Glossary of Rheological Terms¶

This glossary defines 50+ key terms used throughout the RheoJAX documentation.

Core Concepts¶

Rheology: The study of deformation and flow of matter under applied stress.
Viscoelasticity: Material behavior exhibiting both elastic (solid-like) and viscous (liquid-like) characteristics.
Linear Viscoelasticity: Regime where stress is proportional to strain (small deformations).
Deborah Number (De): Ratio of material relaxation time to observation time: \(\text{De} = \tau / t_{\text{obs}}\)

Storage Modulus (\(G'\)): Elastic component of complex modulus; energy stored per cycle (Pa).
Loss Modulus (\(G''\)): Viscous component of complex modulus; energy dissipated per cycle (Pa).
Complex Modulus (\(G^*\)): \(G^* = G' + iG''\); combines storage and loss moduli.
Loss Tangent (\(\tan\delta\)): Ratio of loss to storage modulus: \(\tan\delta = G''/G'\); damping factor.
Relaxation Modulus (\(G(t)\)): Stress response to step strain as a function of time (Pa).
Equilibrium Modulus (\(G_e\)): Long-time plateau in relaxation modulus (viscoelastic solids, Pa).
Compliance (\(J\)): Inverse of modulus; strain per unit stress (\(\text{Pa}^{-1}\)).
Young’s Modulus (\(E\)): Tensile (axial) modulus; ratio of tensile stress to tensile strain (Pa).
Complex Young’s Modulus (\(E^*\)): \(E^* = E' + iE''\); tensile analog of the complex shear modulus \(G^*\).
Storage Modulus (tensile) (\(E'\)): In-phase (elastic) component of the complex Young’s modulus from DMTA (Pa).
Loss Modulus (tensile) (\(E''\)): Out-of-phase (viscous) component of the complex Young’s modulus from DMTA (Pa).
Poisson’s Ratio (\(\nu\)): Ratio of transverse to axial strain; relates \(E\) and \(G\) via \(E = 2(1+\nu)G\). Typical values: rubber \(\approx 0.5\), glassy polymer \(\approx 0.35\).

DMTA (Dynamic Mechanical Thermal Analysis): Oscillatory technique measuring \(E^*(\omega, T)\) under tensile, bending, or compression deformation. Widely used for polymer glass transitions and temperature sweeps.
DMA (Dynamic Mechanical Analysis): Synonym for DMTA; sometimes specifically refers to isothermal frequency sweeps in tensile geometry.
Deformation Mode: The type of mechanical loading applied to the sample: shear, tension, bending, or compression. In RheoJAX, set via deformation_mode='tension' in fit() / predict().

Viscosity (\(\eta\)): Resistance to flow under shear; stress/shear-rate ratio (Pa·s).
Zero-Shear Viscosity (\(\eta_0\)): Viscosity at vanishingly small shear rates (Newtonian plateau, Pa·s).
Complex Viscosity (\(\eta^*\)): Frequency-dependent viscosity from SAOS: \(\eta^* = |G^*|/\omega\) (Pa·s).
Shear Thinning: Decrease in viscosity with increasing shear rate (pseudoplastic).
Shear Thickening: Increase in viscosity with increasing shear rate (dilatant).

Relaxation Time (\(\tau\)): Characteristic timescale for stress to decay to \(1/e\) (~37%) of initial value (s).
Crossover Frequency (\(\omega_c\)): Frequency where \(G' = G''\); related to relaxation time by \(\omega_c \approx 1/\tau\) (rad/s).

Fractional Order (\(\alpha\)): Exponent in fractional derivative; characterizes breadth of relaxation spectrum (\(0 < \alpha < 1\)).
SpringPot: Fractional viscoelastic element interpolating between spring (\(\alpha=0\)) and dashpot (\(\alpha=1\)).
Mittag-Leffler Function (\(E_\alpha\)): Generalization of exponential function for fractional models.

SAOS (Small-Amplitude Oscillatory Shear): Sinusoidal strain input; measures \(G'(\omega)\) and \(G''(\omega)\) in frequency domain.
Stress Relaxation: Step strain input; measures \(G(t)\) in time domain.
Creep: Step stress input; measures compliance \(J(t)\) in time domain.
Steady Shear Flow: Constant shear rate; measures viscosity \(\eta(\dot{\gamma})\) (nonlinear regime).

Viscoelastic Liquid: Material with zero equilibrium modulus; flows at long times (\(G'' > G'\) at low \(\omega\)).
Viscoelastic Solid: Material with finite equilibrium modulus; does not flow (\(G' > G''\) everywhere).
Gel: Material with power-law relaxation (\(G' \approx G'' \sim \omega^\alpha\)).
Yield Stress (\(\sigma_y\)): Critical stress below which material behaves as solid, above which it flows (Pa).

Maxwell Model: Simplest viscoelastic liquid model; single relaxation time.
Zener Model (SLS): Standard Linear Solid; viscoelastic solid with single relaxation time.
Fractional Models: Models using fractional derivatives to capture distributed relaxation spectra.
PowerLaw Model: Shear thinning/thickening flow model: \(\eta = K\dot{\gamma}^{n-1}\).

Mastercurve: Superposition of multi-temperature data to extend frequency range.
Time-Temperature Superposition (TTS): Principle that temperature and timescale are equivalent for thermorheologically simple materials.
WLF Equation: Williams-Landel-Ferry equation for temperature-dependent shift factors.

Posterior Distribution: Probability distribution of parameters given data and priors.
Credible Interval: Bayesian analog of confidence interval; range containing parameter with specified probability.
MCMC (Markov Chain Monte Carlo): Sampling method for Bayesian inference.
NUTS (No-U-Turn Sampler): Efficient MCMC algorithm (Hamiltonian Monte Carlo variant).
ESS (Effective Sample Size): Number of independent samples in MCMC chain; measure of sampling efficiency.
R-hat (Gelman-Rubin Statistic): Convergence diagnostic for MCMC (should be < 1.01).

NLSQ: Nonlinear Least Squares optimization backend (GPU-accelerated).
JAX: Just-After-eXecution library for automatic differentiation and GPU acceleration.
JIT (Just-In-Time Compilation): Runtime compilation for performance optimization.

Symbol	Meaning	Units	Typical Range
\(G'\)	Storage modulus	Pa	\(10^2 - 10^8\)
\(G''\)	Loss modulus	Pa	\(10^2 - 10^8\)
\(\eta\)	Viscosity	Pa·s	\(10^{-3} - 10^{10}\)
\(\tau\)	Relaxation time	s	\(10^{-6} - 10^4\)
\(\alpha\)	Fractional order	dimensionless	0 - 1
\(\omega\)	Angular frequency	rad/s	\(10^{-2} - 10^3\)
\(\gamma\)	Strain	dimensionless	0 - 1
\(\dot{\gamma}\)	Shear rate	\(\text{s}^{-1}\)	\(10^{-3} - 10^3\)
\(\sigma\)	Stress	Pa	\(10^0 - 10^6\)
\(E'\)	Storage modulus (tensile)	Pa	\(10^6 - 10^{10}\)
\(E''\)	Loss modulus (tensile)	Pa	\(10^6 - 10^{10}\)
\(\nu\)	Poisson’s ratio	dimensionless	0.3 - 0.5