# A Damage Model for Concrete under Fatigue Loading
## 3 Thermodynamics Based Continuum Damage Mechanics Model
### Thermodynamics Interpretation
#### 3.1.3 Damage Variable Definition and Its Thermodynamics Interpretation  
Based on the above thermodynamics interpretation and the damage variable definition method in Appendix A, the elastic deformation fatigue damage variable in the elastic strain space and total strain space (\(D_{n}{}^{\mathrm{E}}\) and \(D_{n}{}^{\mathrm{e}}\)), and the irreversible deformation fatigue damage variable (\(D_{n}{}^{\mathrm{i}}\)) are defined, respectively, as follows (see Figure 7):  
\[D_{n}^{\mathrm{E}}=\frac{\psi_{n}^{\mathrm{de}}}{\psi_{n}^{\mathrm{e}}+\chi_{ n}}=\frac{\psi_{n}^{\mathrm{de}}+\gamma_{n}^{\mathrm{de}}}{\left(\psi_{0,n}^{ \mathrm{e}}+\psi_{n}^{\mathrm{de}}+\gamma_{n}^{\mathrm{de}}+\gamma_{n}^{ \mathrm{di}}\right)+\chi_{n}}=\frac{\varepsilon_{n}^{\mathrm{de}}}{\varepsilon _{n}^{\mathrm{E}}} \tag{37}\]  
\[D_{n}^{\mathrm{e}}=\frac{\psi_{n}^{\mathrm{de}}}{\psi_{n}^{\mathrm{f}}+\chi_{ n}}=\frac{\psi_{n}^{\mathrm{de}}+\gamma_{n}^{\mathrm{de}}}{\left(\psi_{0,n}^{ \mathrm{e}}+\psi_{n}^{\mathrm{de}}+\gamma_{n}^{\mathrm{de}}+\gamma_{n}^{ \mathrm{di}}\right)+\chi_{n}}=\frac{\varepsilon_{n}^{\mathrm{de}}}{\varepsilon _{n}} \tag{38}\]  
\[D_{n}^{\mathrm{i}}=\frac{\psi_{n}^{\mathrm{di}}}{\psi_{n}+\chi_{n}}=\frac{ \gamma_{n}^{\mathrm{di}}}{\left(\psi_{0,n}^{\mathrm{e}}+\psi_{n}^{\mathrm{de}} +\gamma_{n}^{\mathrm{de}}+\gamma_{n}^{\mathrm{di}}\right)+\chi_{n}}=\frac{ \varepsilon_{n}^{\mathrm{di}}}{\varepsilon_{n}} \tag{39}\]  
\[D_{n}=\frac{\psi_{n}^{\mathrm{d}}}{\psi_{n}+\chi_{n}}=\ D_{n}^{\mathrm{e}}+D_{n }^{\mathrm{i}}=\frac{\varepsilon_{n}^{\mathrm{de}}+\varepsilon_{n}^{\mathrm{ di}}}{\varepsilon_{n}}=\ \frac{\varepsilon_{n}^{\mathrm{d}}}{\varepsilon_{n}}=\ \frac{\varepsilon_{n}-\varepsilon_{0,n}}{\varepsilon_{n}}=\ \frac{ \varepsilon_{n}-\sigma_{\max}/E_{0}}{\varepsilon_{n}} \tag{40}\]  
where \(\varepsilon_{n}^{\mathrm{d}}\), \(\varepsilon_{n}^{\mathrm{de}}\), and \(\varepsilon_{n}^{\mathrm{di}}\) denote the strain development caused by the total damage, the elastic deformation damage and the irreversible deformation damage, respectively (related to the total cracking, the mode-I cracking and the irreversible deformation cracking discussed in micro-mechanical description in this work, respectively), when the material is subjected to the \(n\)th cyclic loading (Figure 7); \(\varepsilon_{n}{}^{\mathrm{E}}\) denotes the elastic strain, \(\varepsilon_{n}{}^{\mathrm{E}}=\varepsilon_{0,n}+\varepsilon_{n}^{\mathrm{de}}\) (\(\varepsilon_{0,n}=\varepsilon_{0}\)); \(\varepsilon_{n}\) denotes the total strain, \(\varepsilon_{n}=\varepsilon_{0,n}+\varepsilon_{n}^{\mathrm{de}}+\varepsilon_{n}^{ \mathrm{de}}+\varepsilon_{n}^{\mathrm{de}  
\(\varepsilon_{n}\textsuperscript{di}\); and \(\chi_{n}\) denotes the energy dissipation exists as a typical characteristics of the elastic behaviors, it accompanies and equals the initial strain energy \(\psi_{0,n}\textsuperscript{e}\) (see Figures 7 and A1), that is,  
\[\chi_{n}=\frac{1}{2\upsilon}\sigma_{\text{max}}\varepsilon_{0,n}=\frac{1}{2 \upsilon}\sigma_{\text{max}}\varepsilon_{0}=\psi_{0,n}^{\text{e}}=\psi_{0}^{ \text{e}} \tag{41}\]  
Note that Equation (40) illustrates that the damage evolution \(D_{n}\) depending on the total strain \(\varepsilon_{n}\) will be varied, when \(\sigma_{\text{max}}\) is changed during different fatigue loading processes. The nonlinear stress-strain relation is described as follows:  
\[ \begin{array}{ll}\sigma&=\left(1-D_{n}^{\text{E}}\right)E_{0}\Big{(} \varepsilon_{n}-\varepsilon_{n}^{\text{di}}\Big{)}=\left(1-D_{n}^{\text{E}} \right)\left(1-D_{n}^{\text{i}}\right)E_{0}\varepsilon_{n}\\ &=\left[1-\left(D_{n}^{\text{e}}+D_{n}^{\text{i}}\right)\right]E_{0} \varepsilon_{n}=\left(1-D_{n}\right)E_{0}\varepsilon_{n}\end{array}