# A Large-Deformation Gradient Damage Model for Single Crystals Based on Microdamage Theory
## 5 Specialization of the Constitutive Equations  
Considering the crystallographic structure of a single crystal, the damage evolution is formalized as an addition of each damage system such as:  
\[\dot{\mathbf{F}}^{d}\mathbf{F}^{d-1}=\mathbf{D}^{d}=\sum_{s=1}^{N_{planes}^{d}} \dot{\delta}_{c}^{s}\mathbf{n}^{s}\otimes\underline{\mathbf{n}}^{s}+\dot{ \delta}_{1}^{s}\underline{\mathbf{n}}^{s}\otimes\underline{\mathbf{l}}_{1}^{s} +\dot{\delta}_{2}^{s}\underline{\mathbf{n}}^{s}\otimes\underline{\mathbf{l}}_{2}^ {s} \tag{40}\]  
For three mode of fracture, damage evolution is calculated separately as \(\delta_{c}^{s}\)\(\delta_{1}^{s}\), \(\delta_{2}^{s}\) and the number of damage planes \(N_{planes}^{d}\) is determined from a given crystal structure. The opening \(\delta_{s}\) of crystallographic planes with the normal vector \(\underline{\mathbf{n}}^{s}\) can be defined as cleavage damage in the system and once cleavage has started, other damage systems are allowed to initiate for the inplane accommodation along orthogonal directions \(\underline{\mathbf{l}}_{1}^{s}\) and \(\underline{\mathbf{l}}_{2}^{s}\), (Figure 2). Three damage criteria are associated to one opening and two accommodation systems:  
\[f_{c}^{s}=|\underline{\mathbf{n}}^{s}\cdot\mathbf{M}^{c}\cdot\underline{ \mathbf{n}}^{s}|-Y^{s} \tag{41}\]  
\[f_{i}^{s}=|\underline{\mathbf{n}}^{s}\cdot\mathbf{M}^{c}\cdot\underline{ \mathbf{l}}_{i}^{s}|-Y^{s}\;\;(i=1,2) \tag{42}\]  
where \(Y\) is the damage threshold and the scalar damage is simply the accumulation of damage generated in all systems. \[\dot{d}=|\dot{\delta}_{c}^{s}|+|\dot{\delta}_{1}^{s}|+|\dot{\delta}_{2}^{s}| \tag{43}\]  
The correspond equivalent stresses projected on to the damage system takes the form:  
\[\bar{\sigma}_{s}=\mathbf{M}^{c}:\mathbf{N}^{s} \tag{44}\]  
\[\mathbf{N}_{c}^{s}=\underline{\mathbf{n}}^{s}\cdot\mathbf{M}^{c}\cdot \underline{\mathbf{n}}^{s}\quad,\quad\mathbf{N}_{i}^{s}=\underline{\mathbf{n }}^{s}\cdot\mathbf{M}^{c}\cdot\underline{\mathbf{l}}^{s} \tag{45}\]  
For this specific model, the free energy function is taken as a quadratic potential of elastic strain, \(\mathbf{E}^{c}\), damage, \(d\), microdamage, \(\dot{x}d\) and its gradient, \(\
abla^{\dot{x}}d\)  
\[\psi(\mathbf{E}^{c},d,\dot{x}d,\
abla^{\dot{x}}d)=\mu|\mathbf{E}_{0}^{c}|^{2} +\frac{1}{2}\kappa(tr\mathbf{E}^{c})+\frac{1}{2}B(d-\dot{x}d)^{2}+\frac{1}{2} \beta|\
abla^{\dot{x}}d|^{2} \tag{46}\]  
Figure 2: Schematic representation of the damage systems associated to the crystallographic planes. through higher order continuum can be ignored. Accordingly, the energetic terms become:  
\[\pi_{en} = \mathcal{I}(d-^{X}d) \tag{54}\] \[\mathbf{p}_{en} = -\mathcal{I}(d-^{X}d)\] (55) \[f_{c}^{s}=|\underline{\mathbf{n}}^{s}\cdot\mathbf{M}^{c}\cdot\underline{ \mathbf{n}}^{s}|-Y^{s} \tag{41}\]0  
and the generalized balance law takes the form:  
\[f_{c}^{s}=|\underline{\mathbf{n}}^{s}\cdot\mathbf{M}^{c}\cdot\underline{ \mathbf{n}}^{s}|-Y^{s} \tag{41}\]1  
An important observation from Equation (57) is the Laplacian form constructed in the equation provides regularized solutions only for the microvariable\({}^{X}d\) and \(d\) acts only as a source term. Therefore, in order to attain a similar solution on \(d\), a dissipative penalization is introduced in the microforce balance, such as:  
\[f_{c}^{s}=|\underline{\mathbf{n}}^{s}\cdot\mathbf{M}^{c}\cdot\underline{ \mathbf{n}}^{s}|-Y^{s} \tag{41}\]2  
Considering the stress units in Pascal, for the majority of mechanical problems, \(Y(d)\) is in the order of MPa and the \(B\) is a penalization constant in the order of elastic moduli. For a high value of \(B\), a satisfactory solution for the local integration is possible only if \(d\) follows\({}^{X}d\), otherwise the iterative values of \(\tilde{\sigma}\) overshoots the elastic trial stresses \(\tilde{\sigma}^{tr}\) and the microforce balance will be violated. In that sense \(B\) is selected according to the desired difference between \(d\) and \({}^{X}d\).