> For the complete documentation index, see [llms.txt](https://dante-solutions-inc.gitbook.io/dante-6.3-help-documentation/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://dante-solutions-inc.gitbook.io/dante-6.3-help-documentation/readme/material-models-for-steels/mechanical-plasticity-model.md).
# Mechanical Plasticity Model
DANTE uses BCJ internal state variable (ISV) plasticity model. This model contains kinematic hardening and recrystallization features. The material properties per phase are defined as functions of carbon content, temperature, strain, and strain rate. During a heat treatment process with phase transformations, the stress state of any given point in the part may shift from tension to compression or from compression to tension (An example is shown below).
The traditional plasticity model based on effective strain is not effective to catch the yield with stress reversals due to the Bauschinger effect. The Bauschinger effect describes the yield stress difference when the material goes through loading direction changes, which is caused by kinematic hardening. The plasticity model used in DANTE has kinematic hardening effect, which is necessary for modeling the heat treatment process effectively. The Figure below shows the Bauschinger effect plotted using MEC\_Fit utility tool using the DANTE material database.
The stress constitutive formula of the BCJ internal state variable plasticity model is shown below.
$$\sigma = \alpha + \kappa + Y + V \cdot \log\left(\frac{|\dot{\epsilon}| + \sqrt{|\dot{\epsilon}|^2 + f^2}}{f}\right)$$
where,
* *Y, V and f are calculated from the ISV model parameters (detailed formula is defined later)*
* *Sigma (σ) is the stress tensor, with 6 components*
* *Alpha (α) is the directional ISV variable, with 6 components*
* *Kappa (κ) is the isotropic ISV variable, with 1 component*
* *ė is strain rate* *Note: All stress, strain and strain rate are deviatoric*
The formula for *Y*, *V* and *f* are described in equations below.
$$Y = \frac{(0.5 \cdot (1.0 + \tanh(C\_{19} \cdot (C\_{20} - T))) \cdot (C\_3 + \text{Carb} \cdot (C\_{26} - C\_{27} \cdot T))}{(C\_{21} + \exp\left(\frac{-C\_4}{T}\right))}$$
$$V = C\_1 \cdot \exp\left(\frac{-C\_2}{T}\right) \quad f = C\_5 \cdot \exp\left(\frac{-C\_6}{T}\right)$$
Where *Cx* are plasticity model parameters, *T* is temperature, and *Carb* is carbon.
Material is under yield if the following equation is true.
$$|\sigma - \alpha| > \kappa + Y + V \cdot \log\left(\frac{|\dot{\epsilon}| + \sqrt{|\dot{\epsilon}|^2 + f^2}}{f}\right)$$
The meaning of variable Y and V are shown in the Figure below. DANTE utility tool, [**Mec Fit**](https://github.com/DANTE-Solutions/DANTE-6.3-Docs/blob/main/docs/utility-tools/mec-fit/README.md) is used to plot the strain-stress curves for different conditions.
The plasticity model parameters for each phase is defined in the material data \*.MEC file using the keyword ***\*STEEL\_PLASTIC***. Please reference [**Steel Material Mechanical Data File**](/dante-6.3-help-documentation/readme/material-database/steel-alloy-data/steel-material-mechanical-data-file.md).
During tempering of medium or high carbon steels, the volume of tempered martensite reduces. This phenomenon is more significant with increased tempering temperature. The DANTE mechanical model can effectively describe the volume change of tempered martensite, which is calculated by carbide coarsening and the defined strain change due to carbide coarsening. The Figure below shows an example modeled dilatometry strain curve with martensite (snaped tempered martensite) as initial phase using DANTE Mat\_Simulator tool.
## Hardness Factor of Tempered Martensite
For tempered martensite phase, the yield property is affected by the conditions of the iron carbide and alloy precipitate through the combined hardness factor. The **hardness factor** is calculated as the ratio of current hardness to hardness with size class-10 iron carbide. the hardness factor range is \[1.0, 5.0]. The effect of hardness factor on the plasticity of tempered martensite is implemented as a multiplier to the ***C3*** as shown below.