High Temperature Vibratory Response of Hastelloy-X: Stereo-DIC Measurements and Image Decomposition Analysis
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The mechanical behavior of solids in combined high-temperature and vibratory environments, such as those experienced during hypersonic flight, are historically not well explored. In this work on Hastelloy-X plates, elevated temperatures were achieved by induction heating and periodic vibratory loading was applied using a shaker. Surface displacements and strains were measured using stereo digital image correlation (DIC) in the blue spectrum to alleviate issues associated with thermal radiation. Through the use of image decomposition techniques the resultant high-quality experimental data were used to validate numerical simulations of combined thermoacoustic loading. The simulations were based on the deformed shape and the corresponding temperature distributions measured experimentally as well as taking into account the thermal dependence of Hastelloy-X mechanical properties.
KeywordsStereo digital image correlation High temperature measurement Thermo-acoustic loading Induction heating Image decomposition analysis
Digital image correlation (DIC) is a versatile measurement technique, capable of obtaining full-field displacements and strains at a range of length scales, temperatures, and loading rates. Motivated by the increasing interest in applications involving significant thermal effects, such as aero-engines, spacecraft re-entry, aircraft hypersonic flight, and nuclear power applications, recent efforts have extended the upper temperature limit of the DIC technique by the use of optical bandpass filters and narrow band lighting, which reduces the influence of light radiated by the specimen at high temperatures [1, 2, 3, 4, 5, 6]. Grant et al. demonstrated this approach for high-temperature DIC by using blue light along with blue range bandpass filters to measure the coefficient of thermal expansion of RR1000 (a nickel-based alloy) at temperatures up to 1000 °C with two-dimensional (2D)-DIC . Using a similar method, Novak and Zok demonstrated the blue-filtering technique on a C/SiC composite at temperatures up to 1500 °C . Later, Chen et al. and Pan et al. applied the blue-filtering technique to stereo-DIC – in which two cameras are used to make three-dimensional (3D) measurements – at temperatures up to 1100 and 1200 °C, respectively [3, 4]. Most recently, Berke and Lambros showed that the temperature range can be extended even further by using ultraviolet lighting and optics, which operate at an even shorter wavelength than blue light .
However, in many of the high temperatures applications mentioned above (aero-engines, spacecraft re-entry, hypersonic flight) it is the coupled effects of thermal and mechanical loading, often at high frequencies, that provide an extreme operating environment which can critically affect structural performance and cause failure by the interaction of coupled, non-linear, failure modes that are otherwise not present . Specifically, the combined high-temperature and vibration environment, as is experienced by aircraft during hypersonic flight and which is of interest to this study, is historically not well explored. It is only recently that dynamic experiments have been attempted at high temperature. Abotula et al. used a pair of high-speed cameras to capture the deformation of Hastelloy (a nickel-based superalloy) samples at temperatures up to 900 °C under shock wave loading . To illuminate their specimens, Abotula et al.  used a 450 nm filter with a full width at half maximum (FWHM) of 40 nm and a transmission efficiency of 45 %, which significantly reduced the amount of light reaching the camera sensors – potentially a critical drawback in high-speed imaging applications which require limited image exposure times. To overcome this limit, they used a high energy flash lamp which delivered 220 kW to the specimen for 5 ms. In another high-speed study, Vautrot et al. used a 500 frames per second (fps) camera to study the plastic deformation of high-carbon steels at up to 720 °C under strain rates ranging between 10−3/s to 400/s using a split Hopkinson pressure bar . They used a much broader filter which passed the entire 400–700 nm visible wavelength range while screening out infrared, and were able to conduct their tests using a lighting system of white light LEDs.
develop procedures to obtain high quality experimental data suitable for validation of numerical results in a combined thermal and mechanical/vibratory environment,
study the influence of elevated temperature on the vibratory response of a metallic plate, and specifically investigate the (potential) change of resonant modes/frequencies at elevated temperatures above room temperature, and
quantify the (potential) discrepancies between full-field experimental measurements and their counterpart numerical simulations.
To achieve these goals, thermo-vibratory loading conditions are combined on a thin plate of a nickel-based superalloy using magnetic induction for heating and a mechanical shaker for resonant vibratory loading. Combined high-temperature/vibration experimental measurements are based on the optical technique of stereo-DIC that measures the full-field surface displacements and strains of the plate. A companion computational model is also used to perform a modal analysis by assuming an elastic plate but allowing for the variation in elastic properties with temperature. Resonant modes are then compared between experiments and modeling using an image decomposition technique involving 2D Tchebichef polynomials.
Experiments were conducted on 120 × 80 × 1 mm plates of Hastelloy-X, a nickel-based superalloy, purchased from American Special Metals Inc. [Miami, FL, USA]. A 5 mm diameter hole was drilled through the center of the plate for attaching one end of a threaded stinger rod to the plate while attaching the other end to a mechanical shaker. The front surface of the plate was speckled using high temperature refractory paints purchased from Aremco [Valley Cottage, NY, USA]. To help the paints adhere, the plate’s surface was pre-oxidized in a box furnace by heating the specimen to 800 °C (i.e. higher than the expected maximum in the experiments and below the heat treatment temperature during manufacture) and maintaining the temperature for a dwell time of about 20 min before cooling. The now-oxidized front surface was then speckled white with Pyropaint 634-ZO, which has a maximum temperature rating of 1800 °C. The white paint was cured at room temperature for 2 h and at 100 °C for 2 h as described in the manufacturer’s curing instructions. The surface was then additionally speckled with a random black speckle pattern, to provide contrast, using Hi-E Coat 840-CM, which has a maximum temperature rating of 1371 °C. The black paint was cured for 1 h at room temperature, followed by 30 min at 100 °C and 1 h at 260 °C as described by the manufacturer’s instructions. The heating and cooling rates for all thermal cycles during paint application were about 20 °C/min.
The plate was heated by magnetic induction using a pancake-type coil . The coil was positioned behind the plate so as not to obstruct image capture of the front surface of the plate. An opening was included in the center of the coil for the stinger rod to connect to the specimen without either the rod or the specimen coming into contact with the coil. The temperature was monitored using a single K-type thermocouple bonded to the back of the plate at the position where the threaded rod was attached. Like the rod, the thermocouple wire also passed through the center of the coil without contacting the coil.
For each of the resonant frequencies, a series of stereo images were taken using a pair of 1024 × 1024 pixel CCD cameras [Prosilica GX1050, Allied Vision Technologies GmbH, Stradtroda, Germany] and 50 mm lenses with blue-range bandpass optical filters [BP470, Midwest Optical, Palatine, IL, USA] resulting in an average magnification of 0.13 mm/pixel. The cameras were triggered using the Fulcrum module of the Vic-Snap image acquisition software provided by Correlated Solutions Inc. [Columbia, SC, USA]. To prevent motion blur, the blue LED ring lights were lit as brightly as possible and the exposure time for the cameras was set as low as possible. The minimum exposure time of the cameras was 10 microseconds. The maximum frame rate of the cameras was 112 fps, which was much slower than the oscillation of the plate. To correct for the slower frame rate, the Fulcrum software uses phase-locking to time image acquisition such that images taken from successive periods of oscillation can later be reconstructed into one representative period. Image pairs were acquired every 5° in the period of the oscillation, resulting in a total of 71 image pairs for each resonant mode. Displacements and strains were computed using Vic-3D. The analysis was performed with a 29 × 29 pixel subset at an interval of 7 pixels, which were found to work satisfactorily based on rigid-body motion experiments of the specific patterns in each case. Pixel subsets which overlapped with the attachment nut were excluded from all DIC computations. These experimental parameters were chosen based on prior work in which detailed assessment of uncertainties of 3-dimensional dynamic measurements were made .
A modal analysis was performed to reproduce the experiments using commercially available finite element analysis (FEA) software . Simulations were performed both at room temperature and at high temperature. At both temperatures the initial shape of the plate as obtained from the DIC measurements, shown in Fig. 3, was used as the starting FEA model geometry. The effect of temperature on material properties was taken into account in the high-temperature simulation by assigning the elastic modulus of the material according to the estimated temperature distribution in Fig. 4. No thermal expansion or conduction was explicitly modeled in the high-temperature simulation and both simulations assumed a small strain linear elastic response (albeit with the inhomogeneous elastic modulus as described above for the high temperature case). Note that since the experimental data naturally cannot extend all the way to the edges of the plate, as the DIC measurements are made at the center pixel of subsets, a small amount of extrapolation, i.e. equivalent to approximately half a subset or 1.82 mm, had to be performed to extend the estimated temperature distribution over the finite element (FE) mesh which did cover the entire area of the plate.
Results and Discussion
Resonant frequencies (Hz) of mode shapes at room temperature (RT) and high temperature (HT)
There are no significant changes in the simulated frequencies with the change in temperature. One possible explanation is that the model used in the simulation was too simplistic. Although the material properties in each element were informed by the temperature distribution, the discrepancy in frequencies might also be due to stresses caused by uneven heating which were not included in the model. Another explanation might be that temperatures in the plate fluctuated as the distance between the induction coil and the plate changed throughout the period of oscillation, but these fluctuations were neither detected experimentally nor included in the simulation. Nonetheless, the model does capture well the main features of the nine modes.
Image Decomposition Analysis
Each plot also includes a dashed line of gradient one, which is indicative of perfect agreement between the simulation and the experimental results. The closer a data point lies to this line, the better the agreement is for that coefficient between simulation and experiment [16, 18]. In the comparison, it is likely that there will be noise in the experimental data and errors in the simulation which will cause the points to deviate from this line. Therefore an acceptable area can be defined around the ideal line as +/−2u(S E ). If all of the data points from the comparison of the Tchebichef feature vectors fall within this area then the simulation can be deemed an acceptable representation of the experiment .
In general, the uncertainty bands are smaller for the high temperature results (Fig. 15) than for the room temperature results (Fig. 16). This is a result of a lower average residual in the reconstruction of the high temperature experimental results because the larger displacements have less noise. For the majority of the plots, all of the data points fall within the acceptable area, and many of the data points fall on the ideal line, indicating agreement between the simulation and the experiment. For example for Mode 3 in Fig. 15, only one Tchebichef kernel is dominant – specifically #6 in Fig. 13 – which corresponds to a shape produced by bending about the vertical axis. Most other kernels have coefficients clustered at the origin of the plot in Fig. 15, i.e., they are near zero. Overall Figs. 15 and 16 show that there is a very good comparison of the spatial distribution of out-of-plane displacements between the predicted and measured resonant modes, although as was seen in Table 1 the corresponding frequencies are not predicted as closely. One notable exception to this is mode 3 at high temperature. In this mode there is a kernel, #2, which falls outside the acceptable area defined by +/−2u(S E ). This kernel has a significant non-zero value for the simulation data but is zero-valued for the DIC data. Kernel #2 corresponds to a rigid-body out-of-plane rotation about a horizontal axis, and is the dominant kernel in mode 2. The simulation predicts mode 2 and mode 3 to occur very close to each other in the frequency domain, i.e. 217 and 218 Hz, respectively, and hence perhaps there is significant interaction between these modes in the model, which is not seen in the data from experiment when the gap between the modes in the frequency domain is 174 and 134 Hz respectively for the room and high temperature.
There was some concern that support conditions, i.e., the connections between the stinger and plate could be responsible for some of the discrepancies observed between the measured and predicted results. The effect of the length and stiffness of stinger were explored and found to change the natural frequencies but to have no effect on the resulting displacement fields.
The work described here has combined, for the first time, high frequency vibratory loading (up to about 1300 Hz) at elevated temperature (up to 600 °C) with the full-field optical diagnostics of stereo-DIC. This combination has provided high quality experimental data suitable for the validation of numerical results. Using this combined loading experimental set-up with induction heating and shaker-induced vibration, the thermo-acoustic behavior of a rectangular plate of Hastelloy-X was explored. By investigating the first 9 resonant modes at both room temperature and high temperature, and measuring mode shapes for each of the resonant frequencies using stereo-DIC, the influence of temperature on the thermomechanical vibratory response of the plate was assessed. A decrease in resonant frequencies was seen with increasing temperature, although the first 9 mode shapes themselves were similar. Companion finite element numerical simulations that accounted for the temperature dependence of elastic modulus also produced very similar mode shapes, although the influence of temperature on modal frequencies was not as significant as observed experimentally. As a model process for validating simulation results based on this type of experimental data collection an image decomposition technique based on 2D Tchebichef polynomials  was used. The image decomposition, which allows comparison between numerics and experiments using the fitted Tchebichef coefficients rather than the entire images, confirmed quantitatively the qualitative observation that as temperature increased the resonant mode shapes remained very similar. Overall, the results of this combined thermal-mechanical study point to the conclusion that the simulation is a good representation of the experiment both at room temperature and at high temperature (in fact a little better at high temperature as the signal to noise ratio in that experiments is larger). Further study would include a continuous recording of data rather than just at resonant frequencies, so that the transition from one mode to the other may be better studied. Since the DIC technique is not material or specimen dependent, it should be relatively simple to apply these methods to other scenarios involved thermo-vibratory excitation.
This effort was sponsored by the Air Force Office of Scientific Research, Air Force Material Command, USAF under grant numbers FA8655-11-3083 and FA9550-12-1-0386. The U.S. Government is authorized to reproduce and distribute reprints of Governmental purpose notwithstanding any copyright notation thereon. Major Matt Synder (EOARD) and Dr. David Stargel (AFOSR) respectively are the program officers for these grants. EAP is the recipient of a Royal Society Wolfson Research Merit Award. The collaboration that was central to this study was supported by a Royal Academy of Engineering Distinguished Visiting Fellowship awarded to JL.
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