\prime, \tensorI{e}_z^\prime\}\).

With the transformation tensor \(\tensorII{Q}\) one can determine components with given basis into the wanted basis.

\[\begin{split}\begin{split} \tensorII{Q} &= \begin{bmatrix} \tensorI{e}_x^\prime \cdot \tensorI{e}_x & \tensorI{e}_x^\prime \cdot \tensorI{e}_y & \tensorI{e}_x^\prime \cdot \tensorI{e}_z \\ \tensorI{e}_y^\prime \cdot \tensorI{e}_x & \tensorI{e}_y^\prime \cdot \tensorI{e}_y & \tensorI{e}_y^\prime \cdot \tensorI{e}_z \\ \tensorI{e}_z^\prime \cdot \tensorI{e}_x & \tensorI{e}_z^\prime \cdot \tensorI{e}_y & \tensorI{e}_z^\prime \cdot \tensorI{e}_z \\ \end{bmatrix} \\ &= \begin{bmatrix} \cos\theta_{xx} & \cos\theta_{xy} & \cos\theta_{xz} \\ \cos\theta_{yx} & \cos\theta_{yy} & \cos\theta_{yz} \\ \cos\theta_{zx} & \cos\theta_{zy} & \cos\theta_{zz} \\ \end{bmatrix} \end{split}\end{split}\]

2 Stress and strain tensor - symmetric second-order tensor

Stress tensor in matrix notation

\[\begin{split}\tensorII\sigma = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\ \sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\ \sigma_{zx} & \sigma_{zy} & \sigma_{zz} \\ \end{bmatrix}\end{split}\]

Stress tensor in Voigtnotation

\[\begin{split}\tensor\sigma = \begin{bmatrix} \sigma_{xx} \\ \sigma_{yy} \\ \sigma_{zz} \\ \sigma_{yz} \\ \sigma_{xz} \\ \sigma_{xy} \\ \end{bmatrix}\end{split}\]

Stain tensor in matrix notation

\[\begin{split}\tensorII\varepsilon = \begin{bmatrix} \varepsilon_{xx} & \varepsilon_{xy} & \varepsilon_{xz} \\ \varepsilon_{yx} & \varepsilon_{yy} & \varepsilon_{yz} \\ \varepsilon_{zx} & \varepsilon_{zy} & \varepsilon_{zz} \\ \end{bmatrix}\end{split}\]

Strain tensor in Voigtnotation

\[\begin{split}\tensor\varepsilon = \begin{bmatrix} \varepsilon_{xx} \\ \varepsilon_{yy} \\ \varepsilon_{zz} \\ \gamma_{yz} \\ \gamma_{xz} \\ \gamma_{xy} \\ \end{bmatrix}\end{split}\]

with \(\gamma_{xy}=2\varepsilon_{xy}\), \(\gamma_{yz}=2\varepsilon_{yz}\) and \(\gamma_{zx}=2\varepsilon_{zx}\).

The difference of stress and strain tensor in the Voigtnotation is explained by the energy expression

\[\psi = \frac{1}{2} \tensorII\varepsilon : \tensorII\sigma = \frac{1}{2} \tensor\varepsilon \cdot \tensor\sigma\]

or by the constitutive relations

\[\begin{split}\begin{split} \tensorII\sigma &= \tensorIV{C} : \tensorII\varepsilon \\ \tensor\sigma &= \tensor{C} \cdot \tensor\varepsilon \end{split}\end{split}\]

2.1 Transformation

\[\begin{split}\begin{split} \tensorII\sigma^\prime &= \tensorII{Q}\tensorII\sigma\tensorII{Q}^T \\ \tensorII\varepsilon^\prime &= \tensorII{Q}\tensorII\varepsilon\tensorII{Q}^T \end{split}\end{split}\]

3 Stiffness tensor - symmetric fourth-order tensor

Stiffness tensor in Voigtnotation

\[\begin{split}\tensor{C} = \begin{bmatrix} C_{xxxx} & C_{xxyy} & C_{xxzz} & C_{xxyz} & C_{xxxz} & C_{xxxy} \\ C_{yyxx} & C_{yyyy} & C_{yyzz} & C_{yyyz} & C_{yyxz} & C_{yyxy} \\ C_{zzxx} & C_{zzyy} & C_{zzzz} & C_{zzyz} & C_{zzxz} & C_{zzxy} \\ C_{yzxx} & C_{yzyy} & C_{yzzz} & C_{yzyz} & C_{yzxz} & C_{yzxy} \\ C_{xzxx} & C_{xzyy} & C_{xzzz} & C_{xzyz} & C_{xzxz} & C_{xzxy} \\ C_{xyxx} & C_{xyyy} & C_{xyzz} & C_{xyyz} & C_{xyxz} & C_{xyxy} \\ \end{bmatrix}\end{split}\]

3.1 Transformation

\[\tensorII{C}^\prime = \tensorII{Q}\tensorII{Q}\tensorII{C}\tensorII{Q}^T\tensorII{Q}^T\]

In Voigtnotation it simplifies to

\[\tensor{C}^\prime = \tensor{Q} \tensor{C} \tensor{Q}^T\]

where the rotation matrix \(\tensor{Q}\) is defined as

\[\begin{split}\tensor{Q} = \begin{bmatrix} \tensor{Q}\ho{I} & 2\tensor{Q}\ho{II} \\ \tensor{Q}\ho{III} & \tensor{Q}\ho{IV} \end{bmatrix} \quad \text{with}\quad \left.\begin{array}{@{}lr@{}} \tensor{Q}\ho{I} = \tensorII{Q}^{\circ 2} \\ Q\ho{II}_{ij} = Q_{i \, \mathrm{mod}(j+1,3)} \, Q_{i \, \mathrm{mod}(j+2,3)} \\ Q\ho{III}_{ij} = Q_{\mathrm{mod}(i+1,3) \, j} \, Q_{\mathrm{mod}(i+2,3) \, j} \\ Q\ho{IV}_{ij} = Q_{\mathrm{mod}(i+1,3) \, \mathrm{mod}(j+1,3)} \, Q_{\mathrm{mod}(i+2,3) \, \mathrm{mod}(j+2,3)} \\ \qquad\qquad + Q_{\mathrm{mod}(i+1,3) \, \mathrm{mod}(j+2,3)} \, Q_{\mathrm{mod}(i+2,3) \, \mathrm{mod}(j+1,3)} \end{array}\right\} \quad i,j \in \{1,2,3\}\end{split}\]

\[\begin{split}\mathrm{mod}(k,3) = \begin{cases} k & k \leq 3 \\ k-3 & k > 3 \end{cases}\end{split}\]

\[\begin{split}\tensor{Q}\ho{I} = \begin{bmatrix} Q_{11}^2 & Q_{12}^2 & Q_{13}^2 \\ Q_{21}^2 & Q_{22}^2 & Q_{23}^2 \\ Q_{31}^2 & Q_{32}^2 & Q_{33}^2 \\ \end{bmatrix}\end{split}\]

\[\begin{split}\tensor{Q}\ho{II} = \begin{bmatrix} Q_{12}Q_{13} & Q_{13}Q_{11} & Q_{11}Q_{12} \\ Q_{22}Q_{23} & Q_{23}Q_{21} & Q_{21}Q_{22} \\ Q_{32}Q_{33} & Q_{33}Q_{31} & Q_{31}Q_{32} \\ \end{bmatrix}\end{split}\]

\[\begin{split}\tensor{Q}\ho{III} = \begin{bmatrix} Q_{21}Q_{31} & Q_{22}Q_{32} & Q_{23}Q_{33} \\ Q_{31}Q_{11} & Q_{32}Q_{12} & Q_{33}Q_{13} \\ Q_{11}Q_{21} & Q_{12}Q_{22} & Q_{13}Q_{23} \\ \end{bmatrix}\end{split}\]

\[\begin{split}\tensor{Q}\ho{IV} = \begin{bmatrix} Q_{22}Q_{33}+Q_{23}Q_{32} & Q_{23}Q_{31}+Q_{21}Q_{33} & Q_{21}Q_{32}+Q_{22}Q_{31} \\ Q_{32}Q_{13}+Q_{33}Q_{12} & Q_{33}Q_{11}+Q_{31}Q_{13} & Q_{31}Q_{12}+Q_{32}Q_{11} \\ Q_{12}Q_{23}+Q_{13}Q_{22} & Q_{13}Q_{21}+Q_{11}Q_{23} & Q_{11}Q_{22}+Q_{12}Q_{21} \\ \end{bmatrix}\end{split}\]