## 2 A Conventional Micromorphic Approach for Gradient Plasticity  
### Kinematics  
Considering the position of a body denoted by \(\mathbf{X}\) in the reference configuration is mapped to the spatial point \(\mathbf{x}\) at time \(t\), the motion of that body can be represented by the smooth function \(\mathbf{x}=\mathbf{\chi}(\mathbf{X},t)\), where velocity, velocity gradient and deformation gradient take the following form:  
\[\mathbf{F}=\
abla\mathbf{\chi},\quad\mathbf{v}=\mathbf{\chi},\quad\mathbf{L}=\text{ grad}(\mathbf{v})=\mathbf{F}\mathbf{F}^{-1}, \tag{1}\]  
We consider the multiplicative decomposition of Kroner [27] and the deformation gradient is decomposed into elastic and plastic part  
\[\mathbf{F}=\mathbf{F}^{e}\mathbf{F}^{p} \tag{2}\]  
where \(\mathbf{F}^{e}\) is the elastic distortion which represents the stretch and rotation and \(\mathbf{F}^{p}\) represents the local plastic deformation. Please note that \(\mathbf{F}^{p}\) is invariant with respect to rigid body motions that are carried by \(\mathbf{F}^{e}\). The velocity gradient can be also decomposed into elastic and plastic components in the following form:  
\[\mathbf{L}=\mathbf{L}^{e}+\mathbf{F}^{e}\mathbf{L}^{p}\mathbf{F}^{-1} \tag{3}\]  
with,  
\[\mathbf{L}^{e}=\dot{\mathbf{F}}^{e}\mathbf{F}^{e-1},\quad\mathbf{L}^{p}=\dot{ \mathbf{F}}^{p}\mathbf{F}^{p-1} \tag{4}\]  
The volume ration is expressed as:  
\[J=\text{det}\mathbf{F}>0. \tag{5}\]  
Since the plasticity is assumed as incompressible (5), the Jacobian can also be decomposed as  
\[J=J^{e}J^{p},\quad J^{e}=\text{det}\mathbf{F}^{e}>0,\quad J^{p}=\text{det} \mathbf{F}^{p}>0. \tag{6}\]  
The right polar decomposition of \(\mathbf{F}^{e}\) yields  
\[\mathbf{F}^{e}=\mathbf{R}^{e}\mathbf{U}^{e} \tag{7}\]where \(\mathbf{U}^{e}\) is the right stretch tensor and \(\mathbf{R}^{e}\) is the rotation tensor. Then, the right elastic Cauchy-Green tensor and the stretch tensor read the following  
\[\mathbf{C}^{e}=(\mathbf{U}^{e})^{2}=\mathbf{F}^{eT}\mathbf{F}^{e} \tag{8}\] \[\mathbf{U}^{e}=\sum_{i=3}^{3}\lambda_{i}^{e}\,r_{i}^{e}\otimes r_ {i}^{e} \tag{9}\]  
As a strain measure, [PERSON]'s logarithmic strain [28] is chosen due to its good agreement with experiments on a wide class of materials [29; 30]  
\[\mathbf{E}^{e}=\ln(\mathbf{U}^{e})=\sum_{i=1}^{3}E_{i}^{e}\mathbf{r}_{i}^{e} \otimes r_{i}^{e} \tag{10}\]  
The elastic and inelastic stretching and spin tensors are  
\[\mathbf{D}^{e}=\text{sym}\mathbf{L}^{e},\quad\mathbf{D}^{p}= \text{sym}\mathbf{L}^{p} \tag{11}\] \[\mathbf{W}^{e}=\text{skew}\mathbf{L}^{e},\quad\mathbf{W}^{p}= \text{skew}\mathbf{L}^{p} \tag{12}\]  
Assuming irrotational plastic evolution i.e.,  
\[\mathbf{W}^{p}=0 \tag{13}\]  
the rate of plastic deformation gradient would be  
\[\dot{\mathbf{F}}^{p}=\mathbf{D}^{p}\mathbf{F}^{p}\,\ \ \mathbf{F}^{p}(\mathbf{X}, \mathbf{0})=\mathbf{1} \tag{14}\]  
Introducing a positive-valued equivalent tensile stress and corresponding plastic strain rate  
\[\tilde{\sigma}=\sqrt{3/2}|\mathbf{M}_{\mathbf{0}}^{e}|\quad\text{and}\ \ \ \tilde{\epsilon}^{p}\sqrt{2/3}|\mathbf{D}^{p}| \tag{15}\]  
where \(\mathbf{M}_{\mathbf{0}}^{e}\) is the deviatoric component of the Mandel stress and the direction of the plastic flow becomes  
\[\mathbf{N}^{p}=\frac{\mathbf{M}_{\mathbf{0}}^{e}}{|\mathbf{M}_{\mathbf{0}}^{e }|} \tag{16}\]