# Non-Linear Creep-Relaxation Constitutive Damage Model for Aging Concrete
## 3 Application of the Constitutive Model and Discussion of Results
### Relaxation Case Study  
A particle subjected to a constant uniaxial strain, as shown in Figure 6, is given by \(\varepsilon_{11}^{v}=\varepsilon_{0}H_{t}\) (and \(\varepsilon_{ij}^{v}=0\), otherwise) through the Heaviside function \(H_{t}\), and the immediate strain can be considered. It is easily proven that the positive orthogonal projection components are \(P_{111}^{\varepsilon+1}=1\) and \(P_{ijl}^{\varepsilon+1}=0\), otherwise. Thus, the positive strain tensor is equal to the strain tensor, i.e., \(\varepsilon_{ij}^{v+}=\varepsilon_{ij}^{v}\). Then, by substitution into the constitutive model equations, Equations (36)-(38), (46), (49), and (53), and the relaxation functions in Equation (56), the field equations that describe the relaxation case study are determined by assuming the following constant values for simplification: \(v(0)=1\), \(\varepsilon_{0}^{v}=1\), \(k_{do}=20\), \(C_{\lambda i}=C_{\mu i}=C_{\chi_{Ii}}=1\), \(k_{i}^{\lambda}=k_{i}^{\mu}=k_{i}^{\lambda}=k_{0}=\frac{1}{8},\frac{1}{4},\frac {1}{2}\), \(1\). The graphical behavior of the state variables is shown in Figures 7 and 8. 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.  
Figure 7a,b shows the relaxation of the state of stress, and the behavior is typical of a viscoelastic material. The parameter \(\sigma_{11}\) represents the stress due to the application of the short-term creep strain \(\varepsilon_{11}^{v}\), while the stresses \(\sigma_{22}\) and \(\sigma_{33}\) are developed due to the confinement of the specimen, as shown in Figure 6. On the other hand, Figure 7c,d shows the active damage variables in the material. As shown in Figure 7c, the maximum damage occurs in the direction where the strain is applied, which is consistent with the type of boundary conditions specified. Figure 7d shows that damage also occurs in the perpendicular directions of the strain. This is because the constitutive equations consider a coupling in damage in other directions. Although the micro-cracks open and grow driven by the tensile strains, in general, the micro-cracks are not perfectly planar in nature. This feature was considered first by [PERSON] [8]. Nonetheless, the developed model considers this feature as a natural consequence of the representation theorem utilized to define the  
thermodynamic potential, Equation (35). It is worth noting that a healing mechanism develops inside the material as time advances, as shown in Figure 7c,d. This behavior is due to the fact that the unloading process is being analyzed. Although this should be omitted in the calculations, the obtained results are relevant, since they agree with the definition of damage in viscoelastic materials, in which the damage is time-dependent [43]. Furthermore, from Figure 8e, the loading-unloading condition takes a positive value only when \(t=0\), and negative or zero when \(t>0\), and thus, damage only grows when the loading-unloading condition is positive definite. Figure 8a-d are consistent with the damage and stress behavior shown.  
Figure 6: State of strain considered; (**a**) Specim of study; (**b**) Constant strain at any given time \(t>0\).  
Figure 7: State of stress and damage variables for the relaxation case study; (**a**) Stress in the direction of the acting strain \(\sigma_{11}\); (**b**) Stresses acting perpendicular to the acting strain \(\sigma_{22}\) and \(\sigma_{33}\); (**c**) Damage variable in the direction of the acting strain \(D_{11}\); (**d**) Damage variable acting perpendicular to the acting strain \(D_{22}\) and \(D_{33}\).  
Figure 8: Damage strengthening and its associated variable, Energy release rate, and loading condition for the relaxation case study; (**a**) Damage strengthening; (**b**) Variable \(B\); (**c**) Energy release rate \(Y_{11}\); (**d**) Energy release rate \(Y_{22}\) and \(Y_{33}\); (**e**) Loading condition.