Appendix E — Viscoelastic constitutive laws
This appendix summarizes the main viscoelastic constitutive models used in micromechanical homogenization with Echoes.
Linear viscoelasticity framework
A linear viscoelastic material is characterized by a relaxation function \(\uuuu{C}(t)\) or a creep function \(\uuuu{J}(t)\) such that the stress and strain histories are related by a Stieltjes convolution
\[ \sig(t) = \int_0^t \uuuu{C}(t-\tau) : \dot{\eps}(\tau) \, \ud\tau \qquad\text{or equivalently}\qquad \eps(t) = \int_0^t \uuuu{J}(t-\tau) : \dot{\sig}(\tau) \, \ud\tau \tag{E.1}\]
For non-ageing materials, \(\uuuu{C}\) and \(\uuuu{J}\) depend only on the elapsed time \(t-\tau\).
Correspondence principle
The Laplace-Carson transform of a function \(f(t)\) is defined as
\[ f^*(p) = p \int_0^\infty f(t)\, e^{-pt} \, \ud t \tag{E.2}\]
Under this transform, the convolution (E.1) becomes an algebraic product
\[ \sig^*(p) = \uuuu{C}^*(p) : \eps^*(p) \]
This correspondence principle allows the direct reuse of elastic homogenization schemes in the Laplace-Carson domain: the effective relaxation function \(\uuuu{C}^{*,hom}(p)\) is obtained by replacing elastic moduli by their Laplace-Carson transforms in the homogenization formula.
For ageing materials, the correspondence principle does not apply directly since the relaxation function \(\uuuu{C}(t,t')\) depends on both the current time \(t\) and the loading time \(t'\). In this case, a direct time-stepping approach must be used.
Rheological models
Maxwell model
A Maxwell element consists of a spring (modulus \(\mu\)) in series with a dashpot (viscosity \(\eta\)). In the Laplace-Carson domain, the complex shear modulus is
\[ \mu^*(p) = \frac{\mu \, p \, \tau}{1 + p \, \tau} \qquad\text{with}\quad \tau = \frac{\eta}{\mu} \tag{E.3}\]
where \(\tau\) is the relaxation time.
Kelvin-Voigt model
A Kelvin-Voigt element consists of a spring in parallel with a dashpot:
\[ \mu^*(p) = \mu + \eta \, p = \mu\left(1 + p\,\tau\right) \tag{E.4}\]
Generalized Maxwell (Prony series)
A generalized Maxwell model is a parallel combination of \(N\) Maxwell elements plus an optional equilibrium spring \(\mu_\infty\):
\[ \mu(t) = \mu_\infty + \sum_{i=1}^{N} \mu_i \, e^{-t/\tau_i} \tag{E.5}\]
In the Laplace-Carson domain:
\[ \mu^*(p) = \mu_\infty + \sum_{i=1}^{N} \frac{\mu_i \, p \, \tau_i}{1 + p \, \tau_i} \tag{E.6}\]
This representation is the most common for numerical computations. The relaxation modulus at \(t=0\) is \(\mu_0 = \mu_\infty + \sum_i \mu_i\) and the long-term modulus is \(\mu_\infty\) (zero for a fluid).
Generalized Kelvin-Voigt
Dually, a generalized Kelvin-Voigt model is a series combination of \(N\) Kelvin-Voigt elements plus an instantaneous compliance \(J_0\):
\[ J(t) = J_0 + \sum_{i=1}^{N} J_i \left(1 - e^{-t/\tau_i^r}\right) \tag{E.7}\]
where \(\tau_i^r\) are the retardation times. This form is convenient for describing creep.
2S2P1D model
The 2 Springs, 2 Parabolic elements, 1 Dashpot (2S2P1D) model is widely used for bituminous materials. Its complex modulus is defined as
\[ E^*(p) = E_0 + \frac{E_\infty - E_0}{1 + \delta\,(p\,\tau)^{-k} + (p\,\tau)^{-h} + (p\,\beta\,\tau)^{-1}} \tag{E.8}\]
where:
- \(E_0\) and \(E_\infty\) are the static and glassy moduli
- \(\delta\), \(k\), \(h\) (\(0 < k < h < 1\)) are shape parameters of the parabolic elements
- \(\tau\) is a characteristic time (related to the shift factor \(a_T\) for thermorheologically simple materials)
- \(\beta\) is the Newtonian viscosity parameter
In the frequency domain (\(p = i\omega\)), this model produces realistic Cole-Cole and Black diagrams for bituminous binders and mastics (see Chapter 16).
Ageing viscoelastic materials
For ageing materials such as hydrating cement paste, the relaxation function depends on both the current time \(t\) and the loading time \(t'\):
\[ \sig(t) = \int_0^t \uuuu{C}(t,\tau) : \dot{\eps}(\tau) \, \ud\tau \]
The effective behavior cannot be obtained by the correspondence principle. Instead, a time-stepping approach discretizes the time interval and solves the homogenization problem at each time step, updating the internal variables (e.g., Maxwell chain stiffnesses that depend on the hydration degree).
In Echoes, ageing viscoelastic materials are handled through the visco_prop and visco_law objects that encode the time-dependent evolution of the moduli, and homogenize_visco performs the incremental homogenization (see Chapter 17).
Laplace inversion algorithms
Recovering the time-domain relaxation or creep function from the Laplace-Carson solution requires a numerical inversion. Common algorithms include:
- Gaver-Stehfest algorithm: uses real-valued evaluations of the Laplace-Carson transform at \(M\) points. The time-domain function is approximated by
\[ f(t) \approx \frac{\ln 2}{t} \sum_{k=1}^{M} V_k \, \hat{f}\!\left(\frac{k\ln 2}{t}\right) \tag{E.9}\]
where \(V_k\) are combinatorial weights and \(\hat{f}\) is the Laplace transform. This method is fast and easy to implement but can be inaccurate for oscillatory functions.
De Hoog algorithm: uses complex-valued evaluations along a contour in the complex plane, providing better accuracy for oscillatory or multi-scale responses at the cost of complex arithmetic.
Collocation on Prony series: the Laplace-Carson transform is evaluated at \(N\) chosen values of \(p_i = 1/\tau_i\) and the resulting system is inverted to identify the Prony series coefficients \((\mu_i, \tau_i)\) in (E.5). This yields an explicit time-domain representation that can be efficiently used for incremental computations.
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