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

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


Simulation 1
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The first simulation (solution that we are trying to validate) consists in applying a temperature variation to a material point, by blocking the following deformations :math:`x`: :math:`{\varepsilon }_{\mathrm{xx}}=0`. The imposed temperature increases linearly as a function of time.

Unless otherwise stated, the temperature varies from :math:`{T}_{0}=20°C` to :math:`{T}_{\mathrm{max}}=500°C`. The material parameters are chosen so that part of the transient is in the non-linear domain of the law of behavior, except in elasticity. The transition consists of NCAL steps. The reference temperature is :math:`{T}_{\mathrm{Ref}}={T}_{0}`.


Simulation 2
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The second simulation (which must be equivalent to the first) consists in applying, in pure mechanics, without thermal, and at each moment, an imposed deformation following :math:`x` equivalent to the thermal deformation of the first simulation: It is therefore a question of performing a loop on mechanical NCALcalculs. For each calculation :math:`i`, the imposed load consists of the thermal deformation :math:`{\varepsilon }_{\mathrm{xx}}=-{\varepsilon }_{\mathrm{th}}=-\alpha (T)({T}_{i}-{T}_{\mathrm{Ref}})`. The initial load consists of the deformations, stresses and internal variables from the previous mechanical calculation.



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

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

in the first case, :math:`{\sigma }_{\mathrm{xx}}=E(T)(0-{\varepsilon }^{\mathrm{th}}-{\varepsilon }_{\mathrm{xx}}^{p})`, and in the second: :math:`{\sigma }_{\mathrm{xx}}=E(T)(\varepsilon -{\varepsilon }_{\mathrm{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 STAT_NON_LINE.