## 4 Constitutive Model and Finite Element Implementations
### Numerical Implementation
#### 4.2.2 Consistent Tangent Stiffness Matrix  
The finite element implementation of the rate-independent AF-OW II model required the consistent tangent stiffness matrices, which could accelerate convergence. The specific derivation process was described below:  
Figure 10: Flow diagram of radial return algorithm.  
Differentiating the aforementioned intrinsic model yields:  
\[\Delta\sigma_{n+1}=C:\Big{(}\Delta\varepsilon_{n+1}-\Delta\varepsilon_{n+1}^{p} \Big{)} \tag{39}\]  
\[\Delta\varepsilon_{n+1}^{p}=\Delta p_{n+1}n_{n+1} \tag{40}\]  
\[dn_{n+1}=\sqrt{\frac{3}{2}}\frac{(d\Delta s_{n+1}-d\Delta a_{n+1})}{\sigma_{y} }-\frac{n_{n+1}}{\sigma_{y}}\frac{d\sigma_{y}}{dp_{n+1}}d\Delta p_{n+1} \tag{41}\]  
\[n_{n+1}^{T}:n_{n+1}=\frac{3}{2} \tag{42}\]  
Differentiating Equation (33) yields the following:  
\[d\Delta\sigma_{n+1}=C:\Big{(}d\Delta\varepsilon_{n+1}-d\Delta\varepsilon_{n+1 }^{p}\Big{)} \tag{43}\]  
Substituting (22) into (37), the differentiation of the bias stress increment was obtained by using the fourth-order bias operation tensor \(I_{d}\) expressed as the following:  
\[d\Delta s_{n+1}=2G\Big{(}I_{d}:d\Delta\varepsilon_{n+1}-[M_{1}]d\Delta \varepsilon_{n+1}^{p}\Big{)} \tag{44}\]  
Differentiating Equation (34) yields the following:  
\[d\Delta\varepsilon_{n+1}^{p}=[M_{2}](d\Delta p_{n+1}n_{n+1}+\Delta p_{n+1}dn_{ n+1}) \tag{45}\]  
Differentiating Equation (36) yields the following:  
\[n_{n+1}^{T}:dn_{n+1}=0 \tag{46}\]  
\[n_{n+1}^{T}:d\Delta\varepsilon_{n+1}^{p}=n_{n+1}^{T}:[M_{2}](d\Delta p_{n+1}n_ {n+1}+\Delta p_{n+1}dn_{n+1}) \tag{47}\]  
Substituting (40) into (41) gave the following:  
\[d\Delta p_{n+1}=\frac{2}{3}n_{n+1}^{T}:d\Delta\varepsilon_{n+1}^{p} \tag{48}\]  
Substituting (35) and (42) into (39) gave the following:  
\[\frac{2\sigma_{y}}{3\Delta p_{n+1}}d\Delta\varepsilon_{n+1}^{p}=\frac{4}{9} \Big{(}n_{n+1}:n_{n+1}^{T}\Big{)}\bigg{(}\frac{\sigma_{y}}{\Delta p_{n+1}}- \frac{d\sigma_{y}}{dp_{n+1}}\bigg{)}d\varepsilon_{n+1}^{p}+[M_{2}](d\Delta s_ {n+1}-d\Delta\alpha_{n+1}) \tag{49}\]  
The differentiation of the backstress increment \(d\Delta\alpha\) could usually be expressed in terms of the fourth-order constitutive parameters \(H_{n+1}^{(i)}\):  
\[d\Delta\alpha_{n+1}=\sum_{i=1}^{M}H_{n+1}^{(i)}:d\Delta\varepsilon_{n+1}^{p} \tag{50}\]  
Substituting Equations (38) and (44) into Equation (43) yields the following:  
\[L_{n+1}:d\Delta\varepsilon_{n+1}^{p}=2 GI_{d}:d\Delta\varepsilon_{n+1} \tag{51}\]  
\[L_{n+1}=\bigg{(}\frac{2\sigma_{y}}{3\Delta p_{n+1}}+2G\bigg{)}:I+[M_{2}]\sum _{i=1}^{M}H_{n+1}^{(i)}-\frac{4}{9}\bigg{(}\frac{\sigma_{y}}{\Delta p_{n+1}}- \frac{d\sigma_{y}}{dp_{n+1}}\bigg{)}\Big{(}n_{n+1}:n_{n+1}^{T}\Big{)} \tag{52}\]where \(I\) was the unit matrix.  
\[H_{n+1}^{(i)}=\frac{\frac{2}{3}\bigg{(}r_{i}\gamma_{i}[M_{2}]-\gamma_{i}\dot{n}_{ n+1}^{i}n_{n+1}^{T}\bigg{[}\mu_{i}+\bigg{(}\frac{\overline{a_{n+1}^{i}}}{r_{i}} \bigg{)}^{m_{i}}\bigg{\langle}n_{n+1}^{T}:\frac{\dot{a}_{n+1}^{i}}{r_{i}}-\mu_{ i}\bigg{\rangle}\bigg{]}\bigg{)}}{1+\mu_{i}\Delta p_{n+1}+\bigg{(}\frac{ \overline{a_{n+1}^{i}}}{r_{i}}\bigg{)}^{m_{i}}\bigg{\langle}\Delta\varepsilon_{ n+1}^{p}:\frac{\dot{a}_{n+1}^{i}}{a_{n+1}^{i}}-\mu_{i}\Delta p_{n+1}\bigg{\rangle}} \tag{53}\]  
Substituting Equation (45) into Equation (37) yields the consistent elastic-plastic tangent stiffness matrix:  
\[\frac{d\Delta\sigma_{n+1}}{d\Delta\varepsilon_{n+1}}=C-4G^{2}L_{n+1}^{-1}:I_{d} \tag{54}\]