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+# 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.