# A Damage Model for Concrete under Fatigue Loading
## Appendix A Damage Variable Definition Based on Thermodynamics
Thus, the Helmholtz free energy is derived as follows,  
\[\chi=\frac{1}{2v}\sigma\varepsilon_{0}=\frac{1}{2v}\sigma\varepsilon=\psi_{0}^{ \text{e}} \tag{12}\]6due to the energy dissipation \(\chi=\psi_{0}\mathrm{e}=0\). The irreversible deformation damage variable is obtained, as follows:  
\[\chi=\frac{1}{2v}\sigma\varepsilon_{0}=\frac{1}{2v}\sigma\varepsilon=\psi_{0}^{ \text{e}} \tag{12}\]7  
And the irreversible strain is derived, as follows:  
\[\chi=\frac{1}{2v}\sigma\varepsilon_{0}=\frac{1}{2v}\sigma\varepsilon=\psi_{0}^{ \text{e}} \tag{12}\]8  
Furthermore, the nonlinear stress-strain relation is described as follows:  
\[\left\{ \begin{array}{l}\sigma=\sigma\\ \varepsilon=\varepsilon^{\mathrm{di}}\end{array} \right. \tag{19}\]
4. Quasi-brittle materials (Elastic-irreversible deformation damage materials) The Helmholtz free energy of a quasi-brittle material is obtained as follows (see Figure 1e): \[\Psi=\psi+\gamma=\left(\psi_{0}^{\mathrm{e}}+\psi^{\mathrm{de}}\right)+\left( \gamma^{\mathrm{de}}+\gamma^{\mathrm{di}}\right)\] (20)  
where  
\[\psi^{\mathrm{de}}=\gamma^{\mathrm{de}}=\frac{1}{2\upsilon}\sigma\varepsilon^ {\mathrm{de}} \tag{21}\]  
And the energy dissipation \(\chi\) is derived in this work (see Figure 1), that is,  
\[\chi=\frac{1}{2\upsilon}\sigma\varepsilon_{0}=\psi_{0}^{\mathrm{e}} \tag{22}\]  
The elastic deformation damage variable in the elastic strain space is defined in this work by two methods considering two different energy dissipation aspects, respectively, as follows:  
\[D^{\mathrm{E}}=\frac{\psi^{\mathrm{de}}}{\psi}=\frac{\psi^{\mathrm{de}}}{\psi _{0}^{\mathrm{e}}+\psi^{\mathrm{de}}}=\frac{\varepsilon^{\mathrm{de}}}{ \varepsilon_{0}+\varepsilon^{\mathrm{de}}}=\frac{\varepsilon^{\mathrm{de}}}{ \varepsilon^{\mathrm{E}}} \tag{23}\]  
\[D^{\mathrm{E}}=\frac{\psi^{\mathrm{de}}}{\psi^{\mathrm{e}}+\chi}=\frac{\psi^{ \mathrm{de}}+\gamma^{\mathrm{de}}}{\left(\psi_{0}^{\mathrm{e}}+\psi^{\mathrm{ de}}+\gamma^{\mathrm{de}}\right)+\chi}=\frac{\varepsilon^{\mathrm{de}}}{ \varepsilon^{\mathrm{E}}} \tag{24}\]  
The elastic deformation damage variable in the total strain space is also defined in this work by the method considering energy dissipation, as follows:  
\[D^{\mathrm{e}}=\frac{\psi^{\mathrm{de}}}{\psi+\chi}=\frac{\psi^{\mathrm{de}}+ \gamma^{\mathrm{de}}}{\left(\psi_{0}^{\mathrm{e}}+\psi^{\mathrm{de}}+\gamma^{ \mathrm{de}}+\gamma^{\mathrm{di}}\right)+\chi}=\frac{\varepsilon^{\mathrm{de} }}{\varepsilon} \tag{25}\]  
where the total strain \(\varepsilon=\varepsilon_{0}+\varepsilon^{\mathrm{de}}+\varepsilon^{\mathrm{di}}\), and the elastic strain \(\varepsilon^{\mathrm{E}}=\varepsilon_{0}+\varepsilon^{\mathrm{de}}\). The irreversible deformation damage variable is defined in this work by the method considering energy dissipation, as follows:  
\[D^{\mathrm{i}}=\frac{\psi^{\mathrm{di}}}{\psi+\chi}=\frac{\gamma^{\mathrm{di}} }{\left(\psi_{0}^{\mathrm{e}}+\psi^{\mathrm{de}}+\gamma^{\mathrm{de}}+\gamma^ {\mathrm{di}}\right)+\chi}=\frac{\varepsilon^{\mathrm{di}}}{\varepsilon} \tag{26}\]  
The nonlinear stress-strain relation is described as follows:  
\[ \begin{array}{l}\sigma\\ =\left(1-D^{\mathrm{E}}\right)E_{0}\left(\varepsilon-\varepsilon^{\mathrm{di} }\right)=\left(1-D^{\mathrm{E}}\right)\left(1-D^{\mathrm{i}}\right)E_{0} \varepsilon\\ =\left[1-\left(D^{\mathrm{e}}+D^{\mathrm{i}}\right)\right]E_{0}\varepsilon =\left(1-D\right)E_{0}\varepsilon\end{array}  \tag{27}\]The damage variable \(D^{\rm E}\) (or \(D^{\rm e}\)) and \(D^{\rm i}\) is able to be employed to characterize the stiffness degradation and irreversible strain development, respectively, as follows (see Figure 11e),  
\[E=\Big{(}1-D^{\rm E}\Big{)}E_{0}=\left(1-\frac{D^{\rm e}}{1-D^{\rm i}}\right)E_{ 0}=\frac{1-D}{1-D^{\rm i}}E_{0} \tag{10}\]  
\[\varepsilon^{\rm di}=D^{\rm i}\varepsilon \tag{11}\]  
Note that the energy dissipation \(\chi\) presents a typical property of materials' elastic behaviors.