====== SGCP ======
===== Description =====
STRAIN GRADIENT CRYSTAL PLASTICITY CONSTITUTIVE LAW \\

Implemented by S. Yuan, L. Duchêne, 2017

==== The model ====
Mechanical analysis of strain gradient crystal plasticity problem \\
Evers, L.P., Brekelmans, W.A.M., Geers, M.G.D., 2004, J. Mech. Phys. Solids. 52, 2379-2401. doi: 10.1016/j.jmps.2004.03.007 \\
Evers, L.P., Brekelmans, W.A.M., Geers, M.G.D., 2004, Int. J. Solids Struct. 41, 5209-5230. doi: 10.1016/j.ijsolstr.2004.04.021 \\
Bayley, C.J., Brekelmans, W.A.M., Geers, M.G.D., 2006, Int. J. Solids Struct. 43, 7268–7286. doi: 10.1016/j.ijsolstr.2006.05.011 \\
==== Files ====
Prepro: T151_V3.F \\
Lagamine: T152_V3.F \\
===== Availability =====
|Plane stress state| NO |
|Plane strain state| NO |
|Axisymmetric state| NO |
|3D state| YES |
|Generalized plane state| NO |
===== Input file (COLAW==>*.lag) =====
==== Parameters defining the type of constitutive law ====
^ Line 1 (2I5, 60A1)^^
|IL|Law number|
|ITYPE| 973|
|COMMENT| Any comment (up to 60 characters) |
^ Line 2 (2I5, 8G10.0)^^
|INRM|iterative method (0, 1, 2)|
|MAXIT|   |
|PREC|   |
|damp1|   |
|damp2|   |
if (INRM .eq. 0) AMOR=damp1=1;
if (INRM .eq. 1) $h_k$=damp1 & $\nu(NS)$=damp2;
if (INRM .eq. 2) damping is computed automatically

==== Real parameters ====
^ Line 1 (3G10.0) ^^
|$C_{11}$| $4^{th}$ order anisotropic elastic tensor component|
|$C_{12}$| $4^{th}$ order anisotropic elastic tensor component|
|$C_{44}$| $4^{th}$ order anisotropic elastic tensor component|
^ Line 2 (5G10.0) ^^
|$\dot{\gamma}_{0}$| reference plastic strain rate|
|$m$| rate sensitivity exponent of the original power-law function|
|$G_{0}$| total free energy needed to move a dislocation to overcome a short-range barrier without external work aid|
|$k$| Boltzmann’s constant|
|$T$| absolute temperature|
^ Line 3 (3G10.0) ^^
|$c$| material constant|
|$\mu$| shear modulus|
|$b$| length of Burgers vector|
^ Line 4 (6G10.0) ^^
|$a_{0}$| interactions coefficient: self-hardening|
|$a_{1}$| interactions coefficient: coplanar system|
|$a_{2}$| interactions coefficient: systems pair leading to Glissile junctions formation|
|$a_{3}$| interactions coefficient: systems pair leading to Lomer-Cottrell sessile locks|
|$a_{4}$| interactions coefficient: collinear system|
|$a_{5}$| interactions coefficient: Hirth-Lock system pair with normal slip directions|
^ Line 5 (3G10.0) ^^
|$y_{c}$| critical annihilation length|
|$\rho_{{SSD}_{0}}$| initial SSD density|
|$K$| dislocation segments length constant in average dislocation segment length of mobile dislocations (SSDs) on system α function|
^ Line 6 (6G10.0) ^^
|$h_{0}$| interactions coefficient|
|$h_{1}$| interactions coefficient|
|$h_{2}$| interactions coefficient|
|$h_{3}$| interactions coefficient|
|$h_{4}$| interactions coefficient|
|$h_{5}$| interactions coefficient|
^ Line 7 (3G10.0) ^^
|$R_{e}$| radius of edge dislocation field|
|$R_{s}$| radius of edge dislocation field|
|$\nu$| Poisson’s coefficient|
^ Line 8 (3G10.0) ^^
|$\Phi_{1}$| Euler rotation angle|
|$\Phi$| Euler rotation angle|
|$\Phi_{2}$| Euler rotation angle|
^ Line 9 (2I5) ^^
|$NS$| number of slip systems|
|$ND$| number of dislocation types|
^ Line 10 (3G10.0) ^^
|$UNIT$| PAR_UNIT|
|$UNIT2$| PAR_UNIT2|
|$UNIT3$| PAR_UNIT3|
^ Line 11 (2G10.0) ^^
|$\rho_{{GND}_{0}}$| initial GND density|
|$E$| Young’s modulus (not used in the model)|
===== Stresses =====
==== Number of stresses ====
6 for 3D state
==== Meaning ====
The stresses are the components of CAUCHY stress tensor in global (X,Y,Z) coordinates. \\
For the 3-D state:
|SIGMB(1)|$\sigma_{11}$|
|SIGMB(2)|$\sigma_{22}$|
|SIGMB(3)|$\sigma_{33}$|
|SIGMB(4)|$\sigma_{12}$|
|SIGMB(5)|$\sigma_{13}$|
|SIGMB(6)|$\sigma_{23}$|
===== State variables =====
==== Number of state variables ====
1839 × 8 (integration points)
==== List of state variables ====
|Q(1:9)|${(F^{B}_{P})}^{-1}$|
|Q(10:21)|$\dot{\gamma}^\alpha$|
|Q(22:33)|${\gamma}^\alpha$|
|Q(34:45)|$\rho^{\xi}_{SSD}$|
|Q(46:54)|$1^{th}$ Piola-Kirchhoff stress $P_{11} P_{22} P_{33} P_{12} P_{13} P_{23} P_{21} P_{31} P_{32}$|
|Q(55:72)|reserved for $\rho^{\xi}_{GND}$|
|Q(73:126)|reserved for $r^{\xi}_{i}$|
|Q(127:135)|reserved for $F_{ij}$|
|Q(136:297)|$(\frac{d\sigma_{ij}}{d\rho^{\xi}_{GND}})_{ij\xi}$|
|Q(298:783)|$(\frac{d\sigma_{ij}}{dr^{\xi}_{k}})_{ijk\xi}$|
|Q(784:981)|$(\frac{d\gamma^{\alpha}}{dF_{kl}})_{\alpha kl}$|
|Q(982:1107)|$(\frac{d\gamma^{\alpha}}{d\rho_{GND}^{\xi}})_{\alpha \xi}$|
|Q(1108:1755)|$(\frac{d\gamma^{\alpha}}{dr_{k}^{\xi}})_{\alpha k \xi}$|
|Q(1756:1764)|$2^{nd}$ Piola-Kirchhoff stress $S_{11} S_{22} S_{33} S_{12} S_{13} S_{23} S_{21} S_{31} S_{32}$|
|Q(1765:1770)|$\epsilon^{u}$ nature strain $e_{11} e_{12} e_{13} e_{22} e_{23} e_{33} $|
|Q(1771:1773)|$\gamma^{norm} \rho_{SSD}^{norm} \rho_{GND}^{norm}$|
|Q(1774:1776)|Blank|
|Q(1777)|Blank but resreved for $\epsilon_{eq}^{p}$|
|Q(1778:1813)|reserved for ${(F^{B}_{e})}$ ${(F^{B}_{e})}^{-1}$ ${(F^{B})}^{-1}$ ${(F^{B}_{P})}$|
|Q(1814:1825)|Schmid stress ($\alpha=1..12$)|
|Q(1826)|Schmid stress (square root norm)|
|Q(1827:1838)|back-stress ($\alpha=1..12$)|
|Q(1839)|back-stress (square root norm)|