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Table of Contents
SGCP
Description
STRAIN GRADIENT CRYSTAL PLASTICITY CONSTITUTIVE
Implemented by S. Yuan 2017
The model
Mechanical analysis of strain gradient crystal plasticity problem
Keller, C., Habraken, A.M., Duchene, L., 2012a. Finite element investigation of size effects on the mechanical behavior of nickel single crystals. Mater. Sci. Eng. A 550, 342–349. https://doi.org/10.1016/j.msea.2012.04.085
Keller, C., Hug, E., Habraken, A.M., Duchene, L., 2012b. Finite element analysis of the free surface effects on the mechanical behavior of thin nickel polycrystals. Int. J. Plast. 29, 155–172. https://doi.org/10.1016/j.ijplas.2011.08.007
Bayley, C.J., Brekelmans, W.A.M., Geers, M.G.D., 2006. A comparison of dislocation induced back stress formulations in strain gradient crystal plasticity. Int. J. Solids Struct. 43, 7268–7286. https://doi.org/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) |
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
1813
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 ${(F^{B}_{e})}$ ${(F^{B}_{e})}^{-1}$ ${(F^{B})}^{-1}$ ${(F^{B}_{P})}$ |
