Methodology
============

It is a double simulation, the first in thermomechanics, the second in pure mechanics. The first will be validated in comparison with the second, assuming of course that the behavior tested provides a correct solution in pure mechanics.

The first simulation (thermo-mechanical solution that we are trying to validate) consists in applying a temperature variation to a material point, with zero imposed deformation along axis :math:`x`


.. math::
 :label: eq-1

    {\ mathrm {\ epsilon}}} _ {\ mathit {xx}} ^ {1} =0


The imposed temperature increases linearly as a function of time. The transient consists of NCALpas .According to the models:


* Models :math:`A`, :math:`C`, :math:`E`, :math:`F`: the temperature varies from :math:`{T}_{0}=0°C` to :math:`{T}_{\mathit{max}}=500°C` and the reference temperature is :math:`{T}_{\mathit{ref}}=0°C`.


* Models :math:`B`, :math:`G`: the temperature varies from :math:`{T}_{0}=20°C` to :math:`{T}_{\mathit{max}}=500°C` and the reference temperature is :math:`{T}_{\mathit{ref}}=20°C`.


* Modeling :math:`H`: the temperature varies from :math:`{T}_{0}=20°C` to :math:`{T}_{\mathit{max}}=800°C` and the reference temperature is :math:`{T}_{\mathit{ref}}=20°C`.


* Modeling :math:`I`: the temperature varies from :math:`{T}_{0}=20°C` to :math:`{T}_{\mathit{max}}=200°C` and the reference temperature is :math:`{T}_{\mathit{ref}}=20°C`.


* Modeling :math:`J`: the temperature varies from :math:`{T}_{0}=700°C` to :math:`{T}_{\mathit{max}}=1000°C` and the reference temperature is :math:`{T}_{\mathit{ref}}=700°C`.

The second simulation (which must be equivalent to the first) consists in applying to the same material point a deformation imposed according to :math:`x` such that:


.. math::
 :label: eq-2

    {\ mathrm {\ epsilon}} _ {\ mathit {xx}}} ^ {x}}} ^ {2} =- {\ mathrm {\ epsilon}}} ^ {\ mathit {th}}} =-\ mathrm {\ alpha} (T) (T- {T}} _ {\ mathit {ref}})


in pure mechanics on the NCALinstants of thermomechanical calculations. For each calculation :math:`i`, the imposed load consists of the thermal deformation


.. math::
 :label: eq-3

    {\ mathrm {\ epsilon}} _ {\ mathit {xx}}} ^ {x}}} ^ {2} =- {\ mathrm {\ epsilon}} ^ {\ mathit {th}}} =-\ mathrm {\ alpha} ({\ alpha}} ({\ alpha}} ({T}}} alpha} ({T}}) ({T} _ {i}) ({T} _ {i})


The initial load consists of the deformations, stresses and internal variables of the previous mechanical calculation, the stresses being corrected for the variation in Young's modulus.

Indeed, for any behavior (assuming the additive decomposition of deformations):


.. math::
: label: eq-4

    {\ sigma} _ {\ mathrm {xx}} =E (T) (T) ({\ varepsilon} _ {\ mathrm {xx}} - {\ varepsilon}} ^ {\ mathrm {th}}} = E (T) = E (T) (T) ({\ varepsilon} _ {\ mathrm {xx}}} ^ {p})


In the first case:


.. math::
: label: eq-5

    {\ mathrm {\ sigma}} _ {\ mathit {xx}}} ^ {xx}}} ^ {xx}}} ^ {\ mathit {th}}} - {\ mathit {th}}} - {\ mathrm {\ epsilon}}} _ {\ mathit {xx}} ^ {th}}} - {\ mathrm {\ epsilon}}} - {\ mathrm {\ epsilon}}} _ {\ mathit {xx}} ^ {p}})


In the second case:


.. math::
 :label: eq-6

    {\ mathrm {\ sigma}} _ {\ mathit {xx}}} ^ {2} =E (T) (\ mathrm {\ epsilon}} - {\ mathrm {\ epsilon}}} _ {\ mathit {xx}}} ^ {p})


It is therefore sufficient, at any moment, to apply :math:`{\varepsilon }_{\mathrm{xx}}=-{\varepsilon }^{\mathrm{th}}=-\alpha (T)(T-{T}_{\mathrm{ref}})` for mechanical calculation.

Moreover, in order to obtain the same results in both cases, it is necessary, at each time step of the second simulation, to perform the pure mechanical calculation with coefficients whose values are interpolated as a function of the temperature at the current moment. This interpolation is performed in the test command file, in a loop outside of SIMU_POINT_MAT/STAT_NON_LINE.