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HILSH
Description
Anisotropic elastoplastic constitutive law, numerically integrated on thickness
For:
in generalized plane state
Implemented by: L. Grisard (1991)
The model
This law is only used for mechanical analysis of elastoplastic anisotropic thin bodies (membranes and shells) in generalized plane state.
Local stresses $\sigma_x$ and $\sigma_z$ are computed at some points across the thickness, and numerically integrated to obtain a law linking global stresses (shear and normal efforts, moments,…) to global strains (membrane shear and bending strains). Locally, the anisotropic HILL criteria in plane stress state is assumed :
\[
\sqrt{\alpha_1 \sigma_x^2 + \alpha_2 \sigma_z^2 - \alpha_{12} \sigma_x \sigma_z + 3 \alpha_z \tau^2} \leq \sigma_0
\]
With
| $\alpha_1 = 1$ | $\alpha_3 = 1$ | $R_x = \frac{1 + r_x}{2 r_x}$ |
| $\alpha_2 = R_z / R_X$ | $\alpha_{12} = 1/R_x$ | $R_x = \frac{1 + r_z}{2 r_z}$ |
Coefficients $r_x$ and $r_z$ are classical parameters which can take into account the anisotropic behaviour, via $\alpha_i$’s coefficients. $r_x$ and $r_z$ have the following physical meaning :
$r_x = \frac{d\varepsilon_z^p}{d\varepsilon_y^p}$ when we do a tensile test in the local x‑direction
$r_z = \frac{d\varepsilon_x^p}{d\varepsilon_y^p}$ when we do a tensile test in the local z‑direction
$\varepsilon_y^p$ is, here, the plastic strain in the thickness direction. Isotropic hardening is assumed for $\sigma_0$, taken into account by an equivalent strain $\bar{\varepsilon}^p$.
Files
Prepro: LHILSH.F
Lagamine: HILSH.F
Availability
| Plane stress state | NO |
| Plane strain state | NO |
| Axisymmetric state | NO |
| 3D state | NO |
| Generalized plane state | YES |
Input file
Parameters defining the type of constitutive law
Integer parameters
| Line 1 (3I5) | |
|---|---|
| NPI | number of integration points across the thickness (less than or equal to 10) |
| N | number of points defining the uniaxial constitutive law $\sigma_0(\varepsilon)$ (2 or greater) |
| ITAU | = 0 if shear stress is not taken into account in plasticity |
| = 1 if shear stress is taken into account in plasticity (this parameter has no meaning if LTYPE is not equal to 22). | |
Remarks:
- A trapezoidal rule is assumed for the integration across the thickness. This rule has been performed for moments.
- The integration points are equally spaced across the thickness, with a point on both sides (upper and lower layers).
- The constitutive law will be stored in central memory. So, we must have the following condition : $2\times N + 6 \leq MPARA$
- For a membrane law (ITYPE = 20), NPI is always equal to one.
Real parameters
| Line 1 - Global Data (3G10.0) | |
|---|---|
| ANU | Poisson's coefficient |
| RT | $r_x$ anisotropic coefficient (default = 1) |
| RL | $r_z$ anisotropic coefficient (default = RT) |
| Line 2:N+1 - Uniaxial Constitutive Law (2G10.0) | |
| SIG | $\sigma_0$ value at the considered point |
| EPS | $\varepsilon$ value at the considered point |
Warning: The first point is the end of the elastic behaviour.
After the last point, tangent modulus is kept.
Results
Stresses
| SIG(1) | N1, normal effort in x‑direction |
| SIG(2) | N2, normal effort in z‑direction |
| SIG(3) | M1, moment associated to the x‑direction |
| SIG(4) | M2, moment associated to the z‑direction |
| SIG(5) | T, shear effort |
State variables
| Q(1) | $b$, out‑of‑plane thickness (along z‑direction), according to the generalized plane state |
| Q(2) | $e$, in‑plane thickness of the shell (updated at the end of each step) |
| Q(3) | $W_t$, total work associated to this integration point (by unit of shell area) |
| Q(4) | $W_p$, plastic work associated to this integration point (by unit of shell area) |
| Q(5) | $\bar{\varepsilon}_0^p$, equivalent membrane plastic strain |
| Q(6) | $\bar{\chi}^p$, equivalent bending plastic strain (curvature) |
| Q(7) | $N_p$, membrane yield stress |
| Q(8) | $M_p$, bending yield stress |
For i = 1 to NPI :
| Q(9+5(i‑1)) | $\sigma_x$, local stress in x‑direction at the $i^e$ integration point across the thickness e |
| Q(10+5(i‑1)) | $\sigma_z$, local stress in z‑direction at the $i^e$ integration point across the thickness e |
| Q(11+5(i‑1)) | $\tau$, local shear stress at the $i^e$ integration point across the thickness e |
| Q(12+5(i‑1)) | $\sigma_0$, local yield stress at the $i^e$ integration point across the thickness e |
| Q(13+5(i‑1)) | $\bar{\varepsilon^p}$, local equivalent plastic strain at the $i^e$ integration point across the thickness e. |
The total number of state variables is so : 8 + 5 * NPI.
Remarks : the shear stress $\tau$ is only stored in Q in the case of a MINDLIN shell behaviour. In the other cases, only four parameters ($\sigma_x$, $\sigma_z$, $\sigma_0$, $\bar{\varepsilon}$) are stored in Q at each integration point, and the total number of state variables is so : 8 + 4 * NPI.
