# Non-Linear Creep-Relaxation Constitutive Damage Model for Aging Concrete
## 2 Theoretical Formulation
### Instantaneous Helmholtz Free Energy
#### 2.2.2 Helmholtz Free Energy for a Short-Term Linear Viscoelastic Aging Material  
Based on [45], the first term of the Helmholtz free energy stated in Equation (7), \(\rho\Psi^{vx}\), is obtained. The proof given in [45] addresses the thermodynamic potential for a unidimensional viscoelastic material. Nevertheless, the herein formulation leads to a further generalization of the equations, by extending the concepts to a three-dimensional case, assuming isotropy in the material, and the non-linear behavior by considering the development of the damage due to higher levels of stress. Accordingly, the material is assumed to be viscoelastic damageable material. In other words, the material is a viscoelastic recoverable material until the point where damage due to microcracking begins to develop, similar to that in [24]. Hence, assuming one solidification process to be sufficient for concrete [45], the thermodynamic potential for short-term viscoelastic aging material is obtained, which is expressed in its convolution Stieltjes integral form, i.e.,  
\[\rho\Psi^{vx}=\rho\int_{\theta=0}^{t}\psi(t,\theta)d\psi(\theta)=\frac{1}{2} \int_{\theta=0}^{t}\int_{\tau^{\prime}=\theta}^{t}\int_{\tau=\theta}^{t}G_{ ijkl}\big{(}2t-\tau-\tau^{\prime}\big{)}de_{ij}^{\eta}(\tau)d\epsilon_{kl}^{ \
u}(\tau^{\prime})d\psi(\theta) \tag{8}\]  
where \(G_{ijkl}\) is the age-independent relaxation tensor; \(\theta\) is the time variable from which the time is zero, meaning that no-aging changes in the mechanical properties have occurred, up to the current time \(t\); and \(\tau^{\prime}\) is an auxiliary variable that represents the loading time, similarly as \(\tau\). If the most general form of an isotropic fourth-order tensor is considered,  
\[G_{ijkl}=\frac{1}{3}(G_{2}-G_{1})\delta_{ij}\delta_{kl}+\frac{1}{2}G_{1}\left( \delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}\right) \tag{9}\]  
where \(G_{1}\) and \(G_{2}\) are isotropic age-independent relaxation functions related to the deviatoric and dilatational strain, respectively, and \(\delta_{ij}\) is the Kronecker delta. Furthermore, if the strain is split into dilatational and deviatoric, i.e.,  
\[\epsilon_{ij}^{\
u}=\frac{1}{3}\epsilon_{ik}^{\
u}\delta_{ij}+\epsilon_{ij}^{\
u} \tag{10}\]  
where the first term in Equation (10) represents the dilatational part of strain, and the second is the deviatoric part. Then, by the substitution Equation (9) into Equation (8), performing the changes in limit and order of integration represented in Figure 3, from Figure 3a-e, to discard the integration with respect to \(v(\theta)\), and by substituting Equation (10) and considering the symmetry of the strain tensor, one finds  
\[\rho\Psi^{vx}=\int_{\tau^{\prime}=0}^{t}\int_{\tau=\tau^{\prime}}^{t}\frac{1} {3}\hat{G}_{2}\big{(}2t-\tau,\tau^{\prime}\big{)}d\epsilon_{ij}^{\
u}(\tau)d \epsilon_{ij}^{\
u}(\tau^{\prime})+\int_{\tau^{\prime}=\tau^{\prime}}^{t} \hat{G}_{1}\big{(}2t-\tau,\tau^{\prime}\big{)}de_{ij}^{\
u}(\tau)d\epsilon_{ ij}^{\
u}(\tau^{\prime}) \tag{11}\]  
where \(\hat{G}_{1}\) and \(\hat{G}_{2}\) are isotropic age-dependent (aging) relaxation functions related to the deviatoric and dilatational strain, given by Equation (12). It is important to note that, from this point forward, the circumflex accent \(\hat{*}\) will represent aging functions that depend on the variable \(v(t)\). Hence, \(\hat{G}_{1}\) and \(\hat{G}_{2}\) are written as  
\[\hat{G}_{1}(2t-\tau,\tau^{\prime})=v(\tau^{\prime})G_{1}(2t-\tau-\tau^{\prime}) \tag{12}\] \[\hat{G}_{2}(2t-\tau,\tau^{\prime})=v(\tau^{\prime})G_{2}(2t-\tau- \tau^{\prime})\]  
Equation (11) gives the energy stored in the isotropic linear viscoelastic aging material due to the application of a three-dimensional state of strain with dilatational and deviatoric components for a non-symmetric temporal region of integration.