Constitutive law for metallurgical phase transformations with mechanical interaction in solids.

This law is used for prediction of metallurgical phase transformations for a given evolution of the temperature field and of the mechanical field.

Prepro: LMETAM.F

ALAMB and RHOC are to be given at the initial temperature and for the initial metallurgical composition of the solid.

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The special file is the number 35, generally called IN.MET and read by METLAW in PREPRO.

This file contains all the data necessary to use effectively the laws METAMEC, THMET and ARBTHMEC or ELAMET. It must always exist to perform a metallurgic mechanic thermal analysis. Sections 1. to 8. are repeated with increasing ILAWN if more than one steel is described.

Any comment that will be reproduced on the output listing. Try to characterise your steel (60NCD11, ARBED, 42CD4, …).

If IET=1, then read NEPS (number of strain levels) (1I5).

Remarks : Some additional parameters can occur depending on the steel and its plasticity transformation formula or the modification of the formula of the sift ($D=C\sigma$). If you want to change, you must adopt:

1. NTPCA (section 7.2) ;
2. Subroutine METLAW that read and write with comments the parameters of section 7.3 ;
3. Subroutine ARMEA that read the great vector PAMET where are stocked the parameters of section 7.3 and where the formulae of $D$ and $\varepsilon^{pt}$ are implemented.

If IPOLY = 1 : the conductivity $\lambda$, the mass density $\rho$, the heat capacity $C$ and the hardness $H_v$ have to be defined for each phase. The latent heat is defined for each transformation. You can choose the order in which you want to define these parameters :

• Firstly you give five letters: 'A1^A2' where:
  1. A1 defines the thermal parameter for $\lambda$, $\rho$, $C$ and $H_v$
     • $\lambda \rightarrow$ LA
     • $\rho \rightarrow$ RO
     • $C \rightarrow$ CA
     • $H_v \rightarrow$ HV
  2. A2 defines the phase
     • Austenite $\rightarrow$ AU
     • Bainite $\rightarrow$ BA
     • Proeutectoïd $\rightarrow$ PR
     • Martensite $\rightarrow$ MA
     • Pearlite $\rightarrow$ PE
  3. for L (latent heat) : A1 = TR and A2 = PR, PE, BA or MA to define in which phase the austenite is transformed.
• Secondly you give the polynomial coefficient (A(I), I=1, NDPO+1) which defines the following polynomial: \[A(1)+\left(A(2)T+A(3)T^2+\ldots+A(NDPO+1)T^{NDPO}\right)\]
• Finally the non-defined parameters will be initialized to zero. So if you know that only certain phases will be present you do not need to define the other phase parameters. The end of this section is detected by A1=FI followed by a blank card.
  

If IPOLY = 0: One must write FI followed by a blank card.

1. Firstly you give five letters : 'A1^A2' and NTER where :
   • A1 defines the mechanical parameter:
     • YOUNG modulus: YO
     • POISSON ratio: NU
     • Thermal dilatation: AC or AP (AC for the $\alpha$ coefficient of classical type and AP for the $\alpha$ coefficient of partial type)
     • Yield stress $\sigma_y$: SY
     • Plastic slope: ET
   • If IPOLY=0:
     • Thermal conductivity: LA
     • Mass density: RO
     • Heat capacity: CA
     • Vickers hardness: HV
     • Latent heat of transformation: TR
   • '^' is a space;
   • A2 defines the phase concerned;
     • Austenite: AU (Except for A1=TR)
     • Proeutectoïd: PR
     • Pearlite: PE
     • Bainite: BA
     • Martensite: MA
   • NTER as the number of temperature used to describe the evolution of the parameter.
     
2. Secondly, ONLY IF AC is chosen:
   (A5,G10.0) 3 spaces and 'TO' or 'T0'
   VALUE: Value of $T_0$ (usually $T_0$ is the room temperature so 20°C or 293K, be careful there is no default value for this parameter, so you should enter a value, otherwise the preprocessor will crash)
3. Thirdly, you repeat NTER times:
   • TEMPE: Temperature
   • VALUE: Value of the parameter

Remarks :

1. No defined tables are initialized to zero
2. A1 = FI followed by a blank card indicates the end of section 5.
3. For the table describing ALPHA, the first temperature must be zero otherwise the integration of $\int_0^{T_{\alpha}} dT$ will not be correct.
   

Remarks about the thermal coefficient $\alpha$:

1. The $\alpha_C$ coefficient of classical type is defined by : \[\alpha_C(T) = \frac{1}{L_0}\frac{L_{(T)}-L_0}{T-T_0}\] where $L_0=L(T_0)$.
   The $\alpha_P$ coefficient of partial type is defined by : \[\alpha_P(T)=\frac{1}{L_{(T)}}\frac{dL}{dT}\] The user could give the classical type or the partial type $\alpha$ coefficient (with AC or AP).
   
2. If $\alpha$ is of classical type ($\alpha_C$) :
   • The unity, chosen by the user, of $T_0$ has to be the same as the unity of the temperatures at which the $\alpha_C$ coefficient is given.
   • The temperatures, at which the $\alpha_C$ coefficient is given, have to be given in increasing order.
   • It is necessary to give at least 2 values of $\alpha_C$ at two temperatures.
   • Each $\alpha_C$ coefficient has to verify the following relation : \[1+\alpha_C(T).(T-T_0)>0\;\text{ and not equal to 0}\]
3. If the user gives the classical type $\alpha_C$ coefficient (AC), the pre-processor will calculate the partial type $\alpha_P$ coefficient. If the user gives the partial one, the pre-processor keeps these values. In the case of the calculus of the partial type $\alpha_P$ coefficient, the following relations are used (see the file ALPHAPARTIEL.F) for given couples ($T_i,\alpha_{Ci}$) with $i=1,\ldots,n$ : \[\alpha_{P1} = \frac{\alpha_{C1}+(T_1-T_0)\frac{\alpha_{C2}-\alpha_{C1}}{T_2-T_1}}{1+\alpha_{C1}(T_1-T_0)}\]\[\alpha_{Pn} = \frac{\alpha_{Cn}+(T_n-T_0)\frac{\alpha_{Cn}-\alpha_{Cn-1}}{T_n-T_{n-1}}}{1+\alpha_{Cn}(T_n-T_0)}\] and for $i$ such as $1<i<n$: \[\alpha_{Pi}=\frac{\alpha_{Ci}+\frac{1}{2}(T_i-T_0)\left(\frac{\alpha_{Ci}-\alpha_{Ci-1}}{T_i-T_{i-1}}+\frac{\alpha_{Ci+1}-\alpha_{Ci}}{T_{i+1}-T_i}\right)}{1+\alpha_{Ci}(T_i-T_0)}\]
4. For more information about the $\alpha$ coefficient of classical and partial type, the reader could see the internal report N° M&S/2002-8 of the 5$^{th}$ December 2002 entitled “Comparaison des coefficients de dilatation thermique classique et partiel”.
5. An example of what the user should give is:

  AC AU    5
      T0       0.0
         0.0   10.E-06
       200.0   12.E-06
       400.0   16.E-06
       700.0   25.E-06
       900.0   30.E-06
  AC PE    2
      T0       0.0
         0.0   10.E-06
       900.0   30.E-06
  AP MA    5
         0.0   30.E-06
       200.0   25.E-06
       400.0   16.E-06
       700.0   12.E-06
       900.0   10.E-06     

The three phases : proeutectoïd (PROEU), pearlite (PERLI) and bainite (BAINI) have to be described successively by sections 8.1. to 8.4. The order PROEU, then PERLI, then BAINI must be respected.

PROEU or PERLI or BAIN from columns 1 to 5

N.B. These data are used to compute $n$ and $b$, the coefficients of the Johnson-Mehl-Avrami law.

Remark : The NTR number must be limited by the data NT1, NT2 or NT3 given in section 7.2. for each phase (PROEU, PEARLI or BAINI).