Hopkinson bar experiments: test setup and method of data evaluation

The dynamic experiments were performed at the Fraunhofer Institute for High-Speed Dynamics, Ernst-Mach-Institute (EMI). The Hopkinson bar setup used in this investigation was the spall configuration. It consists of an acceleration facility for the generation of the loading pulse and a long slender incident bar, at the end of which a cylindrical specimen is positioned; see Figure 4.15. The loading is caused by accelerating a striker bar with defined length and diameter, which then strikes the incident bar. The length of the specimens was adopted from numerous previous studies on various types of concrete at the EMI (Schuler et al. 2006; Millon et al. 2009; Mechtcherine et al. 2011a). Both the specimen length and material properties influence the wave shape and its amplitude due to damping effects. To

crosshead b)

a)

LVDT

70 40

40 270

load cell

reinforcing glue bar

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minimize such influence, the wave properties were measured directly on the specimen. In each experiment, the impact velocity is derived from the recordings of a two-channel light barrier. The system is installed close to the impact zone, providing realistic impact velocities.

Since the distance between the two gauges is known, the impact velocity can be calculated from the measured time difference. The impact induces a longitudinal propagating pulse which consists of a compression portion and a subsequent decompression portion. The pulse advances along the incident bar until it reaches the sample, where, due to different impedances of the aluminum incident bar and the specimen, it is partially transmitted by the sample and is partially reflected to the bar. When the incident pulse (compressive wave) reaches the free end of the specimen, it reflects completely into a tensile wave. Due to the high magnitude of the impact wave, the tensile strength of the tested material is exceeded already in the overlapping zone of the decompression part of the compression wave with the reflected tensile wave, causing spall fracture; see Figure 4.15.

Figure 4.15. Hopkinson bar schema and wave propagation in spall experiments in time.

incident bar incident bar L = 5.5 m

d = 75 mm striker bar

apparatus specimen

L = 250 mm

fractured specimen

compressive wave in incident bar

partial reflection at contact with specimen

resultant tensile wave t1

t2

t3

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The wave propagation in the bar and the specimen is controlled by longitudinal strain gauges.

Their strain-time signals are taken directly for the evaluation of the experiments. Recording systems with high sample rate in the MHz range are required for these stress-wave experiments with loading times in the millisecond range.

Unnotched samples are used for evaluating the dynamic tensile strength and Young's modulus because undisturbed wave propagation through the specimen is required. Notched samples are used for determining the dynamic fracture energy, since it is important to ensure fragmentation of the sample into two parts. Furthermore, the notch makes it possible to monitor the fragmentation process closely by focusing the optical measuring equipment on a predefined failure region.

Figure 4.16. Longitudinal wave propagation in the material and particle speed recording on an M1-PE sample.

The specimens are equipped with two longitudinally arranged strain gauges and a thin reflector on the transverse face at the free end. The strain gauges are necessary for evaluating the strains and propagation velocity of the longitudinal wave in the sample, while the reflector serves the laser measuring system to record the time variation of the particle speed; see Figure 4.16. By compiling the particle velocity-time signal with the strain-time signals of the strain gauges, the time differences (time of wave propagation) t are derived. Under consideration

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of the distances s between the specimen’s free end and the strain gauges an average velocity is calculated.

The evaluation of spall experiments is based on the law of conservation of momentum and energy and uses the theory of longitudinal wave propagation in a solid long thin bar (Schuler et al. 2006). For determining the dynamic Young's modulus Edyn, it is necessary to know the longitudinal wave's velocity CL in the tested material and the density ρ of the material; see Eq. 4.2. The velocity of the longitudinal wave is determined according to Eq. 4.1 from the signals recorded by the strain gauges as shown in Figure 4.16. For this, steady wave propagation is assumed.

𝐶_𝐿 = ^∆𝑠

∆𝑡 (4.1)

𝐸_𝑑𝑦𝑛 = 𝐶_𝐿² ∙ 𝜌 (4.2)

As mentioned above, a laser measuring system monitors the particle velocity on the free face of the sample. From the resulting curve (Figure 4.16) the pull-back velocity ∆𝑢_𝑝𝑏 can be determined and implemented in the evaluation of the dynamic tensile strength ft,dyn, as in Eq. 4.3:

𝑓_{𝑡,𝑑𝑦𝑛} = ¹

2∙ 𝐶_𝐿∙ 𝜌 ∙ ∆𝑢_𝑝𝑏 (4.3)

The dynamic specific fracture energy 𝐺_𝐹^′ of a material defines the energy which is dissipated during the fracture process (here spallation). It can be determined on notched specimens and is based on the momentum transfer from the second fragment to the first one (I1-2) and by involving the velocities of the fragments before and after the crack formation (mean crack opening velocity 𝛿̇₁₋₂) (Schuler et al. 2006; Millon et al. 2009) according to Eq. 4.4:

𝐺_𝐹^′ = ¹

𝐴𝑒𝑓𝑓∙ 𝐼₁₋₂∙ 𝛿̇₁₋₂ (4.4)

where Aeff is the effective (net) cross-section of the specimen.

The velocities of the fragments before fracture can be computed analytically according to Schuler et al. (2006), while the speeds of the fragments after crack initiation are measured with an analogous high-speed extensometer. Figure 4.17 presents an example of the displacement-time relationship for each fragment formed.

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Figure 4.17. Displacement-time measurement with an optical high-speed extensometer.

Before the dynamic tensile strength is reached in a spall experiment, elastic material behavior is assumed. Thus, the strain gauge recordings enable to estimate stress waves from which stress-time profiles are calculated for each point of the specimen (Millon et al. 2009).

According to Eq. 4.5 the strain rate for a certain position and a certain time can be calculated as

𝜀̇ = ^𝑑

𝑑𝑡∙^{𝜎(𝑥,𝑡)}

𝐸 (4.5)

with 𝜎(𝑥, 𝑡) the calculated stress-time profile and 𝐸 the dynamic Young’s modulus.