Notation and conventions
This page fixes the notation used throughout the documentation. It is deliberately short: every symbol that is not listed here is redefined on the page where it appears.
Tensor order is carried by the typeface
The convention is that of the Echoes manual, so that formulas can be compared side by side with it.
| Object | Typeset as | Example |
|---|---|---|
| scalar | italic | $k$, $\mu$, $\nu$, $\omega$, $\eta$, $\chi$ |
| vector (order 1) | underlined | $\underline{u}$, $\underline{n}$, $\underline{\xi}$ |
| tensor of order 2 | bold | $\boldsymbol{A}$, $\boldsymbol{B}$, $\boldsymbol{K}$, $\boldsymbol{\sigma}$, $\boldsymbol{\varepsilon}$ |
| tensor of order 4 | blackboard bold | $\mathbb{C}$, $\mathbb{P}$, $\mathbb{Q}$, $\mathbb{H}$, $\mathbb{S}$ |
| set, geometry, shape function | calligraphic | $\mathcal{E}_{\boldsymbol{A}}$, $\mathcal{G}_i$, $\mathcal{T}_a$, $\mathcal{K}_\eta$ |
The single most useful thing to remember: an underline is a vector, bold is order 2, blackboard bold is order 4. This is what makes an expression such as
\[\mathbb{H} = \tfrac{3}{4}\, \underline{n}\stackrel{s}{\otimes}\boldsymbol{B}\stackrel{s}{\otimes}\underline{n}\]
readable at a glance: an order-4 tensor $\mathbb{H}$ is built from an order-2 tensor $\boldsymbol{B}$ and a unit vector $\underline{n}$.
Operators
| Operator | Meaning |
|---|---|
| $\underline{u}\cdot\underline{v}$, $\boldsymbol{A}\cdot\underline{u}$ | simple contraction (one index) |
| $\boldsymbol{A}:\boldsymbol{B}$, $\mathbb{C}:\boldsymbol{\varepsilon}$ | double contraction (two indices) |
| $\underline{u}\otimes\underline{v}$ | tensor product |
| $\underline{u}\stackrel{s}{\otimes}\boldsymbol{A}\stackrel{s}{\otimes}\underline{u}$ | symmetrized tensor product (order 4, both symmetries) |
| $\boldsymbol{A}\stackrel{s}{\boxtimes}\boldsymbol{B}$ | symmetrized box product |
| $\boldsymbol{A}^{\!T}$ | transpose |
| $[\![\,\underline{u}\,]\!]$ | jump across an interface, $\underline{u}^+-\underline{u}^-$ |
| $\langle\,\cdot\,\rangle_\Omega$ | volume average over $\Omega$ |
| $\mathrm{d}S_\xi$ | surface element in the $\underline{\xi}$ parametrization |
Isotropic and transversely isotropic bases
Any isotropic order-4 tensor is a combination of the spherical and deviatoric projectors, with $\mathbb{I}$ the order-4 identity:
\[\mathbb{J} = \tfrac{1}{3}\,\boldsymbol{1}\otimes\boldsymbol{1}, \qquad \mathbb{K} = \mathbb{I} - \mathbb{J}, \qquad \mathbb{C}^{\mathrm{iso}} = 3k\,\mathbb{J} + 2\mu\,\mathbb{K}.\]
A transversely isotropic order-4 tensor needs five coefficients. The documentation uses the Walpole basis $(\mathbb{W}_1,\dots,\mathbb{W}_6)$, in which a major-symmetric tensor has $\mathbb{W}_3$ and $\mathbb{W}_4$ sharing a single coefficient — hence five independent numbers written $(P_1, P_2, P_3, P_5, P_6)$, with no $P_4$:
\[\mathbb{P} = P_1\,\mathbb{W}_1 + P_2\,\mathbb{W}_2 + P_3\,(\mathbb{W}_3+\mathbb{W}_4) + P_5\,\mathbb{W}_5 + P_6\,\mathbb{W}_6 .\]
This is the storage used by TensND.TensTI{4} and by hill_tensor on a transversely isotropic matrix ([1], [2]).
Ellipsoid geometry
An ellipsoid is described by an invertible order-2 shape tensor $\boldsymbol{A}$ such that $\boldsymbol{A}^{\!T}\!\cdot\boldsymbol{A}$ is symmetric positive definite, with eigenvalues the squared semi-axes:
\[\underline{x}\in\mathcal{E}_{\boldsymbol{A}} \iff \underline{x}\cdot(\boldsymbol{A}^{\!T}\!\cdot\boldsymbol{A})^{-1} \cdot\underline{x}\le 1, \qquad \boldsymbol{A}^{\!T}\!\cdot\boldsymbol{A} = \sum_{i=1}^{3}\rho_i^{2}\, \underline{e}^{\boldsymbol{A}}_i\otimes\underline{e}^{\boldsymbol{A}}_i .\]
Semi-axes are ordered $a = \rho_1 \ge b = \rho_2 \ge c = \rho_3$, with $(\underline{e}^{\boldsymbol{A}}_i)$ the orthonormal eigenvectors. Two aspect ratios recur:
- $\eta = b/a$ — in-plane aspect ratio (of a crack, in particular);
- $\omega$ — the flatness $c/a$, or the axial-to-transverse ratio of a spheroid, depending on the page. It is always redefined locally, because the two usages differ.
Storage: Kelvin–Mandel
Order-2 and order-4 tensors are stored in the Kelvin–Mandel convention (orthonormal 6-dimensional basis, off-diagonal components carrying $\sqrt{2}$ and $2$ factors), not in the engineering Voigt convention. This makes a double contraction an ordinary matrix product and a tensor inverse an ordinary matrix inverse. The practical consequences when reading printed components are spelled out in A storage convention worth knowing.
$\mathrm{Mat}(\mathbb{C})$ denotes the $6\times 6$ Kelvin–Mandel matrix of an order-4 tensor, and $\mathrm{Mat}\bigl(\mathbb{C}, \underline{e}^{\boldsymbol{A}}_i\bigr)$ the same matrix expressed in the ellipsoid's principal frame.
Two rules this documentation follows
Every symbol is defined where it is used. A page never relies on a symbol introduced only on another page, even at the cost of repeating a definition.
Every expression in this documentation is either (i) accompanied by a citation to published work, or (ii) derived explicitly on the page from expressions that are. Where a convention differs between references — the crack opening displacement tensor $\boldsymbol{B}$ is the notable case, see Crack opening displacement — the competing conventions are named and the one implemented by MeanFieldHom is stated.