## 2 Methodology
### Interconversion Relationship between Time-Domain and Frequency-Domain Functions
#### 2.2.2 Interconversion from Creep Compliance to Relaxation Modulus  
As noted above, Equation (6) reveals the relationship between creep compliance and relaxation modulus in Laplace transform domain. Accordingly, relaxation modulus can be converted by creep compliance based on this formula in the following.  
Substituting the Prony series representations of relaxation modulus and creep compliance into Equation (6), the integral can be expressed by:  
\[\int_{0}^{t}E_{e}+\sum\limits_{i}^{m}E_{i}e^{-[(t-\tau)/\rho_{i}]}\Bigg{(}D_{g }\delta(t)+\sum\limits_{j}^{n}\frac{D_{j}}{\tau_{j}}e^{-(t/\tau)}\Bigg{)}d\tau=1 \tag{12}\]  
where \(\delta(\ )\) is the Dirac delta function.  
According to the literature [15], further simplifying the integral can derive the following equations:  
\[\left\{\begin{array}{l}\sum\limits_{i=1}^{m}\left[\sum\limits_{j=1}^{n} \frac{\rho_{i}D_{j}}{\rho_{i}-\tau_{j}}\left(e^{-(t/\tau_{j})}-e^{-(t/\tau_{j}) }\right)+D_{g}e^{-(t/\tau_{j})}\right]E_{i}=1-\Bigg{(}D_{g}+\sum\limits_{j=1}^ {n}D_{j}\Big{(}1-e^{-(t_{k}/\tau_{j})}\Big{)}\Bigg{)}E_{e},\rho_{i}\
eq\tau_{j }\\ \sum\limits_{i=1}^{m}\left[\sum\limits_{j=1}^{n}\frac{\rho_{i}D_{j}}{\tau_{j} }e^{-(t/\rho_{i})}+D_{g}e^{-(t/\rho_{i})}\right]D_{j}=1-\Bigg{(}D_{g}+\sum \limits_{j=1}^{n}D_{j}\Big{(}1-e^{-(t_{k}/\tau_{j})}\Big{)}\Bigg{)}E_{e}\ \ \ \ \ \ \ \ \ \,\rho_{i}=\tau_{j}\end{array}\right. \tag{13}\]  
where,  
\[E_{e}\equiv\underset{t\rightarrow\infty}{\lim}E(t)=\underset{s\to 0}{\lim} \widetilde{E}(s)=\frac{1}{\underset{s\to 0}{\lim}\widetilde{D}(s)}=\frac{1}{ \underset{t\rightarrow\infty}{\lim}D(t)}=\frac{1}{D_{g}+\sum\limits_{j=1}^{n} D_{j}} \tag{14}\]  
Adjusting them into linear system of equations, Equation (15) can be given as:  
\[[A][E]=[B]A_{ki}=\left\{\begin{array}{l}D_{g}e^{-(t_{k}/\rho_{i})}+\sum \limits_{j=1}^{n}\frac{\rho_{i}D_{j}}{\rho_{i}-\tau_{j}}\Big{(}e^{-(t_{k}/\rho _{i})}-e^{-(t_{k}/\tau_{j})}\Big{)},\rho_{i}\
eq\tau_{j}\\ D_{g}e^{-(t_{k}/\rho_{i})}+\sum\limits_{j=1}^{n}\frac{t_{k}D_{j}}{\tau_{j}} \Big{(}e^{-(t_{k}/\rho_{i})}\Big{)}\ \ \ \ \ \ \ \ \,\rho_{i}=\tau_{j}\\ B_{kj}=1-\Bigg{(}D_{g}+\sum\limits_{j=1}^{n}D_{j}\Big{(}1-e^{-(t_{k}/\tau_{ j})}\Big{)}\Bigg{)}/\Bigg{(}D_{g}+\sum\limits_{j=1}^{n}D_{j}\Bigg{)}\end{array}\right. \tag{15}\]  
In Equation (15), matrix \([E]\) denotes the collection of \(E_{i}\) for relaxation modulus, which can be derived from matrix operation between matrix \([A]\) and matrix \([B]\). Time spectrum \(t_{k}(k=1,2,...,p)\) denotes the upper limit of integral in Equation (12) and can be identified through the collocation method or the least squares method. In the collocation method, the number of \(t_{k}\) must be equal to \(E_{i}\), whereas the number of \(t_{k}\) could be more than \(E_{i}\) in the least squares method. In that case, the linear system may be overdetermined equations, and the exact solution cannot be reached. Thus, the optimal solution for it may be effective by calculating in MATLAB to minimize \(\|AE-B\|^{2}\).  
Consequently, the procedure of the converted relaxation modulus from creep compliance can be given as:  
1. Pre-smooth the experimental data of creep compliance and fit its Prony series coefficients \(D_{g}\) and \(D_{j}\);
2. Calculate the Prony series of relaxation modulus based on \([E]=[A]^{-1}[B]\) by converting \(D_{g}\) and \(D_{j}\) to \(E_{e}\) and \(E_{i}\);Substitute \(E_{e}\) and \(E_{i}\) into the relaxation modulus representation to plot the master curve of it over a wide time region.