art/grounding/sor-nomenclature.md

# Notation and conventions — Official Nomenclature of the US and European Societies of Rheology

*Scope: the joint SoR/ESR nomenclature, 2018-10 edition.* This file gives the
symbols, definitions, and SI units that The Society of Rheology asks authors to
use, so that an AI model reading or writing rheology follows them. Paste it as
the first message of a chat, or keep it in a project so a coding assistant
(Claude Code, Copilot, Cursor, and the like) reads it. Paste it again when a
new conversation starts. Notation is written in LaTeX.

Source: J. M. Dealy and D. Vlassopoulos, with advice from G. G. Fuller,
R. G. Larson, J. Mewis, D. J. Read, J. Vermant, N. J. Wagner, and G. McKinley,
"Official Nomenclature of US and European Societies of Rheology (2018-10),"
https://www.rheology.org/sor1/Publications/SoR_Nomenclature_201810.pdf.
Earlier version: "Official symbols and nomenclature of The Society of
Rheology," J. Rheol. 57, 1047 (2013), https://doi.org/10.1122/1.4811184.
The symbol tables below transcribe the 2018 edition; the conventions section
is a summary written for this primer. Where the source is inconsistent or
appears to contain a typographical error, the entry is given as printed and
the issue is listed under "Notes on the source".

If you only have room for part of this file, the "Conventions" section alone
covers the rules most often broken.

## How to use this notation

- Write rheological quantities with the symbols below, in SI units.
- When a paper, textbook, or user uses different symbols (for example $\tau$
  for shear stress, $\lambda$ for a relaxation time, $\mu$ for viscosity),
  translate to this notation and say once that you have done so, e.g.
  "the paper's $\tau_{xy}$ is the shear stress $\sigma$ here."
- When quoting a source verbatim, keep its symbols and mark the quotation.
- Do not invent a symbol for a quantity that has one here. For a quantity not
  covered here, define your symbol explicitly the first time it is used.
- If a symbol is ambiguous in context (several are reused below), say which
  quantity is meant.

## Conventions

- **Stress is $\sigma$, not $\tau$.** Shear stress is $\sigma$ (the shear
  component of the total stress tensor $\boldsymbol{\sigma}$). The symbol
  $\boldsymbol{\tau}$ is reserved for the extra stress tensor; scalar $\tau$
  denotes relaxation times ($\tau_i$, $\tau_d$, $\tau_R$, ...).
- **Simple shear coordinates:** $x_1$ flow direction, $x_2$ velocity-gradient
  direction, $x_3$ neutral direction. Shear strain $\gamma = dx_1/dx_2$, shear
  rate $\dot{\gamma} = |dv_1/dx_2|$.
- **Viscosity is $\eta$.** $\eta(\dot{\gamma}) = |\sigma|/\dot{\gamma}$;
  zero-shear viscosity $\eta_0$. The viscosity of a suspending medium or matrix
  is $\eta_m$ (the source discourages $\mu$ and, where the medium is itself
  complex, the subscript $f$). Solvent viscosity is $\eta_s$.
- **Normal stresses:** $N_1 = \sigma_{11} - \sigma_{22}$,
  $N_2 = \sigma_{22} - \sigma_{33}$; coefficients
  $\Psi_1 = N_1/\dot{\gamma}^2$, $\Psi_2 = N_2/\dot{\gamma}^2$.
- **Relaxation times are $\tau$, not $\lambda$.** Discrete spectrum:
  strengths $g_i$ and times $\tau_i$; continuous spectrum $H(\tau)$. In the
  molecular table, $\lambda$ is chain stretch (stretch ratio).
- **Linear viscoelasticity:** relaxation modulus $G(t)$, creep compliance
  $J(t)$. In oscillatory shear, $\gamma(t) = \gamma_0 \sin\omega t$ and
  $\sigma(t) = \sigma_0 \sin(\omega t + \delta)$; storage and loss moduli
  $G'$, $G''$; $G^* = G' + iG''$; loss angle $\delta$ (so the loss tangent is
  $\tan\delta = G''/G'$). Complex viscosity $\eta^* = \eta' - i\eta''$ with
  $\eta' = G''/\omega$ and $\eta'' = G'/\omega$. Complex compliance
  $J^* = J' - iJ''$. Plateau modulus $G_N^0$.
- **Transient (nonlinear) functions:** superscript $+$ for start-up of flow
  (growth functions, e.g. $\eta^+(t,\dot{\gamma})$, $\eta_E^+(t,\dot{\varepsilon})$),
  superscript $-$ for cessation (decay functions, e.g. $\eta^-(t,\dot{\gamma})$).
- **Extension:** Hencky strain $\varepsilon = \ln(L/L_0)$, Hencky strain rate
  $\dot{\varepsilon}$, net tensile stress $\sigma_E = \sigma_{zz} - \sigma_{rr}$,
  extensional viscosity $\eta_E(\dot{\varepsilon})$ (the long-time limit of
  $\eta_E^+$). Biaxial quantities carry subscript $B$. Young's modulus is $E$;
  $K$ is recommended for compression (dilatational) moduli.
- **Yield:** yield stress $\sigma_y$, yield strain $\gamma_y$.
- **Rate-of-strain tensor without the 1/2:**
  $\dot{\boldsymbol{\gamma}} = \nabla\mathbf{v} + (\nabla\mathbf{v})^T$ and
  vorticity tensor $\boldsymbol{\Omega} = \nabla\mathbf{v} - (\nabla\mathbf{v})^T$.
  If the half-sum is wanted, call it
  $\mathbf{D} = \tfrac{1}{2}[\nabla\mathbf{v} + (\nabla\mathbf{v})^T]$; do not
  reuse $\dot{\boldsymbol{\gamma}}$ for it.
- **Weissenberg vs. Deborah.** $Wi$ = (characteristic time of the fluid) x
  (rate of deformation), e.g. $\tau\dot{\gamma}$ or $\tau\dot{\varepsilon}$.
  $De$ = (characteristic time of the fluid) / (duration of the deformation),
  e.g. $\tau\omega$. $N_1/\sigma$ is the *stress ratio* $SR$, a property of
  the data, not a Weissenberg number. Subscripts name the time used, e.g.
  $Wi_R$ (Rouse time) and $Wi_d$ (longest relaxation time).
- **Suspensions:** volume fraction $\phi$, maximum packing fraction
  $\phi_{max}$, relative viscosity $\eta_r$, particle contribution $\eta_p$.
  The bulk stress is $\boldsymbol{\Sigma} = \boldsymbol{\Sigma}^f + \boldsymbol{\Sigma}^p$,
  split further into hydrodynamic $\Sigma^H$, interparticle $\Sigma^{IP}$, and
  Brownian $\Sigma^B$ parts; particle pressure $\Pi$. Péclet number
  $Pé = \dot{\gamma}a^2/D_0$ with $a$ the particle radius.
- **Interfacial rheology:** superscript $s$ (e.g. $\eta^s$, $G^{s\prime}$),
  with units per unit length (Pa m = N m$^{-1}$). Interfacial tension
  $\sigma_{\alpha,\beta}$; surface pressure $\Pi$; interfacial dilatational
  strain $\alpha^s = \ln(A/A_0)$; interfacial concentration $\Gamma$.
- **Polymer solutions:** relative, specific, reduced, and intrinsic viscosity
  $\eta_r$, $\eta_{sp}$, $\eta_{red}$, $[\eta]$; overlap concentration $c^*$.
- **Entangled polymers:** entanglement molecular weight
  $M_e = \rho RT/G_N^0$ (Ferry); in tube-model work use
  $M_e = \tfrac{4}{5}\rho RT/G_N^0$ and say so. $Z = M/M_e$, reptation time
  $\tau_d$, Rouse time $\tau_R$.
- **Units:** SI throughout, with a space (or a dot) between units: Pa s,
  Pa s$^2$, s$^{-1}$, rad. The source notes that "molecular weight" in
  kg mol$^{-1}$ is formally a molar mass, but keeps the familiar term.

## Symbols

### Table I. Steady simple shear (viscometric flow)

| Symbol | Meaning | Definition | SI units |
|---|---|---|---|
| `x_1` | direction of flow (simple shear) | | m |
| `x_2` | direction of velocity gradient (simple shear) | | m |
| `x_3` | neutral direction (simple shear) | | m |
| `F` | shear or normal force | `ma` | N |
| `v_1` | velocity | `dx_1/dt` | m s^-1 |
| `a_1` | acceleration | `dv_1/dt` | m s^-2 |
| `\sigma` | shear stress | `F/A` (A = area) | Pa |
| `\gamma` | shear strain | `dx_1/dx_2` | – |
| `\dot{\gamma}` | shear rate | `\lvert dv_1/dx_2 \rvert` | s^-1 |
| `\omega_3` | vorticity (fluid dynamics adds a prefactor 1/2) | `-dv_1/dx_2` | s^-1 |
| `\eta(\dot{\gamma})` | viscosity | `\lvert\sigma\rvert/\dot{\gamma}` | Pa s |
| `\sigma_y` | yield stress | | Pa |
| `\gamma_y` | yield strain | | – |
| `N_1` | first normal stress difference | `\sigma_{11}-\sigma_{22}` | Pa |
| `N_2` | second normal stress difference | `\sigma_{22}-\sigma_{33}` | Pa |
| `\Psi_1` | first normal stress coefficient | `N_1/\dot{\gamma}^2` | Pa s^2 |
| `\Psi_2` | second normal stress coefficient | `N_2/\dot{\gamma}^2` | Pa s^2 |
| `\Psi(\dot{\gamma})` | normal stress ratio | `-N_1/N_2` as printed (see notes) | – |
| `\eta_0` | zero-shear viscosity | `\eta(\dot{\gamma}\to 0)` | Pa s |
| `\Psi_{1,0}` | zero-shear first normal stress coefficient | `\Psi_1(\dot{\gamma}\to 0)` | Pa s^2 |
| `n` | power-law exponent | `\eta = K\dot{\gamma}^{n-1}` | – |
| `K` | power-law prefactor (consistency) | see above | Pa s^n |

### Table II. Linear viscoelasticity

Simple shear:

| Symbol | Meaning | Definition | SI units |
|---|---|---|---|
| `G` | shear modulus of a solid | `\sigma/\gamma` | Pa |
| `G(t)` | relaxation modulus | `\sigma(t)/\gamma` | Pa |
| `g_i` | relaxation strength in discrete spectrum | | Pa |
| `\tau_i` | relaxation time in discrete spectrum | | s |
| `H(\tau)` | continuous relaxation spectrum | `G(t)=\int_{-\infty}^{\infty} H(\tau)\, e^{-t/\tau}\, d\ln\tau` | Pa |
| `G^*_\perp(\omega,\dot{\gamma})` | orthogonal superposition complex modulus | | Pa |
| `G^*_\parallel(\omega,\dot{\gamma})` | parallel superposition complex modulus | | Pa |
| `m(s)` | memory function | `-dG(s)/ds` | Pa s^-1 |
| `J(t)` | creep compliance (shear) | `\gamma(t)/\sigma` | Pa^-1 |
| `J_e` | equilibrium compliance of solid | `J(t\to\infty)` | Pa^-1 |
| `J_r(t)` | recoverable compliance | `J(t)-t/\eta_0` | Pa^-1 |
| `J_s^0` | steady-state compliance of fluid | `J(t)-t/\eta_0\ (t\to\infty)` | Pa^-1 |
| `L(\tau)` | continuous retardation spectrum | `J(t)=\int_{-\infty}^{\infty} L(\tau)[1-e^{-t/\tau}]\, d\ln\tau` (see notes) | Pa^-1 |

The superposition subscripts $\perp$ and $\parallel$ apply equally to storage
and loss moduli and viscosities.

Small-amplitude oscillatory shear:

| Symbol | Meaning | Definition | SI units |
|---|---|---|---|
| `\gamma_0` | strain amplitude | `\gamma(t)=\gamma_0\sin\omega t` | – |
| `\delta` | loss angle (phase angle) | `\sigma(t)=\sigma_0\sin(\omega t+\delta)` | rad |
| `\sigma_0` | stress amplitude | as above | Pa |
| `G^*` | complex modulus | `G'+iG''` | Pa |
| `\lvert G^*\rvert` or `G_d` | absolute magnitude of `G^*` | `\sigma_0/\gamma_0` | Pa |
| `G'` | storage modulus | `G_d\cos\delta` | Pa |
| `G''` | loss modulus | `G_d\sin\delta` | Pa |
| `\eta^*` | complex viscosity | `\eta'-i\eta''` | Pa s |
| `\lvert\eta^*\rvert` | absolute magnitude of `\eta^*` | `\sigma_0/\omega\gamma_0` | Pa s |
| `\eta'` | dynamic viscosity (in phase with strain rate) | `G''/\omega` | Pa s |
| `\eta''` | out-of-phase (with strain rate) component of `\eta^*` | `G'/\omega` | Pa s |
| `J^*` | complex compliance | `J'-iJ''` | Pa^-1 |
| `\lvert J^*\rvert` | absolute magnitude of `J^*` | `\gamma_0/\sigma_0 = 1/G_d` | Pa^-1 |
| `J'` | storage compliance | `(\gamma_0/\sigma_0)\cos\delta` | Pa^-1 |
| `J''` | loss compliance | `(\gamma_0/\sigma_0)\sin\delta` | Pa^-1 |
| `G_N^0` | plateau modulus (inferred from `G'` and `G''`; there is no true plateau) | | Pa |

Tensile (uniaxial) extension (also applies to nonlinear extension):

| Symbol | Meaning | Definition | SI units |
|---|---|---|---|
| `\sigma_E` | net tensile stress | `\sigma_{zz}-\sigma_{rr}` | Pa |
| `\varepsilon` | Hencky strain | `\ln(L/L_0)` | – |
| `\dot{\varepsilon}` | Hencky strain rate | `d(\ln L)/dt` | s^-1 |
| `E(t)` | tensile relaxation modulus | `\sigma_E(t)/\varepsilon_0` | Pa |
| `D(t)` | tensile creep compliance | `\varepsilon_0(t)/\sigma_E` | Pa^-1 |

### Table III. Shift factors for time-temperature superposition

| Symbol | Meaning | Definition | SI units |
|---|---|---|---|
| `b_T` | vertical shift factor | `b_T G'(T,\omega a_T) = G'(T_0,\omega)` | – |
| `a_T` | horizontal shift factor | as above | – |
| `c_1` | first WLF coefficient | `\log_{10} a_T = -c_1(T-T_0)/[c_2+(T-T_0)]` | – |
| `c_2` | second WLF coefficient | as above | K |
| `E_a` | activation energy for flow | `a_T = \exp[(E_a/R)(1/T - 1/T_0)]` | kJ mol^-1 |

For rubbers, and polymers generally, $b_T = T_0\rho_0/T\rho$ with $\rho$ the
mass density and $\rho_0 = \rho(T_0)$ (Ferry, 3rd ed., 1980).

### Table IV. Nonlinear viscoelasticity in shear

Stress relaxation (step strain):

| Symbol | Meaning | Definition | SI units |
|---|---|---|---|
| `\gamma_0` | strain amplitude | | – |
| `G(t,\gamma_0)` | relaxation modulus (nonlinear) | `\sigma(t)/\gamma_0` | Pa |
| `h(\gamma_0)` | damping function in shear | `G(t,\gamma_0)/G(t)` | – |
| `N_1(t,\gamma_0)` | first normal stress relaxation function | | Pa |
| `N_2(t,\gamma_0)` | second normal stress relaxation function | | Pa |
| `\Psi_1(t,\gamma_0)` | first normal stress relaxation coefficient | `N_1(t,\gamma_0)/\gamma_0^2` | Pa |
| `\Psi_2(t,\gamma_0)` | second normal stress relaxation coefficient | `N_2(t,\gamma_0)/\gamma_0^2` | Pa |

Start-up of steady shear (at fixed shear rate):

| Symbol | Meaning | Definition | SI units |
|---|---|---|---|
| `\sigma^+(t,\dot{\gamma})` | shear stress growth function | | Pa |
| `\eta^+(t,\dot{\gamma})` | shear stress growth coefficient | `\sigma^+(t,\dot{\gamma})/\dot{\gamma}` | Pa s |
| `N_1^+(t,\dot{\gamma})` | first normal stress growth function | `\sigma_{11}-\sigma_{22}` | Pa |
| `\Psi_1^+(t,\dot{\gamma})` | first normal stress growth coefficient | `N_1^+(t,\dot{\gamma})/\dot{\gamma}^2` | Pa s^2 |
| `N_2^+(t,\dot{\gamma})` | second normal stress growth function | `\sigma_{22}-\sigma_{33}` | Pa |
| `\Psi_2^+(t,\dot{\gamma})` | second normal stress growth coefficient | `N_2^+(t,\dot{\gamma})/\dot{\gamma}^2` | Pa s^2 |
| `SR` | stress ratio | `N_1(\dot{\gamma})/\sigma(\dot{\gamma})` | – |

Cessation of steady shear ($\dot{\gamma} = 0$ from $t = 0$):

| Symbol | Meaning | Definition | SI units |
|---|---|---|---|
| `\sigma^-(t,\dot{\gamma})` | shear stress decay function | | Pa |
| `\eta^-(t,\dot{\gamma})` | shear stress decay coefficient | `\sigma^-(t,\dot{\gamma})/\dot{\gamma}` | Pa s |
| `N_1^-(t,\dot{\gamma})` | first normal stress decay function | `\sigma_{11}-\sigma_{22}` | Pa |
| `\Psi_1^-(t,\dot{\gamma})` | first normal stress decay coefficient | `N_1^-(t,\dot{\gamma})/\dot{\gamma}^2` | Pa s^2 |
| `N_2^-(t,\dot{\gamma})` | second normal stress decay function | `\sigma_{22}-\sigma_{33}` | Pa |
| `\Psi_2^-(t,\dot{\gamma})` | second normal stress decay coefficient | `N_2^-(t,\dot{\gamma})/\dot{\gamma}^2` | Pa s^2 |

Creep and creep recovery (recoil):

| Symbol | Meaning | Definition | SI units |
|---|---|---|---|
| `J(t,\sigma)` | creep compliance | `\gamma(t,\sigma)/\sigma` | Pa^-1 |
| `J_s(\sigma)` | steady-state compliance | `J(t\to\infty,\sigma)` | Pa^-1 |
| `\gamma_r(t',\sigma)` | recoverable strain after `t_0`, when `\sigma\to 0`; `t' = t - t_0` | `\gamma[t_0,\sigma]-\gamma[t,\sigma],\ t>t_0` | – |
| `\gamma_\infty(\sigma)` | ultimate recoil | `\gamma_r(t'\to\infty,\sigma)` | – |
| `J_r(\sigma)` | steady-state recoverable compliance | `\gamma_\infty(\sigma)/\sigma` | Pa^-1 |

By the Boltzmann principle the steady-state compliance and the steady-state
recoverable compliance should be equal, although they are measured
differently.

### Table V. Nonlinear viscoelasticity in extension

Tensile (uniaxial) extension:

| Symbol | Meaning | Definition | SI units |
|---|---|---|---|
| `\varepsilon` | engineering strain | `(L-L_0)/L_0` | – |
| `\sigma` | engineering stress | `F/A_0` | Pa |
| `E` | Young's modulus of a solid | `\sigma/\varepsilon` | Pa |
| `\sigma_\varepsilon` | net tensile stress (true) | `\sigma_{zz}-\sigma_{rr}` | Pa |
| `\varepsilon` or `\epsilon` | Hencky strain | `\ln(L/L_0)` | – |
| `\dot{\varepsilon}` or `\dot{\epsilon}` | Hencky strain rate | `d(\ln L)/dt` | s^-1 |
| `\sigma_E^+(t,\dot{\varepsilon})` | tensile stress growth function | | Pa |
| `\eta_E^+(t,\dot{\varepsilon})` | tensile stress growth coefficient | `\sigma_E^+(t,\dot{\varepsilon})/\dot{\varepsilon}` | Pa s |
| `\eta_E(\dot{\varepsilon})` | extensional viscosity | `\eta_E^+(t,\dot{\varepsilon}),\ t\to\infty` | Pa s |
| `D(t,\sigma_E)` | tensile creep compliance | `\varepsilon(t)/\sigma_E` | Pa^-1 |
| `\varepsilon_r(t',\varepsilon)` | recoverable strain after `t_0`, when `\sigma_E\to 0`; `t'=t-t_0` | `\varepsilon[t_0,\sigma_E]-\varepsilon[t,\sigma_E],\ t>t_0` | – |

In mechanics the same symbols are often used for engineering and true stress
and strain; they agree only for very small deformations.

Biaxial extension (symmetrical):

| Symbol | Meaning | Definition | SI units |
|---|---|---|---|
| `\varepsilon_B` | biaxial strain | `\ln(R/R_0)` | – |
| `\dot{\varepsilon}_B` | biaxial strain rate | `d(\ln R)/dt` | s^-1 |
| `\sigma_B` | net stretching stress | `\sigma_{rr}-\sigma_{zz}` | Pa |
| `\sigma_B^+(t,\dot{\varepsilon}_B)` | biaxial stretch growth function | | Pa |
| `\eta_B^+(t,\dot{\varepsilon}_B)` | biaxial stretch growth coefficient | `\sigma_B^+(t,\dot{\varepsilon}_B)/\dot{\varepsilon}_B` | Pa s |
| `\eta_B^-(t,\dot{\varepsilon}_B)` | biaxial stress decay coefficient | `\sigma_B^-(t,\dot{\varepsilon}_B)/\dot{\varepsilon}_B` | Pa s |
| `\eta_B(\dot{\varepsilon}_B)` | biaxial extensional viscosity | `\sigma_B^+(t\to\infty,\dot{\varepsilon}_B)/\dot{\varepsilon}_B` | Pa s |
| `D(t,\sigma_B)` | biaxial creep compliance | `\varepsilon_B(t)/\sigma_B` | Pa^-1 |

### Table VI. Rheometry

Capillary rheometers:

| Symbol | Meaning | Definition | SI units |
|---|---|---|---|
| `\sigma_A` | apparent wall shear stress | `P_d R/2L` (`P_d` = driving reservoir pressure, exit pressure neglected) | Pa |
| `\dot{\gamma}_A` | apparent wall shear rate | `4Q/\pi R^3` | s^-1 |
| `\sigma_W` | wall shear stress | `-\sigma_{rz}\ (r=R)` | Pa |
| `\dot{\gamma}_W` | wall shear rate | `-dv_z/dr\ (r=R)` | s^-1 |

Cone-plate rheometers (cone radius $R$, cone angle $\beta$ measured from the
plate, rotation $\varphi$ about the axis, normal force $F$ on the plate):

| Symbol | Meaning | Definition | SI units |
|---|---|---|---|
| `\beta` | cone angle | | rad |
| `\varphi` | angular rotation | | rad |
| `\Omega` | angular velocity | `d\varphi/dt` | rad s^-1 |
| `M` | torque | `2\pi R^3\sigma_{\varphi\theta}/3` (valid for `\beta < 0.1` rad) | N m |
| `F_z` | normal thrust | | N |

### Table VII. Solutions

| Symbol | Meaning | Definition | SI units |
|---|---|---|---|
| `c` | concentration | | kg m^-3 (g mL^-1 often used) |
| `c^*` | overlap concentration | | kg m^-3 (see notes) |
| `\eta_s` | solvent viscosity | | Pa s |
| `\eta_r` | relative viscosity | `\eta/\eta_s` | – |
| `\eta_{sp}` | specific viscosity | `\eta_r - 1` | – |
| `\eta_{red}` | reduced viscosity | `\eta_{sp}/c` | m^3 kg^-1 (mL g^-1 often used) |
| `[\eta]` | intrinsic viscosity | `\lim(\eta_{red};\ \dot{\gamma}\to 0,\ c\to 0)` | m^3 kg^-1 (mL g^-1 often used) |
| `\eta_m` | viscosity of matrix (e.g. nanocomposites) | | Pa s |
| `\boldsymbol{\sigma}^s` | solvent contribution to the stress tensor | | Pa |
| `\boldsymbol{\sigma}^p` | polymeric contribution to the stress tensor | | Pa |

### Table VIII. Suspensions

| Symbol | Meaning | Definition | SI units |
|---|---|---|---|
| `\phi` | volume fraction solid | `V_{solid}/V_{suspension}` | – |
| `\phi_{max}` | maximum packing fraction | | – |
| `\eta_m` | suspending medium viscosity | | Pa s |
| `\eta` | effective viscosity of suspension | `\Sigma_{12}/\dot{\gamma}` | Pa s |
| `\eta_r` | relative viscosity of suspension | `\eta/\eta_f` as printed (see notes) | – |
| `\eta_p` | particle contribution to `\eta` | | Pa s |
| `\boldsymbol{\sigma}(\mathbf{x},t)` | local stress tensor | | Pa |
| `\boldsymbol{\Sigma}` | total bulk stress | `\boldsymbol{\Sigma}^f + \boldsymbol{\Sigma}^p` | Pa |
| `\boldsymbol{\Sigma}^f` | fluid (suspending medium) contribution | | Pa |
| `\boldsymbol{\Sigma}^p` | particle contribution | | Pa |
| `N_1` | first normal stress difference | `\Sigma_{11}-\Sigma_{22}` | Pa |
| `N_2` | second normal stress difference | `\Sigma_{22}-\Sigma_{33}` | Pa |
| `\Upsilon_1` | dimensionless first normal stress difference | `-N_1/\eta_f\dot{\gamma}` | – |
| `\Upsilon_2` | dimensionless second normal stress difference | `-N_2/\eta_f\dot{\gamma}` | – |
| `\Pi` | particle pressure | `-\tfrac{1}{3}(\Sigma^p_{11}+\Sigma^p_{22}+\Sigma^p_{33})` | Pa |
| `\Sigma^H` | hydrodynamic particle stress | | Pa |
| `\Sigma^{IP}` | interparticle stress | | Pa |
| `\Sigma^B` | Brownian stress | | Pa |

The source prefers subscript $m$ ("medium" or "matrix") to $f$ for the
suspending fluid's viscosity, particularly when that fluid is itself
rheologically complex, and notes that $\mu$ is often used for it.

### Table IX. Interfacial and surface rheology

| Symbol | Meaning | Definition | SI units |
|---|---|---|---|
| `\sigma_{\alpha,\beta}` | interfacial or surface tension (`\alpha` or `\Gamma` are sometimes used; avoid) | | Pa m |
| `\Pi` | surface pressure | `\sigma^0_{\alpha,\beta}-\sigma_{\alpha,\beta}(F^s/A)` | Pa m |
| `\sigma^s` | interfacial shear stress | `F^s/L` | Pa m |
| `\gamma^s` | interfacial shear strain | `dx_1/dx_2` | – |
| `\dot{\gamma}^s` | interfacial shear rate | `dv_1/dx_2` | s^-1 |
| `\alpha^s` | interfacial dilatational strain | `\ln(A/A_0)` | – |
| `\dot{\alpha}^s` | interfacial dilatational strain rate | `d(\ln A)/dt` | s^-1 |
| `\Gamma` | interfacial concentration | | kg m^-2 |

Steady shear and dilatation:

| Symbol | Meaning | Definition | SI units |
|---|---|---|---|
| `\eta^s` | interfacial shear viscosity | `\sigma^s/\dot{\gamma}^s` | Pa s m |
| `\kappa^s` | interfacial dilatational viscosity | `\sigma^s/\dot{\alpha}^s` | Pa s m |

Simple shear:

| Symbol | Meaning | Definition | SI units |
|---|---|---|---|
| `G^s` | interfacial shear modulus | `\sigma^s/\gamma^s` | Pa m |
| `G^s(t)` | relaxation modulus (shear) | `\sigma^s(t)/\gamma^s` | Pa m |

Pure dilatation ($E$ is sometimes used for $K$; $K$ is recommended, to avoid
confusion with Young's modulus):

| Symbol | Meaning | Definition | SI units |
|---|---|---|---|
| `K^s` | interfacial dilatational modulus | `\sigma^s/\alpha^s` | Pa m |
| `K^s(t)` | dilatational storage modulus (as printed) | `\sigma^s(t)/\alpha^s` | Pa m |
| `K_\Pi` | Gibbs elasticity (surfactants) | `d\sigma_{\alpha,\beta}/d(\ln A)` | Pa m |

Small-amplitude oscillatory shear:

| Symbol | Meaning | Definition | SI units |
|---|---|---|---|
| `\gamma^s_0` | strain amplitude | `\gamma^s=\gamma^s_0\sin(\omega t)` | – |
| `\delta` | phase angle (loss angle) | `\sigma^s(t)=\sigma^s_0\sin(\omega t+\delta)` | rad |
| `\sigma^s_0` | stress amplitude | `F^s/L` | Pa m |
| `G^{s*}` | complex interfacial shear modulus | `G^{s\prime}+iG^{s\prime\prime}` | Pa m |
| `\lvert G^{s*}\rvert` | absolute magnitude of `G^{s*}` | `\sigma^s_0/\gamma^s_0` | Pa m |
| `G^{s\prime}` | storage modulus | `\lvert G^{s*}\rvert\cos\delta` | Pa m |
| `G^{s\prime\prime}` | loss modulus | `\lvert G^{s*}\rvert\sin\delta` | Pa m |
| `\eta^{s*}` | complex shear viscosity | `\eta^{s\prime}-i\eta^{s\prime\prime}` | Pa s m |
| `\lvert\eta^{s*}\rvert` | absolute magnitude of `\eta^{s*}` | `\sigma^s_0/\omega\gamma^s_0` | Pa s m |
| `\eta^{s\prime}` | dynamic shear viscosity (in phase with strain rate) | `G^{s\prime\prime}/\omega` | Pa s m |
| `\eta^{s\prime\prime}` | out-of-phase component of `\eta^{s*}` | `G^{s\prime}/\omega` | Pa s m |

Small-amplitude oscillatory dilatation:

| Symbol | Meaning | Definition | SI units |
|---|---|---|---|
| `\alpha^s_0` | dilatational strain amplitude | `\alpha^s=\alpha^s_0\sin(\omega t)` | – |
| `K^{s*}` | complex dilatational modulus | `K^{s\prime}+iK^{s\prime\prime}` | Pa m |
| `\lvert K^{s*}\rvert` | absolute magnitude of `K^{s*}` | `\sigma^s_0/\alpha^s_0` | Pa m |
| `K^{s\prime}` | storage dilatational modulus | | Pa m |
| `K^{s\prime\prime}` | loss dilatational modulus | | Pa m |
| `\kappa^{s*}` | complex dilatational viscosity | `\kappa^{s\prime}+i\kappa^{s\prime\prime}` as printed (see notes) | Pa s m |
| `\lvert\kappa^{s*}\rvert` | absolute magnitude of `\kappa^{s*}` | `\sigma^s_0/\omega\alpha^s_0` (see notes) | Pa s m |
| `\kappa^{s\prime}` | dynamic dilatational viscosity | `K^{s\prime\prime}/\omega` | Pa s m |
| `\kappa^{s\prime\prime}` | out-of-phase component of `\kappa^{s*}` | `K^{s\prime}/\omega` | Pa s m |

Other properties:

| Symbol | Meaning | Definition | SI units |
|---|---|---|---|
| `J^s(t)` | creep compliance (shear) | `\gamma^s(t)/\sigma^s` | Pa^-1 m^-1 |
| `J^s_e` | equilibrium compliance of solid | `J^s(t\to\infty)` | Pa^-1 m^-1 |
| `J^s_r(t)` | recoverable compliance | `J^s(t)-t/\eta^s_0` | Pa^-1 m^-1 |
| `J^s_0` | steady-state compliance of fluid | `J^s(t)-t/\eta^s_0\ (t\to\infty)` | Pa^-1 m^-1 |
| `\eta^s_E(\dot{\varepsilon}^s)` | extensional viscosity | `\eta^s_E(t,\dot{\varepsilon}^s)\ (t\to\infty)` | Pa s m |

### Table X. Molecular description of entangled polymers

| Symbol | Meaning |
|---|---|
| `a` | tube diameter (typically nm); average entanglement spacing or mesh size, `\sqrt{\langle R^2\rangle_0 M_e/M}` |
| `b_K` | Kuhn segment length, with `\langle R^2\rangle = b_K^2 N_K` and `R_{max} = b_K N_K` |
| `f` | tension in a chain segment (typically N) |
| `f_{max}` | maximum tension in a chain segment |
| `k_B` | Boltzmann's constant |
| `L` | mean tube contour length |
| `M` | molecular weight, kg mol^-1 (formally a molar mass) |
| `M_C` | critical molecular weight for the effect of entanglements on `\eta_0`, kg mol^-1 |
| `M_C'` | molecular weight for the effect of entanglements on `J_s^0`, kg mol^-1 |
| `M_e` | molecular weight between entanglements, `\rho RT/G_N^0` (Ferry); in tube-model work `\tfrac{4}{5}\rho RT/G_N^0` |
| `N_K` | number of Kuhn segments in the equivalent freely jointed chain |
| `p` | packing length, `M/[\langle R^2\rangle_0 \rho N_A]` (Fetters et al., Macromolecules 27, 4639 (1994)) |
| `p^2` | degree of branch-point hopping within the tube (fraction of tube diameter) |
| `R` | end-to-end distance of a polymer molecule |
| `R_{max}` | fully extended chain length |
| `s` | tube contour variable (curvilinear coordinate along the tube) |
| `\mathbf{S}` | tube orientation tensor |
| `Z` | number of entanglement segments per molecule, `M/M_e` (in tube theory, "number of tube segments", using the 4/5 definition of `M_e`) |
| `\alpha` | dilution exponent for `M_e` |
| `\zeta` | friction coefficient |
| `\zeta_0` | monomer friction coefficient |
| `\lambda` | chain stretch; stretch ratio |
| `\xi` | correlation length; characteristic size scale (blob) |
| `\tau_d` | reptation (tube disengagement) time |
| `\tau_e` | Rouse time of an entanglement strand, `\tau_R/3\pi^2 Z^2` as printed (see notes) |
| `\tau_p` | relaxation time of the `p`th mode (`p` is the mode index) |
| `\tau_r` | Rouse reorientation relaxation time (Doi and Edwards' rotational relaxation time; conjectured to equal `2\tau_R`) |
| `\tau_R` | Rouse stress relaxation time, `\zeta N^2 b^2/(6\pi^2 k_B T)` |
| `\tau_s` | stretch relaxation time |

### Table XI. Stress and strain tensors

| Symbol | Meaning |
|---|---|
| `\boldsymbol{\sigma}` | total stress tensor |
| `\boldsymbol{\tau}` | extra stress tensor |
| `\boldsymbol{\gamma}` | strain tensor for linear viscoelasticity |
| `\mathbf{C}` | Cauchy tensor |
| `\mathbf{B}` or `\mathbf{C}^{-1}` | Finger tensor |
| `\mathbf{Q}` | Doi-Edwards strain tensor |
| `\dot{\boldsymbol{\gamma}}` | rate-of-strain tensor, `\nabla\mathbf{v}+(\nabla\mathbf{v})^T` |
| `\boldsymbol{\Omega}` | vorticity tensor, `\nabla\mathbf{v}-(\nabla\mathbf{v})^T` |
| `\mathbf{D}` | `\tfrac{1}{2}[\nabla\mathbf{v}+(\nabla\mathbf{v})^T]`, the fluid-mechanics alternative to `\dot{\boldsymbol{\gamma}}` (fluid dynamics also puts 1/2 in front of `\boldsymbol{\Omega}`) |

### Table XII. Dimensionless groups used to describe experimental regimes

| Symbol | Name | Definition |
|---|---|---|
| `Bn` | Bingham number | (yield stress)/(shear stress); sometimes the wall shear stress, or (viscosity x velocity)/length, is used in the denominator |
| `Bq` | Boussinesq number | (surface shear stress)/[(bulk subphase shear stress) x (perimeter length along which the surface shear stress acts)] |
| `Ca` | capillary number | `\eta v/\sigma_{\alpha,\beta}` or `\eta\dot{\gamma}R/\sigma_{\alpha,\beta}`; (viscous forces)/(surface forces) |
| `De` | Deborah number | (characteristic time of fluid)/(duration of deformation), e.g. `\tau\omega`; specific forms `De_R` (Rouse time), `De_d` (longest relaxation time) |
| `Pé` | Péclet number | `\dot{\gamma}a^2/D_0`; (rate of advection)/(rate of diffusion); `a` particle radius, `D_0` particle diffusion coefficient |
| `\nu` | Poisson ratio | `-d\varepsilon_{transverse}/d\varepsilon_{axial}`; in three dimensions also `E/2G - 1` |
| `Re` | Reynolds number | `\rho v d/\eta`; (inertial forces)/(viscous forces); `d` characteristic length |
| `Wi` | Weissenberg number | (characteristic time of fluid) x (rate of deformation), e.g. `\tau\dot{\varepsilon}` or `\tau\dot{\gamma}`; specific forms `Wi_R`, `Wi_d`. Not `N_1/\sigma`, which is the stress ratio |

Definitions and uses of $Wi$ and $De$: "Weissenberg and Deborah Numbers —
Their definition and use," Rheol. Bulletin 79(2), 14 (2010).

### Table XIII. Frequently used constants

| Symbol | Meaning | Value |
|---|---|---|
| `N_A` | Avogadro constant | 6.023 x 10^23 mol^-1 as printed; exact SI value 6.02214076 x 10^23 mol^-1 |
| `R` | ideal gas constant | 8.314 J mol^-1 K^-1 |
| `k_B` | Boltzmann constant | 1.38 x 10^-23 J K^-1 (exact SI value 1.380649 x 10^-23 J K^-1) |

## Notes on the source

These are places where the 2018-10 edition is inconsistent or appears to
contain a typographical error. They are recorded so that a model does not
"correct" the official notation on its own, and so that a user can decide.

- **Normal stress ratio (Table I)** is printed as $-N_1/N_2$. Most of the
  literature reports $-N_2/N_1$, which is small and positive for polymer
  melts. State which is meant.
- **Power-law names (Table I):** the source labels $n$ "consistency index"
  and $K$ "constant". In common usage $K$ is the consistency index and $n$
  the power-law (flow behaviour) index. The symbols and the equation
  $\eta = K\dot{\gamma}^{n-1}$ are unambiguous.
- **Retardation spectrum (Table II, footnote d)** is printed with
  $1 - \exp(t/\tau)$; the physically meaningful form, used above, is
  $1 - \exp(-t/\tau)$.
- **Net tensile stress** is $\sigma_E$ in Table II and $\sigma_\varepsilon$
  in Table V. Prefer $\sigma_E$, which matches $\eta_E$ and $\sigma_E^+$.
- **Overlap concentration (Table VII)** has units printed as kg m$^3$; it is
  a concentration, kg m$^{-3}$.
- **Suspension viscosities (Table VIII):** the medium viscosity is $\eta_m$,
  but $\eta_r$ and $\Upsilon_{1,2}$ are defined with $\eta_f$. Read $\eta_f$
  as $\eta_m$.
- **$K^s(t)$ (Table IX)** is labeled "dilatational storage modulus" but
  defined as $\sigma^s(t)/\alpha^s$, a dilatational relaxation modulus.
- **Complex dilatational viscosity (Table IX)** is printed as
  $\kappa^{s\prime} + i\kappa^{s\prime\prime}$, unlike the bulk and
  interfacial shear cases ($\eta' - i\eta''$); and $|\kappa^{s*}|$ is printed
  as $\sigma_0^s/\omega\delta_0^s$, where $\alpha_0^s$ (used above) is
  evidently meant.
- **$\tau_e$ (Table X)** is printed as $\tau_R/3\pi^2Z^2$. Prefactors
  relating $\tau_e$ and $\tau_R$ differ between tube models (in
  Likhtman-McLeish, $\tau_R = Z^2\tau_e$), so check against the model in use.
- **Reused symbols.** $\Pi$ is both particle pressure (Table VIII) and
  surface pressure (Table IX). $\sigma$ is both shear stress and engineering
  tensile stress. Superscript $s$ marks both the solvent stress tensor
  $\boldsymbol{\sigma}^s$ (Table VII) and interfacial quantities (Table IX).
  $\alpha$ is the dilution exponent (Table X) and, with superscript $s$, the
  interfacial dilatational strain. Say which is meant when both could appear.
- **Not covered.** The list has no symbols for medium- or large-amplitude
  oscillatory shear (MAOS, LAOS) analysis, thixotropy, or granular and dense
  suspension friction (for example a friction coefficient $\mu$ or the
  viscous number). For these, use the notation of the paper being discussed
  and define it on first use.

END OF NOTATION PRIMER. Please confirm you have received the complete primer
(it ends with this sentence) and will use this notation in the rest of the
conversation.