# Non-Linear Creep-Relaxation Constitutive Damage Model for Aging Concrete
## 4 Conclusions  
A thermodynamic constitutive model has been formulated based on the Bazant theory of solidification and the theory of continuum damage mechanics. The model is strain-based and can predict the anisotropy of the material due to the microcracks growth and the time-dependent damage (known as tertiary creep). To develop the formulation, several steps and considerations were adopted. These steps and considerations, along with their advantages and shortcomings, are listed next. These also serve to point out the main conclusion of this research. 1. A second-order tensor damage variable was considered to account for the anisotropy induced. The principal directions of the damage variable are assumed to coincide with those of the viscoelastic strain tensor. No initial damage was considered. Therefore, the formulation herein is not suitable for damage presented before strain occurs, which is the case of cracking due to drying shrinkage, poor curing, thermal, and others. 2. The solidification is assumed to be independent of the position vector \(\mathbf{x}\). This is consistent with the theory of solidification of basic creep proposed by [PERSON]. The solidification process is dependent on the position. The conjugate of the solidification process was found to be zero when no strain had been applied. This is because the solidification process is chemical in nature. 3. The creep strain was separated into two main components, viscoelastic strain and viscous flow strain, consistently with the [PERSON] rheological model. The immediate strain is not considered. However, depending on the definition of the input strain, the immediate strain can be considered. 4. The thermodynamics functions of aging viscoelasticity were extended to a three-dimensional case coupled with anisotropic damage, by assuming the material as viscoelastic damageable material (which is analogous to the theory for brittle elastic materials in continuum damage mechanics) and the aid of a representation theorem. 5. The relationship between the strength to sustain damage and its conjugate variable was considered linear, which is consistent with the assumption of the damage. However, experimental results are needed to define whether the proportionality function \(\hat{k}_{D}\) can be assumed as a constant \(k_{do}\). 6. The constitutive equations were determined from the thermodynamic potential formulas. The dissipation inequality was found from the Clausius-Duhem inequality along with the thermodynamic potential. The dissipation inequality found shows three dissipating processes: damage state, damage strengthening, and viscous flow. The damage and damage strengthening were considered as coupled, while the viscous flow is assumed as independent since it occurs at any level of strain. The determination of the evolution equations was based on the loading surface as a dissipation potential assumption. The potential for the damage dissipative process was assumed to be similar to the \"potential theory\" in the theory of plasticity, which has been proven suitable for concrete. A loading-unloading condition was considered, which leads to the possibility of analyzing tertiary creep. * The thermodynamic restrictions on the relaxation function were obtained based on the positive energy assumption and the Clausius-Duhem inequality. The former governs the semi-positive definite form of the relaxation functions, while the latter leads to the negative (decreasing behavior) rate of change of the relaxation functions. * The second Cauchy equation of functional equations is suitable for the thermodynamic restrictions obtained. In addition, the second Cauchy equation is equivalent to the Dirichlet series utilized in the development of the relaxation functions. * To determine the applicability of the formulation, two practical case studies were carried out. From the graphical representation, we can conclude that the formulation developed can account for time-dependent (rate-type) damage as long as the loading-unloading condition remains positive.