Relationship META_LEMA_ANI in Code_Aster
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Generalities
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The model presented is elastoviscous, without threshold (the elastic limit is zero), taking into account the metallurgical transformations of this material (described in document R4.04.04) and taking into account the anisotropy of the phase. Viscosity is described by a Lemaître law.

The model is introduced in*Code_Aster* in 3D, plane deformations (D_ PLAN), and axisymmetry (AXIS) under the name META_LEMA_ANI.

Anisotropy is taken into account by a fourth-order tensor (Hill matrix :math:`M`) affecting the laws of evolution of viscous deformation and the equivalent stress (Von Mises stress in the Hill sense).

The speed equations are integrated numerically by an implicit Euler diagram. The system obtained is solved by Newton's method.




Restriction on the use of the model
-----------------------------------

The equations of the model can be written either in Cartesian coordinates, or in the cylindrical coordinate system associated with tube :math:`(1\mathrm{=}{e}_{r},2\mathrm{=}{e}_{\theta }\mathrm{,3}\mathrm{=}z)`. This is because the coefficients of the Hill matrix, :math:`M`, are known in this coordinate system.

At the implementation level in *code_aster*, in this case, we change the variables of the tensor fields (another choice would have been to change the variable to the Hill :math:`M` tensor, but it is simpler to proceed the other way around)


* For a 3D calculation or in plane deformation, the stress tensor known in the global coordinate system :math:`(1\mathrm{=}x,2\mathrm{=}y,3\mathrm{=}z)` is transformed into the local coordinate system :math:`(1\mathrm{=}{e}_{r},2\mathrm{=}{e}_{\theta }\mathrm{,3}\mathrm{=}z)`;


* For a 2D, axisymmetric calculation, the stress tensor known in the global coordinate system :math:`(1\mathrm{=}{e}_{r},2\mathrm{=}z\mathrm{,3}\mathrm{=}{e}_{\theta })` is calculated in the local coordinate system :math:`(1\mathrm{=}{e}_{r},2\mathrm{=}{e}_{\theta }\mathrm{,3}\mathrm{=}z)`; in this case, the change of variables is simple since it is only a question of inverting the indices 2 and 3.

**Limitation: it was assumed that the axis** :math:`z` **of the cylindrical coordinate system associated with the tube corresponded to that of the global coordinate system.** **If several tubes must be modelled or if the axis of the tube does not correspond to that of the global coordinate system,** **it is currently necessary to use a capitalized script in the hsnv134b test case to take this difference into account.**




Use
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In operator STAT_NON_LINE, this mechanical model is accessed using the keyword RELATION = 'META_LEMA_ANI' in the keyword factor COMPORTEMENT.

The material data relating to model META_LEMA_ANI is entered in the DEFI_MATERIAU operator using the key words factor META_LEMA_ANI.

**Note**: **the Hill matrices for the phases** :math:`\alpha` **and** :math:`\beta` **are given in the cylindrical coordinate system** :math:`(1\mathrm{=}{e}_{r},2\mathrm{=}{e}_{\theta }\mathrm{,3}\mathrm{=}z)` **, even for an axisymmetric 2D calculation where the indices 2 and 3 are interchanged.**




Internal variables
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The internal variables of model META_LEMA_ANI are:

:math:`V1\to \mathit{VN}`: the components of the symmetric tensor of elastic deformations (:math:`N` is 6 in 3D and 4 in 2D)

:math:`\mathit{VN}+1`: cumulative viscous deformation :math:`p`

:math:`\mathit{VN}+2`: beta phase proportion :math:`{Z}_{\mathrm{\beta }}`

:math:`\mathit{VN}+3`: thermal deformation :math:`{\mathrm{\epsilon }}^{\mathit{th}}`

:math:`\mathit{VN}+4`: equivalent Hill stress :math:`{A}_{\mathit{eq}}`

:math:`\mathit{VN}+5,+6,+7`: viscous stresses :math:`{\mathrm{\sigma }}_{v1}`, :math:`{\mathrm{\sigma }}_{v2}` and :math:`{\mathrm{\sigma }}_{v3}`, respectively from phases :math:`\alpha` pure, :math:`\alpha \beta` and :math:`\beta`

:math:`\mathit{VN}+8`: phase change indicator (0 or 1)

:math:`\mathit{VN}+9`: moment at which the temperature is set to TDEQ (initialized to 0 at the start of the calculation)

:math:`\mathit{VN}+10`: moment at which the temperature is set to TFEQ (initialized to 0 at the start of the calculation)