# Laboratory Evaluation of Tensile Creep Behavior of Concrete at Early Ages
## 4 Test Results
### Dirichlet Series Creep Function  
The creep compliance data shown in Figure 6 were fitted with the Dirichlet series functions given in Equation (9) by the collocation method. The method of fitting the data with the Dirichlet series may be found elsewhere [32]. In this study, six collocation points were selected from \(10^{-3}\) to \(10^{2}\) h at decimal spacing. This means the Kelvin model used in this study consists of six Kelvin units in addition to the initial spring unit. The initial compliance (\(D_{0}\)) was selected; as such, it corresponds to the instantaneous response of the specimen to loading within 1 s (\(2.8\times 10^{-4}\) h). The last two collocation values were  
Figure 5: Strain components of the test concrete at different ages. Figure 6: Creep compliance of the test concrete at different ages. extrapolated from the test data referring to the creep rate reported in the previous studies for the long-term creep in compression [39; 40]. The material retardation time (\(\tau_{i}\)) for each Kelvin unit was determined at six points from \(5\times 10^{-4}\) to \(50\) h at decimal intervals. With these pre-selected variables, the internal compliance (\(D_{i}\)) of the Kelvin model was determined using Equation (15), which is a matrix formulation of Equation (9) rearranged for \(D_{i}\). Since the model has six Kelvin units, the number of collocation points (\(i,j\)) varies from \(1\) to \(6\). \[\left\{\,D_{i}\,\right\}\,=\,\left[\,1-e^{-\,i\,/\,\tau_{i}}\,\right]^{-1}\left\{ \,D\left(t_{j}\right)-D_{0}\,\right\} \tag{15}\]  
where \(\left\{D_{i}\right\}\) = the compliance vector to be determined for the respective Kelvin units, and \(D(t_{j})\) = compliance at the \(j\)th collocation point (\(t_{j}\)). Figure 7 shows the fitted creep compliance curves compared to the respective test data. The internal parameters, \(D_{0}\) and \(D_{i}\), determined from the fitting process, are presented in Table 3. The aging effect, represented by the dependency of the response to the initial loading time, may be explained by the changes in the internal compliance of the concrete at different ages. The conversion procedure can be simplified with the Laplace transform (\(L\)). The Laplace transform with respect to time is defined by  
\[L\left[f\left(x,\ t\right)\right]=\overline{f}(x,s)=\int\limits_{0}^{\infty}f(x, t)\cdot e^{-st}dt \tag{16}\]  
where \(s\) is the transform parameter. The transform of the relationship, Equation (10), is  
\[\overline{E}(s)\,s^{2}\,\overline{D}(s)=1 \tag{17}\]  
By introducing the Carson transform, \(\overline{f}=s\cdot\overline{f}\), Equation (17) becomes  
\[s\cdot\overline{E}=\frac{1}{s\cdot\overline{D}}\quad\text{or}\ \widetilde{E}=\frac{1}{ \widetilde{D}} \tag{18}\]  
The Carson transform of the uniaxial relaxation modulus (Equation (8)) and creep compliance (Equation (9)) are  
\[\widetilde{E}(s)=E_{\infty}+\sum\limits_{i=1}^{m}\Biggl{(}\,\frac{s\cdot E_{i} }{s+\frac{1}{\rho_{i}}}\,\Biggr{)} \tag{19}\]  
\[\widetilde{D}(s)=D_{0}+\sum\limits_{i=1}^{m}\Biggl{(}\,\frac{1}{\eta_{i}}\cdot \frac{1}{(s+\frac{1}{\eta_{i}})}\,\Biggr{)} \tag{20}\]  
The substitution of Equations (19) and (20) into Equation (18) gives  
\[E_{\infty}+\sum\limits_{i=1}^{m}\biggl{(}\,\frac{s\cdot\rho_{i}}{1+s\cdot\rho _{i}}\cdot E_{i}\,\Biggr{)}=\frac{1}{D_{0}+\sum\limits_{i=1}^{m}\Bigl{(}\, \frac{D_{i}}{1+s\cdot\tau_{i}}\,\Bigr{)}} \tag{21}\]  
By the final value theorem, \(s\cdot\overline{f}\,\Bigr{|}_{s\to 0}=f(\infty)\), the infinite modulus can be found from Equation (21) as  
Figure 6:0  
Therefore, when the creep compliance is defined, the material parameters \(D_{0}\), \(D_{i}\), and \(E_{\infty}\) are known, and the only unknown is the internal spring moduli, \(E_{i}\). This set of unknown variables may be effectively defined by the collocation method within the Carson domain. The collocation points, \(s_{i}\), in the Carson domain may be selected at \(1/t_{i}\), where \(t_{i}\) represents the properly selected collocation points in the time domain.