The concept of a general relation between the complex modulus of a linear viscoelastic system and its spectrum is well-known in the literature. We follow [
24] and relate the relaxation function
G to the continuous relaxation spectrum
h via
$$\begin{aligned} G(t)=\int _{0}^{\infty }h(v)\mathrm {e}^{-vt}\mathrm {d}v. \end{aligned}$$
(3.17)
In practical applications, where the function
G must be calculated numerically, it is more convenient to introduce the so-called cumulative relaxation spectrum:
$$\begin{aligned} h_c(v)=\int _{0}^{v}h(z)\mathrm {d}z, \end{aligned}$$
(3.18)
which is monotonically an increasing function such that
\(h_c(0)=0\) and
\(h_c(\infty )<\infty \). Then, integrating by parts (
3.17), we obtain an alternative representation of the relaxation function:
$$\begin{aligned} G(t)=\int _{0}^{\infty }h_c\left( \frac{v}{t}\right) \mathrm {e}^{-v}\mathrm {d}v. \end{aligned}$$
(3.19)
However, to take advantage of this formula, we have to first answer the question how to determine
\(h_c\) corresponding to the fractional Zener model and its analytical solution (
3.13), which specifies the one-dimensional stress–strain relation.
In order to describe the relaxation properties of viscoelastic material, the dynamical tests are performed in which the harmonic oscillations are applied to a material specimen, and the resulting stress is measured until a steady-state value is reached. It is common practice in engineering to use complex variables to present the harmonic response. To this end, we assume the complex harmonic function
$$\begin{aligned} f(t)=\varDelta f\cdot \mathrm{e}^{\mathrm {i}\omega t}H(t), \end{aligned}$$
(3.20)
where
\(\omega \) is the angular frequency,
\(\varDelta f\) is the amplitude and
H is the Heaviside function. Applying the Caputo integral operator (
2.16) and calculating the fractional derivative of the function
f in the stationary case, i.e. for large values of
t (for details see [
1,
4,
22]), we have
$$\begin{aligned} \frac{\mathrm{{d}}^{\alpha }f(t)}{\mathrm{{d}}t^{\alpha }}=(\mathrm {i}\omega )^{\alpha }\, f(t). \end{aligned}$$
(3.21)
Without loss of generality, we assume
\(\varepsilon _0=0\) and consider the complex and stress as
$$\begin{aligned} \overline{\varepsilon }(t)=\varDelta \varepsilon \cdot \mathrm {e}^{\mathrm {i}\omega t}H(t),\ \ \ \overline{\sigma }(t)=\varDelta \sigma \cdot \mathrm {e}^{\mathrm {i}\omega t}H(t). \end{aligned}$$
(3.22)
Then, by inserting a complex-valued deformation process
\(\overline{\varepsilon }\) into Eq. (
3.13), we have
$$\begin{aligned} \overline{\sigma }(t)=G^{*}(\mathrm {i}\omega )\,\overline{\varepsilon }(t), \end{aligned}$$
(3.23)
with the quantity
$$\begin{aligned} G^{*}(\mathrm {i}\omega )=\mu _0+\mathrm {i}\omega \int _{0}^{\infty }G(\tau )\mathrm {e}^{-\mathrm {i}\omega \tau }\mathrm {d}\tau , \end{aligned}$$
(3.24)
which defines the so-called complex dynamic modulus for a stationary stress response. Taking into account the representation (
3.17) of the relaxation function
G and carrying out the integration with respect to the variable
\(\tau \) yields
$$\begin{aligned} G^{*}(\mathrm {i}\omega )=\mu _0+\mathrm {i}\omega \int _{0}^{\infty }\frac{h(v)}{v+\mathrm {i}\omega }\mathrm {d}v. \end{aligned}$$
(3.25)
The formula (
3.25) shows that the complex modulus
\(G^{*}\) and relaxation spectrum
h are related to each other. If a relaxation spectrum is given, the complex dynamic modulus is determined. On the other hand, we notice that the function
\((G^{*}(i\omega )-\mu _0)/i\omega \) is the Stieltjes transformation of the relaxation spectrum
h, i.e.
$$\begin{aligned} S_h(\mathrm {i}\omega )=\frac{G^{*}(\mathrm {i}\omega )-\mu _0}{\mathrm {i}\omega }=\int _{0}^{\infty }\frac{h(v)}{v+\mathrm {i}\omega }\mathrm {d}v. \end{aligned}$$
(3.26)
Assuming that
h is an absolutely integrable function on [0,
R], for every finite
\(R>0\), one may calculate the inverse of the Stieltjes transformation using the formula
$$\begin{aligned} h(v)&=\frac{1}{2\pi \mathrm {i}}\lim _{\xi \rightarrow 0^{+}}\left( S_h(-v-\mathrm {i}\xi )-S_h(-v+\mathrm {i}\xi )\right) \nonumber \\&=\frac{1}{2\pi \mathrm {i}}\lim _{\xi \rightarrow 0^{+}}\left[ \frac{G^{*}(-v-\mathrm {i}\xi )}{-v-i\xi }-\frac{G^{*}(-v+\mathrm {i}\xi )}{-v+\mathrm {i}\xi } +\mu _0\left( \frac{1}{-v+\mathrm {i}\xi }-\frac{1}{-v-\mathrm {i}\xi }\right) \right] , \end{aligned}$$
(3.27)
where
v is a positive value for which the function
h is defined. It is convenient to express the complex arguments
\(-v\pm \mathrm {i}\xi \) in the form of the complex exponential functions
$$\begin{aligned} -v\pm \mathrm {i}\xi =\sqrt{v^2+\xi ^2}\,\mathrm {e}^{\pm (\pi -\arctan (\xi /v))}. \end{aligned}$$
(3.28)
Then, calculating the limit
\(\xi \rightarrow 0^{+}\), we obtain the general formula
$$\begin{aligned} h(v)&=\frac{1}{2\pi \mathrm {i}}\left( \frac{G^{*}(v\mathrm{e}^{-\mathrm {i}\pi })}{v\mathrm{e}^{-\mathrm {i}\pi }}-\frac{G^{*}(v\mathrm{e}^{\mathrm {i}\pi })}{v\mathrm{e}^{\mathrm {i}\pi }}\right) =\frac{1}{2v\pi \mathrm {i}}\left( G^{*}(v\mathrm{e}^{\mathrm {i}\pi })-G^{*}(v\mathrm{e}^{-\mathrm {i}\pi })\right) , \end{aligned}$$
(3.29)
which can be used to compute the continuous relaxation spectrum if the complex dynamic modulus is known.
Moreover, bearing in mind (
3.21) and inserting the harmonic deformation
\(\overline{\varepsilon }\) in combination with the harmonic stress
\(\overline{\sigma }\) into (
3.1), we transform the fractional Zener model into the frequency domain. Thus, we obtain again the expression (
3.23), where the complex dynamic modulus has a form
$$\begin{aligned} G^{*}(\mathrm {i}\omega )=\frac{B_0+B_1(\mathrm {i}\omega )^{\alpha }}{A_0+(\mathrm {i}\omega )^{\alpha }}. \end{aligned}$$
(3.30)
The formula (
3.29) in combination with (
3.30) can be applied to calculate the relaxation spectrum
h of the function
G given by the Mittag-Leffler function (
3.14). We obtain
$$\begin{aligned} h(v)=\frac{1}{2\pi v \mathrm {i}}\left( \frac{B_0+B_1\,v^{\alpha }\mathrm {e}^{\mathrm {i}\alpha \pi }}{A_0+v^{\alpha }\mathrm {e}^{\mathrm {i}\alpha \pi }}-\frac{B_0+B_1\,v^{\alpha }\mathrm {e}^{-\mathrm {i}\alpha \pi }}{A_0+v^{\alpha }\mathrm {e}^{-\mathrm {i}\alpha \pi }}\right) , \end{aligned}$$
(3.31)
which finally leads to
$$\begin{aligned} h(v)=\frac{A_0\mu _1\sin (\alpha \pi )v^{\alpha }}{\pi v\left( A_0^2+2A_0\cos (\alpha \pi )v^{\alpha }+v^{2\alpha }\right) }. \end{aligned}$$
(3.32)
For further calculations, it is useful to consider the cumulative relaxation spectrum
\(h_c\) given by (
3.18). Then, analytical integration of function (
3.32) yields the following formula
$$\begin{aligned} h_c(v)=\frac{\mu _1}{\alpha \pi }\left[ \arctan \left( \frac{v^{\alpha }}{A_0\sin (\alpha \pi )}+\cot (\alpha \pi )\right) +\pi \left( \alpha -\frac{1}{2}\right) \right] . \end{aligned}$$
(3.33)
This result provides an interesting formula for the efficient numerical calculation of the Mittag-Leffler function
\(E_{\alpha ,1}\) for large negative arguments. Indeed, in combination with (
3.19) and (
3.33), we can formulate the following corollary.