Fractional viscoelastic models

Remarks

Example: for the springpot, the strain step $\varepsilon = \varepsilon_0 H(t)$ has $D^\alpha H(t) = t^{-\alpha}/\Gamma(1 - \alpha)$, so the relaxation modulus is the power law $$G(t) = \frac{E}{\Gamma(1 - \alpha)} \left(\frac{t}{\tau}\right)^{-\alpha}.$$ The limits $\alpha \to 0$ and $\alpha \to 1$ recover a spring and a dashpot of viscosity $E\tau$. Example: the fractional Maxwell model is a spring of modulus $E$ in series with a springpot of modulus $E$ and time $\tau$. It satisfies $D^\alpha \varepsilon = D^\alpha \sigma / E + \sigma / (E \tau^\alpha)$. By the Laplace transform its relaxation modulus is $G(t) = E \, E_\alpha(-(t/\tau)^\alpha)$, where $E_\alpha$ is the Mittag-Leffler function. Non-example: the law $\sigma = E \varepsilon + c \varepsilon^2$ with $c \neq 0$ contains no derivatives of non-integer order and is not linear, so it is not a fractional viscoelastic model. For histories with $\varepsilon(0) = 0$ the Riemann–Liouville and Caputo derivatives of order $\alpha \in (0, 1)$ agree, so either can be used.