# A Large-Deformation Gradient Damage Model for Single Crystals Based on Microdamage Theory
## 2 Microdamage Model for Single Crystals  
### Basic Kinematics  
Considering a body in the reference configuration represented by \(\mathbf{X}\) is mapped to the spatial point \(\mathbf{x}\) at time \(t\), the motion of the body is represented by the smooth function \(\mathbf{x}=\boldsymbol{\chi}(\mathbf{X},t)\). Accordingly, the deformation gradient, the velocity and the velocity gradient are given by  
\[\mathbf{F}=\
abla\boldsymbol{\chi},\quad\mathbf{v}=\boldsymbol{\dot{\chi}}, \quad\mathbf{L}=\text{grad}(\mathbf{v})=\dot{\mathbf{F}}\mathbf{F}^{-1}, \tag{1}\]  
Following the [PERSON]'s decomposition of the deformation gradient [20], the plastic deformation and damage in a crystal is represented in the inelastic distortion and accordingly the decomposition takes the following form:  
\[\mathbf{F}=\mathbf{F}^{\text{e}}\mathbf{F}^{i} \tag{2}\]  
where \(\mathbf{F}^{\text{e}}\) represents the rotation and stretch as elastic distortion and \(\mathbf{F}^{i}\) is the inelastic distortion capturing the plasticity and damage in the crystal. Figure 1 represents such a deformation for initial and final configurations.  
Figure 1: Decomposition of the deformation gradient.  
However, for the sake of simplicity and the clarity of the proposed theory, the plastic part of the inelastic distortion is neglected throughout this work. Then the deformation gradient simply reads  
\[\mathbf{F}=\mathbf{F}^{\epsilon}\mathbf{F}^{d} \tag{3}\]  
\(\mathbf{F}^{d}\) represents the local inelastic deformation due to damage generated in the material and invariant under a change in frame. In this framework, the velocity gradient can be decomposed in the following form:  
\[\mathbf{L}=\mathbf{L}^{\epsilon}+\mathbf{F}^{\epsilon}\mathbf{L}^{d}\mathbf{F }^{\epsilon-1} \tag{4}\]  
where  
\[\mathbf{L}^{\epsilon}=\dot{\mathbf{F}}^{\epsilon}\mathbf{F}^{\epsilon-1}, \quad\mathbf{L}^{d}=\dot{\mathbf{F}}^{d}\mathbf{F}^{d-1} \tag{5}\]  
volumetric contribution to elastic and inelastic deformation is defined as  
\[J=J^{\epsilon}J^{d},\quad J^{\epsilon}=\text{det}\mathbf{F}^{\epsilon}>0, \quad J^{d}=\text{det}\mathbf{F}^{d}>0. \tag{6}\]  
Considering the right polar decomposition of the elastic distortion  
\[\mathbf{F}^{\epsilon}=\mathbf{R}^{\epsilon}\mathbf{U}^{\epsilon} \tag{7}\]  
in terms of the right stretch tensor \(\mathbf{U}^{\epsilon}\) and the rotation tensor \(\mathbf{R}^{\epsilon}\). The right elastic Cauchy-Green tensor and the stretch tensor are defined as  
\[\mathbf{C}^{\epsilon}=(\mathbf{U}^{\epsilon})^{2}=\mathbf{F}^{\epsilon T} \mathbf{F}^{\epsilon} \tag{8}\]  
\[\mathbf{U}^{\epsilon}=\sum_{i=3}^{3}\lambda_{i}^{\epsilon}\;r_{i}^{\epsilon} \otimes r_{i}^{\epsilon} \tag{9}\]  
Hencky strain [21] is chosen as a strain measure due to its good agreement with the mechanical response of a wide class of materials [22; 23].  
\[\mathbf{E}^{\epsilon}=\ln(\mathbf{U}^{\epsilon})=\sum_{i=1}^{3}E_{i}^{\epsilon }\mathbf{r}_{i}^{\epsilon}\otimes\mathbf{r}_{i}^{\epsilon} \tag{10}\]  
The elastic and inelastic spin and stretching tensors are  
\[\mathbf{D}^{\epsilon}=\text{sym}\mathbf{L}^{\epsilon},\quad\mathbf{D}^{d}= \text{sym}\mathbf{L}^{d} \tag{11}\]  
\[\mathbf{W}^{\epsilon}=\text{skew}\mathbf{L}^{\epsilon},\quad\mathbf{W}^{d}= \text{skew}\mathbf{L}^{d} \tag{12}\]  
Assuming irrotational damage evolution, i.e.,  
\[\mathbf{W}^{d}=0 \tag{13}\]  
The inelastic deformation gradient rate becomes  
\[\dot{\mathbf{F}}^{d}=\mathbf{D}^{d}\mathbf{F}^{d}\;,\;\;\mathbf{F}^{d}( \mathbf{X},\mathbf{0})=\mathbf{1} \tag{14}\]  
The damage stretching tensor is defined as  
\[\mathbf{D}^{d}=\sum_{i=1}^{3}\dot{\delta}_{s}\mathbf{N}_{s}^{d}\;, \tag{15}\]where \(\dot{\delta}_{s}\) is the damage rates initiated at each damage system which is a scalar internal variable and \(\mathbf{N}_{s}^{d}\) is the damage flow direction tensor. Finally, the scalar damage rate takes the following form  
\[\dot{d}=\sum|\dot{\delta}_{s}| \tag{16}\]