# Non-Linear Creep-Relaxation Constitutive Damage Model for Aging Concrete
## 2 Theoretical Formulation
### Components of Strain and Thermodynamic Processes  
#### 2.1.1 Components of Strain  
To formulate the Helmholtz free energy that accounts for the storage of energy in the plain concrete in the proposed formulation, let us first enlist the processes that take place in the concrete and the components of strain to be considered.  
Firstly, to account for the aging process, a state variable \(v(t)\) known as solidification [43; 45] is considered in the formulation. Under high levels of stress (i.e., stress higher than \(0.45\prime^{\prime}c-0.50\prime^{\prime}c\) under compression), concrete develops microcracks that cause irreversible strain. Furthermore, under sustained load, the concrete undergoes time-dependent creep strain. Thus, under the former considerations and assuming the small strain regime,where the state of the strain at a point is given by \(\epsilon=\epsilon(\mathbf{x},t)\), the state of the strain may be split into  
\[\epsilon=\epsilon^{ep}+\epsilon^{c} \tag{1}\]  
where \(\epsilon^{ep}\) accounts for the elastoplastic strain and \(\epsilon^{c}\) represents the creep strain, both representing the strain for any level (low or high) of stress. It should be mentioned that \(\epsilon^{ep}\) and \(\epsilon^{c}\) can remain coupled if necessary (as it will be assumed for the case studies in Section 3), depending on the definition of the state of the strain, i.e., if the strain tensor contains the immediate and the temporal strain, respectively [54]. However, it is advantageous if the strain remains uncoupled, to measure them independently as well as their corresponding cracking (damage).  
Secondly, since it has been well established in [43], creep strain for concrete has a short-term and a long-term component, where the latter is irreversible. Then, the creep strain may be further split into  
\[\epsilon^{c}=\epsilon^{v}+\epsilon^{f} \tag{2}\]  
where \(\epsilon^{v}\) is the strain related to the short-term creep and \(\epsilon^{f}\) is the viscous flow strain associated with long-term creep due to the migration of CSH particles. It shall also be noted that the short-term creep \(\epsilon^{v}\) will represent the viscoelastic strain \(\epsilon^{ve}\), viscoplastic strain \(\epsilon^{ep}\), and visco-damage strain \(\epsilon^{vd}\) components. In other words,  
\[\epsilon^{v}=\epsilon^{v}\left(\epsilon^{ve},\epsilon^{vp},\epsilon^{vd}\right) \tag{3}\]  
These latter strain components will remain coupled in the constitutive equation, as implied by Equation (3). The remaining subsections of this article will address the constitutive model related to the creep strain component \(\epsilon^{c}\), and the remaining strains are proposed for further research. Figure 1 shows an approximated rheological chain model proposed for the complete set of strain components for the split state of the strain at any point, including current and future research models, where the springs represent conservative processes and the dashpots are dissipative.  
Figure 1: Strain components for the split state of strain at any point: \(\epsilon^{0}\) shrinkage and thermal strain, \(\epsilon^{c}\) creep strain, \(\epsilon^{rp}\) elastoplastic strain, \(\epsilon^{e}\) elastic strain, \(\epsilon^{p}\) plastic strain, \(\epsilon^{d}\) damage strain, \(\epsilon^{v}\) short-term  
creep strain, \(\epsilon^{pv}\) viscoelastic strain, \(\epsilon^{vp}\) viscoplastic strain, \(\epsilon^{pd}\) visco-damage strain, and \(\epsilon^{f}\) viscous flow strain (long-term creep).