Nomenclature
This page lists the symbols of the formulas of the documentation, grouped by subject. A symbol that means different things in different chapters is listed once for each meaning, with the pages where that meaning holds; a meaning listed without pages holds everywhere else. Hovering an equation on any page shows the symbols it holds, with their meaning on that page.
Only the symbols that recur are listed. Following the rule of Notation and conventions, a symbol used on one page only is defined on that page, and the typography — the typeface that carries the order of a tensor, the position of an index, the products and their Julia operators — is set out there.
Identities and projectors
| Symbol | Meaning | Unit | Where |
|---|---|---|---|
| order-2 identity tensor | 1 | theory/notation, theory/tensor_algebra, theory/bases_variance, theory/isotropic, theory/walpole, manual/structured_tensors, tutorials/generated/bases_variance, tutorials/generated/cluster, tutorials/generated/green_function, tutorials/generated/hill_tensors, tutorials/generated/operator_identities, tutorials/generated/sphere_problems, tutorials/generated/tensor_algebra, tutorials/generated/walpole | |
| identity matrix (6×6 in Kelvin–Mandel form, or the 3×3 block 1₃) | 1 | theory/kelvin_mandel, theory/rotations, theory/cubic, developer/testing_conventions | |
| transverse projector 1 − n ⊗ n | 1 | ||
| axial projector n ⊗ n | 1 | ||
| order-4 identity on symmetric order-2 tensors | 1 | ||
| spherical (hydrostatic) projector | 1 | ||
| deviatoric projector 𝕀 − 𝕁 | 1 | ||
| order-4 identity on all order-2 tensors | 1 | ||
| cubic projector onto the two-dimensional E_g subspace | 1 | theory/cubic, theory/projection, manual/structured_tensors | |
| cubic projector onto the three-dimensional T_2g shear subspace | 1 | theory/cubic, manual/structured_tensors | |
| orthotropic projector Pₘ = eₘ ⊗ eₘ | 1 | ||
| covariant Kronecker symbol | 1 | ||
| mixed Kronecker symbol | 1 | ||
| permutation (Levi-Civita) symbol | 1 |
Bases and coordinates
| Symbol | Meaning | Unit | Where |
|---|---|---|---|
| vector of a general (possibly non-orthonormal) basis | theory/notation, theory/bases_variance, theory/tensor_algebra, manual/bases, tutorials/generated/bases_variance | ||
| unit vector of the normalized basis of a chart | 1 | theory/curvilinear | |
| dual (reciprocal) basis vector | |||
| third basis vector, reference or symmetry axis | 1 | ||
| radial unit vector of polar or spherical coordinates | 1 | ||
| natural (covariant) basis vector ∂OM/∂qⁱ of a chart | |||
| dual natural basis vector of a chart | |||
| position vector of the current point | m | ||
| covariant metric components | |||
| contravariant metric components | |||
| metric tensor | |||
| scale factors (Lamé coefficients) of a basis or chart | |||
| curvilinear coordinates of a chart | |||
| basis | |||
| coordinate system | theory/notation | ||
| real coordinate space ℝ^d (or ℝ^N, ℝ⁶) | theory/bases_variance, theory/curvilinear, theory/cubic, theory/projection, theory/walpole_extended, theory/submanifolds, manual/submanifolds, tutorials/generated/submanifolds | ||
| radial coordinate (polar, cylindrical, spherical) | m | ||
| polar angle: polar/cylindrical angle, spherical colatitude, Euler angle | rad | ||
| azimuth: spherical coordinate and second Euler angle | rad | ||
| first Cartesian coordinate | m | ||
| second Cartesian coordinate (rotation axis label in R_y) | m | ||
| third Cartesian coordinate, cylindrical axial coordinate | m | ||
| spheroidal coordinate, constant on a hyperboloid sheet (−1 ≤ p ≤ 1) | 1 | theory/curvilinear | |
| spheroidal coordinate, constant on a spheroid (q ≥ 1 prolate, q ≥ 0 oblate) | 1 | theory/curvilinear | |
| shorthand √(1 − p²) of the spheroidal coordinates | 1 | theory/curvilinear | |
| shorthand √(q² − 1) of the prolate spheroidal coordinates | 1 | theory/curvilinear | |
| shorthand √(q² + 1) of the oblate spheroidal coordinates | 1 | theory/curvilinear | |
| half the focal distance (prolate) or radius of the focal circle (oblate) of the spheroidal coordinates | m | theory/curvilinear | |
| contravariant or covariant components uⁱ, uᵢ of a vector | theory/bases_variance | ||
| first elliptic coordinate | tutorials/generated/christoffel | ||
| second elliptic coordinate | tutorials/generated/christoffel, tutorials/generated/numerical_operators |
Products and operators
| Symbol | Meaning | Unit | Where |
|---|---|---|---|
| column index of a Kelvin–Mandel matrix | theory/kelvin_mandel | ||
| arbitrary order-4 tensor | theory/notation, theory/tensor_algebra, theory/kelvin_mandel, theory/isotropic, tutorials/generated/tensor_algebra | ||
| tensor of arbitrary order | |||
| generic order-2 tensor (first operand) | theory/notation, theory/tensor_algebra, theory/bases_variance, theory/kelvin_mandel, tutorials/generated/tensor_algebra | ||
| generic order-2 tensor (second operand) | theory/notation, theory/tensor_algebra, tutorials/generated/tensor_algebra | ||
| generic order-2 tensor (third operand) | |||
| generic order-2 tensor (fourth operand) | |||
| generic vector | theory/notation, theory/tensor_algebra, theory/bases_variance, tutorials/generated/tensor_algebra | ||
| generic second vector in a product | theory/tensor_algebra, tutorials/generated/tensor_algebra | ||
| components Tᵢⱼₖₗ of an order-4 tensor 𝕋 | theory/notation, theory/tensor_algebra, theory/kelvin_mandel | ||
| row index of a Kelvin–Mandel matrix | theory/kelvin_mandel | ||
| derivative along the coordinate qⁱ, variation of the basis included | |||
| gradient, derivative index appended on the right | |||
| symmetrized gradient | |||
| divergence, contracting the last index | |||
| second gradient (Hessian), grad ∘ grad | |||
| Laplacian, div ∘ grad | |||
| trace | |||
| matrix 6×6 (or 6-vector) of a tensor in Kelvin–Mandel form | |||
| diagonal matrix | |||
| differential (line element ds, measure dg) | |||
| generic order-4 tensor (first operand) | theory/isotropic, theory/kelvin_mandel, theory/cubic, theory/orthotropy, theory/rotations | ||
| generic order-4 tensor (second operand) | theory/isotropic, theory/kelvin_mandel, theory/cubic, theory/orthotropy |
Structured tensors
| Symbol | Meaning | Unit | Where |
|---|---|---|---|
| basis tensors of transverse isotropy (Walpole basis) | 1 | ||
| axial tensor 1ₙ ⊗ 1ₙ of the Walpole basis | 1 | ||
| transverse spherical tensor ½ 1_T ⊗ 1_T of the Walpole basis | 1 | ||
| axial–transverse coupling tensor 1ₙ ⊗ 1_T/√2 | 1 | ||
| transverse–axial coupling tensor 1_T ⊗ 1ₙ/√2 | 1 | ||
| in-plane deviatoric tensor of the Walpole basis | 1 | ||
| axial-shear tensor of the Walpole basis | 1 | ||
| major-antisymmetric axial-shear generator of the extended basis | 1 | ||
| major-antisymmetric in-plane deviatoric generator of the extended basis | 1 | ||
| symmetrized five-element Walpole basis (𝕎^s₃ = 𝕎₃ + 𝕎₄) | 1 | ||
| coefficients of a TI tensor on the Walpole basis | |||
| coefficient on 𝕎₁ (axial) | |||
| coefficient on 𝕎₂ (transverse spherical) | |||
| coefficient on 𝕎₃ (axial–transverse coupling) | |||
| coefficient on 𝕎₄ (transverse–axial coupling) | |||
| coefficient on 𝕎₅ (real part of z₂) | |||
| coefficient on 𝕎₆ (real part of z₁) | |||
| coefficient on 𝕎₇ (imaginary part of z₁) | |||
| coefficient on 𝕎₈ (imaginary part of z₂) | |||
| squared norms of the Walpole basis tensors (diagonal Gram matrix) | 1 | ||
| 2×2 matrix block of the Walpole coefficients ℓ₁ to ℓ₄ | theory/walpole, theory/walpole_extended, manual/structured_tensors, tutorials/generated/walpole, tutorials/generated/walpole_extended | ||
| matrix block (2×2) of the second Walpole operand 𝕄 | |||
| complex coefficient ℓ₆ + iℓ₇ of the axial-shear block | |||
| complex coefficient ℓ₅ + iℓ₈ of the in-plane deviatoric block | |||
| generic transversely isotropic tensor in Walpole notation | |||
| antisymmetric in-plane rotation generator about the axis | 1 | ||
| arbitrary vector acted on by the rotation generator | |||
| number of stored Walpole coefficients (TensTI parameter) | theory/cubic, theory/walpole_extended, developer/testing_conventions | ||
| unit symmetry axis of transverse isotropy | 1 | theory/walpole, theory/walpole_extended, theory/ti_parametrizations, theory/rotations, theory/projection, manual/projection, manual/structured_tensors, tutorials/generated/walpole, tutorials/generated/walpole_extended, tutorials/generated/projection | |
| order-2 TI or orthotropic tensor | theory/walpole, theory/walpole_extended, theory/orthotropy | ||
| three-dimensional irreducible representation of Oₕ (shear block) | theory/cubic | ||
| field of complex numbers | theory/walpole_extended | ||
| coefficient on 𝕁 of an isotropic or cubic tensor | theory/isotropic, theory/cubic, theory/tensor_algebra, theory/index, manual/structured_tensors | ||
| coefficient on 𝕂 (isotropic) or on 𝔼 (cubic) | theory/isotropic, theory/cubic, theory/tensor_algebra, theory/index, theory/projection, manual/structured_tensors | ||
| coefficient on the cubic shear projector 𝕋 | theory/cubic, theory/projection, manual/structured_tensors | ||
| scalar of an isotropic order-2 tensor λ1 | theory/isotropic, manual/structured_tensors | ||
| matrix units E₁₁, E₂₂, E₁₂, E₂₁ of 2×2 matrices | theory/walpole, tutorials/generated/walpole | ||
| two-dimensional irreducible representation of Oₕ | theory/cubic | ||
| octahedral group of the cube |
Elastic moduli
| Symbol | Meaning | Unit | Where |
|---|---|---|---|
| number of concentric layers of the sphere | tutorials/generated/sphere_problems, tutorials/index | ||
| remote uniform strain applied to the sphere | 1 | ||
| linearized strain components (ε₂₃, ε_θθ, …) | 1 | theory/walpole_extended, tutorials/generated/symbolic_operators | |
| unit separation direction between the two points | 1 | tutorials/generated/cluster | |
| displacement field of the elastic sphere | m | tutorials/generated/sphere_problems | |
| radial displacement profile u(r), uᵣ | m | tutorials/generated/sphere_problems, tutorials/generated/numerical_operators | |
| strain-concentration tensor | 1 | theory/walpole_extended, tutorials/generated/ad_interop | |
| elastic stiffness tensor | Pa | theory/isotropic, theory/kelvin_mandel, theory/cubic, theory/orthotropy, theory/ti_parametrizations, theory/projection, manual/parametrizations, tutorials/index, tutorials/generated/ad_interop, tutorials/generated/green_function, tutorials/generated/hill_tensors | |
| elastic compliance tensor | Pa⁻¹ | ||
| polarization (Hill) tensor of the inclusion | Pa⁻¹ | ||
| elastic Green tensor (Kelvin point-force solution) | |||
| real-space Green operator, symmetrized −Hess(G) | tutorials/generated/green_function, tutorials/generated/cluster, tutorials/index | ||
| wave-vector form of the Green operator ξ ⊗^s K⁻¹ ⊗^s ξ | tutorials/generated/hill_tensors | ||
| kernel ℂ : 𝚪 : ℂ of Hill-type integrals | |||
| acoustic tensor ξ · ℂ · ξ | |||
| direction (wave) vector of the acoustic tensor | 1 | tutorials/generated/hill_tensors, tutorials/index | |
| axisymmetric displacement field ξ(r, z) | m | tutorials/generated/symbolic_operators | |
| order-2 (stress) tensor field in divergence formulas | |||
| exponent of the radial power ansatz r^α | 1 | tutorials/generated/sphere_problems | |
| first Lamé parameter | Pa | tutorials/generated/hill_tensors | |
| shear modulus | Pa | theory/isotropic, theory/cubic, tutorials/generated/ad_interop, tutorials/generated/cluster, tutorials/generated/green_function, tutorials/generated/hill_tensors, tutorials/generated/numerical_operators, tutorials/generated/sphere_problems, tutorials/generated/tensor_algebra | |
| isotropic Poisson's ratio | 1 | tutorials/generated/cluster, tutorials/generated/green_function | |
| in-plane Poisson's ratio (engineering TI form) | 1 | ||
| axial–transverse Poisson's ratio (engineering TI form) | 1 | ||
| in-plane Poisson's ratio (Hoenig form) | 1 | ||
| axial–transverse Poisson's ratio (Hoenig form) | 1 | ||
| transverse Young's modulus (Hoenig form) | Pa | ||
| transverse Young's modulus (engineering TI form) | Pa | ||
| axial Young's modulus (engineering TI form) | Pa | ||
| axial shear modulus (engineering TI form) | Pa | ||
| constant C₁₁ = C₁₁₁₁ of a cubic or orthotropic tensor | Pa | theory/cubic, theory/orthotropy, manual/structured_tensors | |
| constant C₁₂ = C₁₁₂₂ of a cubic or orthotropic tensor | Pa | theory/cubic, theory/orthotropy, manual/structured_tensors | |
| shear constant C₄₄ = C₂₃₂₃ of a cubic or orthotropic tensor | Pa | theory/cubic, theory/orthotropy, manual/structured_tensors | |
| in-plane normal component of a TI stiffness | Pa | ||
| in-plane coupling component of a TI stiffness | Pa | ||
| axial–transverse coupling component of a TI stiffness | Pa | ||
| axial normal (longitudinal) component of a TI stiffness | Pa | ||
| axial shear component of a TI stiffness | Pa | ||
| components Sᵢⱼₖₗ of a TI compliance | Pa⁻¹ | theory/ti_parametrizations, tutorials/generated/walpole | |
| axial-to-transverse modulus ratio (Hoenig form) | 1 | theory/ti_parametrizations, manual/parametrizations, tutorials/generated/walpole | |
| bulk modulus | Pa | theory/cubic | |
| shear anisotropy ratio (Hoenig form) | 1 | theory/ti_parametrizations, manual/parametrizations, tutorials/generated/walpole | |
| offset of the source frame along e₃ | m | tutorials/generated/cluster | |
| outer radius Rᵢ of layer i | m | tutorials/generated/sphere_problems | |
| angular functions f^θ, f^r of the remote loading | 1 | tutorials/generated/sphere_problems | |
| elastic stiffness of the matrix | Pa | ||
| elastic stiffness of the inclusion | Pa |
Differential geometry
Rotations and projection
| Symbol | Meaning | Unit | Where |
|---|---|---|---|
| dimension of the symmetry-class subspace | theory/projection | ||
| basis tensors 𝕋ᵢ spanning a symmetry class | theory/projection | ||
| relative tolerance of the symmetry-detection cascade | 1 | theory/projection, tutorials/generated/projection | |
| symmetry class, a linear subspace at fixed orientation | theory/projection | ||
| order-4 rotation tensor R ⊠ˢ R | 1 | theory/rotations, theory/kelvin_mandel | |
| third Euler angle (spin about the axis) | rad | ||
| orientation (rotation in SO(3)) optimized in the projection | |||
| closest isotropic tensor (isotropization) | |||
| tensor to be projected onto a symmetry class | theory/projection, theory/walpole | ||
| closest tensor of the symmetry class (projection result) | theory/projection | ||
| coefficients αᵢ of a tensor on the basis of a symmetry class | theory/projection | ||
| rotation group integrated over in a group average | |||
| entry (1,1) of the Kelvin–Mandel matrix C = Mat(𝔸) | theory/projection | ||
| entry (1,2) of the Kelvin–Mandel matrix C = Mat(𝔸) | theory/projection | ||
| entry (4,4) of the Kelvin–Mandel matrix C = Mat(𝔸) | theory/projection | ||
| special orthogonal (rotation) group; SO(2) about the axis | theory/rotations, theory/kelvin_mandel, theory/cubic, theory/projection, theory/walpole_extended | ||
| rotation matrix 3×3 (Z–Y–Z Euler angles) | 1 | theory/rotations, theory/kelvin_mandel, theory/index, theory/curvilinear | |
| rotation (Bond) matrix 6×6 in Kelvin–Mandel form | 1 |