# Elasto-plastic Materials

## Johnson-Cook (/MAT/LAW2)

In LAW2 there are three parts to the stress calculation.
• Influence of plastic strain
• Influence of strain rate
• Influence of temperature change

### Material Parameters

There are two ways to input material parameter for LAW2.
• Iflag=0: Classic input for Johnson-Cook parameter $a$, $b$, $n$ is active
• Iflag=1: New, simplified input with yield stress, UTS (engineering stress), or strain at UTS
Iflag= 0
(1) $\sigma =a+b\cdot {\epsilon }_{p}{}^{n}$
Where,
$a$
The yield stress which could be read from material test and converted to true stress.
$b$ and $n$
The material parameters. Fitting the material stress-strain curve (for example, Altair Compose script) can result in these two parameter.
Iflag = 1
With this new input, you will need yield stress (${\sigma }_{y}$), Ultimate tensile engineer stress (UTS) and engineer strain (${\epsilon }_{UTS}$) at necking point. With this new input, Radioss automatically calculates the equivalent value for $a$, $b$ and $n$.

### Strain Rate

Strain rate has a major effect of material character on crash performance in tensile or in fracture. In Johnson-Cook theory, the yield stress is affected directly by the strain rate and is described as:(2) $\sigma =\left(a+b\cdot {\epsilon }_{p}{}^{n}\right)\left(1+c\mathrm{ln}\frac{\stackrel{˙}{\epsilon }}{{\stackrel{˙}{\epsilon }}_{0}}\right)$
Generally, yield stress increases with increasing the test strain rate. With the strain rate coefficient, $c$, you can scale the factor of yield stress increase. No effect of strain rate could also be defined, if $c$=0; or with ${\stackrel{˙}{\epsilon }}_{0}={10}^{30}$ or $\stackrel{˙}{\epsilon }\le {\stackrel{˙}{\epsilon }}_{0}$.

### Temperature Change

Yield stress decreases with increasing temperature. In LAW2 influence is considered with $\left(1-{T}^{*m}\right)$.(3) $\sigma =\left(a+b\cdot {\epsilon }_{p}{}^{n}\right)\left(1+c\mathrm{ln}\frac{\stackrel{˙}{\epsilon }}{{\stackrel{˙}{\epsilon }}_{0}}\right)\left(1-{T}^{*m}\right)$ with(4) ${T}^{*}=\frac{T-{T}_{r}}{{T}_{melt}-{T}_{r}}$
Where,
${T}_{melt}$
Melt temperature in unit Kelvin.
${T}_{r}$
Room temperature in unit Kelvin.
With $T$ computed as:(5) $T={T}_{i}+\frac{{E}_{\mathrm{int}}}{\rho {C}_{p}\left(Volume\right)}$
Where,
${E}_{\mathrm{int}}$
Internal energy.
Change of internal energy will affect the yield stress in Johnson-Cook law.

### Hardening Coefficient

Metal deformed up to yield and then generally hardened (yield stress increased). Different materials show different ways of hardening (isotropic hardening, kinematic hardening, etc.). This is also a very important material character (for spring-back).

In LAW2, use option Chard (hardening coefficient) to describe which hardening model is used for the material. This feature is also available in material LAW36, 43, 44, 57, 60, 66, 73 and 74.

The value of Chard is from 1 to 0. Chard =0 for isotropic model, Chard=1 for kinematic Prager-Ziegler model, or between 1 and 0 for hardening between the above two models.
Chard = 0: Isotropic Model
In a one dimension case, material strengthens after yield stress. The maximum stress of the last tension is the yield in the subsequent loading, and this new yield stress is the same in subsequent tension and compression.
Chard = 1: Kinematic Prager-Ziegler Model
To model the Bauschinger effect (after hardening in tension, there is softening in a subsequent compression which mean yield in compression is decreased), use kinematic hardening.

## Elastic Plastic Piecewise Linear Material (/MAT/LAW36)

In LAW36, the numbers of plastic stress-strain curves can be directly defined for different strain rates.

Plastic stress-strain of high strain rate should always be above the lower plastic stress-stain curve.

### Young's Modulus

Young's modulus can be updated (decreased) in unloading with options fct_IDE, Einf and CE. Using this feature improves the accuracy of spring-back (in unloading phase) for high strength steel. This feature is also available in material LAW43, LAW57, LAW60, LAW74 and LAW78.
• Use fct_IDE to update the Young's modulus (fct_IDE ≠ 0):
• Use Einf and CE to update the Young's modulus (fct_IDE = 0):

### Material Behavior

fct_IDp is used to distinguish the behavior in tension and compression for certain materials (pressure dependent yield). The effective yield stress is then obtained by multiplying the nominal yield stress by the yield factor corresponding to the actual pressure.

## HILL Materials

In Radioss material laws LAW32, LAW43, LAW72, LAW73, LAW74, LAW78 and LAW93 use HILL criteria.

### HILL Criteria

The typical HILL criteria is:
• 3D equivalent HILL stress:(6)
• Shell element:(7) $f=\sqrt{F{\sigma }_{yy}^{2}+G{\sigma }_{xx}^{2}+H{\left({\sigma }_{xx}-{\sigma }_{yy}\right)}^{2}+2N{\sigma }_{xy}^{2}}=\sqrt{\underset{}{\underbrace{\left(G+H\right)}}{\sigma }_{xx}^{2}+\underset{}{\underbrace{\left(F+H\right)}}{\sigma }_{yy}^{2}-\underset{}{\underbrace{2H}}{\sigma }_{xx}{\sigma }_{yy}+\underset{}{\underbrace{2N}}{\sigma }_{xy}^{2}}$

Where, $F$, $G$, $H$, $L$, $M$ and $N$ are six HILL anisotropic parameters. For shell elements, only $F$, $G$, $H$ and $N$ are the four HILL parameters needed.

In LAW78, the HILL criteria is:(8) $\phi \left(A\right)=\underset{G+H}{\underbrace{1}}\cdot {A}_{xx}^{2}-\underset{2H}{\underbrace{\frac{2{r}_{0}}{1+{r}_{0}}}}{A}_{xx}{A}_{yy}+\underset{F+H}{\underbrace{\frac{{r}_{0}\left(1+{r}_{90}\right)}{{r}_{90}\left(1+{r}_{0}\right)}}}{A}_{yy}^{2}+\underset{2N}{\underbrace{\frac{{r}_{0}+{r}_{90}}{{r}_{90}\left(1+{r}_{0}\right)}\left(2{r}_{45}+1\right)}}{A}_{xy}^{2}$
There are two ways to determine HILL parameters by using Lankford parameters.
• Strain ratio ${r}_{00},{r}_{45},{r}_{90}$ (LAW32, LAW43, LAW72, LAW73)
• Yield stress ratio ${R}_{11},{R}_{22},{R}_{33},{R}_{12},{R}_{13},{R}_{23}$ (LAW74, LAW93)

### Strain Ratio

The Lankford parameters ${r}_{\alpha }$ is the ratio of plastic strain in plane and plastic strain in thickness direction ${\epsilon }_{33}$.(9) ${r}_{\alpha }=\frac{d{\epsilon }_{\alpha +\pi /2}}{d{\epsilon }_{33}}$

Where, $\alpha$ is the angle to the orthotropic direction 1.

${r}_{\alpha }$ could be measured with different samples which cut in different angle with orthotropic direction 1. Like ${r}_{00}$ measured from tensile test in which the loading direction is along the orthotropic direction 1. ${r}_{90}$ measured from tensile test in which the loading is perpendicular to orthotropic direction 1.

The strain ratio is the strain in width direction of sample to strain in thickness direction of sample.
In this case, the HILL parameters are:(10) $F=\frac{{r}_{00}}{{r}_{90}\left({r}_{00}+1\right)}$ (11) $G=\frac{1}{\left({r}_{00}+1\right)}$ (12) $H=\frac{{r}_{00}}{\left({r}_{00}+1\right)}$ (13) $N=\frac{\left(1+2{r}_{45}\right)\left({r}_{00}+{r}_{90}\right)}{2{r}_{90}\left({r}_{00}+1\right)}$

Here, $G+H=1$.

In LAW32, LAW43, and LAW73, the HILL criteria is:(14) ${\sigma }_{eq}=\sqrt{{A}_{1}{\sigma }_{1}^{2}+{A}_{2}{\sigma }_{2}^{2}-{A}_{3}{\sigma }_{1}{\sigma }_{2}+{A}_{12}{\sigma }_{12}^{2}}$
 $R=\frac{{r}_{00}+2{r}_{45}+{r}_{90}}{4}$ $H=\frac{R}{1+R}$ ${A}_{1}=H\left(1+\frac{1}{{r}_{00}}\right)$ ${A}_{2}=H\left(1+\frac{1}{{r}_{90}}\right)$ ${A}_{3}=2H$ ${A}_{12}=2H\left({r}_{45}+0.5\right)\left(\frac{1}{{r}_{00}}+\frac{1}{{r}_{90}}\right)$

They all request Lankford parameter (strain ratio) ${r}_{00},{r}_{45},{r}_{90}$ and the HILL parameter ${A}_{i}$ is automatically computed by Radioss.

### Yield Stress Ratio

In LAW93, the yield stress ratio used is:(15) ${R}_{ij}=\frac{{\sigma }_{ij}}{{\sigma }_{0}}$
To get yield stress ratio ${R}_{ij}$, yield stress in two loading cases need to be measured.
• Yield stress ${\sigma }_{11},{\sigma }_{22},{\sigma }_{33}$ from tensile test
• Yield shear stress ${\sigma }_{12},{\sigma }_{13},{\sigma }_{23}$ from shear test

In LAW93, if parameter input is used, then take initial stress parameter ${\sigma }_{y}$ as reference yield stress ${\sigma }_{0}$. If curve input is used, then take the yield stress from curve as reference yield stress ${\sigma }_{0}$.

Four HILL parameters for shell are automatically computed by Radioss.(16) $F=\frac{1}{2}\left(\frac{1}{{R}_{22}^{2}}+\frac{1}{{R}_{33}^{2}}-\frac{1}{{R}_{11}^{2}}\right)$ (17)$G=\frac{1}{2}\left(\frac{1}{{R}_{33}^{2}}+\frac{1}{{R}_{11}^{2}}-\frac{1}{{R}_{22}^{2}}\right)$ (18) $H=\frac{1}{2}\left(\frac{1}{{R}_{22}^{2}}+\frac{1}{{R}_{11}^{2}}-\frac{1}{{R}_{33}^{2}}\right)$ (19) $N=\frac{3}{2{R}_{12}^{2}}$
In LAW74, yield stress ratio ${R}_{ij}$ is used with yield stress ${\sigma }_{11},{\sigma }_{22},{\sigma }_{33}$ and ${\sigma }_{12},{\sigma }_{13},{\sigma }_{23}$ input directly, and then six HILL parameters for solid are automatically computed by Radioss.
 $F=\frac{1}{2}\left(\frac{1}{{\sigma }_{22}^{2}}+\frac{1}{{\sigma }_{33}^{2}}-\frac{1}{{\sigma }_{11}^{2}}\right)$ $G=\frac{1}{2}\left(\frac{1}{{\sigma }_{22}^{2}}+\frac{1}{{\sigma }_{33}^{2}}-\frac{1}{{\sigma }_{11}^{2}}\right)$ $H=\frac{1}{2}\left(\frac{1}{{\sigma }_{22}^{2}}+\frac{1}{{\sigma }_{33}^{2}}-\frac{1}{{\sigma }_{11}^{2}}\right)$ $L=\frac{1}{2{\sigma }_{23}^{2}}$ $M=\frac{1}{2{\sigma }_{31}^{2}}$ $N=\frac{1}{2{\sigma }_{12}^{2}}$

For shell element, take $M=N$ and $L=N$.