Time-Resolved Force Measurements to Determine Positioning Tolerances for Impulse-Based Indentations

High-throughput experimentation methods determine characteristic values, which are correlated with material properties by means of mathematical models. Here, an indentation method based on laser-induced shock waves is presented, which predicts the material properties, such as hardness and tensile strength, by the induced plastic deformation in the substrate material. The shock wave pushes a spherical indenter inside a substrate material. For reproducible indentations, the applied load is of importance. To compare different processes and process parameters, the measured plastic deformation is normalized by the applied load. However, eccentric irradiation leads to altered beam profiles on the surface of spherical indenters and the angle of incidence is changed. Thus, the influence of eccentric irradiation is studied with an adapted time-resolved force measurement setup to determine the required positioning tolerances. The spherical indenter is placed inside a cylindrical pressure cell to increase the laser-induced shock pressure. From the validated time-resolved force measurement method we derive that deviations from the indentation forces are acceptable, when the lateral deviation of the beam center, which depends only on the alignment of the setup, does not exceed ± 0.4 mm. A vertical displacement from the focus position between -3.0 mm and + 2.0 mm still leads to acceptable deviations from the indentation force.


Introduction
The exact determination of the material properties of new alloys still requires numerous testing methods although computer aided predictions of the material properties exist since the 1970s [1]. Hence, cost-efficient, and high-throughput material development processes are needed to meet the demands to efficiently identify novel structural materials [2]. A possibility to decrease the processing time is to produce smaller samples instead of large casts [3]. However, high throughput material development does not only imply the fast production of new alloys. It also requires rapid material experimentation and characterization. Moreover, the constant urge for better and well-adapted material properties increases the complexity of materials and the scientific ability for the rational materials design. As a result of this complexity, high-throughput material development has been recognized as a new scientific approach to generate knowledge [4]. Instead of applying time-consuming conventional measurement methods, such as hardness or tensile tests, on macroscopic samples, the characterization is performed on small samples with suitable testing methods [5]. These testing methods determine characteristic values, which can be rapidly identified and correlated with material properties [6]. Nevertheless, the characteristics of these novel methods must be understood to be able to operate at the same accuracy level as existing methods.
Here, an indentation method is investigated, which is based on laser-induced shock waves. The novel laser-induced shock wave indentation method is further referred to as LiSE. Instead of using conventional indentation testing machines, a shock wave is induced on top of a spherical indenter with a high intensity pulsed transversely excited atmospheric-pressure (TEA)-CO 2 laser. The pressure of the shock wave pushes the indenter within 20 µs inside the test sample and creates an indentation [7]. From the indentation profile, different characteristic values are determined. These values are correlated with mechanical material properties. So far, it has been shown that the measured characteristic values (indentation depth, indentation diameter and pile-up as well as sink-in behavior) with LiSE strongly correlate with the material hardness [8] and tensile strength [9]. Pre-treatment and ablation layers of the indenter are not necessary with the wavelength of the laser beam (λ = 10.6 µm) because a plasma is quasi-instantly formed on top of the irradiated surface, which absorbs nearly all the irradiation of the laser beam [10]. Although the plasma temperature increases up to 19,000 K [11] above the indenter during the indentation process, no significant heating of the tested substrate material is observed underneath the indenter [12]. When the intensity of the CO 2 laser pulse exceeds a critical threshold (> 10 7 W/cm 2 ), the plasma results in a shock wave [13]. Barchukov et al. observed that the shock wave initiates approximately 5 mm above the metallic surfaces for pulsed CO 2 laser [14]. Studies from Sai Shiva et al. showed that the shock wave transforms during propagation from planar, to cylindrical and finally to spherical [15].
To achieve high-throughput experimentation, several indentations must be induced into the substrate material in the shortest possible time. The shock waves can be generated by a scanner system above several indenters, which are located on one large sample or on many micro samples. The investigated TEA-CO 2 laser has a repetition rate of 50 Hz. When dealing with high-throughput processes, one challenge is the accurate positioning in the given time range. For the implementation of a high-throughput method, it is therefore necessary to identify the required positioning tolerances between the laser beam and the indenter. Eccentric irradiation leads to altered beam profiles on the surface of spherical indenters and the angle of the resulting force is changed. Thus, the reproducible application of a defined load is of great importance and the influence of eccentric irradiation should be investigated. We present and investigate a time-resolved measurement method for short-time indentations to determine the maximum indentation force and transferred momentum of the indenter and to derive acceptable positioning tolerances. We imply that a shift of focal position or shift in x-and y-direction (lateral shift) is still acceptable, if the maximum force at the shifted position is still within the range of standard deviation of the maximum force at the reference position. This and the time-resolved measurement method are validated by conventional indentation tests, pendulum experiments, drop tests and LISE experiments.

Laser-Induced Shock Wave Indentation Method
The experiments are conducted with a pulsed TEA-CO 2 laser from SLCR. A maximum pulse energy of 6 J was used for the experiments. The temporal laser energy distribution measurements reveal a pulse duration of 100 ns. A focusing mirror with a focal length of 200 mm is used, resulting in a minimal quadratic beam spot of 4 mm 2 and in a laser intensity of 1.5 GW/cm 2 at maximum pulse energy. The Rayleigh length of the laser beam is z R = 10.8 mm. The spatial energy distribution of the laser beam is a top-hat with a near-uniform fluence. The indentations were created with spherical Al 2 O 3 indenter with an indenter radius of R I = 1.5 mm. A cylindrical pressure cell made of brass was used to increase the acting shock wave pressure on the indenter (see Fig. 1). Findings by Corsi et al. suggest that the reflected shock wave front can move faster in cylindrical cavities through the heated and accelerated air mass of the preceding direct shock wave [16]. Finally, this leads to a superposition of the preceding and the reflecting shock wave, which increases the acting shock wave pressure. Moreover, the pressure cell was designed according to the findings from [17] identifying the center of the shock wave above the surface. Thus, the cylindric pressure cell has a height of 10 mm and an inner diameter of 3.02 mm. The indenter is placed inside the pressure cell.
Additionally, a reference setup without a pressure cell was investigated to compared time-resolved force measurements (see Fig. 2). The position unit was used for the reference setup without pressure cell. The mounting of the test samples is based on the design guidelines for hardness testing according to [18]. The setups were flushed with compressed air to remove particles above the irradiated surface. Without a gas flushing of the pressure cell, the laser-induced plasma tends to be ignited randomly in mid-air, which lowers the pressure at the irradiated surface. The indentation experiments were conducted on Aluminum EN AW-1050 samples (further referred to as Al99.5) with a 5 mm thickness and on steel S235 samples with a thickness of 3 mm. The samples were embedded and prepared by grinding and polishing for the LiSE experiments and the Brinell hardness testing. For the LiSE experiments, the pulsed laser beam was irradiated on the indenters placed coaxially in the pressure cell and on top of the sample's surface. The position in focus   Fig. 2. The evaluation of the indentation diameter and indentation depth was carried out at 20 × magnification with the microscope VK9710 from Keyence.

Brinell Hardness
The Brinell hardness of the samples was determined with a hardness testing machine from Struers. According to [19], an indentation load of 25 N was applied on a 1 mm indenter at a dwelling time of 15 s. The Al99.5 samples have a Brinell hardness of 35.5 HBW 1/2.5 ± 0.2 HBW 1/2.5 and the steel S235 samples have a Brinell hardness of 105 HBW 1/2.5 ± 2 HBW 1/2.5. To compare the Brinell test results with the LiSE experiments, the measured indentation diameter d i , the indenter diameter (D I = 3 mm) and the maximum indentation force F max are used to estimate the Brinell hardness. For spherical indenter, the correlation between the indentation diameter d i and the maximum indentation force F max is given by the following equation:

Conventional Compression Tests
Conventional compression tests are performed on the Zwick/Roell Z250 stationary testing machine (see Fig. 3 left). This machine can apply a maximum load of 250 kN. The crosshead speed can be variably adjusted between 5 . 10 -5 mm/min and 600 mm/min. For the indentation experiments the crosshead speed is set to 0.1 mm/ min to allow a quasistatic indentation. The quasistatic indentation test is based on the Brinell hardness testing method. For the conventional indentation tests a spherical indenter is attached to a die (see Fig. 3 right). The positioning accuracy of the crosshead is ± 2 µm. For the compression tests a load cell from Zwick/Roell with a maximum test load of 200 N is used. The maximum deviation of the measured value is 0.25% above an applied load of 0.8 N.

Drop Tests
Drop tests are performed to determine whether the necessary forming energy to create indentations is comparable to the LiSE experiments. The drop height h is changed to influence the potential energy E Pot and accordingly, the forming energy. The drop tests setup is described in more detail in [20]. An increasing drop height may cause spinning of the indenter. Thus, the impact velocity in dependence of the drop height is verified with the high-speed camera Phantom V5.1 from Vision Research. A frame rate of 1000 fps with an exposure time of 990 µs is used. The indentation results are compared with LiSE and the derived forming energy.
g is the gravitational constant.

Pendulum Experiments
A pendulum setup as described in detail in [20] was used to determine the maximal deflection velocity of the indenter v A and to estimate whether the LiSE indentation is in the quasistatic or high-speed range. The impact velocity is determined for the assembly with a cylindrical pressure cell and without a pressure cell. The shock wave is generated directly on the indenter with an indenter diameter of D I = 3 mm. The indenter is attached to a thread with a pendulum length of l p = 450 mm. The thread has a diameter of d p = 0.09 mm and a tensile strength of σ m = 7100 N/mm 2 .
Blind holes with a depth of 0.5 mm and a diameter of 0.2 mm are laser-drilled in the indenter with a nanosecond laser from IPG. The thread is glued inside the blind holes. The total mass m G is composed of the mass of the thread and the mass of the indenter. The kinetic energy of the laser-induced shock wave is transmitted on the indenter. The resulting deflection of the pendulum mass is recorded with the highspeed camera Phantom V5.1 from Vision Research. A frame rate of 1000 fps with an exposure time of 990 µs is used. By calibrating the pixels via the diameter of the spherical indenter, a resolution of 0.077 mm/px is determined. The kinetic energy E kin is derived by the maximum deflection velocity of the indenter v A . This is determined by extrapolating the measured deflection velocity of the indenter over time and calculating the y-axis intercept.

Measurement Setup
Force measurements were performed with a piezoelectric polyvinylidene fluoride (PVDF) shock gauge sensor from Piezotech (piezo strain constant of -22.0 pC/N). The sensor is embedded in PTFE-foil and has an additional polyester protection on both sides (sensor model S_25CP). PVDF-sensors have a nanosecond resolution and are suitable for recording measurements with impact loading [21]. As shown in Fig. 4 Schematic setup for time-resolved force measurements for LiSE Fig. 4, the shock gauge sensor was placed below the indenter. The shock gauge sensor was fixated on a ceramic blank. The sensor has a sensing area of 1 mm 2 , which was positioned coaxially to the laser beam and indenter. An Al99.5 foil with a thickness of 50 µm was placed between the indenter and PVDF shock gauge sensor as a protective foil to avoid damage of the sensor by the laser irradiation. The protective foil is also supposed to prevent excessive point loading and allow an accurate force measurement. To guarantee a contact between protective film and sensor with as little air gap as possible, the use of a hold-down device is necessary. The pressure cell functions as the hold-down device. The signal from the shock gauge was fed into the charge amplifier from Kistler (type 5015A). The charge amplifier possesses a sample rate of 1 000 000 data points per second with a bandwidth (-3 dB) of 200 kHz. To gain a time resolved force measurement, the charge amplifier was connected to a LeCroy waverunner LT374 oscilloscope. For every condition, 10 force measurements were conducted. The maximum force and transferred momentum were measured from each force-time profile. The transferred momentum is determined by the integration of the force-time curve between the first force increase and the maximum. Further force maxima are not considered within the integral because experiments revealed that the measured maximum force correlates with the indentation geometry [7]. The average of the determined values and the standard deviation were calculated.

Theory
Since there is a protective foil between the indenter and the sensor, it must be clarified whether the momentum and the maximum force transferred by the spherical indenter through the protective foil can be accurately measured and calculated based on the time-resolved force-measurement system. The validation of the time-resolved force-measurement system is performed in three steps: -Firstly, it is tested whether the conditions of a quasistatic normal impact are fulfilled. In case of a quasistatic impact, the contact forces correspond to the static forces [22] and can be derived from the equation of motion. In case of a quasistatic normal impact, the measurements on the compression testing machine can be compared with those of LiSE. -Secondly, the influence of the protective film is examined from a material engineering point of view by analyzing the effect of multiple indentations on the measured force-time curve and the transferred momentum at the same position. The measurement setup is not changed to ensure that the indenter always indents at the same position of the protective foil. It is suggested that the determined momentum must not change over the number of indentations. On the contrary, the maximum measured force must change after each indentation because the foil experiences deformation. -Thirdly, it is presumed that the measured maximum force changes with an increasing number of indentations, up to the point where the foil can no longer be plastically deformed. At this point, the indenter presses into an elastic half space and the measured and derived (from the momentum) forces of the LiSE process are comparable with a conventional quasistatic indentation process. Thus, indentations are induced with the same time-resolved measurement setup with a compression testing machine and compared to LiSE. The impact of a rigid indenter into an elastic halfspace is comparable to the impact of two elastic indenters [22].
The impact velocity v 0 can be calculated from the transferred momentum M and the mass of the indenter m I .
The impulse shape is implied to be ideal. Therefore, the following applies for the initial conditions where t is the indentation depth and v the velocity. The maximum force F max is derived by solving the equation of motion according to [22] with the initial conditions from Eq. (5).
The effective Young's modulus E* is calculated using the material characteristics of the protective aluminum foil (E Al = 70 GPa, ν Al = 0.34) [23].
To fulfill the condition of a quasistatic indentation, the impact velocity v 0 must be much smaller than the characteristic propagation velocity of elastic waves v el in the elastic half space [22]. According to Hunter, it is sufficient that the following relationship is fulfilled [24]: The characteristic propagation velocity of elastic waves v el correlates with the Young's modulus and the density of the material. For aluminum, the density is ρ Al = 2.7 g/cm 3 .
The impact velocity is derived from pendulum experiments. The impact velocity v 0 corresponds to the maximum deflection velocity of the pendulum v A . Figure 5 shows the time-resolved force measurement below the indenter for the cylindrical pressure cell in comparison with the reference setup (without pressure cell). The profile of the reference setup (without pressure cell) reveals a double-peak profile. In contrast, the force-time curve of the cylindrical pressure cell displays one strong maximum and then decreases down to zero after 20 µs. For each measured force-time curve, the signal-to-noise ratio of the charge-voltage amplifier results in a standard deviation of U = 0.3 N. The determined maximum deflection velocity is shown in Fig. 6 for the assembly with a cylindrical pressure cell and without a pressure cell. The maximum deflection speed is observed with the cylindrical cell. Figure 7 shows the development of measured maximum force and the determined momentum as well as the maximum force calculated from Eq. (6) in dependence of the number of indentations at the same position of the protective foil. The measured maximum force increases with increasing number of indentations. This behavior is explained by the deformation of the protective foil with each indentation. With increasing number of indentations, the deformed indentation volume by the indenter decreases, which also decreases forming energy and accordingly, increases the measured maximum force. After 100 indentations, the measured maximum force approaches a value of F max,me = 27.7 N. A standard deviation of ΔF max = 0.5 N is measured in this range. The measurement of the momentum shows no significant change in dependence of indentations. On average, a momentum of M = 97.7 µNs is From the momentum a maximum force of F max,cal = 40.9 N ± 2.7 N is calculated. The deviation of the measured maximum force F max,me from the calculated maximum force F max,cal is 30%. Moreover, the change in indentation diameter and indentation depth is plotted against the number of indentations. Figure 8 shows that the change of the indentation depth and the indentation diameter decreases with increasing number of indentations. After 90 indentations, no significant change of the indentation diameter and the indentation depth is observed.

Validation of Time-Resolved Force Measurement Method
For indentations by means of a compression testing machine, the same indenter is attached to a die and pressed into the piezoelectric sensor with protective foil. A defined force of F max = 30 N is applied with the tensile-compression testing machine Fig. 6 Determination of the pendulum speed by means of the pendulum test with and without pressure cell Fig. 7 Development of the measured maximum force, the transferred momentum, and the calculated maximum force over the number of indentations Z250. In Fig. 9 the first measured value is minimally lower than the maximum force set at the compression testing machine. With increasing number of indentations, the measured maximum force is reduced until a plateau is reached after 70 indentations. The relative deviation between set maximum force and measured maximum force is 30% in this range.
Additionally, the maximum force F I is varied at the compression testing machine and compared to the measured maximum force F G at the pressure sensor (see Fig. 10). The indenter is pushed in same position as before. Three measurements are performed at each set maximum force. Only a small scattering is detected for each measured maximum force. Between the set maximum  Moreover, indentations were induced in Al99.5 and the steel S235 and compared with the drop test with the LiSE. Figure 11 compares the measurement results of the drop test with those of LiSE. The drop height was varied, and a trend was determined. For LiSE a pulse energy of E P = 6 J and a focal position of f p = + 10 mm were used. For both methods, no significant differences can be obtained between the indentation volume induced with LiSE and the indentation Fig. 10 Difference between the measured maximum force F G with piezo sensor and the set maximum force F I with conventional testing machine and LiSE   Fig. 11 Comparison between drop tests and LiSE of the induced indentation volume volume induced with drop tests at similar potential energy. This is behavior also observed for both investigated materials.

Force and Momentum Measurements
The distribution of maximum force does not show significant changes between the shift in x-and y-direction. Thus, the mean value was taken from the two respective values in x-and y-direction and referred to as lateral shift, as shown in Fig. 12. By positioning the laser beam away from the center of the indenter, the maximum transmitted force and momentum on the indenter is decreased. Moreover, the standard deviation increases with increasing lateral shift. A significant decrease in maximum force and momentum is observed at the lateral shift of Δx = ± 0.5 mm.
For the cylindrical pressure cell, the shift of focal position f p increases the acting maximum force until f p = -4 mm and f p = 12 mm (see Fig. 13). Further shifting the focus away from the surface leads to decrease in transmitted force and momentum. By shifting the focus inside the indenter (f p < 0), the increase in force and momentum is not as dominant compared to the shift in focal position above the indenter. Figure 14 shows indentations measured at 20 × magnification with the microscope VK9710 from Keyence. The reference position in focus and center is compared to different focal positions and lateral shifts.

Indentations
In Fig. 15, the influence of lateral shift is shown. The shift leads to a decrease in induced indentation diameter and indentation depth (see Fig. 15). A significant decrease in indentation diameter is observed at a lateral shift of Δx = 0.5 mm. Moreover, the lateral shift leads to an increase in standard deviation.  Figure 16 shows the influence of shift in focal position on indentation geometry for the cylindrical pressure cell. The measured indentation diameter increases when the focus is shifted above or inside the indenter surface. A significant change is observed at the focal positions f p ≥ -3 mm and f p ≤ 2 mm in reference to the position in focus (f p = 0).

Discussion
The derived maximum deflection velocities in Fig. 5 reveal that the conditions of a quasistatic normal impact are fulfilled according to the findings of Hunter [24]. Since the highest deflection velocities are observed with the cylindrical pressure cell, it  Both ratios are clearly below v' max < < 1, which indicates that the condition of a quasistatic indentation is fulfilled. Thus, the LiSE indentation process is quasistatic, wherefore the present contact forces correspond with the static ones. When the linear trend line is continued in Fig. 9, the calculated maximum force in dependence of the measured maximum force of LiSE lies on the trend line, which validates the derived maximum force. Still, it must be considered that the indentation behavior might differ for impulse-based indentation processes in dependence of the material properties such as young's modulus and strain hardening coefficient. Hence, it is important that the rebound speed must be additionally measured to derive the acting forces, which are relevant for plastic deformation. Consequently, the determined maximum forces from the measurement method cannot be compared with the Brinell hardness from Eq. (1) without additional data about the rebound behavior.
No significant differences can be obtained in Fig. 11 between the indentation volume induced with LiSE and the trend of the indentation volume induced with drop tests, which additionally validates the determined momentum with the time-resolved measurement setup. It is demonstrated that the measured maximum force is influenced in dependence of the number of indentations (see Fig. 8 and Fig. 9). However, the transmitted momentum and the derived maximum force from Eq. (6) remain constant. Thus, the introduced time-resolved measurement system is valid to obtain the momentum of the indenter and maximum transmitted maximum force of the indenter on the workpiece. The validation of the measurement setup allows to use the maximum force or momentum as a reference value and to derive position tolerances.
A shift of the focal position away from the focus (reference position) leads to an increase in maximum transmitted force in both directions (see Fig. 13). The indentation experiments in Fig. 16 underline these findings. When shifting the focal position, the increase in maximum force can be explained by plasma plume investigations shown in [25]: The origin of the shock wave is shifted closer to the indenter surface. Additionally, the pulse energy predominantly governs the shock wave propagation velocity [26]. Both mechanisms increase the transmitted force on the indenter. By shifting the focus inside the indenter (f p < 0), the increase in maximum force and momentum is not as dominant compared to the shift in focal position above the indenter. When the focus is shifted inside the indenter (f p < 0), the laser beam partially irradiates on the upside of the pressure cell (height 10 mm). This causes the formation of a plasma on top of the pressure cell, which absorbs the laser irradiation. However, this effect cannot be accounted for the decrease in maximum force when the focus is shifted above the indenter (f p > 12 mm). The beam diagonal is not large enough to interact with the upside of the pressure cell. The decrease in maximum force can be explained by the Brewster angle θ B of the pressure cell made of brass in dependence of the laser wavelength. The Brewster angle is θ B = 89° [27] for the wavelength of the laser and brass. This Brewster angle is in the range of the angle of incidence of the laser beam on the inside wall of the pressure cell when the focal position is shifted above the indenter. Thus, the shock wave is created further away from the indenter surface, which decreases the transmitted force on the indenter.
Regarding the positioning tolerances of the vertical displacement (shift in focal position), we obtain acceptable positioning tolerances between f p ≥ -3.0 mm and f p ≤ + 2.0 mm for the calculated maximum forces in Fig. 13 and the measured indentation diameter in Fig. 16. The relatively large vertical positioning tolerances are attributed to the Rayleigh length of the laser beam (z R = 10.8 mm). Moreover, no differences are obtained for positioning tolerances of the lateral shift between the calculated maximum forces in Fig. 12 and the measured indentation diameter in Fig. 15. A lateral shift of the laser beam leads to a changed beam profile on the surface. The resulting maximum force as well as indentation diameter do not seem to change significantly up to a lateral shift of Δx = ± 0.4 mm (see Fig. 12 and Fig. 15). The cylindrical pressure cell seems to compress the plasma formation to a small location above the indenter. Only when the plasma formation absorbs a large amount of the laser irradiation (like the shift in focus inside the indenter), the maximum force transmitted on the indenter is reduced.

Conclusions
A measurement method was introduced to determine the momentum and maximum forces transmitted on spherical indenters by laser-induced shock waves. The influence of eccentric irradiation and shift in focal position were investigated with this method to determine the required positioning tolerances. The conclusions are summarized as follows: • The introduced time-resolved measurement system is valid to obtain the momentum of the indenter and maximum transmitted force of the indenter. • From the validated time-resolved force measurement method and indentation experiments we derive that deviations from the maximum forces are still acceptable, when the lateral deviation of the beam center, which depends only on the alignment of the setup, does not exceed ± 0.4 mm. The rather small lateral positioning tolerances are caused by plasma shielding above the cylindrical pressure cell. A vertical displacement from the focus position between f p ≥ -3.0 mm and f p ≤ + 2.0 mm still leads to acceptable deviations from the indentation force. The larger vertical positioning tolerances are attributed to the large Rayleigh length of the laser beam. • The LiSE indentation process is quasistatic, wherefore the present contact forces correspond with the static ones. However, the indentation behavior differs for impulse-based indentation processes in dependence of the material properties such as young's modulus and strain hardening coefficient. Thus, the rebound speed must be additionally measured to derive the acting forces during plastic deformation.
Acknowledgement Financial support of the subproject D02 "Laser induced hardness measurements" funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -Project number 276397488 -SFB 1232 is gratefully acknowledged.
Funding Open Access funding enabled and organized by Projekt DEAL.
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