## 3 Spectra Determination for Various Models
### B3 Model (RILEM)  
According to the B3 model developed by [PERSON] and coworkers at Northwestern University [33] and recommended by RILEM, the compliance function (Equation (A34) of Appendix C) is expressed as  
\[J(t,t_{0})=q_{1}+C_{0}(t,t_{0})+C_{d}(t,t^{\prime},t_{0}) \tag{54}\]  
Comparing the compliance functions in Equation (38) to the previous one in Equation (54), one gets  
\[d_{0}=q_{1}\qquad\quad\text{and}\qquad\quad C^{*}(t,t_{0})=C_{d}(t,t^{\prime},t _{0}) \tag{55}\]  
Since the term \(C_{0}(t,t_{0})\) needs a specific different calculation respect to the previous formulations, it does not appear in Equation (55). Recalling the B3 formulation, reported in Appendix C, the previous expressions in Equations (55) and (54) can be written as  
\[C_{0}(t,t_{0})=q_{2}Q(t,t_{0})+q_{3}\text{ln}\left[1+(t-t_{0})^{0.1}\right]+q_{4}\text{ln}\frac{t}{t_{0}} \tag{56}\] \[C_{d}(t,t^{\prime},t_{0})=q_{5}[e^{\mathcal{B}(1-(1-h)S(t))}-e^{ \mathcal{B}(1-(1-h)S(t^{\prime}_{0}))}]^{1/2} \tag{57}\]  
The function \(C_{0}(t,t_{0})\) describes basic creep by a log-power law with aging incorporated through the solidification theory [22; 34] with an additional logarithmic term that reflects viscous flow. This term can be described by a dashpot with age-dependent viscosity and can be treated directly in the rate form, without the need to construct a Dirichlet series approximating its compliance function. Therefore, to describe the basic creep of B3 model, only the spectrum of the log-power \(\Phi(t-t_{0})=\text{ln}(1+(t-t_{0})^{0.1})\) is needed. Its third order approximation (\(k=3\) in Equation (31)) with the continuous retardation spectrum (Equation (32)) is given by  
\[A(\tau)=-\frac{(-3\tau)^{3}}{2}\ \frac{200(3\tau)^{0.2}+369(3\tau)^{0.1}+171}{(3 \tau)^{2\theta}[(3\tau)^{0.1}+1][1000(3\tau)^{0.2}+2000(3\tau)^{0.1}+1000]} \tag{58}\]  
with \(A_{0}=0.29209\) (from Equation (35)) for \(\tau_{1}=10^{-4}\)days. The quality of the approximation of the B3 compliance function by Dirichlet series with the coefficient obtained from Equation (58) is shown in Figure 3a for different \(t_{0}\) and with \(q_{3}=1.5*10^{-5}\). The approximated curves exhibit a maximum error, plotted in Figure 3b, below 1% with the adjustment factors \(\psi_{j}=1.2\) for \(j=1,2,3...9\) and \(\psi_{10}=0\). For the numerical calculation of the basic creep function of the B3 model, \(C_{0}(t,t_{0})\) in Equation (56), reference must be made to the solidification theory [22; 34] for which the visco-elastic strain rate is given by  
\[\dot{\epsilon}(t)=\left(q_{2}\sqrt{\frac{1}{t}}+q_{3}\right)\dot{\gamma}(t) \qquad\quad\text{with}\qquad\quad\gamma(t)=\int_{0}^{t}\Phi(t-\tau)\text{d} \sigma(\tau) \tag{59}\]  
Substituting in the second expression of Equation (59) the Dirichlet approximation of the compliance function, \(\Phi\), expressed through the coefficient \(A(\tau)\) given by Equation (58) we have  
\[\gamma(t)=A_{0}\sigma(t)+\sum  
Rewriting the first expression of Equation (59) in discrete form, \(\Delta\epsilon=(q_{2}\sqrt{1/t}+q_{3})\Delta\gamma\), and substituting the \(\Delta\gamma\) obtained from Equations (61) and (60), one gets the contribution of the basic creep, \(\Delta\sigma=E_{b}^{*}(\Delta\epsilon-e_{b}^{*})\), which must be added to the stress-strain relations in Section 2.5, in the following form  
\[E_{b}^{*}(t_{i+1})^{-1}=\left(q_{2}\sqrt{\frac{1}{t}}+q_{3}\right)\left\{A_{0} +\sum_{n=1}^{N}A(\tau_{n})\left[1-\left(1-e^{-\frac{\Delta t}{n}}\right)\frac {\Delta t}{\tau_{n}}\right]\right\} \tag{62}\]  
\[e_{b}^{*}(t_{i+1})=\left(q_{2}\sqrt{\frac{1}{t}}+q_{3}\right)\left\{\sum_{n=1 }^{N}\left(1-e^{-\frac{\Delta t}{\tau_{n}}}\right)\left[A(\tau_{n})\sigma(t_{ i})-\gamma_{n}(t_{i})\right]\right\}+q_{4}\frac{\Delta t\ \sigma(t_{i})}{t} \tag{63}\]  
where \(t=t_{i}+\Delta t_{i}/2\), \(\gamma_{n}(t_{i})\) are the internal variable obtained from Equation (61), and the last term represents an additional logarithmic term that reflects the purely viscous flow.