ELASTO VISCO PLASTIC CONSTITUTIVE LAW FOR SOLID ELEMENTS AT VARIABLE TEMPERATURE (Norton-Hoff)
Implemented by: Pascon F (1998), Charles JF (1997 - 1999)
Project: continuous casting research for ARBED (RW2748)
Coupled dynamic recrystallisation-thermo-mechanical analysis of elasto-visco-plastic solids undergoing large strains.
JAUMANN stress rate is used
IANA= 2, 3, 5:
See intermediate report RW2748 (1, 8, 17, 24) and intermediate report of April 1998
For details on equations used in analytical compliance matrix computation, see appendix D of April 1998
IANA= 4:
See intermediate report RW2748 (17, 24)
Prepro: LNHC2.F
Lagamine: NHIC2E.F (IANA= 2, 3 or 5) or NHIC3D.F (IANA= 4)
| File | Subroutine | Description |
|---|---|---|
| CALMAT.F | CALMAT | Computes material data at temperature T |
| NHIMAT.F | CALSIGY | |
| MATMSGS2 | Used for analytical compliance matrix | |
| MATMSGL2 | Used for analytical compliance matrix | |
| MATMSGS | Used for analytical compliance matrix (3D case) | |
| MATMSGL | Used for analytical compliance matrix (3D case) | |
| EIGVECT | Computes eigen vectors | |
| CMATINV | Inverse complex matrix | |
| VGMOYEN | Computes the constant velocities gradient matrix | |
| CALPNH.F | CALPNH | Computes $K_0, P_1, P_2, P_3, P_4$ at temperature T |
| RECRYDYN.F | RECRYDYN | Dynamic recrystallization computation |
| Plane stress state | NO |
| Plane strain state | YES |
| Axisymmetric state | YES |
| 3D state | YES |
| Generalized plane state | YES |
| 1 Line (2I5, 60A1) | |
|---|---|
| IL | Law number |
| ITYPE | 270 |
| COMMENT | Any comment (up to 60 characters) that will be reproduced on the output listing |
| 1 Line (7I5) | |
|---|---|
| NINTV | number of sub-steps used to integrate numerically the constitutive equation in a time step. If NINTV < 0 or = 0, then the number of sub-steps will be computed automatically |
| NTEMP | number of temperatures at which material data (E, ANU and ALPHA) are given |
| IDYN | = 1: if recrystallisation computation |
| = 0: else | |
| ICHP2 | = 2: if parameters $K_0, P_1, P_2, P_3, P_4$ are given at several temperatures |
| = 1: if $P_2= e^{-(\frac{T-C_4}{C_5})}.T^{C_6}$ (only if nodes temperature in Kelvin !!!) | |
| any other value if $P_2= (\frac{C_4}{T})^2 - \frac{C_5}{T} + C_6$ (only if nodes temperature in Kelvin !!!) | |
| IALG | = 1: if enthalpic formulation for ALPHA |
| = 0: if classical formulation for ALPHA | |
| MAXITER | maximum number of iteration in elastic field ≤ 0: set default value = 50 |
| NTEMP2 | number of temperatures at which parameters $K_0, P_1, P_2, P_3, P_4$ are given (only if ICHP2 = 2) |
| 1 Line repeated NTEMP times (4G10) Note: parameters introduced by increasing temperature order |
|
|---|---|
| T | Temperature |
| E | YOUNG’s elastic modulus at temperature T |
| ANU | POISSON’s ratio at temperature T |
| ALPHA | Thermal expansion coefficient (α) at temperature T. Even if IALG = 1, ALPHA must be introduced at temperature T. In this case, $\int_0^T\alpha(T).dT$ will be automatically computed |
| If ICHP2 = 2: 1 Line repeated NTEMP2 times (6G10) Note: parameters introduced by increasing temperature order |
|
|---|---|
| T | Temperature |
| $K_0$ | See further |
| $P_1$ | See “Information about EVP-NH” |
| $P_2$ | |
| $P_3$ | |
| $P_4$ | |
| If ICHP2 ≠ 2: 2 Lines (5G10/4G10) (only if nodes temperature in Kelvin !!!) | |
|---|---|
| $AK_0$ | |
| $C_1$ | See further |
| $C_2$ | See “Information about EVP-NH” |
| $C_3$ | |
| $C_4$ | |
| $C_5$ | |
| $C_6$ | |
| $P_3$ | (be careful: 0 < $P_3$ < 1) |
| $P_4$ | |
| 1 Line (4G10) | |
|---|---|
| TQ | Taylor-Quinney’s coefficient. Absolute value between 0 and 1 : < 0: when thermal analysis within a semi-coupled analysis > 0: for other cases (total coupled analysis or mechanical analysis within a semi-coupled analysis) |
| PRECVG | precision in VGMOY calculation (3D state only) ≤ 0: set default value = $1.10^{-5}$ |
| PRECELA | precision in elastic computation ≤ 0: set default value = $1.10^{-4}$ |
| EPSINC | increment of deformation for the automatic computation of NINTV ≤ 0: set default value = $1.10^{-3}$ |
| If IDYN = 1: 4 Lines (3I5/4G10.0/4G10.0/2G10.0) | |
|---|---|
| ICOUPL | = 1: the recrystallisation is coupled |
| = 0: the recrystallisation is uncoupled | |
| ITYPEPS | = 0: the equations defining the beginning and the end of the recryst. have the form : $\varepsilon= Q_1. Q_4^{Q_2} . [LN(Zener)]^{Q_3}$ |
| = 1: the equations defining the beginning and the end of the recryst. have the form : $\varepsilon= Q_1.ATAN[Q_3.[LN(Zener)-Q_2]]+ Q_4$ | |
| = 2: the equations defining the beginning and the end of the recryst. have the form : $\varepsilon= Q_1.[LN(Zener)]^{Q_2}+ Q_3.LN(Zener) + Q_4$ | |
| NSSMAX | used if ICOUPL = 1: Maximum number of sub-structures The precision on the recryst. fraction is 1/NSSMAX |
| $Q_1$ | parameters for the beginning of the recrystallisation: $\varepsilon_c$ |
| $Q_2$ | parameters for the beginning of the recrystallisation: $\varepsilon_c$ |
| $Q_3$ | parameters for the beginning of the recrystallisation: $\varepsilon_c$ |
| $Q_4$ | parameters for the beginning of the recrystallisation: $\varepsilon_c$ |
| $Q_1$ | parameters for the end of the recrystallisation: $\varepsilon_s$ |
| $Q_2$ | parameters for the end of the recrystallisation: $\varepsilon_s$ |
| $Q_3$ | parameters for the end of the recrystallisation: $\varepsilon_s$ |
| $Q_4$ | parameters for the end of the recrystallisation: $\varepsilon_s$ |
| ACTIV | Activation energy for Zener computation : $Z=\dot{\varepsilon}.EXP(\frac{ACTIV}{R.T})$ with T the temperature and R the Boltzman gas constant |
| EXPO | Exponent for the AVRAMI law : $X= 1- EXP[-3.(\frac{\bar{\varepsilon}-\varepsilon_c}{\varepsilon_s-\varepsilon_c})^{expo}]$ |
NOTE: ISTRA(3) parameter of the execution file:
Units:
= 0: analytical compliance matrix used (default value)
= 1: perturbation method
Tens:
= 0: mean velocities gradient (default value)
= 1: initial velocities gradient
Hundreds:
= 0: yield limit given by intersection between N-H curve and Young’s straight line
= 1: yield limit given by K0 (given parameter – see below)
Information about EVP-NH:
For the 1D case, we have:
$\bar{\sigma}= A. K_0. \bar{\varepsilon}^{P_4}. exp(-P_1.\bar{\varepsilon}). P_2. \sqrt{3}. (\sqrt{3}. \bar{\dot{\varepsilon}})^{P_3}$ with $P_1 \geq 0$
The parameters $K_0, P_1, P_2, P_3, P_4$ can be given at several temperatures (ICHP2 = 2)
Otherwise, if ICHP2 ≠ 2: (see the law in section: “integer parameters”)
$P_1= (\frac{T}{C_1})^{C_2} + C_3$
$P_2= f(C_4, C_5, C_6, T)$
$P_3, P_4, K_0= constants$
6 for 3D state
4 for the other cases
The stresses are the components of CAUCHY stress tensor in global (X,Y,Z) coordinates.
For the 3-D state:
| SIG(1) | $\sigma_{xx}$ |
| SIG(2) | $\sigma_{yy}$ |
| SIG(3) | $\sigma_{zz}$ |
| SIG(4) | $\sigma_{xy}$ |
| SIG(5) | $\sigma_{xz}$ |
| SIG(6) | $\sigma_{yz}$ |
For the other cases:
| SIG(1) | $\sigma_{xx}$ |
| SIG(2) | $\sigma_{yy}$ |
| SIG(3) | $\sigma_{xy}$ |
| SIG(4) | $\sigma_{zz}$ |
27
| Q(1) | thickness |
| Q(2) | equivalent stress (effective if icoupl=1) |
| Q(3) | equivalent strain |
| Q(4) | equivalent strain rate |
| Q(5) | instantaneous thermal flow (effective if icoupl=1) |
| Q(6) | plastic dissipation (effective if icoupl=1) |
| Q(7) | ΔT |
| Q(8) | RHOC capacity |
| Q(9) | LN (ZENER) |
| Q(10) | recrystallised fraction since the beginning of the simulation |
| Q(11) | recrystallised fraction on this step |
| Q(12) | elastic part on this step – in percent (>0 : loading ; <0 : unloading) (effective if icoupl=1) |
| Q(13) | number of sub-structures |
| Q(14) | volumic fraction of the unrecrystallised sub-structure |
| Q(15) | effective equivalent strain |
| Q(16) | equivalent strain standard deviation |
| Q(17) | = 0 if always elastic state since the beginning |
| = 1 if any previous step has been performed in visco-plastic domain | |
| Q(18) | recrystallised fraction during previous step |
| Q(19) | |
| Q(20) | |
| Q(21) | |
| Q(22) | |
| Q(23) | |
| Q(24) | |
| Q(25) | triaxiality (BLZ2T) |
| Q(26) | shape parameter of the element (BLZ2T) |
| Q(27) | Remeshing parameter (BLZ2T) |