Classical Viscoelastic Models¶

This section documents the classical linear viscoelastic models that form the foundation of rheological analysis.

Quick Reference¶

Model	Parameters	Use Case
Maxwell (Classical)	2 (G, \(\eta\))	Stress relaxation, simple viscoelastic liquids
Zener (Standard Linear Solid)	3 (\(G_1, G_2, \eta\))	Solids with single relaxation time, standard linear solid
SpringPot (Scott-Blair Element)	2 (\(c_{\alpha, \alpha}\))	Power-law behavior, fractional element, broad spectra

Overview¶

Classical viscoelastic models represent the historical foundation of rheology, built from combinations of ideal mechanical elements:

Spring (Hookean): Instantaneous elastic response, \(\sigma = G\gamma\)
Dashpot (Newtonian): Viscous flow, \(\sigma = \eta\dot{\gamma}\)
SpringPot (Fractional): Power-law intermediate behavior

These models provide closed-form analytical solutions for standard rheological tests and serve as building blocks for more complex constitutive equations.

Model Hierarchy¶

Classical Models
│
├── Maxwell (Series)
│   └── Spring ── Dashpot
│   └── Liquid-like (terminal flow)
│   └── Single relaxation time τ = η/G
│
├── Zener (Standard Linear Solid)
│   └── Spring ── [Spring ∥ Dashpot]
│   └── Solid-like (equilibrium modulus)
│   └── Kelvin-Voigt element + spring
│
└── SpringPot (Fractional Element)
    └── Intermediate between spring and dashpot
    └── Power-law kernel: G(t) ∼ t^(-α)
    └── Foundation for fractional models

When to Use Which Model¶

Material Behavior	Maxwell	Zener	SpringPot
Single exponential relaxation	✓ Best choice	✓ With plateau	Overkill
Terminal flow (liquid)	✓ Best choice	✗	✗
Equilibrium modulus (solid)	✗	✓ Best choice	✗
Power-law relaxation	✗	✗	✓ Best choice
Broad relaxation spectrum	Poor fit	Poor fit	✓ Best choice
Simple teaching example	✓ Best choice	✓ Good	More complex

Decision Guide:

Start with Maxwell for viscoelastic liquids (polymer melts, solutions)
Use Zener for viscoelastic solids (rubbers, gels with crosslinks)
Use SpringPot when log-log plots show power-law slopes (polymeric glasses, biological tissues, complex fluids)

Key Parameters¶

Parameter	Symbol	Units	Physical Meaning
Shear modulus	G	Pa	Elastic stiffness (energy storage)
Viscosity	\(\eta\)	Pa·s	Resistance to flow (energy dissipation)
Relaxation time	\(\tau\)	s	\(\tau = \eta/G\), characteristic timescale
SpringPot constant	\(c_{\alpha}\)	Pa·s\(^{\alpha}\)	Quasi-property (fractional element)
Fractional order	\(\alpha\)	—	0 = elastic, 1 = viscous, 0.5 = critical gel

Quick Start¶

Maxwell model:

from rheojax.models import Maxwell
import numpy as np

model = Maxwell()
model.fit(t, G_t, test_mode='relaxation')

# Get fitted relaxation time
tau = model.parameters.get_value('eta') / model.parameters.get_value('G0')

Zener model:

from rheojax.models import Zener
import numpy as np

model = Zener()
model.fit(omega, G_star, test_mode='oscillation')

# Equilibrium modulus
G_eq = model.parameters.get_value('Ge')

SpringPot element:

from rheojax.models import SpringPot

model = SpringPot()
model.fit(omega, G_star, test_mode='oscillation')

# Fractional order indicates spectrum breadth
alpha = model.parameters.get_value('alpha')

Model Documentation¶

References¶

Maxwell, J.C. (1867). “On the dynamical theory of gases.” Philosophical Transactions, 157, 49-88. https://www.jstor.org/stable/108968
Zener, C.M. (1948). Elasticity and Anelasticity of Metals. University of Chicago Press.
Ferry, J.D. (1980). Viscoelastic Properties of Polymers, 3rd ed. John Wiley & Sons. ISBN: 978-0471048947.
Tschoegl, N.W. (1989). The Phenomenological Theory of Linear Viscoelastic Behavior. Springer-Verlag. https://doi.org/10.1007/978-3-642-73602-5
Koeller, R.C. (1984). “Applications of fractional calculus to the theory of viscoelasticity.” J. Appl. Mech., 51, 299-307. https://doi.org/10.1115/1.3167616