# Non-Linear Creep-Relaxation Constitutive Damage Model for Aging Concrete
## 2 Theoretical Formulation
### Dissipative Processes
#### 2.4.3 Evolution Equations  
To determine the evolution equations, the relationship between thermodynamic fluxes and forces using a dissipation potential [58; 59] is considered, i.e., \(\mathbf{J}=\dot{\hat{v}}\frac{\partial\Omega}{\partial\mathbf{X}}\), where \(\mathbf{J}\) are the thermodynamic fluxes, \(\mathbf{X}\) are the thermodynamic forces, \(\Omega\) is the dissipation potential, and \(\dot{\hat{z}}\) is a multiplier that needs to be determined from the consistency condition. Hence, from the first three terms in Equation (41), the thermodynamic forces are \(Y_{ij}\), \(B\), and \(s_{ij}\), while the thermodynamic fluxes are \(\dot{D}_{ij}\), \(\dot{\beta}\), and \(\dot{e}^{f}_{ij}\). If two dissipation inequalities are considered, one related to the damage process and the other related to the viscous flow, i.e.,  
\[\Phi^{D}+\Phi^{f}\geq 0 \tag{45}\]  
where \(\Phi^{D}=Y_{ij}\dot{D}_{ij}+B\dot{\beta}\), and \(\Phi^{f}=s_{ij}\dot{e}^{f}_{ij^{\prime}}\) then each term in Equation (45) must satisfy the inequality, i.e., \(\Phi^{D}\geq 0\) and \(\Phi^{f}\geq 0\). This is valid since the viscous flow occurs at any level of stress (or strain), without damage. In addition, the strength to sustain damage (i.e., hardening) must be coupled with the damage experienced by the material, as previously stated. To account for the long-term creep strain of the uniaxial model and the viscous flow, \(\sigma=\hat{\eta}(t)\dot{e}^{f}\) is implemented in [43], where \(\hat{\eta}(t)\) is the aging viscosity function given by \(\hat{\eta}(t)=\eta_{0}v(t)\), with \(\eta_{0}\) known as the effective viscosity of the solidified matter. Following the same line of thought [57],  
\[s_{ij}=\hat{\eta}(t)\dot{e}^{f}_{ij} \tag{46}\]  
Equation (46) provides the evolution equation that relates the thermodynamic force \(s_{ij}\) and the thermodynamic flux \(\dot{e}^{f}_{ij^{\prime}}\) through the multiplier \(\hat{\eta}\). Furthermore, if Equation (46) is substituted into \(\Phi^{f}=s_{ij}\dot{e}^{f}_{ij^{\prime}}\), the inequality \(\Phi^{f}\geq 0\) is satisfied. Furthermore, the restriction of the second law of thermodynamics is imposed to the viscosity as \(\dot{\eta}(t)\geq 0\), which implies that \(v(t)\geq 0\), and by differentiating both sides with respect to \(t\), one finds \(\dot{v}(t)\geq 0\). This is consistent with the positive nature of both \(v(t)\) and its rate of change, as described earlier [43]. For the damage and damage strengthening evolution,  
\[\dot{D}_{ij}=\dot{\xi}^{D}\frac{\partial\Omega^{D}}{\partial\Omega ^{D}_{ij}} \tag{47}\] \[\dot{\beta}=\dot{\xi}^{D}\frac{\partial\Omega^{D}}{\partial\Omega ^{D}}\]  
where \(\Omega^{D}\) is the dissipation potential and \(\dot{\xi}^{D}\) is a multiplier defined through the consistency condition. \(\Omega^{D}\) and \(\dot{\xi}^{D}\) must be considered in both equations, Equation (47), to couple damage and the strength to sustain damage. In general, the dissipation potential \(\Omega^{D}\) has been defined as a loading surface \(f^{D}\), which in turn is defined as a function of \(Y_{ij}\) and \(B\)[25, 29]. Hence, the loading surface,  
\[\Omega^{D}=f^{D}=Y_{EQ}-(B+B_{o})\leq 0 \tag{48}\]  
where \(Y_{EQ}\) is the equivalent energy release rate, defined as \(Y_{EQ}=\left[\frac{1}{2}Y_{ij}Y_{ij}\right]^{\frac{1}{2}}\), \(B\) is the damage strengthening, and \(B_{o}\) is a constant that defines the threshold value for cracks to start growing inside the material. This former assumption is valid since the onset of cracks occurs when a certain value of the stress is attained. For instance, in specimens subjected to compression, the onset occurs when the stress reaches about 45 to 50 percent of the compressive strength, i.e., \(0.45f^{\prime}c-0.50f^{\prime}c\). It is important to note that Equation (48) was defined through an analogy to plasticity theory [25, 29, 60].