## 2 Definition and Properties of Distributed Order Calculus
### Establishment of the Distributed Order Rheological Model and Operator Analysis  
(1) Establishment of the model  
The three-parameter solid model is one of the essential models of viscoelastic materials. It is made of one spring and one Kelvin model in series, as shown in Figure 2a. It can be used to show instantaneous elastic deformation, deceleration creep deformation, and asymptotic creep deformation, as well as the relaxation properties of materials. With its advantages, it has been used to analyse the viscoelastic characteristics of many materials [28]. Therefore, the distributed order rheological model was established based on the three-parameter solid model. The damper components in the integer-order model were changed into the distributed order damper, as shown in Figure 1b. Thus, the distributed order three-parameter solid model was obtained, as shown in Figure 2b. According to the stress-strain relationships of the components and the series-parallel relationships among the components, the three-parameter solid model equation of state could be expressed by Equation (21). \[\left\{ \begin{array}{l}\sigma(t)=E_{1}\epsilon_{1}(t),\,\sigma_{1}(t)=E_{2} \epsilon_{2}(t),\,\sigma_{2}(t)=\int_{a}^{b}\omega(\tau)_{0}D_{t}^{\gamma}[ \epsilon_{2}(t)]d\gamma\\ \epsilon(t)=\epsilon_{1}(t)+\epsilon_{2}(t),\,\sigma(t)=\sigma_{1}(t)+ \sigma_{2}(t)\end{array} \right. \tag{21}\]  
By Equation (21), the constitutive equation of the distributed order three-parameter solid model could be obtained as follows:  
\[ \begin{array}{l}\int_{a}^{b}\omega(\gamma)_{0}D_{t}^{\gamma}[\sigma(t)]d \gamma+(E_{1}+E_{2})\sigma(t)\\ =E_{1}\int_{a}^{b}\omega(\gamma)_{0}D_{t}^{\gamma}[\epsilon(t)]d\gamma+E_{1}E _{2}\epsilon(t)\end{array}  \tag{22}\]  
Let \(\
u=E_{1}+E_{2}\) and \(\kappa=E_{1}E_{2}\), Equation (22) could be changed into Equation (23). \[ \begin{array}{l}\int_{a}^{b}\omega(\gamma)_{0}D_{t}^{\gamma}[\sigma(t)]d \gamma+\
u\sigma(t)=\\ E_{1}\int_{a}^{b}\omega(\gamma)_{0}D_{t}^{\gamma}[\epsilon(t)]d\gamma+\kappa \epsilon(t)\end{array}  \tag{23}\]  
Figure 2: The three-parameter solid model. (**a**) The integer-order three-parameter solid model, (**b**) The distributed order three-parameter solid model. where \(E_{1}>0\), \(\kappa>0\), \(\
u>0\), and \(E_{1}\
u\)-\(\kappa\geq 0\). The parameter values depend on the properties of the materials. Equation (23) could be changed into Equation (24) by the Laplace transform. \[\
u\sigma(s)+\int_{a}^{b}\omega(\gamma)s^{\gamma}d\gamma\sigma(s)=E_{1}\int_{a }^{b}\omega(\gamma)s^{\gamma}d\gamma\varepsilon(s)+\kappa\varepsilon(s) \tag{24}\]  
On this basis, Equation (25) could be obtained. \[\sigma(s)=\frac{E_{1}\int_{a}^{b}\omega(\gamma)s^{\gamma}d\gamma+\kappa}{\
u+ \int_{a}^{b}\omega(\gamma)s^{\gamma}d\gamma}\varepsilon(s) \tag{25}\]  
Thus, the creep compliance of the distributed order three-parameter solid model could be obtained. \[ \begin{array}{l}J(t)=L^{-1}\{I(s)\}=L^{-1}\Big{\{}\frac{\varepsilon(s)}{s \sigma(s)}\Big{\}}\\ =L^{-1}\Bigg{\{}\frac{1}{E_{1}s}+\frac{E_{1}\
u-\kappa}{sE_{1}^{2}}\frac{1}{ \int_{a}^{b}\omega(\gamma)s^{\gamma}d\gamma+\frac{\kappa}{s_{1}}}\Bigg{\}} \end{array}  \tag{26}\]  
Similarly, the relaxation modulus of the distributed order three-parameter solid model could be calculated. \[ \begin{array}{l}G(t)=L^{-1}\{G(s)\}=L^{-1}\Big{\{}\frac{\sigma(s)}{s\sigma( s)}\Big{\}}\\ =L^{-1}\Bigg{\{}\frac{E_{1}}{s}+\frac{\kappa-E_{1}\lambda}{\
u+\int_{a}^{b} \omega(\gamma)s^{\gamma}d\gamma}\frac{1}{s}\Bigg{\}}\end{array}  \tag{27}\]  
The inverse Laplace transform of \(\frac{1}{\
u+\int_{a}^{b}\omega(\gamma)s^{\gamma}d\gamma}\) could be obtained by calculating Equations (26) and (27). 2. Properties of model operators  
Let  
\[\widetilde{f}(s)=\frac{1}{\
u+\int_{a}^{b}\omega(\gamma)s^{\gamma}d\gamma} \qquad(\widetilde{f}\to f:s\to t) \tag{28}\]  
Equation (29) could be obtained by the inverse Laplace transform of Equation (28). \[