# Materials

Different material tests could result in different material mechanic character.

The typical material test for metal is tensile test. With this test strain-stress curve,
yield point, necking point and failure point of material could be observed.

Engineer strain-stress curve could be generated by:

$${\sigma}_{e}=\frac{F}{{S}_{0}}$$

$${\epsilon}_{e}=\frac{\text{\Delta}l}{{l}_{0}}$$

Where,

- ${S}_{0}$
- Section area in the initial state
- ${l}_{0}$
- Initial length

In this Force-elongation curve or engineer stress-strain curve, three
points are important.

- Yield point: where material begin to yield. Before yield you can assume
material is in elastic state (the Young's modulus
`E`could be measured) and after yield, material plastic strain which is non-reversible.- Some material in this test will first reach the upper yield point
(
`R`_{eH}) and then drop to the lower yield point (`R`_{eL}). In engineer stress-strain curve, lower yield stress (conservative value) could be taken. - Some material can not easily find yield point. Take the stress of 0.1 or 0.2% plastic strain as yield stress.

- Some material in this test will first reach the upper yield point
(
- Necking point: where the material reaches the maximum stress in engineer stress-strain curve. After this point, the material begins to soften.
- Failure point: where material failed.

`R`_{m}- Maximum resistance
`F`_{max}- Maximum force
`R`_{eH}- Upper yield level
`R`_{eL}- Lower yield level
`A`_{g}- Uniform elongation
`A`_{gt}- Total uniform elongation
`A`_{t}- Total failure strain

True stress-strain curve which is requested in most materials in Radioss, except in LAW2, where both engineer stress-strain and true stress-strain are possible to input material data.

In Figure 3, find engineer stress-strain curve (blue) by using:

$${\sigma}_{tr}={\sigma}_{e}\text{exp}\left({\epsilon}_{tr}\right)$$

$${\epsilon}_{tr}=\text{ln}\left(1+{\epsilon}_{e}\right)$$

The result is true stress-strain curve (red). Plastic true
stress-strain curve is shown in green, which plastic strain begin from 0. This green
plastic true stress-strain curve is what you need, as in LAW36,
LAW60, LAW63, and so on.

The true stress-strain curve is valid until the necking point of the
material. After the necking point, the material curve has to be defined manually for
hardening. Using a different material law, Radioss will
extrapolation the true stress-strain curve to 100%.

Here, $\alpha $
is weight of Swift hardening and Voce hardening. Here, is one
Compose script as an example to fit the Swift hardening parameters
$A$
,
${\epsilon}_{0}$
,
$n$
and Voce hardening parameters
${k}_{0}$
,
$Q$
,
$B$
with input stress-strain curve.- Linear extrapolation: If stress-strain curve is as function input (LAW36), then stress-strain curve is linearly extrapolated with a slope defined by the last two points of the curve. It is recommended that the list of abscissa value be increased to a value greater than the previous abscissa value.
- Johnson-Cook: After necking point, Johnson-Cook hardening is one of the most
commonly used to extrapolate the true stress-strain curve.$${\sigma}_{y}=a+b{\epsilon}_{p}{}^{n}$$
However, it may overestimate strain hardening for automotive steel, In this case, combination of swift-voce hardening is more accurate.

- Swift and Voce: After necking point, use one of the following equations to
extrapolate the true stress-strain curve.
- Swift model
- ${\sigma}_{y}=A{\left({\overline{\epsilon}}_{p}+{\epsilon}_{0}\right)}^{n}$
- Voce model
- ${\sigma}_{y}={k}_{0}+Q\left[1-\mathrm{exp}\left(-B{\overline{\epsilon}}_{p}\right)\right]$
- Combination of Swift and Voce model (LAW84 and LAW87)
- ${\sigma}_{y}=\alpha \underset{Swift\begin{array}{c}\end{array}hardening}{\underbrace{\left[A{\left({\overline{\epsilon}}_{p}+{\epsilon}_{0}\right)}^{n}\right]}}+(1-\alpha )\underset{Voce\begin{array}{c}\end{array}hardening}{\underbrace{\left\{{k}_{0}+Q\left[1-\mathrm{exp}\left(-B{\overline{\epsilon}}_{p}\right)\right]\right\}}}$