# 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:(1)
${\sigma }_{e}=\frac{F}{{S}_{0}}$
(2)
${\epsilon }_{e}=\frac{\text{Δ}l}{{l}_{0}}$
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.
1. 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 (ReH) and then drop to the lower yield point (ReL). 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.
2. Necking point: where the material reaches the maximum stress in engineer stress-strain curve. After this point, the material begins to soften.
3. Failure point: where material failed.
Rm
Maximum resistance
Fmax
Maximum force
ReH
Upper yield level
ReL
Lower yield level
Ag
Uniform elongation
Agt
Total uniform elongation
At
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:(3)
${\sigma }_{tr}={\sigma }_{e}\text{exp}\left({\epsilon }_{tr}\right)$
(4)
${\epsilon }_{tr}=\text{ln}\left(1+{\epsilon }_{e}\right)$
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%.
• 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.(5)
${\sigma }_{y}=a+b{\epsilon }_{p}{}^{n}$

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
${\sigma }_{y}=A{\left({\overline{\epsilon }}_{p}+{\epsilon }_{0}\right)}^{n}$
$A$ and $n$ are positive.
Voce model
${\sigma }_{y}={k}_{0}+Q\left[1-\mathrm{exp}\left(-B{\overline{\epsilon }}_{p}\right)\right]$
${k}_{0}$ , $Q$ and $B$ are positive.
Combination of Swift and Voce model (LAW84 and LAW87)
${\sigma }_{y}=\alpha \underset{Swift\begin{array}{c}\end{array}hardening}{\underbrace{\left[A{\left({\overline{\epsilon }}_{p}+{\epsilon }_{0}\right)}^{n}\right]}}+\left(1-\alpha \right)\underset{Voce\begin{array}{c}\end{array}hardening}{\underbrace{\left\{{k}_{0}+Q\left[1-\mathrm{exp}\left(-B{\overline{\epsilon }}_{p}\right)\right]\right\}}}$

Here, α 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.