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Rheology Nomenclature 2018-10

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Rheology Nomenclature 2018-10 Official Nomenclature of US and European Societies of Rheology (2018-10) Assembled by 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. TABLE I. Steady simple shear (viscometric flow) Name Definition Symbol SI units a Direction of flow (simple shear) m Direction of velocity gradient (simple shear) m Neutral direction (simple shear) m Shear or normal force F N –1 Velocity ⁄ m s –2 Acceleration ⁄ m s Shear stress b ⁄ Pa Shear strain ⁄ – –1 Shear rate |⁄| s c –1 Vorticity ⁄ s Viscosity ||⁄ Pa s Yield stress Pa Yield strain – First normal stress difference Pa Second normal stress difference Pa 2 First normal stress coefficient / Pa s 2 Second normal stress coefficient / Pa s Normal stress ratio ⁄ – Zero-shear viscosity →0 Pa s (limiting low shear rate viscosity) 2 Zero-shear first normal stress coefficient →0 . Pa s Consistency index in power law for viscosity – Constant in power law for viscosity see above equation Pa sn a SI allows either a dot between units or a space, as used here. b A is the area in m2. c In the fluid dynamics community a prefactor of 1/2 is used in the definition of the vorticity. TABLE II. Linear viscoelasticity Name Definition Symbol Units Simple shear Shear modulus of a solid ⁄ Pa Relaxation modulus ⁄ Pa Relaxation strength in discrete spectrum – Pa Relaxation time in discrete spectrum – s Continuous relaxation spectrum – a Pa b,c ∗ Orthogonal superposition complex modulus – , Pa b,c ∗ Parallel superposition complex modulus – ∥ , Pa Memory function ⁄ Pa s–1 Creep compliance (shear) ⁄ Pa–1 –1 Equilibrium compliance of solid →∞ Pa –1 Recoverable compliance ⁄ Pa –1 Steady-state compliance of fluid ⁄ →∞ Pa Continuous retardation spectrum d – d Pa–1 Small-amplitude oscillatory shear Strain amplitude sin – Loss angle (phase angle) sin rad Stress amplitude sin Pa Complex modulus ∗ Pa ∗ ∗ Absolute magnitude of ⁄ | | or Pa Storage modulus cos Pa Loss modulus sin Pa Complex viscosity ∗ Pa s ∗ Absolute magnitude of η* ⁄ | | Pa s Dynamic viscosity (in phase with strain rate) ⁄ Pa s Out-of-phase (with strain rate) component of η* ⁄ Pa s Complex compliance ∗ Pa–1 ∗ –1 Absolute magnitude of J* ⁄ 1⁄ | | Pa –1 Storage compliance ⁄ cos Pa –1 Loss compliance ⁄ sin Pa e e Plateau modulus – Pa Tensile (Uniaxial) Extension c Net tensile stress Pa Hencky strain ln⁄ – Hencky strain rate ln ⁄ s–1 Tensile relaxation modulus ⁄ Pa –1 Tensile creep compliance ⁄ Pa a ⁄ exp ln b The same subscripts apply to definitions of storage and loss moduli and viscosities. c Also applies to nonlinear phenomena in uniaxial extension. 2 d 1exp / ln e Because there is not a true plateau in G(t) or G′(ω), is inferred from G′(ω) and G"(ω) using methods reviewed by Liu et al. [Polymer 47, 4461−4479 (2006)]. TABLE III. Shift factors for time-temperature superposition Name Definition Symbol SI Units a Vertical shift factor , , – Horizontal shift factor , , – First WLF coefficient – log Second WLF coefficient K log –1 Activation energy for flow 1 1 kJ mol exp a For rubbers, T is given by ⁄, where is mass density, with [J. D. Ferry, Viscoelastic properties of polymers, 3rd Ed., Wiley, NY, 1980]. It has also been found to be appropriate for polymers in general. 3 TABLE IV. Nonlinear viscoelasticity in shear Name Definition Symbol SI Units Stress relaxation (step strain) Strain amplitude – Relaxation modulus (nonlinear) ⁄ , Pa Damping function in shear , ⁄ ℎ – First normal stress relaxation function , Pa Second normal stress relaxation function , Pa First normal stress relaxation coefficient , ⁄ , Pa Second normal stress relaxation coefficient , ⁄ , Pa Start-up of steady shear (at fixed shear rate) Shear stress growth function , Pa Shear stress growth coefficient , ⁄ , Pa s First normal stress growth function , Pa 2 First normal stress growth coefficient , ⁄ , Pa s Second normal stress growth function , Pa 2 Second normal stress growth coefficient , ⁄ , Pa s Stress Ratio ⁄ – Cessation of steady shear ( 0 from 0) Shear stress decay function , Pa Shear stress decay coefficient , ⁄ , Pa s First normal stress decay function , Pa 2 First normal stress decay coefficient , ⁄ , Pa s Second normal stress decay function , Pa 2 Second normal stress decay coefficient , ⁄ , Pa s Creep and creep recovery (recoil) Creep compliance , ⁄ , Pa–1 a –1 Steady-state compliance → ∞, Pa Recoverable strain , , , – (after when →0) Ultimate recoil →∞, – a –1 Steady-state recoverable compliance ⁄ Pa a Although measured in different ways, the steady-state compliance and the steady-state recoverable compliance should be equal to each other according to the Boltzmann principle. 4 TABLE V. Nonlinear viscoelasticity in extension. Name Definition Symbol SI Units Tensile (uniaxial) extension a Engineering strain ⁄ – a Engineering stress ⁄ Pa Young’s modulus of a solid ⁄ E Pa Net tensile stress (true) Pa Hencky strain ln⁄ ε or – Hencky strain rate ln ⁄ or s–1 Tensile stress growth function , Pa Tensile stress growth coefficient , ⁄ , Pa s Extensional viscosity , →∞ Pa s –1 Tensile creep compliance ⁄ , Pa Recoverable strain , , , – (after where E →0) Biaxial extension (symmetrical) Biaxial strain ln⁄ – –1 Biaxial strain rate ln ⁄ s Net stretching stress Pa Biaxial stretch growth function , Pa Biaxial stretch growth coefficient , ⁄ , Pa s Biaxial stress decay coefficient , ⁄ , Pa s Biaxial extensional viscosity →∞, ⁄ Pa s –1 Biaxial creep compliance ⁄ , Pa a In the mechanics literature, the same symbols are often used for both engineering and true stress and strain, but they are only equivalent in the limit of very small deformations. TABLE VI. Rheometry Name Definition Symbol SI Units Capillary rheometers a Apparent wall shear stress 2⁄ Pa –1 Apparent wall shear rate 4⁄ s Wall shear stress Pa –1 Wall shear rate ⁄ s Cone-plate rheometers Cone angle Figure 1 rad Angular rotation Figure 1 rad Angular velocity ⁄ rad s–1 b Torque 2 ⁄3 N m Normal thrust Figure 1 N a = driving (reservoir) pressure (exit pressure neglected) b Approximation valid for 0.1 rad. 5 Figure 1. Symbols describing cone-plate geometry. TABLE VII. Solutions Name Definition Symbol SI units Concentration kg m–3 a Overlap ∗ kg m3 a concentration Solvent viscosity Pa s Relative viscosity ⁄ – Specific viscosity 1 – 3 –1 b Reduced viscosity ⁄ m kg 3 –1 b Intrinsic viscosity lim , →0,→0 m kg c Viscosity of matrix Pa s Solvent contribution Pa to the stress tensor Polymeric Pa contribution to the stress tensor a Units of g mL–1 are often used. b Units of mL g–1 are often used. c Often used for nanocomposites, i.e., particles in a viscoelastic matrix. 6 TABLE VIII. Suspensions Name Definition Symb SI ol Units Volume fraction solid ⁄ – Maximum packing fraction – a Suspending medium viscosity Pa s Effective viscosity of suspension ⁄ Pa s Relative viscosity of suspension ⁄ – Particle contribution to η Pa s Local stress tensor , Pa Total bulk stress Pa Fluid (suspending medium) contribution Pa Particle contribution Pa First normal stress difference Pa Second normal stress difference Pa Dimensionless first normal stress difference ⁄ – Dimensionless second normal stress ⁄ – difference Particle pressure Pa Hydrodynamic particle stress Pa Interparticle stress Pa Brownian stress Pa a Sometimes the subscript f is used to indicate the fluid viscosity, but it seems more adequate to use m which stands for “medium” if the suspending fluid is itself rheologically complex or “matrix” in nanocomposites. Also, μ is often used to denote fluid viscosity. TABLE IX. Interfacial and surface rheology Name Definition Symbol SI Units a Interfacial or surface tension , Pa m Surface pressure , , ⁄ Pa m Interfacial shear stress ⁄ Pa m Interfacial shear strain ⁄ – –1 Interfacial shear rate ⁄ s Interfacial dilatational strain ln⁄ – Interfacial dilatational strain rate ln ⁄ s–1 Interfacial concentration – kg m–2 Steady shear and dilatation Interfacial shear viscosity ⁄ Pa s m Interfacial dilatational viscosity ⁄ Pa s m Simple shear Interfacial shear modulus ⁄ Pa m Relaxation modulus (shear) ⁄ Pa m 7 Pure dilatation Interfacial dilatational modulus ⁄ Pa m Dilatational storage modulus ⁄ Pa m b Gibbs elasticity (surfactants) ,⁄ln Pa m Small-amplitude oscillatory shear Strain amplitude sin – Phase angle (loss angle) sin rad Stress amplitude ⁄ Pa m Complex interfacial shear modulus ′′′ ∗ Pa m ∗ ∗ Absolute magnitude of ⁄ | | Pa m Storage modulus |∗| cos ′ Pa m Loss modulus |∗| sin ′′ Pa m Complex shear viscosity ′′′ ∗ Pa s m ∗ ∗ Absolute magnitude of ⁄ | | Pa s m Dynamic shear viscosity (in phase with strain ′′⁄ ′ Pa s m rate) Out-of-phase (with strain rate) component of ′⁄ ′′ Pa s m ∗ Small-amplitude oscillatory dilatation Dilatational strain amplitude sin – Complex dilatational modulus b ′′′ ∗ Pa m ∗ b ∗ Absolute magnitude of ⁄ | | Pa m Storage dilatational modulus b ′ ′ Pa m Loss dilatational modulus b ′′ ′′ Pa m Complex dilatational viscosity ′′′ ∗ Pa s m ∗ ∗ Absolute magnitude of ⁄ | | Pa s m Dynamic dilatational viscosity ′′⁄ ′ Pa s m Out-of-phase component of ∗ ′⁄ ′′ Pa s m Other properties Creep compliance (shear) ⁄ Pa–1 m–1 –1 –1 Equilibrium
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