# RD-E: 0800 Hopkinson Bar

The purpose of this example is to model and predict the responses of very high strain rates on a material during impact.

Precise data for high strain rate materials is necessary to enable the accurate
modeling of high-speed impacts. The high strain rate characterization of materials
is usually performed using the Split-Hopkinson pressure bar within the strain rate
range 100-10000 s^{-1}. It is assumed that during the experiment the
specimen deforms under uniaxial stress, the bar specimen interfaces remain planar at
all times, and the stress equilibrium in the specimen is achieved using travel
times. The Radioss explicit finite element code is used
to investigate these assumptions.

## Options and Keywords Used

- /QUAD: 2D solid elements defined in the global YZ – plane
- /ANALY: Defines the type of analysis and sets analysis flags
- /MAT/LAW1 (ELAST): Isotropic, linear elastic material using Hooke's law
- /MAT/LAW2 (PLAS_JOHNS): Isotropic elasto-plastic material using the Johnson-Cook material model
- /PROP/TYPE14 (SOLID): General solid property set
- /IMPVEL: Imposed velocities on a group of nodes
- /BCS: Boundary conditions)

Low extremity nodes of the output bar are fixed in the Z direction. The axisymmetric condition on the revolutionary symmetry axis requires the blocking of the Y translation and X rotation.

The projectile is modeled using a steel cylinder with a fixed velocity in the
direction Z. The required strain rate is considered by applying two imposed
velocities, 1.7 ms^{-1} and 5.8 ms^{-1} in order to produce strain
rates in the ranges of 80 s^{-1} and 900 s^{-1} (low and high
rates).

In the experiment, the strain gauge is attached to the specimen. In simulation, the true strain will be determined from 9040 and 6 nodes’ relative Z displacements ( ${l}_{0}$ = 3.83638 mm).

The true stress can be given using two data sources. The first methodology consists of using the equation previously presented, based on the assumption of the one-dimensional propagation of bar-specimen forces. The engineering strain ${\epsilon}_{t}$ associated with the output stress wave is obtained from the Z displacement of nodes located on the output bar. The true plastic strain is extracted from the quads on the specimen, saved in the Time Histories file. True stress can also be measured directly from the Time History using the average of the Z stress quads 6243, 6244, 6224 and 6235. It should be noted that the section option is not available for quad elements.

The strain rate can be calculated from either the true plastic strain of quads saved in /TH/QUAD or from the true strain ${\epsilon}_{\mathrm{true}}$ .High Rate Testing | ||
---|---|---|

True stress | ${\sigma}_{true}\left(t\right)=\frac{{S}_{bar}{E}_{bar}}{{S}_{specimen}}{\epsilon}_{T}\left(t\right)\mathrm{exp}\left({\epsilon}_{pl}\left(t\right)\right)$ | Z stress average from quads saved in /TH |

True strain | ${\epsilon}_{true}\left(t\right)=\mathrm{ln}\left(\frac{{l}_{i}\left(t\right)}{{l}_{0}}\right)\text{\hspace{1em}}{l}_{i}\left(t\right)={l}_{0}+\text{\Delta}l={l}_{0}+\left({u}_{9040}\left(t\right)-{u}_{6}\left(t\right)\right)$ | |

True strain rate | $\dot{\epsilon}=\frac{\text{\Delta}{\epsilon}_{pl}\left(t\right)}{\text{\Delta}t}$ | $\dot{\epsilon}=\frac{\text{\Delta}{\epsilon}_{true}\left(t\right)}{\text{\Delta}t}$ |

## Input Files

## Model Description

The purpose of this example is to model and predict the responses of very high strain rates on a material during impact.

The Split-Hopkinson pressure bar is a suitable method to perform experiments with high strain rates.

- an incident bar and a transmission bar of equal length, between which the sample to be tested is clamped.
- a striker is attached to the outer end of the incident bar. When a steel projectile hits the striker, a stress pulse is introduced into the incident bar.

- ${E}_{bar}$
- Modulus of the output bar.
- ${\epsilon}_{T}$
- Strain associated with the output stress wave.
- ${S}_{bar}$
- Cross-section of the output bar.

If the two bars remain elastic and wave dispersion is ignored, then the measured stress pulses can be assumed to be the same as those acting on the specimen.

The engineering stress value in the specimen can be determined by the wave analysis, using the transmitted wave:

Engineering stress can also be found by averaging out the force applied by the incident that is the reflected and transmitted wave, as shown in the equation:

- ${\epsilon}_{I}$ and ${\epsilon}_{R}$
- Strains associated with input stress wave.
- ${\epsilon}_{T}$
- Strain associated with output stress wave.

True stress in the specimen is computed using the following relation (refer to RD-E: 1100 Tensile Test for further details):

The true strain rate is given by:

- Interface 1
- ${F}_{1}={S}_{bar}\left[{\sigma}_{l}\left(t\right)+{\sigma}_{R}\left(t\right)\right]={S}_{bar}\cdot {E}_{bar}\left[{\epsilon}_{l}\left(t\right)+{\epsilon}_{R}\left(t\right)\right]$
- Interface 2
- ${F}_{2}={S}_{bar}\cdot {\sigma}_{T}\left(t\right)={S}_{bar}\cdot {E}_{bar}\cdot {\epsilon}_{T}\left(t\right)$
- Balance in specimen
- ${F}_{1}={F}_{2}$ ; ${\epsilon}_{l}\left(t\right)+{\epsilon}_{R}\left(t\right)={\epsilon}_{T}\left(t\right)$
- Engineering stress in specimen
- ${\sigma}_{specimen}\left(t\right)=\frac{{F}_{1}}{{S}_{specimen}}=\frac{{F}_{2}}{{S}_{specimen}}$

### Strain Rate Filtering

Because of the dynamic load, strain rates cause high frequency vibrations which are
not physical. Thus, the stress-strain curve may appear noisy. The strain rate
filtering option enables to dampen such oscillations by removing the high frequency
vibrations in order to obtain smooth results. A cut-off frequency for strain rate
filtering `F`_{cut} = 30 kHz was used in this example. Refer to RD-E: 1100 Tensile Test for further details.

### Johnson-Cook Model

The Johnson-Cook model describes the stress in relation to the plastic strain and the strain rate using the following equation:

- $\dot{\epsilon}$
- Strain rate.
- ${\dot{\epsilon}}_{0}$
- Reference strain rate.
- ${\epsilon}_{p}$
- Plastic strain (true strain).
- $a$
- Yield stress.
- $b$
- Hardening parameter.
- $n$
- Hardening exponent.
- $c$
- Strain rate coefficient.

The two optional inputs, strain rate coefficient and reference strain rate, must be defined for each material in /MAT/LAW2 in order to take account of the strain rate effect on stress, that is the increase in stress when increasing the strain rate. The constants $a$ , $b$ and $n$ define the shape of the strain-stress curve.

- $\sigma =\left(496+225{\epsilon}^{0.35}\right)$
- Strain rates below 80 s
^{-1} - $\sigma =\left(496+225{\epsilon}^{0.35}\right)\left(1+0.16\mathrm{ln}\left(\frac{\dot{\epsilon}}{0.08}\right)\right)$
- Strain rates exceeding 80 s
^{-1}up to 3000 s^{-1}

**Material Properties****Value**- Young's modulus
- 73000 $\left[\mathrm{MPa}\right]$
- Poisson's ratio
- 0.33
- Density
- 0.0028 $\left[\frac{g}{m{m}^{3}}\right]$

**Material Properties****Value**- Young's modulus
- 210000 $\left[\mathrm{MPa}\right]$
- Poisson's ratio
- 0.33
- Density
- 0.0078 $\left[\frac{g}{m{m}^{3}}\right]$

**Bars**- Length
- 4 m
- Diameter
- 12 mm

**Projectile**- Radius
- 12 mm
- Weight
- 170 g

### Model Method

Considering the geometry’s revolution symmetry the material and the kinematic
conditions, an axisymmetric model is used (`N2D3D` =
1 in /ANALY set up in the Starter
file). Y is the radial direction and Z is the axis of revolution.

## Results

The purpose of the test is to obtain results at high deformation rates. In this model the Johnson-Cook type material law is used. The increase of stress is expected to be approximately 30% above the stress compared to the quasi-static deformation rate.

### Experimental Data

Experimental results show that the variation of the true tensile flow stress compared
with the true strain is approximately equivalent to a strain rate between 80
s^{-1} and 100 s^{-1}. The reference strain,
${\epsilon}_{R}$
in the Johnson-Cook model is set to 0.08
ms^{-1} (correspond to 80 s^{-1}, which represents the
quasi-static deformation rate. At higher deformation rates, the true flow stress
increases significantly with increasing strain rates. The 7010 aluminum alloy
exhibits an increase in the flow stress by a typical value of 30% at high strain
rates (900 s^{-1} – 3000 s^{-1}) compared to the quasi-static
value.

^{1}

^{-1}, the flow stress reaches 850 $\left[\mathrm{MPa}\right]$ at a 0.25 strain.

Strain Rate: 80 s^{-1} |
Strain Rate: 900 s^{-1} |
||||||
---|---|---|---|---|---|---|---|

True strain | 0.02 | 0.05 | 0.1 | 0.02 | 0.05 | 0.1 | 0.25 |

True plastic strain | 0.012 | 0.042 | 0.092 | 0.011 | 0.039 | 0.089 | 0.238 |

True stress (MPa) | 550 | 600 | 610 | 625 | 775 | 800 | 850 |

### Johnson-Cook Model

The speed of wave, $C$ along the bars is calculated using the relation:

- $E$
- Young's modulus.
- $\rho $
- Density of the bars.

The time step element is controlled by the smallest element located in
the specimen. It is set at 5x10^{-5} ms. The stress wave thus reaches the
specimen in 0.77 ms and travels 0.26 mm along the bar for each time step. Obviously,
it remains lower than the element length of the smallest dimension (0.88
mm).

^{-1}produces a strain rate in the specimen of approximately 900 s

^{-1}, while a strain rate of approximately 80 s

^{-1}is achieved using an imposed velocity of 1.7 ms

^{-1}. A simulation is performed for each velocity value.

^{-1}(Figure 15 and Figure 16).

Either data sources used to evaluate the strain rate give similar results.

- the strain rate effect on stress, with or without the cut-off frequency for smoothing (100 kHz);
- the influence of the strain rate coefficient (comparison with experimental data).

These studies are performed for the high strain rate model (
$\epsilon $
= 900 s^{-1}).

More physical flow stress distribution is obtained using filtering. Explicit is an element-by-element method, while the local treatment of temporal oscillations puts spatial oscillations into the mesh.

^{1}CRAHVI, G4RD-CT-2000-00395, D.1.1.1, Material Tests - Tensile properties of Aluminum Alloys 7010T7651 and AU4G Over a Range of Strain Rates.