# Efficient Deep Learning for Gradient-Enhanced Stress Dependent Damage Model
## 2 Continuum Damage Theory  
### Constitutive Equations of Isotropic Damage Models  
Let us assume small strain theory in the context of a scalar/isotropic damage model. The relation between the Cauchy stress tensor \(\sigma\) and the linear strain tensor \(\varepsilon\) is given by  
\[\sigma=(1-\omega)\mathbb{C}:\varepsilon=\mathbb{C}^{eff}:\varepsilon \tag{1}\]  
where \(\omega\) denotes a monotonically increasing scalar damage variable, \(\mathbb{C}\) the fourth-order elasticity tensor and \(\mathbb{C}^{eff}=(1-\omega)\mathbb{C}\) refers to the so-called 'effective' elasticity tensor. A value of \(\omega=1\) indicate a completely damaged material while the material is intact for a value o \(\omega=0\). The evolution of the  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Single-Layer Neural Networks** & **Input Layer-Output Layer** \\ \hline \multirow{3}{*}{Multilayer-Layer} & Shallow & Input Layer-Hidden Layer \\  & Neural Networks & -Output Layer \\ \hline Neural Networks & Deep & Input Layer-Hidden Layers \\  & Neural Networks & -Output Layers \\ \hline \hline \end{tabular}
\end{table}
Table 1: The branches of the neural network depending on the layer architecture.  
damage variable is commonly governed by a scalar state variable \(\kappa\), i.e., \(\omega=\omega(\kappa)\). The Kuhn-Tucker conditions, which finally lead to a convex optimization problem, ensures that the product of the loading function \(f\) and the rate of the state variable is equal to zero:  
\[\left\{\begin{array}{ll}f&\leq 0\\ \hat{\kappa}&\geq 0\\ f\hat{\kappa}&=0\end{array}\right. \tag{2}\]  
\(\varepsilon_{eq}\) indicating the effective strain, which is a projection of a multi-axial strain state onto a single scalar value. We consider a strain-based formulation, where the loading function is expressed in terms of the effective strain (instead of effective stresses), which facilitates the implementation in the context of displacement based finite element analysis:  
\[f=f(\varepsilon_{eq},\kappa)=\varepsilon_{eq}-\kappa=0 \tag{3}\]  
We test two different approaches to compute \(\varepsilon_{eq}\). The first one has been presented by [PERSON] [23] for quasi-brittle materials and is given by  
\[\varepsilon_{eq}=\sqrt{\left\langle\varepsilon\right\rangle:\left\langle \varepsilon\right\rangle}=\sqrt{\left\langle\varepsilon_{1}\right\rangle^{2} +\left\langle\varepsilon_{2}\right\rangle^{2}} \tag{4}\]  
where \(\left\langle\cdot\right\rangle\) indicates a Macaulay bracket while \(\varepsilon_{1}\) and \(\varepsilon_{2}\)--for two-dimensional problems--denote the principal strains. For this approach, we employ an exponential evolution law for the damage variable as suggested by [PERSON] et al. [24]:  
\[\omega(\kappa)=1-\frac{\kappa_{0}}{\kappa}\left[1-\alpha+\alpha e^{-\beta( \kappa-\kappa_{0})}\right] \tag{5}\]  
where \(\kappa_{0}\) stands for an initial damage threshold while the material parameters \(\alpha\) and \(\beta\) needs to be determined through experiments. We also employ an expression for the equivalent strain as suggested by [PERSON] et al. [25] and [PERSON] et al. [26]:  
\[\varepsilon_{eq}=\frac{k-1}{2k(1-\
u)}I_{1}+\frac{1}{2k}\sqrt{\frac{(k-1)^{2} }{(1-2\
u)^{2}}I_{1}^{2}+\frac{12k}{(1+\
u)^{2}}I_{2}} \tag{6}\]  
where \(I_{1}\) indicates the first invariant of the linear strain tensor and \(I_{2}\) refers to the second invariant of the deviatoric part of the strain tensor.  
\[\left\{\begin{array}{ll}I_{1}&=\varepsilon_{11}+\varepsilon_{22}+\varepsilon _{33}\\ J_{2}&=\frac{1}{2}\varepsilon_{ij}\varepsilon_{ij}-\frac{1}{6}I_{1}^{2}\end{array}\right. \tag{7}\]  
The parameter \(k\) in Equation (6) determines the ratio of the compressive and tensile strength.