# 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
To account for this behavior, the positive part of the strain tensor must be assumed as [25]  
\[\rho\psi=\frac{1}{2}\lambda[tr(\epsilon)]^{2}+\mu\left[tr(\epsilon^{2})\right] =\frac{1}{2}\lambda\epsilon_{ii}\epsilon_{jj}+\mu\epsilon_{ij}\epsilon_{ij} \tag{18}\]4  
where \(\mathbb{P}^{\epsilon+}\) is a fourth-order tensor known as the positive orthogonal projection of the strain [56], defined as  
\[\rho\psi=\frac{1}{2}\lambda[tr(\epsilon)]^{2}+\mu\left[tr(\epsilon^{2})\right] =\frac{1}{2}\lambda\epsilon_{ii}\epsilon_{jj}+\mu\epsilon_{ij}\epsilon_{ij} \tag{18}\]5  
where \(\mathbf{Q}^{\epsilon+}\) is the positive strain rotation tensor given by the multiplication of the tensor product of the eigenvectors of the strain tensor \(\mathbf{n}^{(i)\epsilon}\) acting on itself, and a Heaviside function defined in the principal strain space, \(H_{\epsilon}(\epsilon_{i}^{v})\), where \(\epsilon_{i}^{v}\) are the eigenvalues of the short-term creep strain tensor, i.e.,  
\[\rho\psi=\frac{1}{2}\lambda[tr(\epsilon)]^{2}+\mu\left[tr(\epsilon^{2})\right] =\frac{1}{2}\lambda\epsilon_{ii}\epsilon_{jj}+\mu\epsilon_{ij}\epsilon_{ij} \tag{18}\]6  
where \(H_{\epsilon}\) is defined as 1 if \(\epsilon_{i}^{v}\) is positive, and 0 when \(\epsilon_{i}^{v}\) is either negative or zero. Thus, if the spectral decomposition of \(\epsilon^{v}\) is considered, similar to Equation (19), it is easily proven that Equation (25) is  
\[\rho\psi=\frac{1}{2}\lambda[tr(\epsilon)]^{2}+\mu\left[tr(\epsilon^{2})\right] =\frac{1}{2}\lambda\epsilon_{ii}\epsilon_{jj}+\mu\epsilon_{ij}\epsilon_{ij} \tag{18}\]7  
Therefore, upon replacing \(\epsilon^{v}\) by \(\epsilon^{p+}\) in the terms with \(\mathbf{D}\) in Equation (22), the Helmholtz free energy of a viscoelastic aging isotropic material coupled with anisotropic damage with unilateral effect is found as Equation (29)  
\[ \begin{array}{ll}\rho\Psi^{vx}=\frac{1}{2}\int_{\tau^{\prime}=0}^{t}&\int_{ \tau^{\prime}=0}^{t}\widetilde{\lambda}(t,\tau,\tau^{\prime})d\epsilon_{ii}^{v }(\tau)d\epsilon_{jj}^{v}(\tau^{\prime})+\int_{\tau^{\prime}=0}^{t}\int_{t=0} ^{t}\widetilde{u}(t,\tau,\tau^{\prime})d\epsilon_{ij}^{v}(\tau)d\epsilon_{ij} ^{v}(\tau^{\prime})\\ &+\int_{\tau^{\prime}=0}^{t}\int_{\tau=0}^{t}\widetilde{\lambda}_{1}(t,\tau, \tau^{\prime})D_{il}d\epsilon_{jj}^{v}(\tau)d\epsilon_{ik}^{v+}(\tau^{\prime} )\\ &+\int_{\tau^{\prime}=0}^{t}\int_{\tau=0}^{t}\widetilde{\lambda}_{2}(t,\tau, \tau^{\prime})D_{ik}d\epsilon_{ij}^{v+}(\tau)d\epsilon_{ij}^{v+}(\tau^{\prime })+\int_{\tau^{\prime}=0}^{t}\int_{\tau=0}^{t}\widetilde{\lambda}_{3}(t,\tau, \tau^{\prime})D_{ij}d\epsilon_{ik}^{v+}(\tau)d\epsilon_{ij}^{v+}(\tau^{\prime })\\ &+\int_{\tau^{\prime}=0}^{t}\int_{\tau=0}^{t}\widetilde{\lambda}_{4}(t,\tau, \tau^{\prime})D_{jk}d\epsilon_{ij}^{v+}(\tau)d\epsilon_{ik}^{v+}(\tau^{\prime })\\ &\text{where }\epsilon_{ij}^{v+}\text{ and }\epsilon_{ii}^{v+}\text{ are given by the double contraction of the positive orthogonal projection}\\ &\text{components }P_{ijkl}^{v+}\text{ and the short-term creep strain components }\epsilon_{ij}^{v}\text{, defined in Equation (25); hence,}\\ \epsilon_{ij}^{v+}=P_{ijk}^{v+}\epsilon_{kl}^{v}\epsilon_{ii}^{v+}=P_{ikl}^{v+ }\epsilon_{kl}^{v}\end{array}  \tag{30}\]  
Figure 5: Microcracks in concrete specimens: (**a**) Specimen in tension; (**b**) Specimens in compression. Equation (29) provides the first term on the right-hand side of the thermodynamic potential expressed in Equation (7). Hence, it is important to determine the remaining terms in this equation, i.e., \(\rho\Psi^{f}\) and \(\rho\Psi^{D}\). It should be noted that the development followed to determine \(\rho\Psi^{vc}\), Equation (29), has been difficult to handle. However, the remaining terms in Equation (7) are determined in a straightforward manner.