PSI - Issue 42

Sebastian Henschel et al. / Procedia Structural Integrity 42 (2022) 110–117 S. Henschel and L. Kru¨ger / Procedia Structural Integrity 00 (2019) 000–000

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2

All other measures, especially the force F and the displacement u within the bar are directly related to the strain:

F = A · E · ε u = c ε d t

(2)

(3)

In these equations, A , E and c are the cross-sectional area, Young’s modulus and longitudinal wave speed of the bar, respectively. According to Gray (2000), Klepaczko (2007), Chen and Song (2011), and Kariem et al. (2012), it is common practice that the conversion factor K ε is obtained from a dynamic calibration. This is important, since the strain gauges are not perfectly aligned due to the manual application on the bars’ surface. This calibration involves impact of the striker bar on the bar of interest, e.g. the incident bar. In case of identical material and diameter for the striker and incident bars, the strain amplitude of the (nearly) rectangular pressure pulse ε max can be calculated from the striker velocity v S under the assumption of momentum conservation:

v S 2 c

(4)

ε max =

Hence, the measured voltage U can be related to the expected strain ε max and K ε is obtained by Eq. 1. This calibration is known as ‘bars apart’ (Gray (2000)) and has to be done for both incident and transmitted bars. The knowledge of the wave speed c is an essential requirement for using Eqs. 1 and 4. E.g. Koumlis and Lamberson (2019), Casem et al. (2003), Alves et al. (2012), and Santiago and Alves (2013) propose the usage of a laser Doppler interferometer or a high-speed camera for measuring the particle velocity of the incident and transmitted bars. From these velocities, the strain in the bars is calculated (Gray (2000)):

v c

(5)

ε =

However, it is shown by Santiago and Alves (2013) that the measurements of the high-speed camera cannot be directly used to calibrate the strain gauges. The applied camera has a relatively low spatial resolution. Hence, the strain / time relationship of the specimen (calculated by the incident, reflected, and transmitted pulses) is only roughly reproduced by the optical measurement of the camera. The laser interferometer was positioned along the bar axis (Alves et al. (2012); Santiago and Alves (2013)). Hence, this calibration is in analogy to the ‘bars apart’ methodology. Forrestal et al. Forrestal et al. (2002) and Song et al. Song et al. (2009) observe an e ff ect of the sabot mass on the amplitude of the incident pulse. Di ff erent striker geometries are used for special purposes, especially for pulse shaping, e.g. Li et al. (2000); Gerlach et al. (2011). Consequently, the above-mentioned (nearly) rectangular incident pulse is a special case. In the case of a significantly shaped incident pulse, Eq. 4 cannot be accurately applied. The second step of the conventional dynamic calibration is known as ‘bars together’ (Gray (2000)). It is the aim to obtain a coe ffi cient of pulse transmission ( K σ ) while there is no specimen between the incident and transmitted bars:

K σ = ε I , max /ε T , max

(6)

From the two above-mentioned calibration steps, it becomes clear that there are di ff erent assumptions regarding the conservation of momentum. In the first step of calibration (‘bars apart’), it is assumed that the momentum of the striker bar is completely transferred to the bar of interest. Gama et al. (2004) argue that “material non-homogeneity