# A Damage Model for Concrete under Fatigue Loading
## 3 Thermodynamics Based Continuum Damage Mechanics Model
### Thermodynamics Interpretation
#### 3.1.1 Interpretation in First and Second Loading Cycle  
Firstly, considering the stress-strain response during the first loading cycle, the unloading curve A\({}_{1}\)B\({}_{1}\) is simplified by the line A\({}_{1}\)B\({}_{1}\) in Figure 7a,b in this work. At point A\({}_{1}\), the strain \(\varepsilon_{1}\) and irreversible damage strain \(\varepsilon_{1}\)di exist in the specimen. The initial stiffness changes from \(E_{0}\) to \(E_{1}\). Even though these notations are for the uniaxial case, they are able to be used in indicial tensor notation in the equations without loss of generality. The total strain (described by line OB\({}_{1}\)G\({}_{1}\)H\({}_{1}\) in Figure 7a) is given as follows:  
\[\varepsilon_{1}=\varepsilon_{1}^{\text{E}}+\varepsilon_{1}^{\text{I}}=\left( \varepsilon_{0,1}+\varepsilon_{1}^{\text{de}}\right)+\varepsilon_{1}^{\text {di}} \tag{2}\]  
where the subscript \(1\) denotes the cyclic number of the first fatigue loading. The strain energy is expressed as follows (see the area described by points B\({}_{1}\)A\({}_{1}\)H\({}_{1}\) in Figure 7a)  
\[\psi_{1}=\frac{1}{2v}E_{1}\cdot\left(\varepsilon_{1}^{\text{E}}\right)^{2}= \frac{1}{2v}E_{0}\varepsilon_{0,1}\varepsilon_{1}^{\text{E}} \tag{3}\]  
\[\psi_{1}=\psi_{0}^{\text{e}}+\psi_{1}^{\text{de}} \tag{4}\]  
where \(\psi_{0}\)e denotes the initial strain energy (see the area G\({}_{1}\)A\({}_{1}\)H\({}_{1}\) in Figure 7a), and \(\psi_{1}\)de denotes the elastic deformation damage strain energy during the first cycle (see the area B\({}_{1}\)A\({}_{1}\)G\({}_{1}\) in Figure 7a), that is,  
\[\psi_{0}^{\text{e}}=\frac{1}{2v}E_{0}\cdot\left(\varepsilon_{0,1}\right)^{2} \tag{5}\]  
\[\psi_{1}^{\text{de}}=\frac{1}{2v}E_{0}\varepsilon_{0,1}\varepsilon_{1}^{\text {de}}=\psi_{1}-\psi_{0}^{\text{e}}=\frac{1}{2v}E_{0}\varepsilon_{0,1}\left( \varepsilon_{1}^{\text{E}}-\varepsilon_{0,1}\right) \tag{6}\]  
And the irreversible energy is expressed as follows (see the area OA\({}_{0}\)A\({}_{1}\)B\({}_{1}\) in Figure 7a)  
\[\gamma_{1}=\frac{1}{v}\left(\sigma_{\text{max}}\varepsilon_{1}^{\text{di}}+ \frac{1}{2}\sigma_{\text{max}}\varepsilon_{1}^{\text{de}}\right)=\frac{1}{v}E _{0}\varepsilon_{0,1}\left(\varepsilon_{1}^{\text{d}}-\frac{1}{2}\varepsilon_ {1}^{\text{de}}\right) \tag{7}\]  
\[\gamma_{1}=\gamma_{1}^{\text{di}}+\gamma_{1}^{\text{de}} \tag{8}\]  
where \(\gamma_{1}\)di denotes the irreversible-damage irreversible energy (see the area OA\({}_{0}\)I\({}_{1}\)B\({}_{1}\) in Figure 7a), and \(\gamma_{1}\)de denotes the elastic deformation damage irreversible energy (see the area B\({}_{1}\)I\({}_{1}\)A\({}_{1}\) in Figure 7a), that is,\[\gamma^{\rm di}_{1}=\frac{1}{v}\sigma_{\rm max}\epsilon^{\rm di}_{1}=\frac{1}{v}E_{0 }\epsilon_{0,1}\epsilon^{\rm di}_{1} \tag{9}\]  
\[\gamma^{\rm de}_{1}=\frac{1}{2v}\sigma_{\rm max}\epsilon^{\rm de}_{1}=\frac{1}{2 v}E_{0}\epsilon_{0,1}\epsilon^{\rm de}_{1} \tag{10}\]  
In regard to stored energy \(\lambda\) (contains both the purely reversible stored energy \(\psi\) and the irreversible energy \(\gamma\)), one is able to obtain the formula as follows  
\[\lambda_{n}=\psi_{n}+\gamma_{n} \tag{11}\]  
\[\psi_{1}=\frac{1}{2v}E_{1}\cdot\left(\varepsilon_{1}^{\text{E}}\right)^{2}= \frac{1}{2v}E_{0}\varepsilon_{0,1}\varepsilon_{1}^{\text{E}} \tag{3}\]0 Sketch of the mechanical parameter definition for concrete under fatigue loading. (a) Energy dissipation; (b) Energy dissipation; (c) Energy dissipation (\(\sigma_{\rm min}=0\)); (d) Energy dissipation (\(\sigma_{\rm min}\
eq 0\));  
When the material is assumed to be a perfect elastic material, it undergoes a strain \(\varepsilon_{1}\) and obtains a stored energy \(\lambda_{1}{}^{0}\) = 1/2(\(E_{0}\varepsilon_{1}{}^{2}\)) = \(\psi_{1}{}^{0}\) (i.e., the perfect material's purely reversible stored energy or strain energy) due to external loads. However, the material focused on in this work is a quasi-brittle material assumed to undergo both elastic deformation damage and irreversible deformation damage.