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 pth 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.