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# Tempering Model
Temper is one of the most important heat treatment process steps to obtain the targeting mechanical properties. During low temperature temper, martensite transforms to tempered martensite with a small iron carbide coarsening, and the hardness reduction is low. During high temperature temper, the iron carbide coarsening is significant, and the hardness reduction is high. For some alloy steels, such H13, Ferrium C64, etc., the hardness during high temperature temper may increase, which is caused by the precipitation of alloy carbides. DANTE material models have the following features for modeling both low and high temperature temper processes.
* Phase transformation kinetics for transformation from martensite to tempered martensite,
* Iron carbide coarsening and hardness models of tempered martensite accounting for the iron carbide fraction, carbide size, and size distribution,
* Alloy carbide precipitation model, coarsening model, and hardness models of tempered martensite.
**Below is the martensite to tempered martensite transformation equation.**
$$\frac{d\text{TM}}{dt} = v\_1 \cdot \frac{1}{\exp(v\_2 + v\_3 \cdot \ln\left(\frac{v\_4 - T}{v\_5}\right))} \cdot \text{TM}^{v\_6} \cdot (1 - \text{TM})^{v\_7} \cdot M$$
Where *TM* is volume fraction of the tempered martensite, *t* is time, *v1*, *v2*, *v3*, *v4*, *v5*, *v6*, *and v7* are constants, *M* is volume fraction of martensite, and *T* is temperature.
The phase transformation from martensite to tempered martensite is defined by keyword: \***STEEL\_KIN\_MTMART** in the \*.CQT file. For this keyword, there is one data line to define the temperature range of transformation and the transformation rate. An example of data definition is shown below.
```
** Martensite-->Tempered Martensite
*STEEL_KIN_MTMART
** Line 1: Kinetics (9 values)
100.0, 700.0, 0.500, 1.500, 15.50, 450.0, 150.0, 0.350, 0.150
```
The martensite to tempered martensite transformation is carbon dependent, and the material data is defined in the \*.CQT file using the keyword “\*STEEL\_KIN\_MTMART” under specific carbon levels. The tempering rate is time and temperature dependent.
* Parameter 1: Low temperature bound for tempering kinetics
* Parameter 2: High temperature bound for tempering kinetics
* Parameters 3-9: Variable values for *v1*, *v2*, *v3*, *v4*, *v5*, *v6*, *and v7* in the martensite to tempered martensite transformation equation.
The iron carbide and alloy carbide coarsening rate is temperature dependent using the equation below.
$$\frac{dS}{dt} = c\_1 + c\_2 \cdot \exp\left(\frac{T - c\_4}{c\_3}\right)$$
Where *S* is the size of iron carbide or alloy precipitate, *t* is time, *T* is temperature, *c1, c2, c3, c4* are model coefficients. The material data are defined in the material data **\*.PHA file**. The keyword “\*STEEL\_TMCBD\_CCRSN” is used to define the iron carbide coarsening rate, and the keyword “\*STEEL\_TMCBD\_ACRSN” is used to define the coarsening rate of precipitated alloy carbide.
```
**~~Carbides Coarsening Model Parameters
** Temp LBound,Temp UBound, C1, C2, C3, C4
** coarsening coefficient: c1+c2*exp((T-c4)/c3) c3>1.0
*STEEL_TMCBD_CCRSN
50.0, 700.0, 0.0, 0.25, 42.5, 200.0
**~~CBD-A Coarsening Model Parameters
*STEEL_TMCBD_ACRSN
50.0, 700.0, 0.0, 0.100, 4.5, 525.0
```
A modeling example of quench hardening and high temperature tempering of carburized high alloy steel is shown below, including austenitizing, quenching, deep freeze, and high temperature temper steps. During temper, the material softens first due to iron carbide coarsening, followed by hardening due to the precipitation of alloy carbide.
---
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