Hyperelastic constitutive law for solid elements at constant temperature
This law is only used for large strains analysis of rubber-like materials
Prepro: LRUBB.F
Lagamine: RUBB2S.F, RUBB2E.F, RUBB2A.F, RUBB3D.F
| Plane stress state | YES |
| Plane strain state | YES |
| Axisymmetric state | YES |
| 3D state | YES |
| Generalized plane state | NO |
| Line 1 (2I5, 60A1) | |
|---|---|
| IL | Law number |
| ITYPE | 6 |
| COMMENT | Any comment (up to 60 characters) that will be reproduced on the output listing |
| Line 1 (2I5) | |
|---|---|
| MODEL | = 0 : using Mooney-Rivlin model |
| = 1 : using Kilian model | |
| MFUN | = 0 : using penalty function : $ln I_3$ |
| = 1 : using penalty function : $I_3 - l$ | |
Remark: the strain energy density function W for Mooney-Rivlin model : \[ W = C_l (I_l - 3) + C_2 (I_2 - 3) \] For Kilian model : \[ W = G \{ 2 D_m \left[ \ln\left( l \frac{D_l}{D_m}\right) + \frac{D_l}{D_m} \right] + \frac{2}{3} a D_l^{\frac{2}{3}} \} \] \[ D_l = 0.5 (I_l - 3 ) \] ($I_1, I_2, I_3$ are the three strain invariants of the Cauchy-Green deformation tensor)
For Mooney-Rivlin model:
| Line 1 (3G10.0) |
|---|
| $C_1$ |
| $C_2$ |
| ANU |
For Kilian model (4G10.0):
| Line 1 (4G10.0) |
|---|
| $G$ |
| $D_m$ |
| ANU |
| a |
( the range of the Poisson's ratio : $0.489 < ANU < 0.5$ )
= 6 for the 3-D 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) | $\sigma_{zz}$ |
3
| Q(1) | = element thickness (t) in plane stress state |
| = 1 in plane strain state | |
| = circumferential strain rate ($\varepsilon_r$) in axisymmetrical state | |
| = 0 in 3‑D state | |
| = element thickness (t) in generalized plane state. | |
| Q(2) | hydrostatic pressure |
| Q(3) | the third strain invariant of the Cauchy-Green deformation tensor |