# A Damage Model for Concrete under Fatigue Loading
## 3 Thermodynamics Based Continuum Damage Mechanics Model
### Continuum Damage Mechanics Model  
The damage evolution is the variation process of the micro structure in material under external load, i.e., the process of the crack initiation, development and converge, thus, it reveals the physical nature of certain material and it does not impact the stress status [53]. Hence, the fatigue damage evolution of concrete in uniaxial case is presumed to be applied on multi-axial case without loss of generality. Therefore, the constitutive model in scalar form (Equation (42)) is able to be extended into the tensor form, as follows:  
\[\boldsymbol{\sigma}_{n}=[\mathbf{I}-\mathbf{D}_{n}]:\mathbf{E}_{0}:\varepsilon _{n} \tag{50}\]  
where \(\mathbf{D}_{n}\) denotes the tensor fatigue damage variable, which is used to model the nonlinearity of stress-strain response and can be expressed as follows:  
\[\mathbf{D}_{n}=D_{i,n}^{\pm}\mathbf{P}_{i}^{\pm}=D_{1,n}^{+}\mathbf{P}_{1}^{+} +D_{1,n}^{-}\mathbf{P}_{1}^{-}+D_{2,n}^{+}\mathbf{P}_{2}^{+}+D_{2,n}^{-} \mathbf{P}_{2}^{-}+D_{3,n}^{+}\mathbf{P}_{3}^{+}+D_{3,n}^{-}\mathbf{P}_{3}^{-} \tag{51}\]  
\[\left\{ \begin{array}{l}\mathbf{P}_{i}^{+}=\mathrm{H}[\sigma_{i}(n)]m^{(i)} \otimes m^{(i)}\otimes m^{(i)}\\ \mathbf{P}_{i}^{-}=[1-\mathrm{H}[\sigma_{i}(n)]]m^{(i)}\otimes m^{(i)}\otimes m ^{(i)}\end{array} \right. \tag{52}\]  
where \(i\) denotes the number of the principal stress direction, \(i=1,2,3\), for simplicity, in the uniaxial case, it can be omitted; \(\mathrm{H}(x)\) denotes the Heaviside function, if \(x>0\), its value is 1, otherwise 0; \(+\) and \(-\) denote the tensile and compressive loading condition, respectively. In the biaxial stress condition, the fatigue damage constitutive model in the principal stress direction is able to be described as follows:  
\[\left\{ \begin{array}{l}\sigma_{1,n}\\ \sigma_{2,n}\\ \tau_{12,n}\end{array} \right\}=\frac{1}{1-\mu^{2}}\left[ \begin{array}{c} \alpha_{1}\Big{(}1-D_{1,n}^{\pm}\\ \alpha_{2}\Big{(}1-D_{2,n}^{\pm}\\ 1\end{array} \Big{)}\left\| \begin{array}{ccc}E_{0}&\mu E_{0}&\mu E_{0}\\ \mu E_{0}&E_{0}&\mu E_{0}\\ \mu E_{0}&\mu E_{0}&E_{0}\end{array} \right.\right]\left\{ \begin{array}{l} \varepsilon_{1,n}\\ \varepsilon_{2,n}\\ \varsigma_{12,n}\end{array} \right\} \tag{53}\]  
where \(\alpha\) denotes the parameter considering the bia-compressive effects [22; 53], its value can be obtained from [22]; \(\mu\) denotes the Poisson ratio; \(\tau\) denotes the shear stress; \(\zeta\) denotes the shear strain. The cyclic creep and stiffness degradation in a three-stage process are characterized by the tensor fatigue irreversible deformation damage variable \(\mathbf{D}_{n}{}^{i}\) and the tensor fatigue elastic deformation damage variable \(\mathbf{D}_{n}{}^{e}\), respectively, and the post-fatigue stress-strain response is also described by the recently defined damage variables. As a result, the current stiffness and the irreversible/residual strain are able to be described as follows, respectively:  
\[E_{n}=\frac{1-D_{n}^{\pm}}{1-D_{n}^{\pm}}E_{0} \tag{54}\]  
\[\varepsilon_{n}^{\mathrm{di}}=D_{n}^{\pm}\varepsilon_{n} \tag{55}\]  
\[\varepsilon_{n}^{\mathrm{r}}=\ \varepsilon_{n}-\frac{\sigma_{\max}-\sigma_{\min}}{E_{n}} \tag{56}\]  
The post-fatigue stress-strain response of concrete under monotonic uniaxial loading is assumed to be expressed as follows:  
\[\sigma_{n}^{\mathrm{p}\pm}=\left\{ \begin{array}{l}E_{n}\cdot\left( \varepsilon-\varepsilon_{n}^{\mathrm{di}}\right),\hskip 28.452756 pt\varepsilon_{n}^ {\mathrm{di}}\leq\varepsilon\leq\sigma_{\max}/E_{n}\\ \left[1-k_{N}^{\pm}\cdot D_{n}^{f\circ u\pm}\right]\cdot E_{0}\cdot\left( \varepsilon-\varepsilon_{n}^{\mathrm{di}}\right),\hskip 14.226378 pt\sigma_{\max}/E_{n} <\varepsilon\leq\varepsilon_{N}\\ \left(1-D^{\pm}\right)\cdot E_{0}\cdot\varepsilon,\hskip 14.226378 pt\varepsilon> \varepsilon_{N}\end{array} \right.