## 2 Numerical Analysis of Creep Behavior
### Non-Aging Kelvin Chain  
For non-aging material the expressions previously derived are simplified, since the equivalence is imposed with a non-aging Kelvin element (with spring stiffnesses constant). Moreover, the approximation of a non-aging compliance function, \(J(t-t_{0})\), through a Dirichlet (or Prony) series, is obtained using constant moduli \(E_{j}\) as  
\[J(t,t_{0})=\frac{1}{E_{0}}+\sum_{j=1}^{N}\frac{1}{E_{j}}\left(1-e^{-\frac{t-t _{0}}{\gamma}}\right) \tag{24}\]  
The values of the coefficients \(E_{j}\) can be determined by the best fitting a given non-aging compliance function \(J(t-t_{0})\) either using a minimization algorithm (Least Squares or Levenberg-Marquardt algorithm) or passing through the continuous retardation spectrum [24], which is a more general approach that is summarized in the following. As shown in Equation (24), a Kelvin chain model with \(N\) units gives a non-aging creep compliance function \(C(t-t_{0})\) which can be approximated by a Dirichlet series as [22]\[C(t-t_{0})=\sum_{n=1}^{N}A_{n}\left(1-e^{-(t-t_{0})/\tau_{n}}\right) \tag{25}\]  
where \(A_{n}=1/E_{n}\), \(t=\) age of concrete, and \(t_{0}=\) age of concrete at the moment of loading. To deal with a general procedure and to avoid some weak points of other approaches [22; 25], the continuous Kelvin chain model with infinite units (continuous spectrum), in which the retardation times are infinitely close, is used [24]. Passing through the continuous spectrum, the discrete spectrum can be obtained by discretizing the continuous one. The creep compliance function \(C(t-t_{0})\) may be approximated in a continuous form as  
\[C(\xi)\approx\int_{0}^{\infty}A(\tau)\left(1-e^{-\xi/\tau}\right)\mathrm{d}( \ln\tau) \tag{26}\]  
where \(\xi=t-t_{0}\) and \(A(\tau)\) is the continuous retardation spectrum, which has the same meaning in the logarithmic time scale as \(A_{n}\) in the real time scale. Following the method developed by [26] and setting \(\tau=1/\chi\) we have  
\[C(\xi)=\int_{0}^{\infty}A(\chi^{-1})\left(1-e^{-\xi\chi}\right)\chi^{-1} \mathrm{d}\chi \tag{27}\]  
and  
\[C(\xi)=\int_{0}^{\infty}A(\chi^{-1})\chi^{-1}\mathrm{d}\chi-\int_{0}^{\infty} A(\chi^{-1})e^{-\xi\chi}\chi^{-1}\mathrm{d}\chi \tag{28}\]  
The previous Equation (28) can be rewritten as  
\[C(\xi)=f(0)-f(\xi)\quad\text{ with }\quad f(\xi)=\int_{0}^{\infty}A(\chi^{-1} )e^{-\xi\chi}\chi^{-1}\mathrm{d}\chi \tag{29}\]  
where \(f(\xi)\) if the Laplace transform of the function \(A(\chi^{-1})\chi^{-1}\) and \(f(0)\) is a constant. Using the inversion formula of [PERSON] [27] the function \(A(\chi^{-1})\chi^{-1}\) can be obtained asymptotically as  
\[\lim_{k\to\infty}\left[\frac{(-1)^{k}}{k!}\left(\frac{k}{\chi}\right)^{k+1}f^ {(k)}\left(\frac{k}{\chi}\right)\right]=A(\chi^{-1})\chi^{-1} \tag{30}\]  
where \(f^{(k)}\) is the \(k\)th derivative of the function \(f\). Remembering that \(f(\xi)=f(0)-C(\xi)\) and \(\chi=1/\tau\), we obtain the continuous retardation spectrum  
\[A(\tau)=-\lim_{k\to\infty}\frac{(-k\tau)^{k}}{(k-1)!}C^{(k)}\left(k\tau\right) \tag{31}\]  
The approximate spectrum of order \(k\) is obtained by assuming a finite value of \(k\) in Equation (31). For practical purpose an approximate spectrum of third order (\(k=3\)) may be used with enough accuracy [24; 28], i.e.,  
\[A(\tau)=-\frac{(-3\tau)^{3}}{2}\left.\frac{\partial^{3}C(\xi)}{\partial\xi^{3 }}\right|_{\xi=3\tau} \tag{32}\]  
Decomposing the integral by a finite sum over finite time intervals given by Equation (23), the Equation (26) can be rewritten in the time of interest, i.e., \(0<t<\tau_{N}\sqrt{10}\), as  
\[C(\xi)=A_{0}+\sum_{n=1}^{N}\int_{\tau_{n}/\sqrt{10}}^{\tau_{n}\sqrt{10}}A(\tau )\left(1-e^{-\xi/\tau}\right)\mathrm{d}(\ln\tau) \tag{33}\]  
The integrals can be evaluated using the n-point Gaussian quadrature rule.