STRAIN GRADIENT CRYSTAL PLASTICITY CONSTITUTIVE LAW
Implemented by S. Yuan, L. Duchêne, 2017
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
Prepro: T151_V3.F
Lagamine: T152_V3.F
| Plane stress state | NO |
| Plane strain state | NO |
| Axisymmetric state | NO |
| 3D state | YES |
| Generalized plane state | NO |
| 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
| 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) |
6 for 3D state
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) | $\sigma_{23}$ |
1839 × 8 (integration points)
| 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) |