# Non-Linear Creep-Relaxation Constitutive Damage Model for Aging Concrete
## 2 Theoretical Formulation
### Instantaneous Helmholtz Free Energy  
#### 2.2.1 Components of Helmholtz Free Energy  
Based on the aforementioned discussions, the instantaneous Helmholtz free energy at any point can be written as follows:  
\[\rho\Psi=\rho\Psi(\mathbf{x},t)=\mathbf{\sigma}:\epsilon^{c}+\beta B+\mathbf{D}:\mathbf{Y}+\
u V \tag{4}\]  
Figure 2: Thermodynamic processes that take place in the plain concrete during aging and loading. where \(\rho\) is the mass density, \(\Psi\) is the instantaneous Helmholtz free energy per unit mass, \(\rho\Psi\) is the instantaneous Helmholtz free energy per unit volume, \(\sigma:\epsilon^{c}\) is strain energy associated with the creep strain component, \(\beta B\) is the energy associated with the damage strengthening (hardening and softening), \(\mathbf{D}:\mathbf{Y}\) is the energy associated with the degree of cracking (damage), and \(vV\) is the energy associated with the solidification of the concrete. Hence, the state variables are as described below:  
1. \(\mathbf{\sigma}\) is the state of stress at the point;
2. \(\epsilon^{c}\) is the state of creep strain;
3. \(\beta\) is the damage strengthening;
4. \(B\) is the thermodynamic conjugate of the damage strengthening;
5. \(\mathbf{D}\) is the second-order damage tensor variable;
6. \(\mathbf{Y}\) is the energy release rate associated to the damage;
7. \(v\) is the degree of solidification;
8. \(V\) is the thermodynamic conjugate of the degree of solidification. It is important to indicate that all the state variables mentioned above, and their corresponding thermodynamic conjugates, not only depend on the position vector \(\mathbf{x}\), but also on the time variable \(t\). For instance, the state of stress is represented by \(\sigma=\sigma(\mathbf{x},t)\). The only exceptions are \(v\) and \(V\), where position independence will be assumed, to simplify the formulation and to be consistent with the theory proposed in [45], i.e., \(v=v(t)\) and \(V=V(t)\). Also, from this point forward, the authors will also be referring to the instantaneous Helmholtz free energy as thermodynamic potential; and its position and time dependency will be implied but not stated in further equations, i.e., \(\rho\Psi(\mathbf{x},t)=\rho\Psi\), for simplification. As stated above, the creep strain is split into two components, i.e., Equation (2). Hence, by substituting into Equation (4),  
\[\rho\Psi=\mathbf{\sigma}:\epsilon^{v}+\mathbf{\sigma}:\epsilon^{f}+\beta B+\mathbf{D}:\bm {Y}+vV \tag{5}\]  
Since the authors are considering Helmholtz free energy as thermodynamic potential, the independent state variables in the thermodynamic potential are the state of strain \(\epsilon^{v}\) and \(\epsilon^{f}\), the damage variable \(\mathbf{D}\), the damage strengthening variable \(\beta\), and the solidification variable \(v\). The remaining variables \(\sigma\), \(\mathbf{Y}\), \(B\), and \(V\) are the dependent state variables, or thermodynamic conjugates to be found from the thermodynamic potential, i.e.,  
\[\rho\Psi=\rho\Psi(\epsilon^{v},\epsilon^{f},\beta,\mathbf{D},v) \tag{6}\]  
The damage is considered to develop as the state of strain reaches a certain magnitude, which indicates the dependence of damage and strain. Also, although the strength to sustain damage (i.e., damage strengthening) depends on the amount of damage itself in the material, for the time being, it is assumed that the strength is uncoupled from the damage in the thermodynamic potential. However, these two entities (i.e., strength and damage) will be coupled later in the formulation when a dissipation potential is established. Furthermore, the viscous flow strain, although associated with stress, can be assumed to be independent of damage, because it is a process that appears at any level of strain. Finally, it is stipulated that the volume of solidified matter of concrete hardens the material, thereby making the material less susceptible to deformation. Therefore, the change in volume must be coupled to the strain, with both viscous flow strain and short-term creep strain.