1. 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 tested behavior provides a correct solution in pure mechanics.

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

The second simulation (which must be equivalent to the first) consists in applying to the same material point an imposed deformation following \(x\): \({\varepsilon }_{\mathrm{xx}}=-{\varepsilon }^{\mathrm{th}}=-\alpha (T)(T-{T}_{\mathrm{ref}})\), in pure mechanics. Indeed, for any behavior (assuming the additive decomposition of deformations):

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

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

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