Theoretical Background

This page summarises the physics and mathematics behind the algorithms implemented in ViscoAnalysis.jl.

1. Linear viscoelasticity and complex modulus

A linear viscoelastic material subject to a sinusoidal strain $\varepsilon(t) = \varepsilon_0 \, e^{i\omega t}$ responds with a sinusoidal stress $\sigma(t) = \sigma_0 \, e^{i(\omega t + \varphi)}$. The ratio defines the complex modulus:

\[E^*(\omega) = \frac{\sigma_0}{\varepsilon_0} e^{i\varphi} = E_1(\omega) + i\, E_2(\omega)\]

where:

• $E_1 = |E^*|\cos\varphi$ is the storage modulus (elastic part),
• $E_2 = |E^*|\sin\varphi$ is the loss modulus (viscous part),
• $\varphi$ is the phase angle (loss angle),
• $|E^*| = \sqrt{E_1^2 + E_2^2}$ is the complex modulus norm.

DMA experiments directly measure $|E^*|$ and $\varphi$ at discrete frequencies and temperatures.

2. Time-temperature superposition

Bituminous materials are thermorheologically simple: a change in temperature is equivalent to a shift in the frequency axis. This is expressed by the shift factor $a_T$:

\[E^*(\omega, T) = E^*(\omega \cdot a_T(T), T_{\text{ref}})\]

Cole-Cole and Black diagrams

These are standard representation planes for comparing different materials or temperature conditions independently of the frequency axis:

Both are parametrised by frequency; a perfect thermorheologically simple material produces a unique curve independent of temperature.

The continuity of these curves enables verification of the validity of the time–temperature equivalence principle

3. Master curve construction

The validation of the Time–Temperature Superposition principle allows for the construction of a master curve.

Kramers-Kronig verification

For a linear viscoelastic material, the real and imaginary parts of $E^*$ are not independent — they are related by the Kramers-Kronig relations. A practical consequence (Booij & Thoone, 1982) is:

\[\frac{\varphi(\omega)}{90°} \approx \frac{d\log|E^*|}{d\log\omega}\]

ViscoAnalysis computes both sides numerically and displays them on a scatter plot. Points lying close to the identity line $y = x$ indicate consistent, reliable viscoelastic measurements.

Determination of shift factor $a_T$

The shift factor $a_T$ are computed incrementally from one temperature to the next using the phase angle and the modulus gradient:

\[\log(a_T(T_i, T_{\text{ref}})) = \frac{\pi}{2} \left( \sum_{k=i}^{k=\text{ref}} \frac{\log\!\left(|E^*(T_k,\omega)|\right) - \log\!\left(|E^*(T_{k+1},\omega)|\right)} {{\varphi_{\text{avr}}}^{T_{k,k+1}}} \right)\]

Williams-Landel-Ferry (WLF) equation

The shift factor follows the empirical WLF law:

\[\log_{10}(a_T) = \frac{-C_1 \,(T - T_{\text{ref}})}{C_2 + T - T_{\text{ref}}}\]

The constants $C_1$ and $C_2$ are identified by nonlinear least-squares fitting (using LsqFit.jl) from the experimentally-derived shift factors.

Change of reference temperature

WLF coefficients at one reference temperature can be exactly converted to another:

\[C_2^{\text{new}} = C_2^{\text{old}} + (T_{\text{new}} - T_{\text{old}}) \qquad C_1^{\text{new}} = C_1^{\text{old}} \,\frac{C_2^{\text{old}}}{C_2^{\text{new}}}\]

Master curve

Applying the WLF shift to all isotherms sweeps collapses them onto a single master curve at $T_{\text{ref}}$:

\[\omega_{\text{red}} = \omega \cdot a_T(T)\]

where $\omega_{\text{red}}$ is the reduced angular frequency — the measured angular frequency $\omega$ shifted to the reference temperature $T_{\text{ref}}$.

3. Rheological models

All three models are written in the complex (Laplace) domain with $p = i\omega$.

3.1 2S2P1D model

The 2 Springs, 2 Parabolic elements, 1 Dashpot model (Olard & Di Benedetto, 2003) uses 7 parameters:

\[E^*(p) = E_\infty + \frac{E_0 - E_\infty} {1 + \delta\,(p\tau)^{-k} + (p\tau)^{-h} + \dfrac{1}{p\,\beta\,\tau}}\]

3.2 1S2P1D model

The 1 Spring variant removes the high-frequency spring, so $E_\infty$ is absorbed into the denominato numerics (effectively $E_\infty = 0$ for the standalone spring term), using 6 parameters:

\[E^*(p) = \frac{E_0} {1 + \delta\,(p\tau)^{-k} + (p\tau)^{-h} + \dfrac{1}{p\,\beta\,\tau}}\]

3.3 Huet-Sayegh model

The Huet-Sayegh model (Huet, 1963; Sayegh, 1965) drops the dashpot ($\beta \to \infty$), giving 6 parameters:

\[E^*(p) = E_\infty + \frac{E_0 - E_\infty} {1 + \delta\,(p\tau)^{-k} + (p\tau)^{-h}}\]

4. Model identification strategy

All three models are identified by global optimisation of a weighted least-squares objective computed on the complex master curve:

\[J = \sum_k \Delta\log\omega_k \left|1 - \frac{E^*(\omega_k)}{E^*_{\text{exp}}(\omega_k)}\right|^2\]

The $\Delta\log\omega$ weight ensures that every decade of frequency contributes equally regardless of the number of measurement points it contains.

References

• Olard, F., & Di Benedetto, H. (2003). General "2S2P1D" model and relation between the linear viscoelastic behaviours of bituminous binders and mixes. Road Materials and Pavement Design, 4(2), 185–224.
• Huet, C. (1963). Étude par une méthode d'impédance du comportement viscoélastique des matériaux hydrocarbonés. PhD thesis, Université de Paris.
• Sayegh, G. (1965). Contribution à l'étude des propriétés viscoélastiques des bitumes purs et des bétons bitumineux. PhD thesis, Université de Paris.
• Chailleux, E., Ramond, G., Such, C., & de La Roche, C. (2006). A mathematical-based master-curve construction method applied to complex modulus of bituminous materials. Road Materials and Pavement Design, 7(sup1), 75–92.
• Williams, M. L., Landel, R. F., & Ferry, J. D. (1955). The temperature dependence of relaxation mechanisms in amorphous polymers and other glass-forming liquids. Journal of the American Chemical Society, 77(14), 3701–3707.
• Booij, H. C., & Thoone, G. P. J. M. (1982). Generalization of Kramers-Kronig transforms and some approximations of relations between viscoelastic quantities. Rheologica Acta, 21(1), 15–24.