# Investigation of Deformation Inhomogeneity and Low-Cycle Fatigue of a Polycrystalline Material
## 2 Material Model and Crystal Plasticity Model
### Constitutive Model of Crystal Plasticity  
To describe the crystal slip in the grains, the paper adopts the crystal plasticity constitutive model and corresponding integral algorithm suggested in ref. [30]. The fundamentally theoretical framework arises from the works by the pioneer scholars [35; 36; 37]. The relation between the shear strain rate \(\dot{\gamma}^{(\alpha)}\) and the shear stress \(\tau^{(\alpha)}\) on the \(\alpha\)-slip system is described based on [PERSON]'s power law [38], in which the Bauschinger effect is considered through back-stresses, and nonlinear kinematic hardening is introduced [39]  
\[\dot{\gamma}^{(\alpha)}=\dot{\gamma}_{0}\text{sgn}(\tau^{(\alpha)}-x^{(\alpha )})\left|\frac{\tau^{(\alpha)}-x^{(\alpha)}}{g^{(\alpha)}}\right|^{k} \tag{3}\]  
where \(x^{(\alpha)}\) is the resolved back-stress on the \(\alpha\)-slip system; \(\dot{\gamma}_{0}\) denotes the reference shear rate, regarded as a constant for all the slip systems; \(k\) is the rate sensitivity parameter, and \(g^{(\alpha)}\) defines the yield limit or the elastic domain beyond which the material behaves inelastic. It evolves as [40]  
\[\dot{g}^{(\alpha)}(\gamma)=\overset{n}{\underset{\beta}{\overset{\text{-}}{ \sum}}}h_{\alpha\beta}(\gamma)\Big{|}\dot{\gamma}^{(\beta)}\Big{|},\quad \gamma=\int\overset{n}{\underset{\beta}{\overset{\text{-}}{\sum}}}\Big{|}d \gamma^{(\beta)}\Big{|} \tag{4}\]  
where \(h_{\alpha\beta}(\gamma)\) are the hardening moduli. It was proposed that [41]  
\[h_{\alpha\beta}(\gamma)=h(\gamma)[q+(1-q)\delta_{\alpha\beta}] \tag{5}\]  
where \(q\) is a constant, and it was suggested that [42]  
\[h(\gamma)=h_{0}\text{sech}^{2}\bigg{(}\frac{h_{0}\gamma}{\tau_{\text{s}}-\tau_ {0}}\bigg{)} \tag{6}\]where \(h_{0}\) is the initial hardening rate, \(\tau_{0}\) and \(\tau_{\rm s}\) are the critical resolved shear stress and its saturation value, respectively. They are regarded as material constants. The evolution of back-stresses, \(x^{(\alpha)}\), is introduced as [30]  
\[\dot{x}^{(\alpha)}=a\dot{\gamma}^{(\alpha)}-c\left[1-e_{1}\Big{(}1-\exp(-e_{2} \gamma))\right]\,x^{(\alpha)}\Big{|}\dot{\gamma}^{(\alpha)}\Big{|}-\lambda x^ {(\alpha)} \tag{7}\]  
where \(a\), \(c\), \(e_{1}\), \(e_{2}\) and \(\lambda\) are material constants. This equation includes three terms, strain hardening term, dynamic recovery term and a static recovery term. By numerical simulations comparing with cyclic tests, the material constants in Equations (3) and (5)- (7) can be identified [30; 31]. The initial slip direction vector is \(\mathbf{m}^{(\alpha)}\), and the initial normal vector is \(\mathbf{n}^{(\alpha)}\) to the slip plane of the slip system \(\alpha\), respectively. Referring to the literature [35; 36; 37], the Schmid tensor, \(\mathbf{P}^{(\alpha)*}\), written as  
\[\mathbf{P}^{(\alpha)*}=\frac{1}{2}\Big{(}\mathbf{m}^{(\alpha)*}\mathbf{n}^{( \alpha)*}+\mathbf{n}^{(\alpha)*}\mathbf{m}^{(\alpha)*}\Big{)} \tag{8}\]  
in which \(\mathbf{m}^{(\alpha)*}=\mathbf{F}^{*}\cdot\mathbf{m}^{(\alpha)}\), \(\mathbf{n}^{(\alpha)*}=\mathbf{n}^{(\alpha)}\cdot\mathbf{F}^{*-1}\), where \(\mathbf{F}^{*}\) is the elastic part of the deformation gradient tensor \(\mathbf{F}\). Then, the plastic deformation rate tensor can be calculated as  
\[\mathbf{D}^{\rm P}=\sum\mathbf{P}^{(\alpha)*}\dot{\gamma}^{(\alpha)} \tag{9}\]  
While the resolved shear stress can be calculated as  
\[\tau^{(\alpha)}=\mathbf{P}^{(\alpha)*}\mathbf{:}\boldsymbol{\sigma} \tag{10}\]  
Since the elastic strain is small, the rate-constitutive equation can be expressed as  
\[\dot{\boldsymbol{\sigma}}^{\rm J}=\overset{<4>}{\mathbf{C}}:\mathbf{D}^{*}= \overset{<4>}{\mathbf{C}}:(\mathbf{D}-\mathbf{D}^{\rm P}) \tag{11}\]  
where \(\dot{\boldsymbol{\sigma}}^{\rm J}\) is the Jaumann rate of Cauchy stress, \(\overset{<4>}{\mathbf{C}}\) is the fourth-order elasticity tensor with respect to the fixed global coordinate axes, \(\mathbf{D}^{*}\) is elastic part of the deformation rate tensor \(\mathbf{D}\).