# Elasto-plastic Materials

## Johnson-Cook (/MAT/LAW2)

- Influence of plastic strain
- Influence of strain rate
- Influence of temperature change

### Material Parameters

`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) $\sigma =a+b\cdot {\epsilon}_{p}{}^{n}$

### Strain Rate

### Temperature Change

- ${T}_{melt}$
- Melt temperature in unit Kelvin
- ${T}_{r}$
- Room temperature in unit Kelvin

- ${E}_{\mathrm{int}}$
- Internal energy

### 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 `C`_{hard} (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.

`C`

_{hard}is from 1 to 0.

`C`

_{hard}=0 for isotropic model,

`C`

_{hard}=1 for kinematic Prager-Ziegler model, or between 1 and 0 for hardening between the above two models.

`C`_{hard}= 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.
`C`_{hard}= 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.

### Young's Modulus

`fct_ID`

_{E},

`E`

_{inf}and

`C`

_{E}. 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_ID`_{E}to update the Young's modulus (`fct_ID`_{E}≠ 0): - Use
`E`_{inf}and`C`_{E}to update the Young's modulus (`fct_ID`_{E}= 0):

### Material Behavior

`fct_ID`

_{p}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

- 3D equivalent HILL stress:
(8) $$\begin{array}{l}f=\sqrt{F{\left({\sigma}_{yy}-{\sigma}_{zz}\right)}^{2}+G{\left({\sigma}_{zz}-{\sigma}_{xx}\right)}^{2}+H{\left({\sigma}_{xx}-{\sigma}_{yy}\right)}^{2}+\text{}2L{\sigma}_{yz}^{2}+2M{\sigma}_{zx}^{2}+2N{\sigma}_{xy}^{2}}\\ \text{=}\sqrt{\underset{}{\underbrace{\left(G+H\right)}}{\sigma}_{xx}^{2}+\underset{}{\underbrace{\left(F+H\right)}}{\sigma}_{yy}^{2}+\underset{}{\underbrace{\left(F+G\right)}}{\sigma}_{zz}^{2}-\underset{}{\underbrace{2H}}{\sigma}_{xx}{\sigma}_{yy}-\underset{}{\underbrace{2F}}{\sigma}_{yy}{\sigma}_{zz}-\underset{}{\underbrace{2G}}{\sigma}_{zz}{\sigma}_{xx}+\underset{}{\underbrace{\text{}2L}}{\sigma}_{yz}^{2}+\underset{}{\underbrace{2M}}{\sigma}_{zx}^{2}+\underset{}{\underbrace{2N}}{\sigma}_{xy}^{2}}\end{array}$$ - Shell element:
(9) $$f=\sqrt{F{\sigma}_{yy}^{2}+G{\sigma}_{xx}^{2}+H{\left({\sigma}_{xx}-{\sigma}_{yy}\right)}^{2}+2N{\sigma}_{xy}^{2}}=\sqrt{\underset{}{\underbrace{\left(G+H\right)}}{\sigma}_{xx}^{2}+\underset{}{\underbrace{\left(F+H\right)}}{\sigma}_{yy}^{2}-\underset{}{\underbrace{2H}}{\sigma}_{xx}{\sigma}_{yy}+\underset{}{\underbrace{2N}}{\sigma}_{xy}^{2}}$$ 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:(10) $$\phi (A)=\underset{G+H}{\underbrace{1}}\cdot {A}_{xx}^{2}-\underset{2H}{\underbrace{\frac{2{r}_{0}}{1+{r}_{0}}}}{A}_{xx}{A}_{yy}+\underset{F+H}{\underbrace{\frac{{r}_{0}\left(1+{r}_{90}\right)}{{r}_{90}\left(1+{r}_{0}\right)}}}{A}_{yy}^{2}+\underset{2N}{\underbrace{\frac{{r}_{0}+{r}_{90}}{{r}_{90}\left(1+{r}_{0}\right)}\left(2{r}_{45}+1\right)}}{A}_{xy}^{2}$$ 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

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.

Here, $G+H=1$.

$R=\frac{{r}_{00}+2{r}_{45}+{r}_{90}}{4}$ | $H=\frac{R}{1+R}$ |

${A}_{1}=H\left(1+\frac{1}{{r}_{00}}\right)$ | ${A}_{2}=H\left(1+\frac{1}{{r}_{90}}\right)$ |

${A}_{3}=2H$ | ${A}_{12}=2H({r}_{45}+0.5)\left(\frac{1}{{r}_{00}}+\frac{1}{{r}_{90}}\right)$ |

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

- 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 ${\sigma}_{0}$.

$$F=\frac{1}{2}\left(\frac{1}{{\sigma}_{22}^{2}}+\frac{1}{{\sigma}_{33}^{2}}-\frac{1}{{\sigma}_{11}^{2}}\right)$$ | $$G=\frac{1}{2}\left(\frac{1}{{\sigma}_{22}^{2}}+\frac{1}{{\sigma}_{33}^{2}}-\frac{1}{{\sigma}_{11}^{2}}\right)$$ |

$$H=\frac{1}{2}\left(\frac{1}{{\sigma}_{22}^{2}}+\frac{1}{{\sigma}_{33}^{2}}-\frac{1}{{\sigma}_{11}^{2}}\right)$$ | $$L=\frac{1}{2{\sigma}_{23}^{2}}$$ |

$$M=\frac{1}{2{\sigma}_{31}^{2}}$$ | $$N=\frac{1}{2{\sigma}_{12}^{2}}$$ |

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