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EP-ANI
Description
Elasto-plastic constitutive law of anisotropic materials for solid elements at constant temperature (Zhu)
The model
Mechanical analysis of orthotropic elasto-plastic solids (Hill's model) undergoing large strains. Mixed hardening is assumed.
Remark: Take care of using local axes (defined in the element) with this law.
Files
Prepro: LHIL.F
Lagamine: ANI2D.F (METH=0), HIL2E.F, HIL2A.F
Availability
| Plane stress state | YES |
| Plane strain state | YES |
| Axisymmetric state | YES |
| 3D state | NO |
| Generalized plane state | YES |
Input file
Parameters defining the type of constitutive law
| Line 1 (2I5, 60A1) | |
|---|---|
| IL | Law number |
| ITYPE | 60 |
| COMMENT | Any comment (up to 60 characters) that will be reproduced on the output listing. |
Integer parameters
| Line 1 (3I5) | |
|---|---|
| NINTV | number of sub-steps used to integrate numerically the constitutive equation in a time step, if -1, automatic calculation |
| NPOINT | number of point to define the hardening law. = 0, linear anisotropic hardening > 0, nonlinear isotropic hardening |
| METH | method to be used for the time integration of the constitutive law = 0, using the “backward Euler” method (recommended). It is only valuable for the isotropic elastic - anisotropic plastic material with anisotropic linear or isotropic nonlinear hardening. = 1, using the “radial return” method. It is valuable for the anisotropic elastoplastic material with anisotropic linear hardening. |
Real parameters
| Line 1 (3G10.0) | |
|---|---|
| ALPHA | angle between the 1-2 principal axes of material and X-Y axes of coordinates |
| ECROU | parameter for mixed hardening = 0 isotropic hardening = 1 kinematic hardening > 0 and < 1 mixed hardening |
| THICK | |
1. For METH = 1 only
| Line 1 (4G10.0) | |
|---|---|
| E1 | YOUNG's modulus in 1st direction |
| $\nu_{12}$ | POISSON's ratio in 1-2 plane |
| SIG1Y | Yield limit of uniaxial tension in 1st direction |
| ET1 | elasto-plastic tangent modulus in 1st direction |
| Line 2 (4G10.0) | |
| E2 | YOUNG's modulus in 2nd direction if $\leq$ 0, E2 = E1 |
| $\nu_{23}$ | POISSON's ratio in 2-3 plane if $\leq$ 0, $\nu_{23}$ = $\nu_{12}$ |
| SIG2Y | Yield limit of uniaxial tension in 2nd direction if $\leq$ 0, SIG2Y = SIG1Y |
| ET2 | elasto-plastic tangent modulus in 2nd direction, if < 0, ET2 = ET1 |
| Line 3 (4G10.0) | |
| E3 | YOUNG's modulus in 3rd direction if $\leq$ 0, E3 = E2 |
| $\nu_{13}$ | POISSON's ratio in 1-3 plane if $\leq$ 0, $\nu_{13}$ = $\nu_{23}$ |
| SIG3Y | Yield limit of uniaxial tension in 3rd direction if $\leq$ 0, SIG3Y = SIG2Y |
| ET3 | elasto-plastic tangent modulus in 3rd direction, if < 0, ET3 = ET2 |
| Line 4 (4G10.0) | |
| G12 | shear elastic modulus in 1-2 plane if $\leq$ 0, G12 = E1/(1+$<_{12}$) |
| zero | |
| SIG12Y | Yield limit of shear in 1-2 plane, if $\leq$ 0, SIG12Y = $\frac{SIG1Y}{\sqrt{3}}$ |
| GT12 | shear elastoplastic tangent modulus in 1-2 plane |
2. For METH = 0
| Line 1 (2G10.0) |
|---|
| E, $\nu$ |
1. If NPOINT = 0
| Line 1 (2G10.0) |
|---|
| SY11, ET11 |
| Line 2 (2G10.0) |
| SY22, ET22 |
| Line 3 (2G10.0) |
| SY33, ET33 |
| Line 4 (2G10.0) |
| SY12, ET12 |
2. If NPOINT > 0
| Line 1 (4G10.0) |
|---|
| SY11, SY22, SY33, SY12 |
| Line 2 (NPOINT*(2G10.0)) |
| EPS, SIG |
Remarks
For METH = 0 (method implemented in 1992), the sub-routine in LAGAMINE is ANI2D(law number 60 in the code). This sub-routine contains the plane strain, the plane stress and the axisymmetrical states. The tangent stiffness matrix can be computed either analytically or with the perturbation method according to the parameter ISTRA(4) in the execution data file. The hardening method can be either anisotropic linear (NPOINT = 0), with a yield locus shape evolving according to the hardening state, or isotropic non-linear (NPOINT > 0), with a constant shape of the yield locus.
For METH = 1 (implemented in 1989), the sub-routine in LAGAMINE is HIL2A (law number 62 in the code) for the axisymmetrical state and HIL2E (number 61) for the plane strain state. The plane stress state is not available. The tangent stiffness matrix is only computed numerically with the perturbation method (the parameter ISTRA(4) is not used with METH = 1). The hardening method is anisotropic linear, i.e. an energy based evolving yield locus shape with a linear growing size.
Stresses
Number of stresses
4
Meaning
The stresses are the components of CAUCHY stress tensor in global (X,Y,Z) coordinates.
| SIG(1) | $\sigma_{xx}$ |
| SIG(2) | $\sigma_{yy}$ |
| SIG(3) | $\sigma_{xy}$ |
| SIG(4) | $\sigma_{zz}$ |
State variables
Number of state variables
19
List of state variables
| Q(1) | = element thickness (t) in plane stress state or in generalized plane state = 1 in plane strain state circumferential strain rate ($\epsilon_{\theta}$) in axisymmetrical state |
| Q(2) | = current yield limit in tension; its initial value in $\overline{\sigma_{y}} |
| Q(3) | = 0 if the current state is elastic = 1 if the current state is elasto-plastic |
| Q(4) | = equivalent total plastic strain |
| Q(5) | = plastic work |
| Q(6) $\rightarrow$ Q(15) | anisotropic materials coefficients for yield function |
| Q(16) | = $\alpha_{1}$ the back stress |
| Q(17) | = $\alpha_{2}$ the back stress |
| Q(18) | = $\alpha_{12}$ the back stress |
| Q(19) | = $\alpha_{3}$ the back stress |
