# Numerical and Experimental Studies of a Conical Striker Application for the Achievement of a True and Nominal Constant Strain Rate in SHPB Tests

## Abstract

The problem of ensuring both nominal and true constant strain rate in the split Hopkinson pressure bar experiment was considered through the application of the conical striker for 316 L steel specimen. The experimentally confirmed results from numerical analyses indicate that the application of a conical striker with the determined apex angle for the given experimental conditions is a good method for achieving a constant value of the strain rate. Moreover, the results of the study showed that the value of the striker apex angle has the greatest influence on the mechanical response of the specimen material. In turn, the impact velocity slightly affects the value of the striker apex angle.

## Keywords

Split Hopkinson pressure bar High-strain-rate testing Constant strain rate Numerical simulation## Introduction

The basic methodological requirement of a split Hopkinson pressure bar (SHPB) technique is that a specimen needs to deform nearly uniformly at a constant strain rate under dynamically equilibrated stresses, and propagation of elastic waves through the input and output bars is described using a one-dimensional wave theory [1]. Strain acceleration, which is the effect of a non-constant strain rate, can produce additional axial stress and radial stress in a specimen [2, 3, 4, 5], which may influence the measurement results [6]. Several methods are available for solving the problem of maintaining the constant strain rate during SHPB tests: a pulse-shaping technique [7, 8], conical (tapered) striker method [9, 10], and a three-bar technique with a dummy specimen [11].

In the current work, the attention is focused on the incident pulse-forming technique with the use of a conical striker to achieve a constant strain rate. The literature contains a relatively large number of publications on the use of a shaped striker in investigations of brittle materials such as rocks [12, 13, 14], concrete [15] and even bones [9, 10]. However, it is difficult to find publications that investigate the use of a shaped striker for testing of metals and their alloys, particularly for materials with a high strain-hardening coefficient. Only Sato and Takeyama published studies on this topic, however, in a narrow range [16, 17].

*V*values were within the range of 15–25 m/s. The striker geometry was described in Fig. 1. For clarity and simplicity of description, it is assumed that, in the case of the striker impact with the smaller-diameter end, the apex angle is positive, and in the other cases, it is negative (Fig. 1).

Material constants of 316 L austenitic steel

| 8000 |
| 304 |

| 193 |
| 1097 |

| 0.27 |
| 0.492 |

| 0.014 |

## Results and Discussion

Since the process of plastic deformation of a sample under the SHPB test conditions is determined by, among others, the dimensions of the specimen, numerical experiments were conducted to determine an influence of the sample geometry and striker impact velocity. The influence of both the diameter and the length of the specimen on the history of the deformation curve was investigated in the case of strikers with apex angles optimized for the following experimental conditions: diameter and length of the sample = 5 × 5 mm and impact velocity V = 20 m/s.

A significantly larger effect of α angle on \( \dot{\varepsilon}(t) \) can be observed for specimens of different lengths (Fig. 4(b)). For the tested specimens corresponding to *L/D* = 0.5, 1, and 1.5, the effect of length is relatively large. However, it can be observed, as expected, that the strain rate of the short samples (*L* = 2.5 mm) decreases with an increase in strain, whereas in the case of long samples (*L* = 7.5 mm), the opposite relationship is observed. Useful practical guidelines are also derived from the analysis of the curves shown in Fig. 4(c). These curves show that the conical striker with a given apex angle can be used regardless of the accepted range of the striker impact velocity for nominal strain rates curves.

Summary of verification test conditions

Test | V [m/s] | α [°] | Type of strain rate | L | D |
---|---|---|---|---|---|

A | 15.6 | 0.78 | Nominal | 4.95 | 4.81 |

B | 15.6 | 0.78 | Nominal | 5.05 | 4.81 |

C | 20.2 | 0.55 | True | 4.97 | 4.83 |

D | 20.3 | 0.55 | True | 4.99 | 4.77 |

The results of experimental investigations on the selected conical strikers are coincident with the results of numerical analyses, which leads to the conclusion that the constant strain rate (both true and nominal) can be obtained using the appropriately selected geometry of the conical striker. It should be noted that, in real experimental conditions, the waveforms obtained in the usable range, i.e., the corresponding plateau, deviate from the constant value to a greater degree than in the case of the curves obtained based on numerical analyses (Fig. 6(b)).

The fact that the assumed constancy of the strain rate was not achieved in the experimental conditions results from the discrepancy between the real conditions of the SHPB experiment and the adopted conditions of the numerical experiment. According to the authors, the above discrepancies are mainly a consequence of using the JC constitutive model for the considered materials and the friction model. For example, the slight increase in the true strain rate in the plateau range observed in Fig. 6(b) is probably a result of using the JC model. It should be assumed that this model do not adequately describe the mechanical behaviour of the tested material. In addition, a characteristic change (marked with a dotted ellipse) in the profile of the curves on the plateau section can be observed. The probable cause of this phenomenon is variability of the friction forces occurring on the specimen-bar contact surfaces during the dynamic deformation of the specimen. In the initial stage, the friction forces are relatively small due to a lubricant layer between the contact surfaces. As a result, the initial increase in the strain rate is visible. Due to deformation of the specimen and the applied loading, the lubricant layer decreases until it disappears entirely (time less than 60 μs). This effect on the strain rate-time curve is expressed by changing the curve trend from increasing to decreasing or approximately constant. This phenomenon was not included in the numerical model because it was assumed that the coefficient of friction does not depend on the conditions of the experiment.

## Conclusion

The numerical and experimental analyses presented in this article showed that application of a conical striker is a good method for achieving a constant strain rate, both nominal and true, in SHPB experiments. However, this method requires the use of a numerical model of the SHPB experiment to determine the conical striker apex angle for a given specimen material, its geometry (diameter, length) and the impact velocity of the projectile.

Contrary to the initial predictions, the use of the striker with a specific geometry does not limit its use to the given experimental conditions. For a given material group, preparation of a several conical strikers set (3–4 pieces) is sufficient for adjustment of the experimental conditions to achieve a constant strain rate. The results of this research showed that the mechanical response of the specimen materials has the greatest influence on the value of the striker apex angle, whereas the dimensions of the disk sample storage affect it to a lesser extent, with the exception of short samples, i.e., those with an L/D ratio lower than 1. Similarly, the impact velocity has a small effect on the value of the striker apex angle. In the case of a nominal strain rate, this angle is nearly independent of the striker velocity, whereas for the true strain rate, it depends on this velocity and is nearly a linear function.

Based on the observations from the experimental studies, it can be concluded that guaranteeing a constant strain rate requires the development of a more complex numerical model that considers the physics of the specimen dynamic deformation. In such a case, striker geometry may deviate from the shape of the cone with a complex contour which is more difficult to develop and probably has a limited range of the use compared with conical strikers.

## Notes

### Acknowledgements

The support of Military University of Technology grant PBS 23-937 is gratefully acknowledged.

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