# 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
(**a**) Symmetric integration from \(\theta\) to \(t\) in \(\tau-\tau^{\prime}\) domain; (**b**) Integration from \(\tau=\tau^{\prime}\) to \(t\) with respect \(\tau\) in \(\tau-\tau^{\prime}\) domain; (**c**) Integration from \(\theta\) to \(\tau=\tau^{\prime}\) with respect \(\tau\) in \(\tau-\tau^{\prime}\) domain; (**d**) Integration from \(\tau^{\prime}=\theta\) to \(t\) with respect \(\tau^{\prime}\) in \(\theta-\tau^{\prime}\) domain; (**e**) Integration from \(0\) to \(\theta=\tau^{\prime}\) with respect \(\theta\) in \(\theta-\tau^{\prime}\) domain; (**f**) Integration from \(\tau=\tau^{\prime}\) to \(t\) with respect \(\tau\) in \(\tau-\tau^{\prime}\) domain; (**g**) Integration from \(0\) to \(t\) with respect \(\tau^{\prime}\) in \(\tau-\tau^{\prime}\) domain; (**h**) Integration from \(0\) to \(\tau=\tau^{\prime}\) with respect \(\tau\) in \(\tau-\tau^{\prime}\) domain. As a final step, we couple Equation (15) with the Lame relaxation aging functions. Furthermore, we use a Heaviside temporal function, \(H_{t}(*)\), which is defined as \(1\) when the term inside the parenthesis, i.e., \((*)\), is positive, \(\frac{1}{2}\) when \((*)\) is zero, and zero when \((*)\) is negative. It is important to mention that, from this point forward, the tilde \(\widetilde{*}\) will represent relaxation aging functions with a temporal Heaviside function defined inside. Hence, the Lame relaxation aging functions are rewritten as  
\[ \begin{split}\widetilde{\lambda}(t,\tau,\tau^{\prime})& =H_{t}(\tau-\tau^{\prime})\tilde{\lambda}(2t-\tau,\tau^{\prime} )+H_{t}(\tau^{\prime}-\tau)\tilde{\lambda}(2t-\tau^{\prime},\tau)\\ \widetilde{\mu}(t,\tau,\tau^{\prime})&=H_{t}(\tau- \tau^{\prime})\hat{\mu}(2t-\tau,\tau^{\prime})+H_{t}(\tau^{\prime}-\tau)\hat{ \mu}(2t-\tau^{\prime},\tau)\end{split}  \tag{16}\]  
and thus, the symmetry in the integration region is attained in Equation (17). Figure 4 represents the regions of integration for both \(\widetilde{\lambda}\) and \(\widetilde{\mu}\). \[\rho\Psi^{\
u\epsilon}=\frac{1}{2}\int_{\tau^{\prime}=0}^{t}\int_{\tau=0}^{t} \widetilde{\lambda}\left(t,\tau,\tau^{\prime}\right)de_{ii}^{\pi}(\tau)de_{jj }^{\pi}(\tau^{\prime})+\int_{\tau^{\prime}=0}^{t}\int_{\tau=0}^{t}\widetilde{ \mu}\left(t,\tau,\tau^{\prime}\right)de_{ij}^{\pi}(\tau)d\epsilon_{ij}^{\
u}( \tau^{\prime}) \tag{17}\]  
Equation (17) is the thermodynamic potential from which a constitutive relation between stress and strain can be obtained for an isotropic linear viscoelastic aging material. Equation (17) can be further simplified to express the stress-strain relation for a uniaxial material, as later discussed in Section 3.2 of this paper. In addition, and similar to Equations (11) and (14), the thermodynamic potential in Equation (17) provides the energy stored in the material for a symmetric temporal region of integration, from which the integration can be attained in a straightforward manner. Equation (17) is analogous to the Helmholtz free energy for a linear elastic isotropic material, Equation (18), where \(tr(*)\) is the trace of the tensor inside the parenthesis, and \(\lambda\) and \(\mu\) are Lame constants. \[\rho\psi=\frac{1}{2}\lambda[tr(\epsilon)]^{2}+\mu\left[tr(\epsilon^{2})\right] =\frac{1}{2}\lambda\epsilon_{ii}\epsilon_{jj}+\mu\epsilon_{ij}\epsilon_{ij} \tag{18}\]  
Equation (18) is represented in two forms. The first form is represented by a quadratic scalar-valued tensor function in its general form based on the invariants of the strain tensor field [55], i.e., the traces \(tr(\epsilon)\), \(tr(\epsilon^{2})\), and \(tr(\epsilon^{3})\), where only quadratic terms are considered to obtain a linear relation between stress-strain. The second form is the component form, or tensorial notation, from which it is easier to notice the similarity between Equations (17) and (18). In addition, from both forms of Equation (18), it is possible to obtain the stress-strain relation for a linear elastic isotropic material, by the differentiation of Equation (18) with respect to the strain tensor components \(\epsilon_{ij}\).