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.
The conventions that are not a matter of symbols, namely the sign of the analogy between elasticity and conduction, the sign of the Green operator, the bases of isotropic and transversely isotropic tensors, the geometry of an ellipsoid and the storage of tensors, are set out in Conventions.
Typography
The order of a tensor is carried by its typeface, as in the Echoes manual, so that formulas can be compared side by side with it.
| Object | Typeset as | Example |
|---|---|---|
| scalar | italic | |
| vector (order 1) | underlined | |
| tensor of order 2 | bold | |
| tensor of order 4 | blackboard bold | |
| set, geometry, shape function | calligraphic |
One class of object is deliberately outside that table: the column arrays and matrices of an algebraic formalism, which are not tensors and carry no tensor order. Such are the two-component state vector
An underline is therefore a vector, bold is order 2 and blackboard bold is order 4, which is what makes an expression such as
readable at a glance: an order-4 tensor
The decorations follow four rules, so that a quantity is written one way on every page and in every docstring:
the reference medium carries the index
and a phase its index as a subscript, , , , while an effective property carries the upright superscript , as in ; every label made of letters is upright and written with
\mathrm, as in, or the differential ; is a compliance, and the Eshelby tensor is written ( in conduction); a formula in a docstring obeys the same rules and is written in LaTeX; Unicode symbols such as
C₀are kept for the names of arguments and for code.
Operators
| Operator | Meaning |
|---|---|
| simple contraction (one index) | |
| double contraction (two indices) | |
| tensor product | |
| symmetrized tensor product (order 4, both symmetries) | |
| symmetrized box product | |
| transpose | |
| jump across an interface, | |
| volume average over | |
| surface element in the | |
| Volterra product and Volterra inverse of two kernels |
Identities and bases
| Symbol | Meaning | Unit | Where |
|---|---|---|---|
| identity tensor of order 4, on symmetric tensors | 1 | ||
| spherical projector, ⅓ 1 ⊗ 1 | 1 | ||
| deviatoric projector, 𝕀 − 𝕁 | 1 | ||
| identity tensor of order 2 | 1 | ||
| identity matrix | 1 | ||
| element of the Walpole basis of transversely isotropic tensors | 1 | ||
| vector of an orthonormal basis | 1 |
Stiffness and elastic moduli
Fields and loading
| Symbol | Meaning | Unit | Where |
|---|---|---|---|
| strain | 1 | ||
| stress (minus the flux in conduction) | Pa | ||
| macroscopic strain | 1 | ||
| macroscopic stress | Pa | ||
| remote strain of the auxiliary single-inclusion problems | 1 | ||
| remote uniform strain | 1 | ||
| polarization stress | Pa | ||
| eigenstrain (stress-free strain) | 1 | ||
| displacement | m | ||
| position vector | m | ||
| unit normal | 1 | ||
| wave vector, or a unit direction of integration | |||
| temperature, or the potential of the conduction problem | |||
| duration of the incorporation trajectory | s | theory/differential_scheme | |
| flux of the conduction problem | |||
| polarization of the flux | |||
| time (t′ the loading time) | s | ||
| thickness of the interfacial transition zone | m | applications/itz_concrete | |
| stress in the aggregate | Pa | applications/itz_elastic_limit | |
| strain in the aggregate | 1 | applications/itz_elastic_limit | |
| stress in the ITZ at the aggregate surface | Pa | applications/itz_elastic_limit | |
| strain in the ITZ at the aggregate surface | 1 | applications/itz_elastic_limit |
Geometry and amounts
Eshelby problem and concentration
| Symbol | Meaning | Unit | Where |
|---|---|---|---|
| Hill polarization tensor | Pa⁻¹ | ||
| Hill polarization kernel | Pa⁻¹ | theory/viscoelasticity, theory/laplace_carson, manual/viscoelasticity, manual/rheological_models, manual/laplace_inversion, tutorials/viscoelasticity, tutorials/symbolic_viscoelasticity, tutorials/generated/freq_vs_time, tutorials/generated/kelvin_maxwell, tutorials/generated/laplace_inversion, tutorials/generated/rheological_models, tutorials/generated/alv_schemes, tutorials/generated/alv_sensitivities, tutorials/generated/aging_ages_aspect, tutorials/generated/laminate_alv, applications/aging_creep, applications/bituminous | |
| Hill tensor in the reference medium | Pa⁻¹ | ||
| Hill tensor of the shape of phase i | Pa⁻¹ | ||
| Hill tensor of the Maxwell domain Ω, the distribution shape | Pa⁻¹ | ||
| second Hill tensor of the Maxwell domain Ω | Pa | ||
| stiffness contribution of the domain Ω filled with the effective medium | Pa | ||
| Hill tensor of the distribution ellipsoid of Ponte Castañeda and Willis | Pa⁻¹ | ||
| Hill tensor of the distribution ellipsoid of phase j around phase i | Pa⁻¹ | theory/scheme_pcw | |
| mean polarization of the inclusions, Σ fⱼ τⱼ | Pa | theory/scheme_pcw | |
| microstructural tensor of the Hashin–Shtrikman functional for phases i and j | Pa⁻¹ | theory/scheme_pcw | |
| Hashin–Shtrikman functional of the polarization | J/m³ | theory/scheme_pcw | |
| strain energy density | J/m³ | theory/scheme_pcw | |
| strain energy density of the comparison medium | J/m³ | theory/scheme_pcw | |
| effective strain energy density | J/m³ | theory/scheme_pcw | |
| block matrix of the Ponte Castañeda–Willis system | Pa⁻¹ | theory/scheme_pcw | |
| tensor mapping the macroscopic strain onto the field applied to phase i | 1 | theory/scheme_pcw | |
| polarization of phase i weighted by its fraction, fᵢ τᵢ | Pa | theory/scheme_pcw | |
| second Hill tensor, ℂ − ℂ:ℙ:ℂ | Pa | ||
| Eshelby tensor, ℙ:ℂ | 1 | ||
| strain concentration tensor | 1 | ||
| dilute strain concentration tensor | 1 | ||
| dilute strain concentration tensor of phase i | 1 | ||
| strain-to-strain concentration tensor | 1 | ||
| stress-to-strain concentration tensor | Pa | ||
| strain-to-stress concentration tensor | Pa⁻¹ | ||
| stress-to-stress concentration tensor | 1 | ||
| stress concentration tensor | 1 | ||
| average stress of the truncated cell per unit load | Pa | theory/corrected_cell | |
| stiffness contribution tensor | Pa | ||
| stiffness contribution tensor of phase i | Pa | ||
| total dilute stiffness contribution, Σ fᵢ ℕᵢ | Pa | ||
| total dilute compliance contribution, Σ fᵢ ℍᵢ | Pa⁻¹ | ||
| compliance contribution tensor | Pa⁻¹ | ||
| compliance contribution tensor of phase i | Pa⁻¹ | ||
| Green operator of the reference medium | Pa⁻¹ | ||
| Fourier symbol of the Green operator | Pa⁻¹ | ||
| Green tensor of the reference medium | |||
| Newton-potential tensor of the ellipsoid | 1 | ||
| closure operator of the corrected fixed point | 1 | theory/corrected_cell | |
| creep kernel 𝕃(t, t′) | Pa⁻¹ | ||
| acoustic tensor, n·ℂ·n | Pa | applications/itz_elastic_limit | |
| vector of the rank-one strain jump across a plane interface | 1 | applications/itz_elastic_limit | |
| ITZ stress at the aggregate surface per unit macroscopic stress | 1 | applications/itz_elastic_limit | |
| stress concentration tensor of the aggregates | 1 | applications/itz_elastic_limit | |
| dilute strain concentration tensor of the aggregates | 1 | applications/itz_elastic_limit | |
| strain concentration tensor of the grain | 1 | applications/granular_friction | |
| stress concentration tensor of the grain | Pa | applications/granular_friction | |
| strain concentration tensor of the pore | 1 | applications/granular_friction, applications/sandstone_strength | |
| strain concentration tensor of the bonded grain | 1 | applications/sandstone_strength | |
| stress concentration tensor of the bonded grain | Pa | applications/sandstone_strength | |
| tensor mapping the macroscopic strain onto the remote strain applied to the inclusions of a one-site scheme | 1 | theory/homogenization, theory/scheme_bounds, theory/scheme_dilute, theory/scheme_mori_tanaka, theory/scheme_maxwell, theory/scheme_self_consistent | |
| contribution of a phase to the estimated stiffness, relative to a reference stiffness | Pa | theory/homogenization, theory/scheme_dilute | |
| contribution of a phase to the estimated compliance, relative to a reference compliance | Pa⁻¹ | theory/homogenization, theory/scheme_dilute | |
| reference stiffness of a contribution | Pa | ||
| reference compliance of a contribution | Pa⁻¹ | ||
| strain concentration tensor of a phase weighted by its amount, along the differential path | 1 | theory/differential_scheme | |
| stress concentration tensor of a phase weighted by its amount, along the differential path | Pa | theory/differential_scheme |
Shape and Walpole coefficients
| Symbol | Meaning | Unit | Where |
|---|---|---|---|
| geometric tensor of the ellipsoid, integral of ξ⊗ξ⊗ξ⊗ξ over the unit sphere | 1 | theory/hill_tensors, manual/neural_inclusions, theory/viscoelasticity | |
| geometric tensor of the ellipsoid, integral of ξ ⊗ˢ 1 ⊗ˢ ξ over the unit sphere | 1 | theory/hill_tensors, manual/neural_inclusions | |
| acoustic parameter of a transversely isotropic matrix: γ₁, γ₂ square roots of the roots of its acoustic polynomial, γ₃² = (C₁₁₁₁ − C₁₁₂₂)/(2C₂₃₂₃) | 1 | theory/hill_tensors, theory/degenerate_shapes | |
| integral of zⁿ/D(a, z) over [−1, 1], with D(a, z) = a + (1 − a) z² | 1 | theory/hill_tensors, theory/degenerate_shapes | |
| integral of zⁿ(1 − z²)^q/(D(a₁, z) D(a₂, z)) over [−1, 1], two acoustic roots | 1 | theory/hill_tensors, theory/degenerate_shapes | |
| shape coefficient of the mean opening of a crack | 1 | theory/cod_tensors, theory/thermal_cracks | |
| contrast, as in Δℂ_J = ℂ_J − ℂ₀ | theory/cluster_model, theory/differential_scheme | ||
| scalar potential of the Papkovich–Neuber representation | theory/layered_spheroid_elasticity | ||
| vector potential of the Papkovich–Neuber representation | theory/layered_spheroid_elasticity | ||
| second engineering parameter of the transversely isotropic crack kernel, (E, ν₁, ν₂, H, Γ) | 1 | theory/cod_tensors | |
| first engineering parameter of the transversely isotropic crack kernel, (E, ν₁, ν₂, H, Γ) | 1 | theory/cod_tensors |
Conduction
| Symbol | Meaning | Unit | Where |
|---|---|---|---|
| conductivity tensor | |||
| conductivity of the reference medium | |||
| conductivity of phase i | |||
| effective conductivity tensor | |||
| Hill polarization tensor of conduction | |||
| Eshelby tensor of conduction, P·K | 1 | ||
| second Hill tensor of conduction | |||
| average-gradient concentration tensor of a layered particle | 1 | ||
| average-flux concentration tensor of a layered particle | |||
| crack resistivity contribution tensor | theory/thermal_cracks, manual/cracks, manual/conductivity, theory/index, theory/conductivity | ||
| rotation from the principal frame of the inclusion | 1 | developer/adding_inclusion | |
| reflection across a coordinate plane | 1 | theory/corrected_cell |
Cracks
| Symbol | Meaning | Unit | Where |
|---|---|---|---|
| crack, elliptic or ribbon-like | theory/cod_tensors | ||
| elliptic crack | theory/cod_tensors | ||
| ribbon-like crack | theory/cod_tensors | ||
| flatness c/b of the ellipsoid or elliptic cylinder a crack is the limit of | 1 | theory/cod_tensors, tutorials/generated/crack_routes | |
| crack opening displacement (COD) tensor | Pa⁻¹ | ||
| Biot tensor | 1 | manual/poromechanics, theory/levin, tutorials/levin, fe_coupling/materials, fe_coupling/fractured_rock, fe_coupling/poroelastic_coupling, fe_coupling/scale_transition, fe_coupling/arma2011, fe_coupling/thick_cylinder, fe_coupling/backends, fe_coupling/index | |
| shape tensor of the inclusion in the transformed coordinates, A·K₀^−1/2 | theory/thermal_cracks | ||
| COD scalar of an elliptic crack in conduction | |||
| COD scalar of a penny-shaped crack in conduction | |||
| COD scalar of a ribbon crack in conduction | |||
| in-plane unit vector of the crack along its major axis | 1 | ||
| in-plane unit vector of the crack along its minor axis | 1 | ||
| Fourier symbol of the second Hill operator | Pa |
Viscoelasticity
Interfaces and layered inclusions
Eigenstrains and prestresses
| Symbol | Meaning | Unit | Where |
|---|---|---|---|
| eigenstrain (free strain) of a phase | 1 | theory/levin, tutorials/levin | |
| prestress of a phase | Pa | theory/levin, tutorials/levin | |
| macroscopic free strain, the strain under a vanishing macroscopic stress | 1 | theory/levin, tutorials/levin | |
| macroscopic prestress, the stress under a vanishing macroscopic strain | Pa | theory/levin, tutorials/levin | |
| effective thermal expansion tensor | K⁻¹ | theory/levin, tutorials/levin | |
| effective thermal expansion coefficient of an isotropic composite | K⁻¹ | theory/levin, tutorials/levin | |
| temperature change | K | theory/levin, tutorials/levin | |
| eigengradient of a phase in conduction, the gradient at which it conducts no flux | theory/levin | ||
| pre-flux of a phase, the flux it conducts under a vanishing gradient | theory/levin | ||
| macroscopic free gradient | theory/levin | ||
| macroscopic pre-flux | theory/levin | ||
| flux concentration tensor of a phase in conduction | 1 | theory/levin |
Poromechanics and N-body models
| Symbol | Meaning | Unit | Where |
|---|---|---|---|
| pore pressure | Pa | manual/poromechanics, fe_coupling/materials, fe_coupling/fractured_rock, fe_coupling/poroelastic_coupling, fe_coupling/scale_transition, fe_coupling/arma2011, fe_coupling/thick_cylinder, fe_coupling/backends, fe_coupling/index, applications/lamellar_clay | |
| Biot modulus | Pa | manual/poromechanics, fe_coupling/materials, fe_coupling/fractured_rock, fe_coupling/poroelastic_coupling, fe_coupling/scale_transition, fe_coupling/arma2011, fe_coupling/thick_cylinder, fe_coupling/backends, fe_coupling/index | |
| Biot coefficient | 1 | manual/poromechanics, applications/cement_paste | |
| interaction tensor between two inclusions | Pa⁻¹ | ||
| block of the linear system of the cluster model | 1 | theory/cluster_model |
Strength
| Symbol | Meaning | Unit | Where |
|---|---|---|---|
| macroscopic mean stress, tr Σ/3 | Pa | ||
| norm of the macroscopic stress deviator, (Σ^d:Σ^d)^1/2 | Pa | ||
| macroscopic deviatoric stress in the normalization of the article, (Σ^d:Σ^d/2)^1/2 | Pa | applications/sandstone_strength, applications/granular_friction | |
| norm of the stress deviator, (σ^d:σ^d)^1/2 | Pa | ||
| deviatoric stress of the solid in the normalization of the article, (σ^d:σ^d/2)^1/2 | Pa | applications/sandstone_strength | |
| strength of the solid: the bound on the norm of its stress deviator | Pa | tutorials/strength_criteria | |
| largest principal stress | Pa | applications/itz_elastic_limit | |
| shear strength of the von Mises solid | Pa | applications/sandstone_strength | |
| tensile strength of the bond between ITZ and aggregate | Pa | applications/itz_elastic_limit | |
| tensile strength of the ITZ | Pa | applications/itz_elastic_limit | |
| spherical stress concentration factor of the aggregates | 1 | applications/itz_elastic_limit | |
| deviatoric stress concentration factor of the aggregates | 1 | applications/itz_elastic_limit | |
| coordinate along the hydrostatic axis, √3 Σₘ | Pa | applications/itz_elastic_limit | |
| friction coefficient of a contact | 1 | applications/granular_friction, applications/sandstone_strength | |
| macroscopic friction coefficient, slope of the Drucker–Prager criterion | 1 | applications/granular_friction | |
| friction angle of the Mohr–Coulomb criterion | rad | applications/granular_friction | |
| normal traction on a contact | Pa | applications/granular_friction, applications/sandstone_strength | |
| tangential traction on a contact | Pa | applications/granular_friction, applications/sandstone_strength | |
| mean normal traction on the contacts | Pa | applications/sandstone_strength | |
| quadratic mean of the tangential traction on the contacts | Pa | applications/sandstone_strength | |
| quadratic mean of the deviatoric strain in the grains, in the normalization of the article (½ under the root) | 1 | applications/sandstone_strength | |
| ratio Σ_d/ | Σₘ | on the strength envelope | |
| energy coefficient k⁻² ∂k/∂μₛ of the grains | Pa⁻² | applications/sandstone_strength | |
| energy coefficient μ⁻² ∂μ/∂μₛ of the grains | Pa⁻² | applications/sandstone_strength | |
| energy coefficient (kₜ/k)² ∂k/∂kₜ of the contacts | Pa⁻¹ m | applications/sandstone_strength | |
| energy coefficient (kₜ/μ)² ∂μ/∂kₜ of the contacts | Pa⁻¹ m | applications/sandstone_strength |