2. Reference solution#

2.1. Calculation method used for the reference solution#

  • Upon heating, a uniform temperature rise from \(700\) to \(900°C\) in \(200s\) is imposed.

  • Analytical solution for thermal calculation (on cooling since \(900°C\)).

\(T(x,y,t)=\theta (x,y,t)(T(x,y\mathrm{,0})-{T}_{\infty })+{T}_{\infty }\)

where:

\(\theta (x,y,t)\mathrm{=}{\mathrm{\sum }_{i\mathrm{=}1}}^{\mathrm{\infty }}{A}_{i}{e}^{\mathrm{-}{\omega }_{i}^{2}\frac{\lambda }{\rho {C}_{P}}t}\mathrm{cos}{\omega }_{i}x\mathrm{\times }{\mathrm{\sum }_{A}}_{i}^{\mathrm{\infty }}{e}^{\mathrm{-}{\omega }_{i}^{2}\frac{\lambda }{\rho {C}_{P}}t}\mathrm{cos}{\omega }_{i}y\)

with \({\omega }_{i}\) verifying:

\({\omega }_{i}L\mathrm{tg}({\omega }_{i}L)=\frac{hL}{\lambda }=5.00\)

and:

\({A}_{i}=\frac{4\mathrm{sin}({\omega }_{i}L)}{2{\omega }_{i}L\mathrm{sin}({\omega }_{i}L)}\)

  • The reference values for metallurgical evolutions depend on the model and on the integration of behavioral relationships over time. Reference values are not available.

  • Since the hardness of a material point depends on the metallurgical proportions of each phase, reference values are not available.

2.2. Benchmark results#

(Thermal calculation):

  • temperature at points \(A\), \(B\), \(C\) at time \(t=300s\),

  • proportion of bainite at points \(A\), \(B\), \(C\) at times \(t=\mathrm{410,}300\) and \(300s\) respectively,

  • proportion of martensite at points \(A\), \(B\), \(C\) at the moment \(t=\mathrm{410 }s\),

  • proportion of austenite at point \(A\) at times \(t=\mathrm{30 }s\) and \(\mathrm{140 }s\).

  • hardness at point \(O\) at times \(t=\mathrm{30 }s,\mathrm{140 }s,300s\) and \(410s\).

2.3. Uncertainty about the solution#

Less than 1% with 30 modes for each sum.

2.4. Bibliographical references#

  1. F.P. INCROPERA, D.P. DEWITT, J. WILEY. Fundamentals of heat and mass transfer. Third edition. 1990.