# Non-Linear Creep-Relaxation Constitutive Damage Model for Aging Concrete
## 3 Application of the Constitutive Model and Discussion of Results
### Creep Case Study  
In a similar manner, a particle subjected to a constant uniaxial stress is considered, as shown in Figure 9. However, in this case, to avoid solving the non-linear Volterra integral equations that define the constitutive model, a growing function of strain is considered, aiming to obtain a similar behavior as the one shown in Figure 9. Hence, if \(\sigma_{11}>0\), it is expected that the strains are \(\varepsilon_{11}^{v}>0\), \(\varepsilon_{22}^{v}<0\), \(\varepsilon_{33}^{v}=\varepsilon_{22}^{v}\), and \(\varepsilon_{ij}^{v}=0\), otherwise. This consideration results in \(P_{1111}^{\varepsilon+}=1\) and \(P_{ijkl}^{\varepsilon+}=0\), otherwise. Thus, \(\varepsilon_{11}^{v+}=\varepsilon_{11}^{v}\), and \(\varepsilon_{ij}^{v+}=0\), otherwise. If stresses and strains are substituted into the constitutive equations of the model, by some simplifications through the Laplace transform, the stress-strain relation in its Riemann-Stieltjes form is obtained as follows:  
\[ \begin{array}{ll}\sigma_{11}=\int_{\tau=0}^{t}\hat{E}(t,\tau)&\frac{\partial \varepsilon_{11}^{v}}{\partial\tau}d\tau+2\int_{\tau=0}^{t}\hat{\lambda}_{1}(t,\tau)D_{kk}\frac{\partial\varepsilon_{11}^{v}}{\partial\tau}d\tau+2\int_{ \tau=0}^{t}\hat{\lambda}_{2}(t,\tau)D_{mm}\frac{\partial\varepsilon_{11}^{v}} {\partial\tau}d\tau+2\int_{\tau=0}^{t}\hat{\lambda}_{3}(t,\tau)D_{11}\frac{ \partial\varepsilon_{11}^{v}}{\partial\tau}d\tau\\ &+2\int_{\tau=0}^{t}\hat{\lambda}_{4}(t,\tau)D_{11}\frac{\partial\varepsilon_{ 11}^{v}}{\partial\tau}d\tau\end{array}  \tag{57}\]  
where \(\hat{E}(t,\tau)\) is the uniaxial aging relaxation function defined in [4; 5] as \(\hat{E}(t,\tau)=v(\tau)E(t-\tau)\). This uniaxial aging relaxation function was obtained by considering the relation between the aging Lame relaxation functions \(\hat{\lambda}\) and \(\hat{\mu}\), with \(\hat{E}\) through the Laplace transform, in a similar manner as in [54]. Furthermore, it is important to state that, if the damage variable is equal to zero, Equation (57) reduces to the linear aging viscoelastic stress-strain relation under uniaxial stress, given in [43]. In addition, the relationship between the strain \(\varepsilon_{11}^{v}\) and \(\varepsilon_{22}^{v}\) is given through the Laplace transform,  
\[\widetilde{\varepsilon}_{22}^{v}=-s\widetilde{v}\widetilde{\varepsilon}_{11}^ {v} \tag{58}\]  
where \(\widetilde{\varepsilon}_{22}^{v}\) is the Laplace transform of the aging strain \(\varepsilon_{22}^{v}\) defined as \(\varepsilon_{22}^{v}=v(\tau)\frac{\partial\varepsilon_{22}^{v}(\tau)}{ \partial\tau}\).For manipulation of the Laplace transform of the convolution integral equation, \(\widetilde{\varepsilon}_{11}^{v}\) is the Laplace transform of the aging strain \(\varepsilon_{11}^{v}\), defined similarly as \(\varepsilon_{11}^{v}=v(\tau)\frac{\partial\varepsilon_{11}^{v}(\tau)}{ \partial\tau}\): \(\widetilde{v}\) is the Laplace transform of [PERSON] relaxation function; and \(s\) is the Laplace parameter. It is worth mentioning that the aging [PERSON] relaxation function \(\hat{v}\) was obtained through the relation  
Figure 9: State of stress considered; (**a**) Specimen of study; (**b**) Constant stress at any given time \(t>0\); (**c**) Short-term creep strain experienced in the material. between Lame relaxation functions \(\lambda\) and \(\hat{\mu}\), with \(\hat{v}\), similarly as for \(\hat{E}\). The remaining state variables are obtained by substitution as in the relaxation case study. For simplification, we assume the strain \(\varepsilon_{11}^{v}=\varepsilon_{0}H_{t}(t)\left[1-\frac{1}{3}e^{-3k_{0}t}\right]\), and also assume \(v(\tau)=v(\tau^{\prime})=1\), \(\frac{\varepsilon_{0}}{\delta_{0}}=\frac{1}{3}\), \(k_{do}=1\), \(C_{\lambda i}=C_{\mu i}=C_{\chi i}=1\), \(k_{i}^{\lambda}=k_{i}^{\mu}=k_{i}^{\lambda I}=k_{0}=\frac{1}{10}\). The graphical behavior of some of the state variables is shown in Figure 10. It is worth mentioning that the first equation in Equation (53) was not considered to show the time-dependent (rate-type) capabilities of the model.