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

113

4

Second, the displacement of each edge in axial direction, i.e. u 1 and u 2 in Fig. 2b, is measured. The sensitivity and the cuto ff frequency of the extensometer are given by 1 V / mm and 250 kHz, respectively. Both channels of the extensometer are recorded by the above-mentioned transient recorder with the same time scale. In order to obtain di ff erent incident pulses, the tests were performed with or without copper pulse shapers. To this end, cylindrical discs (diameter 5 mm, thickness 0.5 or 1.0 mm) were laser-cut from a copper sheet. The tests were conducted with and without a specimen between the incident and transmitted bars. Tests with a specimen were performed with cylindrical samples (diameter d 0 = 6 mm, height h 0 = 6 mm) from a high-strength steel (G42CrMo4 + QT). Machined specimens were heat treated in order to obtain the quenched and tempered state. Characteristics of the material’s tensile behavior under static and dynamic loading conditions can be found in previous studies (Henschel et al. (2013); Henschel (2018)).

3. Calculations

Fig. 2b shows the definitions of strains in the bars and displacements at the positions 1 and 2. The relationships between the strains and the displacements are in analogy to Eq. 3: u 1 ( t ) = c t 0 ( ε I ( τ − t I ) − ε R ( τ + t I )) d τ (7) u 2 ( t ) = c t 0 ε T ( τ + t T )d τ (8) The displacements u 1 and u 2 are calculated for the time when the pulse propagates along the positions 1 and 2, i.e. 0 ≤ t ≤ t P . In these equations, t P is the duration of the pulse. An ideal impact of the striker with the length l S leads to t P = 2 l S / c . The di ff erent signals were shifted in time due to the di ff erent locations of measurement, see Fig. 1. The shifting times are t I = x I / c and t T = x T / c . It has to be noted that t P , t I , and t T can be measured directly by analyzing the strain / time relationships. The time t = 0 was defined as the time when the beginning of incident pulse passed interface 1. To be exact, the transmitted pulse should be evaluated between t = 0 and t = t P + t Sp . The duration of the wave propagation in the specimen is given by t Sp = h 0 / c Sp , with the longitudinal wave speed in the specimen c Sp . In most cases, this additional time does not have any e ff ect on the results of Eq. 8 since t Sp ≪ t P . Hence, most of the displacement of interest is already accomplished during t P . Following the initially stated goal of determining the calibration factors for the incident and transmitted bars, the strain measurements by the strain gauges (SG) were calibrated by using the optical extensometer (OE). In general, from Eqs. 1 and 3 follows u SG = K · c U SG d t (9)

incident bar (IB)

Stickers with b/w edges

h 1 2

transmitted bar (TB)

0

ε

ε

I

T

IB

TB

ε

R

x E =

0.5–1 mm

u

u

(a)

(b)

1

2

Fig. 2. (a) Locations of black-and-white edges at the distance x E to the interfaces 1 and 2. (b) Definitions: Incident, reflected, and transmitted strain ε I , ε R , ε T , respectively. Displacements u 1 and u 2 at positions (interfaces) 1 and 2. Length of the specimen h 0 .