> FiniteStrainCrystalPlasticity is deprecated. Use the CrystalPlasticityUOBase based crystal plasticity system.

The crystal plasticity material class in TensorMechanics is based on plastic flow on individual slip systems to obtain the inelastic deformation in materials. The effect of grain-scale anisotropy on the overall plastic behavior can be incorporated through this model. The present formulation considers large deformation and is based on a stress update algorithm. Some of the important functions associated with this class are outlined below.

**_Function_** - computeQpStress():
User can override this function.

_Input_:

- 2$$^{nd}$$ Piola-Kirchhoff (PK2) stress in the intermediate configuration at previous increment (variable \_pk2\_old, symbol $$T_n$$);

- Plastic deformation gradient at previous increment (variable \_fp\_old, symbol $$F^p_n$$);

- Slip system resistances at previous increment (variable \_gss\_old, symbol $$g^\alpha_n$$);

- Current deformation gradient (variable \_dfgrd, symbol $$F$$).

_Output_:

- Current PK2 stress (variable \_pk2, symbol $$T$$);

- Current plastic deformation gradient (variable \_fp, symbol $$F^p$$);

- Current slip system resistances (variable \_gss, symbol $$g^\alpha$$).

- Slip system resistances $$g^\alpha$$ are solved iteratively using a predictor corrector algorithm. $$T$$ is solved using Newton Raphson.

_Flowchart_:

_i_: At iteration i=0, $$g^\alpha_i=g^\alpha_n$$

_ii_: Calculate the residual r and Jacobian J=$$\partial r/\partial T$$ from the function **calc\_resid\_jacob**

_iii_: Update T as: $$T_{i+1}=T_{i}-J^{-1}r$$

_iv_: Check $$||r||_2 <$$ \_rtol (Optional user defined parameter)

If False then go to Step _ii_

Else

_v_: Calculate $$g^\alpha_{i+1}$$ from function **update\_gss**

_vi_: Check $$max|g^\alpha_{i+1}-g^\alpha_{i}| <$$ \_gtol  (Optional user defined parameter)

If False then go to Step _ii_

Else

_vii_: Obtain rotation tensor R from function **getmatrot**

_viii_: Update rotation as $$\tilde{R}=RQ$$ where Q is the rotation tensor from Euler angles

Exit Function

**_Function_** - **calc\_resid\_jacob**:
User can override this function.

_Input_:

- PK2 stress at previous iteration (variable \_pk2, symbol $$T_i$$, i - iteration);

- Inverse of plastic deformation gradient at previous increment (variable \_fp\_old\_inv, symbol $$F^{p-1}_n$$);

- Current deformation gradient (variable \_dfgrd, symbol $$F$$).

_Output_:

- Residual (variable resid, symbol r);

- Jacobian (variable jac, symbol J);

- Updated inverse of plastic deformation gradient (variable \_fp\_inv, symbol $$F^{p-1}$$)

Residual r as implemented:

_i_. Elastic component of deformation gradient (variable fe) $$F^e=FF^{p-1}_n$$.

_ii_. Elastic right Cauchy-Green Tensor (variable ce) $$C^e=F^{eT}F^e$$.

_iii_. Elastic Green-Lagrangian strain tensor (variable ee) $$E^e=\frac{1}{2}\left(C^e -I\right)$$.

_iv_. Shear stresses resolved on slip systems (variable tau) $$\tau^\alpha=C^eT_i:\left( m_0^\alpha \otimes n_0^\alpha \right)/J^e$$, where $$m_0^\alpha$$ and $$n_0^\alpha$$ are the slip direction and slip system normal in the intermediate configuration respectively. $$J^e=det(F^e)$$.

_v_. Slip increment on slip systems $$\Delta \gamma^\alpha$$ obtained from function **get\_slip\_incr**.

_vi_. Resultant slip increment (variable eqv\_slip\_incr) $$\Delta \gamma_{res}=I-\sum \limits_\alpha \left( \Delta \gamma^\alpha S_0^\alpha\right)$$ where $$S_0^\alpha=m_0^\alpha \otimes n_0^\alpha$$ is the Schmid tensor.

_vii_. Current plastic component of deformation gradient $$F^{p-1}=F^{p-1}_n\Delta \gamma_{res}$$.

_viii_. Elastic component of deformation gradient in iteration i+1, $$F^e=FF^{p-1}$$.

_ix_: PK2 stress due to $$T_i$$ and associated slip increment in current iteration i+1 (variable pk2\_new), $$T^\prime=C:\frac{1}{2}\left(F^{eT}F^e-I\right)$$.

_x_. Residual $$r=T_i-T^\prime$$.

Jacobian J formulation:

_i_. $$r=T_i-T^\prime$$ and $$J=\frac{\partial r}{\partial T_i}=I-\frac{\partial T^\prime}{\partial T_i}$$.

_ii_. $$T^\prime=C:E^e$$ which gives $$\frac{\partial T^\prime}{\partial T_i}=C: \frac{\partial E^e}{\partial T_i}$$.

_iii_. $$E^e=\frac{1}{2}\left(F^{eT}F^e -I\right)$$ which gives in indicial notation $$\frac{\partial E^e_{ij}}{\partial F^e_{kl}}=\frac{1}{2}\left( \delta_{il}F^e_{kj}+\delta_{jl}F^e_{ki}\right)$$ where $$\delta_{ij}$$ is the Kroneckar delta.

_iv_. $$F^e=FF^{p-1}$$ which gives in indicial notation $$\frac{\partial F^e_{ij}}{\partial F^{p-1}_{kl}}=F_{ik}\delta_{lj}$$.

_v_. $$F^{p-1}=F^{p-1}_n\Delta \gamma_{res}$$ which gives $$\frac{\partial F^{p-1}}{\partial \Delta \gamma_{res}}=F^{p-1}_n$$.

_vi_. $$\Delta \gamma_{res}=I-\sum \limits_\alpha \left( \Delta \gamma^\alpha S_0^\alpha\right) which gives \frac{\partial \Delta \gamma_{res}}{\partial \Delta \gamma^\alpha}=-\sum\limits_\alpha S_0^\alpha$$.

_vii_. $$\Delta \gamma^\alpha=f(\tau^\alpha, g^\alpha)$$  which gives $$\frac{\partial \Delta \gamma^\alpha}{\partial \tau^\alpha}$$.

_viii_. $$\tau^\alpha=C^eT_i:S_0^\alpha/J^e$$ which assuming $$C^e \approx 1$$ and  $$J^e \approx 1$$ gives $$\frac{\partial \tau^\alpha}{\partial T_i}=S_0^\alpha$$.

_ix_. Hence $$\frac{\partial T^\prime}{\partial T_i}=C: \frac{\partial E^e}{\partial F^e} \frac{\partial F^e}{\partial F^{p-1}} \frac{\partial F^{p-1}}{\partial \Delta \gamma_{res}}\frac{\partial \Delta \gamma_{res}}{\partial \Delta \gamma^\alpha}\frac{\partial \Delta \gamma^\alpha}{\partial \tau^\alpha}\frac{\partial \tau^\alpha}{\partial T_i}$$.

Jacobian J as implemented:

_i_. Variable dtaudpk2, $$\frac{\partial \tau^\alpha}{\partial T_i}$$.

_ii_. Variable dfpinvdslip, $$\frac{\partial F^{p-1}}{\partial \Delta \gamma^\alpha}$$.

_iii_. Variable dfedfpinv, $$\frac{\partial F^e}{\partial F^{p-1}}$$.

_iv_: Variable deedfe, $$\frac{\partial E^e}{\partial F^e}$$.

_v_. Variable dfpinvdpk2, $$\frac{\partial F^{p-1}}{\partial T_i}=\frac{\partial F^{p-1}}{\partial \Delta \gamma^\alpha} \frac{\partial \Delta \gamma^\alpha}{\partial \tau^\alpha} \frac{\partial \tau^\alpha}{\partial T_i}$$.

_vi_. Variable dslipdtau, $$\frac{\partial \Delta \gamma^\alpha}{\partial \tau^\alpha}$$ obtained from function **get\_slip\_incr**.

_vii_. Variable jac, $$J=C:\frac{\partial E^e}{\partial F^e} \frac{\partial F^e}{\partial F^{p-1}} \frac{\partial F^{p-1}}{\partial T_i}$$, followed by $$J=\mathfrak{I}-J$$ where $$\mathfrak{I}$$ is the fourth order identity tensor.

Update Cauchy stress (variable sig) $$\sigma=F^e T_i F^{eT}/J^e$$.

_**Function**_ - **update\_gss**:
User can override this function.

_Input_:

- Slip increment (variable slip\_incr, symbol $$\Delta \gamma^\alpha$$);

- Slip system resistance from previous increment (variable \_gss\_old, symbol $$g^\alpha_n$$);

- Current slip system resistance (variable \_gss, symbol $$g^\alpha$$);

- Hardness properties read from .i file (variables \_hprops, \_h0; symbols h,$$h_0$$);

- Saturation and initial slip system resistances read from .i file (variables \_tau\_sat, \_tau\_init, symbols $$\tau_s$$, $$\tau_0$$);

- Latent hardening variable read from .i file (variable \_r, symbol q).

_Output_:

- Current slip system resistance (variable \_gss, symbol $$g^\alpha$$).

Implementation:

_i_. Variable hb $$h_\beta=h_0 \left( 1-\frac{g^\beta}{\tau_s} \right)^a$$

_ii_. Update \_gss as $$g^\alpha=g^\alpha_n+\sum\limits_\beta q_{\alpha \beta}h_\beta$$ where $$q_{\alpha \beta}=q$$ for out of plane slip systems and $$q_{\alpha \beta}=1$$ for coplanar systems.

The hardening model provided in [Kalidindi et. al., J. Mech. Phys. Solids, Vol. 40, pp. 537-569, 1992] has been implemented.

_**Function**_ - **get\_slip\_incr**:
User can override this function.

_Input_:

- Resolved shear stresses (variable tau, symbol $$\tau^\alpha$$);

- Current slip system resistance (variable \_gss, symbol $$g^\alpha$$);

- Time increment (variable \_dt, symbol $$\Delta t$$);

- Flow properties read from .i file (variable \_a0, \_xm; symbols $$a_0, m$$).

_Output_:

- Slip increment (variable slip\_incr, symbol $$\Delta \gamma^\alpha$$);

- Derivative of slip increment w.r.t. resolved shear stress (variable dslipdtau, symbol $$\frac{\partial \Delta \gamma^\alpha}{\partial \tau^\alpha}$$).

Implementation:

_i_. Slip increment $$\Delta \gamma^\alpha=a_0\left|\frac{\tau^\alpha}{g^\alpha}\right|^\frac{1}{m}sign(\tau^\alpha)\Delta t$$.

_ii_. Derivative $$\frac{\partial \Delta \gamma^\alpha}{\partial \tau^\alpha} =\frac{a_0}{m g^\alpha}\left|\frac{\tau^\alpha}{g^\alpha}\right|^{\frac{1}{m}-1} \Delta t$$.

_**Function**_ - **get\_mat\_rot**:
Performs RU decomposition of F where R is orthogonal and U is symmetric postive definite.

_Input_:

- Second order positive definite tensor (variable a, symbol F).

_Output_:

- Orthogonal tensor (variable rot, symbol R).

Implementation:

_i_. Variable C, $$C=F^TF=U^TU=U^2$$. C has the same eigen vectors as U but the eigen values are $$\lambda_C=\lambda_U^2$$.

_ii_. Perform eigen value analysis of C using function **dsyev_** and obtain diagonal matrix $$D_C$$ of eigen values and eigen vector P.

_iii_. Construct diagonal matrix $$D_U=D_C^{1/2}$$.

_iv_. Obtain $$U=PD_UP^T$$.

_v_. Obtain $$R=FU^{-1}$$.

_**Function**_ - **get\_euler\_ang**:

_Input_:

- Euler angle input file name defined in .i file. Euler angle for every element must be prescribed as element no., euler 1, euler 2, euler 3.

- _Optional user input_:
Euler angles can also be read from \_euler\_angle\_1, \_euler\_angle\_2 and \_euler\_angle\_3 variables in FiniteStrainCrystalPlasticity material for every block. Default is 0.0, 0.0, 0.0.

_Output_:

- \_euler\_angle\_1, \_euler\_angle\_2 and \_euler\_angle\_3.

_**Function**_ - **computeQpElasticityTensor**:
Defines tangent moduli K and can be used as preconditioner for JFNK. User can override this function.

_Input_:

- Current plastic deformation gradient (variable \_fp, symbol $$F^p$$);

- Current deformation gradient (variable \_dfgrd, symbol $$F$$)

_Output_:

- Tangent moduli (variable \_Jacobian\_mult,symbol K).

Formulation:

_i_. $$K=\frac{\partial \sigma}{\partial F}$$.

_ii_. $$\sigma=F^eTF^{eT}/J^e$$ which gives $$\frac{\partial \sigma}{\partial F}=\frac{1}{J^e}\left(\frac{\partial F^e}{\partial F}TF^{eT}+F^e\frac{\partial T}{\partial F}F^{eT}+F^eT\frac{\partial F^{eT}}{\partial F}\right)$$.

_iii_. In indicial notation $$\frac{\partial F^e_{ij}}{\partial F_{kl}}=\delta_{ki}F^{p-1}_{lj}$$.

_iv_. $$T=C:E^e$$ which gives $$\frac{\partial T}{\partial F}=\frac{\partial T}{\partial E^e}\frac{\partial E^e}{\partial F^e}\frac{\partial F^e}{\partial F}=C:\frac{\partial E^e}{\partial F^e}\frac{\partial F^e}{\partial F}$$.

_v_. In indicial notation $$\frac{\partial E^e_{ij}}{\partial F^e_{kl}}=\frac{1}{2}\left(\delta_{il}F^e_{kj}+\delta_{jl}F^e_{ki}\right)$$.

Implementation:

_i_. Variable dfedf calculates $$\frac{\partial F^e}{\partial F}$$.

_ii_. Variable deedfe calculates $$\frac{\partial E^e}{\partial F^e}$$.

_iii_. Variable dsigspk2dfe calculates $$F^{eT} C:\frac{\partial E^e}{\partial F^e} F^e$$.