A Novel Imagebased Ultrasonic Test to Map Material Mechanical Properties at High Strainrates
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Abstract
An innovative identification strategy based on high power ultrasonic loading together with both infrared thermography and ultrahigh speed imaging is presented in this article. It was shown to be able to characterize the viscoelastic behaviour of a polymer specimen (PMMA) from a single sample over a range of temperatures and strainrates. The paper focuses on moderate strainrates, i.e. from 10 to 200 s^{−1}, and temperatures ranging from room to the material glass transition temperature, i.e. 110^{∘}C. The main originality lies in the fact that contrary to conventional Dynamic Mechanical Thermal Analysis (DMTA), no frequency or temperature sweep is required since the experiment is designed to simultaneously produce both a heterogeneous strainrate state and a heterogeneous temperature state allowing a local and multiparametric identification. This article is seminal in nature and the test presented here has good potential to tackle a range of other types of high strainrate testing situations.
Keywords
Ultrahigh speed Viscoelasticity PMMA Infrared thermography IdentificationIntroduction
In many instances in life, materials around or within us suffer deformation at high rates. This is the case when engineering structures undergo impact, crash, blast, etc. but also when forming materials like stamping or machining for instance. Another important area concerns biological tissues. For instance, traumatic brain injuries (TBI) involve damage of brain tissues caused by their high rate deformation following impact load of the skull. Thanks to the significant progress in computing power and computational mechanics tools, it is now possible to perform extremely detailed numerical simulations of many complex situations where materials deform at high rates, with the objective to design safer structures, assess tissue injuries or devise more effective manufacturing processes, as mentioned above. However, to deliver their full potential, these computations require the input of reliable and accurate mechanical constitutive models of the materials loaded at high strainrates. This is an extremely challenging problem because of both the dynamic nature of the mechanical fields and the technological difficulties associated with strain metrology. Presently, this represents an important scientific bottleneck for society to fully benefit from such advances in numerical simulation.
A number of testing techniques are available to identify the high strainrate properties of materials, as reviewed in [5]. Most of them rely on very limited experimental information, such as strain gauges or point velocity measurements with VISAR technology. As a consequence, the tests need to follow strict assumptions to relate the measurements to the material behaviour, for instance, uniform strain field and no inertia effects are typical assumptions in split Hopkinson pressure bar (SHPB) testing.
The advent of fullfield optical metrology, such as digital image correlation [35] or the grid method [10], combined with the new developments in ultrahigh speed imaging [2, 29, 36] provides a unique opportunity to revisit high strainrate testing techniques. In particular, fullfield of accelerations can be obtained which provides a powerful imageembedded load cell if the material density is known, which is usually the case. This concept was first proposed in 2009 in [22], published in full in 2011 [23], where a proof of principle experiment was performed on quasiisotropic composite specimens, with and without a free hole. The inplane Young’s modulus and Poisson’s ratio were identified satisfactorily before the onset of damage, within the first 30 μ s of the test, at strainrates up to 1000 s^{−1}. The grid method was used to measure the deformation together with a Cordin 55062 rotating mirror ultrahigh speed camera operating at 30 kfps. However, this test was performed on a tensile Kolsky bar, though the bars were not used to obtain the impact force. The concept was then extended to purely inertial tests which showed to be much more suited for this kind of analysis. The very first example dealt with concrete spalling tests [26], and was then extended to composites [27] and metals [3, 4, 19]. Since then, the idea has spread and several groups worldwide are starting to use the technique [13, 14, 16, 17, 21, 40, 41].
Initially, this idea to use the fullfield acceleration as a load cell with the Virtual Fields Method came from seminal work by Prof. Michel Grédiac [8, 9] using vibration tests. In this case, high speed imaging is not necessary and the acceleration derives easily from the deflection using the harmonic assumption. This was extended later on to include damping [7]. Recently, an article showed that ultrasonic excitation combined with ultrahigh speed and infrared imaging [37] could be used to image the high strainrate deformation of a polymeric foam, though the authors did not use the acceleration to identify stiffness. This was released in a later contribution [25]. The present paper builds up on this to explore the potential of this test in more depth on a homogeneous PMMA specimen. In particular, the original idea of the paper consists in investigating how heterogeneous loadings states, e.g. stationary or transient deformation waves, combined with the measurement of the local strain, strainrate, temperature and stress (from acceleration) states can be used to identify viscoelastic material properties over a wide range of thermomechanical conditions. In other words, this provides a unique opportunity to identify from one single experiment, data which would have normally required a battery of tests and samples. This work is seminal in nature and focuses on the methodology rather than on the analysis of the material behaviour. However, comparison with literature results and Dynamic Mechanical Thermal Analysis (DMTA) performed on a specimen from the same material sheet enables to gain confidence in the obtained results. In addition the present data concern the moderate strainrate range, tens to hundreds of s^{−1}, which is notoriously difficult to access with conventional servohydraulic machines or SPHB.
In a first part, the experimental setup and the theoretical framework are detailed. In a second part, the experimental results are presented and discussed. Finally, the identification process is simulated to provide a first idea of the optimal experimental parameters required to achieve a precise identification.
Experiment and Data Processing
Experimental Setup
A highpower ultrasonic transducer  The NextGen Lab 750 system from SynapTec (France) allows to cyclically deform the sample at 20 kHz up to a peaktopeak displacement amplitude of 120 μ m, depending on the sample design and damping. The actuator is a boltclamped Langevin type transducer composed of a stack of piezoelectric elements resonant at 20 kHz. The displacement amplitude is boosted using a titanium horn together with the transducer, i.e. a mechanical amplifier. The output horn diameter, i.e. the active surface, is 12 mm, so that samples up to the same width can be attached to it.
An ultrahigh speed (UHS) camera  The HyperVision HPVX camera from the Shimadzu Corporation (Japan) is used to capture images of the deforming sample to obtain space and time resolved displacement and acceleration^{3} fields. The UHS camera has a resolution of 400 × 250 pixels and 128 frames can be recorded up to a frame rate of 5 Mfps. Here, a frame rate of 500 kfps was chosen, i.e. 25 frames/cycle at 20 kHz. The sensor noise was measured on stationary images to be in the order of 3.5 % of pixel dynamic range. Details concerning the methodology used to obtain this value are provided in “Measurement Uncertainty”. The specimen was illuminated by a Gemini 1000Pro (1000 W) flash light from Bowens (UK). An example of a frame captured by the camera is shown in Fig. 2(a). The magnification was chosen to fit at best the sample length, the free edge is on the left part of the frame.
A bonded regular grid was used to extract the inplane displacement fields through a phaseshifting algorithm [10]. The grid, from COLOURSENSE Ltd. (UK), was produced using the dry transfer technique, i.e. by outputting a vector artwork to a high resolution film negative and using a photolithographic technique to expose a carrier sheet coated with a photosensitive inkbased film deposited on top of a thin layer of glue. The resulting pressuresensitive adhesive multilayer was carefully aligned on the sample surface, manually pressed, and the carrier sheet simply peeledoff. The total thickness of the grid has been measured to be less than 100 μ m, typically around 40  70 μ m. It was demonstrated in [28] that an equivalently thick layer of epoxy glue did not disturb the strain measurements when compared to a strain gauge. The grid used in the present configuration had a pitch of 1 mm and was imaged through a 28  105 mm Nikkor lens, at a sampling of 7 pixels per period, allowing a field of view of 57 × 35 mm with a working distance of about 180 mm (see Fig. 1).
A sample The material chosen for this first investigation was a 4 mm thick, 55 mm long and 12 mm wide PMMA (PolyMethylMethAcrylate) Acrycast^{Ⓡ} sample sourced from Amari Plastics Plc (UK). The sample length was chosen to ensure that its first longitudinal deformation mode was at 20 kHz. Quasistatic reference material properties were obtained at \(\dot {\varepsilon }=10^{2}\) in uniaxial stress configuration using a standard universal test machine. Backtoback strain gauges were employed to take into account spurious bending. A tangent Young’s modulus value of E = 2.9^{±0.1} GPa and a Poisson’s ratio value of ν = 0.35^{±0.01} were found. The material density was obtained by measurement of water displacement to a value of ρ = 1160 kg.m^{−3}. In addition, the glass transition temperature T _{ g } of the material has been measured using both DMTA and Differential Scanning Calorimetry (DSC), and found to be about 110^{±5} ^{∘}C. DMTA data has been obtained from a Q800 analyser from TA Instruments using the dual cantilever (bending) deformation mode at a loading amplitude and a frequency of 0.1 % and 1 Hz respectively and using a temperature ramp of 1^{∘}C.min^{−1}. In the present work, simultaneous deformation and IR temperature measurements are required. To achieve this, the sample was covered with a black matt paint on one face, to produce a uniform and high emissivity surface suitable for IR measurements and, on the other face, with the regular grid described above (see Fig. 1).
Experimental Procedure and Data Processing
Infrared (IR) data were postprocessed following the socalled pixelbypixel calibration strategy [34]. Sets of 20 IR reference images at different uniform and stabilized temperatures, from 20 to 90^{∘}C, were captured using an extended area blackbody from Infrared System Development Corp. (USA). The mean response of each pixel of the IR focal plane array (FPA) as a function of the blackbody temperature was then fitted using a 5^{ t h } order polynomial function to build up a set of coefficients which fully characterise the relationship between the raw digital level and the effective temperature for each pixel and over the expected experimental temperature range. This calibration was then applied to the experimental measurements to convert the raw data into temperature fields. As kinematic measurements have been obtained over discrete 256 μ s periods while infrared data were continuously recorded at 50 fps, the next step consists in finding the single measured thermal field corresponding at best to each kinematic sequence. In the present work, the thermal field taken at time nearest to half the loading period was simply selected. The impact of this choice on the accuracy of the thermal measurement regarding the considered kinematic sequence is discussed in the next section (“Measurement Uncertainty”). Finally, the selected thermal fields were downscaled using spline interpolation to fit the kinematic measurements resolution, and then, averaged over the sample width.
The procedure for reconstruction of stress fields and material identification will be detailed further in “Theoretical Framework” and “Storage Modulus and Damping Identification”.
Measurement Uncertainty
The sources of uncertainty on the kinematic measurement are multiple. They can arise from the intrinsic sensor noise, the method used to recover displacements, the test conditions etc… This uncertainty can be partly quantified by grabbing a set of stationary undeformed images and applying the phase shifting algorithm [10] to recover displacements, and postprocess these displacements (see “Experimental Procedure and Data Processing”) to derive strains, strainrates and accelerations. The resulting identified noise takes into account all these potential sources of error, at least in static conditions. Unfortunately, it does not include errors arising from a misalignment of the sample with respect to the camera reference system and lens distortions which can create some fictitious deformations. It also does not take into account lighting variations and grid defects. Therefore, it can be considered a lower bound of the deformation uncertainty. It should also be noted that the discretized deformation measurement acts as a low pass spatial filter. The systematic error generated by this is explored in “Synthetic Grid Deformation”.
Experimental parameters and uncertainties obtained from stationary images
Sample  Material  PMMA 
Dimensions (mm)  55 × 12 × 4  
Density (kg.m^{−3})  1160  
(\(\dot {\varepsilon }\)=10^{−2})  Young’s modulus (GPa)  2.9^{±0.1} 
Poisson’s ratio  0.35^{±0.01}  
T _{ g } (^{∘}C)  110^{±5}  
Grid  Thickness (μ m)  < 100 
Grid pitch (mm)  1  
Sampling (pixel.period^{−1})  7  
IR  Model  CEDIP Silver 480M 
Lens  27 mm  
Frame rate (fps)  50  
Integration time (μ s)  509  
Number of pixel (pixel^{2})  320 × 256  
Field of view (mm^{2})  60 × 48  
Uncertainty  < 5^{∘}C  
UHS  Model  Shimadzu HPVX 
Digitization  10  bit  
Lens  28  105 mm Nikkor  
Frame rate (kfps)  500  
Number of pixel (pixel^{2})  400 × 250  
Max. sensor noise (% of grey levels)  3.5  
Field of view (mm^{2})  57 × 35  
Smoothing window (x,y,t)  7 × 7 × 7  
Displacement noise  0.6 μ m or 5 × 10^{−3} pix  
Strain noise (Finite diff. in μ def)  ± 195  
Strainrate noise (Finite diff. in s^{−1})  ± 25  
Acceleration noise (Finite diff. in m.s^{−2})  ± 1.3 × 10^{4}  
Stress noise (MPa)  ± 0.15 
Concerning the infrared measurements, the uncertainty is mainly driven by five parameters: (1) the intrinsic noise of the sensor under perfect conditions, i.e. the NEDT over the studied temperature range  (2) the accuracy of the calibration procedure  (3) the surface emissivity distribution and variations  (4) the setup environment and (5) the triggering mismatch. The impact of the environment and emissivity distribution has been mitigated by applying a high emissivity uniform coating to the sample and covering the entire setup with a black curtain during the whole test to avoid IR reflections from the surroundings. The thickness of the coating can also be a problem since it tends to delay the transmission of the thermal information from the specimen to the paint surface [31]. However, this has not been investigated here as the accuracy requirements for the temperature readings are not as stringent as for thermal stress analysis as used in [31]. As detailed in “Experimental Procedure and Data Processing”, the calibration was performed using a pixelbypixel calibration to minimize the uncertainty. The last point is not exactly a problem of temperature uncertainty but relates to the synchronization of the kinematic and thermal data. Indeed, the temperature was recorded at a lower framerate than the kinematic data, and no synchronization between the UHS and IR cameras had been implemented. One can decompose this uncertainty as the sum of: (1) the unknown temperature variation during the few cycles captured by the UHS camera  less than 5 mK  (2) the temperature variation between 2 IR images  about 500 mK  and (3) the unknown concerning the precise time when the UHS frames were grabbed with respect to the IR timeline  less than 5^{∘}C. The value provided depends on the material selfheating rate and is therefore only valid for the expe rimental conditions described above (see “Experimental Procedure and Data Processing”). It is worth noticing that the last point is not an intrinsic problem of the method and will be solved in further experiments. However, it is important to keep in mind that the 5^{∘}C uncertainty in the present case is acceptable as the material temperature sensitivity remains low over this temperature range.

The grey level noise was obtained by calculating the standard deviation over time at each pixel, and dividing it by the mean grey level value at that pixel. This was then averaged across all grey level values. The sensor noise variation as a function of the sensor dynamic range is discussed in “Synthetic Grid Deformation”.

The strain, strainrate and acceleration noise values were obtained from the standard deviation in space and time of the average value across y (as the strain will be obtained for the stressstrain curves).

The stress noise was calculated from the acceleration noise by averaging over the length and multiplying by the density, as in Equation (3).
Theoretical Framework
This section describes how acceleration maps can be used to derive stress information using a simplified version of the dynamic equilibrium equation. The extent of possible loading ranges accessible with such a setup is then explored.
Acceleration as a loadcell
One understands here that the stress reconstruction precision will depend on three main elements: (1) the accuracy of the material density value which plays the role of load cell factor here, (2) the temporal resolution of the grey level images which will affect the acceleration and (3) the spatial resolution as Equation (3) stems from discrete approximations of spatial integrals [25, 27]. In practice, an additional issue arises from the fact that strains are obtained on average over a grid pitch and expressed in its centre, while stresses are obtained on a slice. An additional step consisting in interpolating the stress values at grid centroids will be required to produce consistent stressstrain curves and identified material properties.
Highpower ultrasonic excitation
One of the key elements of the present methodology is the ability to produce a heterogenous thermodynamical state within the material. Such state could be achieved using different strategies, for instance playing on the sample geometry and/or on the loading itself. The choice here is to develop longitudinal stationary waves within the sample. The frequency, the wavelength and the amplitude of such deformation waves will drive the characteristics of the loading.

u _{0} is the amplitude of loading

\(\displaystyle {\omega ^{s}=\frac {2\pi }{L_{n}}}\) is related to the spatial deformation wave, with \(L_{n}\approx \frac {1}{f}\sqrt {\frac {E}{\rho }}\) the deformation wavelength (f the loading frequency) and \(\sqrt {\frac {E}{\rho }}\) the speed of sound in the material;

ω ^{ f } = 2π f is the angular frequency, with f the loading frequency;

Φ^{ s } and Φ^{ f } are the spatial and temporal phases.
As a consequence, reaching a strainrate of e.g. 200 s^{−1} at a frequency of 20 kHz (standard frequency for high power ultrasonics) requires a peaktopeak amplitude of 20 μ m for a PMMA sample (ρ = 1200 kg.m^{−3}, E = 3 GPa). In other words, covering a large strainrate domain requires highpower ultrasonics, i.e. both high amplitude and high frequency. This is generally opposed to lowpower ultrasound (hundreds of nanometers) which are used for diagnostics and control and does not significantly affect the environment in which the wave propagates. Keeping the values given above, the associated strain amplitude would be of the order of ± 0.2 %, without taking into account any thermal effects, and the acceleration would reach 320 km.s^{−2}, i.e. > 3 × 10^{4} g. Spatially, if the sample is designed to be resonant at its first longitudinal mode, i.e. the sample length is half the wavelength, the sample undergoes high displacement and zero strain on free and fixed edges (see Fig. 4) and high strain and strainrate at its centre. In addition, due to the viscoelastic dissipation, the central part of the sample will heat up cycle after cycle, while the edges will remain almost at room temperature. The last point is due to the significant mismatch between the characteristic conduction time within a PMMA sample [12] and the loading frequency, providing adiabatic conditions during loading.
The above shows that it is possible to reach large strainrates in the material, of the order of hundreds of s ^{−1}, with a heterogeneous state of strain, strainrate and temperature which enables to test the material over a wide range of thermomechanical conditions within a single test. The present article aims at demonstrating the experimental feasibility of this idea and at providing initial results on PMMA.
Results and Discussion
Presentation of the Results
A single PMMA sample was submitted to the experimental procedure detailed above (see “Experimental Procedure and Data Processing”). Successive sequences of grey level images were captured while the temperature was continuously recorded until the sample reached its glass transition temperature. The test was composed of two series of successive ultrasonics runs. In the first series, six ultrasonic runs were successively applied to the sample at low power, leading to a maximum strain of ± 735 μ def. It was then followed by five other runs at a higher actuator power, leading to a maximum strain of ± 1542 μ def. Between both test series, the sample was allowed to cool down back to room temperature.
One observes that, when the sample is loaded at ± 735 μ def (1^{ s t } series), the strain amplitude varies continuously along the sample length from 400 to 735 μ def, the strainrate amplitude from 50 to 115 s^{−1}, the stress amplitude from 0.2 to 4 MPa, and the temperature slightly increases in the sample centre from 23.3 to 34.6^{∘}C while the edges remains at room temperature.
When the sample is loaded at ± 1542 μ def (2^{ n d } series), the strain amplitude varies from 200 to 1542 μ def, the strainrate amplitude from 40 to 220 s^{−1}, the stress amplitude from 0.2 to 7.6 MPa, and the temperature increases in the sample centre from 23.4 to 105^{∘}C. Contrary to the first test series where all curves overlapped almost perfectly, the second series evidences a softening of the material as demonstrated on Fig. 7(c). This is due to significant increase of the temperature cycle after cycle (see Fig. 7(d)). A clear change of the material response can be observed at the 4^{ t h } and 5^{ t h } ultrasonic runs. These two loading cases are characterized by a significant drop of the stress down to 6 then 3 MPa as well as clear change in the profile shapes. Indeed, one can see on Fig. 7(a) a clear strain localization in the sample centre and a significant decrease of the strain level everywhere else. These phenomena are due to a sharp localized change in the stiffness of PMMA due to glass transition. The phenomenon starts at the 4^{ t h } run, i.e. around 95^{∘}C but leads a clear collapse of the material stiffness at the 5^{ t h } run, i.e. around 105^{∘}C. Such a drastic local change in material property also leads to a significant change in the deformation mode of the sample. Indeed, while the material is initially perfectly tuned to be resonant on its first longitudinal mode at 20 kHz, one observes a significant decrease of the wavelength (see Fig. 7(c)) at the 4^{ t h } and 5^{ t h } runs. This point is the reason why finite differences have been used to obtain experimental acceleration fields as the assumption of homogeneous material response cannot be ensured anymore. According to the wavelength formulation, available in “Theoretical Framework”, a decrease of the wavelength relates to a drop of the stiffness which will be evidenced further. In the present case, the temporal resolution (50 Hz) of the temperature signal combined with the heating rate does not allow finely capturing the T _{ g }, but a value around 100^{∘}C is reasonably in line with glass transition temperatures measured on the material using DMTA (see Table 1).
The data can now be combined to identify the material properties as a function of temperature and strainrate.
Storage Modulus and Damping Identification
This parameter is widely used to describe the damping of the material. The real part of Young’s modulus, the storage modulus, is then identified from the ratio \(\frac {\alpha _{1}}{\alpha _{0}}\cos \left (\delta \right )\).
The results show a number of important trends. Looking at Fig. 8(a) of the first run when the temperature in the specimen is still uniform (see Fig. 7(d)), one can see that the central sections provide a stiffer response than the outward one. This is the stiffening effect of strainrate. Based on the data in Fig. 7(b), the material at 10 mm experiences a strainrate of about 50s ^{−1} while the centre responds under a strainrate of about 80s ^{−1}. Both stiffness and strainrate contrasts are not very large but the effect is significant. The second trend is the effect of temperature, which is much more prominent for this test. As the runs progress and the temperature increases at the centre of the sample, the stiffness drops as clearly seen in Fig. 8(c) and consistently over the whole test series. Finally, it is also clear from Fig. 8 that as the temperature increases, the hysteresis loops open up, indicating an increase in damping.
Then, Fig. 9(b) presents the variation of the tan(δ) parameter which is associated to the material damping. A few excessively high and unrealistic damping values have been discarded (less than 10 data points higher than 0.4) and the colormap saturates at 0.3 to visually capture the local gradients. The same trend as for the storage modulus is observed, although the data are noisier. This was expected as tan(δ) derives from small fractions of the total stress and strain. As for the storage modulus, the noise is larger in the low temperature and low strainrate range, for the same reason. It is interesting to compare these data to that available in the literature. Looking for instance at [18], the variation of tan(δ), from ambient to 105^{∘}C, ranges between about 0.03 and 0.3 which is the range also observed here. Another interesting observation is that there is a trend for higher values at low strainrate and temperature. This may be related to the presence of a βtransition around room temperature, but the poor signal to noise ratio there precludes any definite conclusion. More tests are needed to explore this part of the temperature / strainrate space.
From a technical point of view, both graphs evidence a very heterogeneous datapoint density. Indeed, Fig. 9(a) has a large significant datapoint density between 50 and 100 s^{−1} for temperature comprised between ambient and 35^{∘}C whereas very few datapoints are available at higher temperature. This point is not intrinsic to the method and mainly depends on the chosen ultrasonic loading amplitudes. In the present work, only two amplitude steps have been used (± 735 μ def and ± 1542 μ def). Using more steps would have led to a more homogeneous space sampling.
Finally, time temperature superposition has been used in an attempt to collapse all the data in Fig. 9(a) on a single master curve. To achieve this, experimental data points undergoing low strainrates (< 70 s ^{−1}, see “Finite Element Validation” for details), and the ones close to T _{ g } and post T _{ g } have been discarded. For the latter, as the glass transition was not identified accurately, the procedure has consisted in removing data over a threshold damping value of 0.3 (see top yellow region in Fig. 9(b)) and a threshold temperature value of 95^{∘}C.
The reconstructed master curve at 25^{∘}C is also presented in Fig. 10(b) with a set of data from the literature. In addition, DMTA was performed on the same material using a Q800 system from TA Instruments. A stepwise temperature control with a 1^{∘}C interval between each step was used during the measurements. The testing temperature ranged from −80^{∘}C up to 50^{∘}C and the data were recorded at multiple frequencies: 0.1, 0.3, 1, 3, 10 and 30 Hz. The 17.25 × 12.07 × 3 mm^{3} specimen was subjected to a dual cantilever deformation mode^{4} and the strain amplitude was kept lower than 0.1 %. Comparing the present “oneshot” material characterisation to DMTA and literature results, a certain discrepancy can be observed (see Fig. 10(b)). The present results are located between that from [20], where the material was subjected to compressive tests at quasistatic (servohydraulic machines) and high strainrates (SHPB), and [24, 30] with data from DMTA and compressive tests on both servohydraulic machines and SHPB. The trends are similar but the current data lie about 1 GPa above that from [24, 30], while they are about 2 GPa below that in [20]. An interesting thing is the fact that the data in [30] and [20] as well as the current data (partially) evidence an inflection in the behaviour somewhere between 10^{1} and 10^{3} s^{−1}, which is not visible on both DMTA result. Nevertheless, nore experiments, at higher strainrates, or at lower temperatures (through the timetemperature superposition principle) are be necessary to confirm the existence of such a sharp increase of the material stiffness at high strainrate as well as the ability of highspeed tests, contrary to standard DMA, to capture it.
The focus of the present work is not to perform a thorough study of the viscoelasticity of PMMA but to propose a truly new test method to reach strainrates that are generally hard to obtain as they lie between what can be obtained with high speed hydraulic machines on one side and split Hopkinson bars on the other side. The data in Fig. 10(b) shows the benefit of having an alternative to the current test techniques to better explore the behaviour of viscoelastic materials at high rates of strain directly, without having to rely on timetemperature superposition, as for DMTA tests. The lack of transition at high strain on the DMTA data is a hint that this test may be missing some important transition. The difficulties associated with tests on high speed hydraulic machines and SHPB contrast with the relative simplicity of the current configuration and the present authors strongly believe that this new test can be a valuable addition to the toolkit of the mechanics of materials researchers and engineers.
The objective of the last section is to shed some light on the issue of the low spatial resolution of the camera. Indeed, the digitized measurements provided by the camera and the grid method are a spatially filtered version of reality and the only way to understand the effect of this filter on the quality of the identification is through numerical simulation.
Finite Element Validation
The purpose of this section is to validate the current experimental choices by understanding how experimental parameters such as the camera spatial resolution, the acquisition frequency, the grid sampling, the camera sensor noise and the sensitivity of the algorithm used to recover deformations can affect the precision of the identification. The idea is also to propose some guidelines to the reader and make this new technique more accessible. The main concern is the poor camera spatial resolution (400 × 250 pixels) so this validation is essential to give confidence to the previous results.
Finite Element Model
FE model parameters
Geometry  dimensions (mm)  55 x 6 (sym.) x 2 (sym.) 
Mesh  size (mm^{3})  0.1 
element  SOLID186  
Material  density (kg.m^{−3})  1160 
Young’s modulus (GPa)  5.5  
Poisson’s ratio  0.34  
Loading  type  disp. 
amplitude (μ m)  30  
frequency (kHz)  20  
Analysis  FE package  ANSYS 16.2 
type  harmonic / full  
solver  Jacobi Conjugate Gradient iterative equation solver  
num. damping (s)  × 10^{−7} 
Figure 11 shows an example of an FE longitudinal displacement field extracted at the top surface of the model and then symmetrized along the transversal direction.
Synthetic Grid Deformation
The procedure used to evaluate the accuracy of the identification on real data is presented here. First, the sample deformation was simulated with the FE model described above. Then, synthetic grid images were numerically deformed using the FE displacements fields, grey level noise was added and the images were processed exactly as the experimental ones (see “Theoretical Framework”). The identified parameters can then be compared to the FE inputs and both systematic and random errors can be evaluated. A similar approach can be found in [32, 33, 38].
Figure 12(c) shows a zoomed in image of an experimental grid image and Figs. 12(a) and (b) the grey level histogram and noise characteristics of the camera sensor. Figure 12(c) shows a zoom in of the synthetic grid after applying deformation and noise. The noise added to the synthetic images is based on Fig. 12(b) where the standard deviation of the pixel noise (obtained from stationary images) for different pixels from dark to bright is plotted. The maximum noise was about 3.5 % of the pixel grey level and decreases as a function of the grey level intensity. The grid contrast has been selected to reproduce the histogram of Fig. 12(a). Comparing Fig. 12(c) and (d), one can note that the procedure detailed above describes reasonably well the grey level dynamic and the noise intensity. However, the procedure does not take into account any grid defect and any fillfactor issues. Indeed, one observes in Fig. 12(c) that the apparent pitch of the grid slightly varies along the horizontal direction. This is due to the fact that the pixel captures photons only over a small part of its physical domain (40 % for the HPVX) which leads to a crop of the information especially visible in presence of sharp edges. Both these aspects have been neglected here.
where F ^{ F E } and F ^{ g r i d } are the FE (input) and postprocessed (output) quantities of interest, M _{ d }[∗] and s[∗] are the median and the standard deviation respectively. For the error on Young’s modulus, the time has been removed from Equations (13) and (14) to calculate the error, as it does not depend on time.
Let us now look into the impact of the experimental spatial grid sampling, the temporal sampling and the camera sensor noise on measurements and identification. First, four different spatial grid pitches have been tested, namely p = 0.7, 0.8, 0.9 and 1 mm, and the error on kinematic quantities evaluated. No clear impact on the systematic error of displacement and strain has been identified since an error of 0.5^{±0.1} % on both displacement and strain has been found whatever the grid pitch. Therefore, the grid pitch chosen experimentally, 1 mm, has negligible impact on the displacement and strain. The reason for this is certainly the low spatial frequency contents of the deformation when the material undergoes deformation at its first longitudinal mode. Deforming the material at higher deformation modes, or using a strain concentrator like a notch or a hole, would require to run this check again.
Systematic error (in %) between FE and synthetic grid deformation data, for different frame rates. The grid pitch is 1 mm and the displacement amplitude is ± 30 μ m
Frame rate (Mfps)  

0.2  0.5  1  2  5  
Accel. (finite diff.)  11.6  0.9  0.7  1.1  1.2 
Accel. (harm.)  0.3 \(\rightarrow \)  
Stress (finite diff.)  11.4  0.7  1.0  1.7  1.9 
Stress (harm.)  0.14 \(\rightarrow \)  
Young’s modulus (finite diff.)  12.3  1.9  0.3  0.5  1.2 
Young’s modulus (harm.)  0.5 \(\rightarrow \) 
One observes that only the 0.2 Mfps frame rate significantly affects the results. Indeed, an error of about 12 % is found for simulation recorded at 0.2 Mfps, while only 12 % of error was found for higher frame rates. This confirms the choice of 0.5 Mfps for the experiments, and also shows that a slight improvement would have been obtained recording at 1 Mfps or more. It is interesting to notice that the small unexpected increase of the identification error over 1 Mfps is probably due to the fact that the systematic errors presented here are based on spatiotemporal data obtained considering a constant number of frames captured by the camera (128) and not considering a constant number of deformation cycles. This has been chosen to be in line with the physical limitation of the camera. In other words, when an increase of the acquisition frequency is simulated, a decrease of the number of captured deformation cycles occurs. At 1 Mfps, two and a half cycles are captured, at 2 Mfps only one and at 5 Mfps, only half of a cycle. Therefore, increasing the frame rate affects the consistency of the comparison and probably slightly increases the systematic error. However, this error still remains very small. The table also shows that avoiding temporal differentiation by using the harmonic assumption leads to lower errors on the acceleration and stress, as expected.
Systematic and random errors (in %) between FE and synthetic grid deformation data.
no smoothing  smoothed k = 7  

Displacement  0.2^{±1.7}  6.7^{±0.6} 
Strain  0.4^{±11.0}  6.7^{±2.7} 
Acc. (finite diff.)  0.6^{±12.9}  7.2^{±1.5} 
Acc. (harm.)  0.2^{±1.7}  6.7^{±0.6} 
Stress (finite diff.)  0.1^{±5.7}  7.0^{±1.2} 
Stress (harm.)  0.2^{±1.0}  6.6^{±0.8} 
Young’s modulus (finite diff.)  2.8^{±3.1}  1.4^{±1.8} 
Young’s modulus (harm.)  1.5^{±3.2}  0.8^{±1.8} 
Table 4, column two, focuses on the smoothing kernel size selected for the treatment of the experimental data. A kernel size of 7 was applied in space to derive the strains, and in time to derive the acceleration using finite differences. Two things can be noticed: first, smoothing the data introduces a systematic error of the order of 7 % for all quantities except the identified Young’s modulus. Second, it significantly reduces the random error for the differentiated quantities, i.e. the strain (8 %), the acceleration (11 %) and the stress (4.5 %). As expected, the random error does not reduce significantly on acceleration and stress when using the harmonic assumption, the smoothing just increasing the systematic error. The situation for Young’s modulus is somewhat unexpected as not only does the random error decrease, as expected, but also the systematic error by a factor of two. This comes from the fact that both stress and strain systematic errors cancel out when the modulus is calculated. Whether this is a general result or just a fortunate fact arising from the precise set of smoothing parameters used here remains to be confirmed.
Finally, Fig. 14 shows the variation of the systematic and random errors when the smoothing kernel size varies. For all quantities, the systematic error increases when the smoothing kernel increases and the random error decreases when the kernel increases, as expected. This allows to select the smoothing in a rational way to minimize the total error, as in [38]. It is clear from Fig. 14(b), (c) and (e) that a smoothing kernel between 5 and 7 is recommended. Regarding the identified Young’s modulus, it is interesting to notice that the trend of the systematic error is almost flat whereas the systematic error on both strain and stress increases as a function of smoothing kernel size. It means that the systematic errors on both stress and strain increase in similar ways and cancel out in the Young’s modulus identification. This specific point means that depending on the purpose of the study, i.e. identifying a material parameter or measuring accurately mechanical quantities, the choice of the smoothing kernel will not necessarily be the same. In parallel, one observes that the random error on Young’s modulus identification significantly decreases over a smoothing window of 5. Nevertheless, none of the tested parameters allow reaching an error level below 1 %. A wider range of smoothing and grid parameters would be needed to better understand this, but this was beyond the scope of the present validation which focuses on evaluating the expected error for the parameters used in the experimental study. These results also demonstrate the benefit of the synthetic image deformation process to gain insight into the errors than can be expected on the identified quantities, as already pointed out in [32, 33, 38].
Figure 15(b) presents the relative error on the identified Young’s modulus as a function of the strainrate amplitude seen by each sample section. One clearly sees that the relative error on Young’s modulus starts from 1 % at maximum strainrate (i.e. sample centre) then gradually increases as the considered section gets closer to the edges. Within the domain [200250] s^{−1}, the systematic error remains below 2 % and reaches 6 % down to 90 s^{−1}. Below this, Young’s modulus is not identified accurately anymore. Such observations are in line with the experimental observations. Indeed, Fig. 10(a) shows an identification scatter about ± 5 % at the sample centre (high temperature  high strainrate) and about ±10 % close to sample edge which is consistent with Fig. 15(b). Moreover, Fig. 15(b) shows that the systematic error slightly increases close to the sample edges which could partially explain the deviation with [30], observed in Fig. 10(a) at 10^{2} s^{−1}.
The present data lead to the following conclusions. (1) Combining a spatially heterogeneous loading together with kinematic and temperature fullfield measurements, it is possible to identify Young’s modulus over a certain range of thermomechanical loading conditions. (2) The accuracy of the local identification strongly depends on experimental conditions and processing parameter selection, especially the sensor noise, the acquisition frequency, the camera resolution, the grid resolution and the smoothing parameters. Such parameters can be chosen in a rational way by combining finite element modelling and synthetic grid deformation. (3) Nevertheless, with such a ”oneshot” technique, the cost to pay is a variation of the signal to noise ratio as a function of spatial location, i.e. also as a function of temperature, strain and strainrate. Here, the results are acceptable down to strainrates of 90 s^{−1} and up to 250 s^{−1}. Deeper investigations using thermomechanical viscoelastic simulations are required to gain better understanding of the metrological limitations of the experiment.
Outstanding Issues and Scope of the Method

The difficulty to clearly define an apparent strainrate. Such difficulty is actually also present in DMTA and SPHB tests but at least the methodologies are consistent since data are also averaged in space. In the present case, the following issue can introduce a horizontal shift of the data in Fig. 9. As the strainrate for result reported here is the maximum strainrate, effective properties may be slightly shifted towards lower strainrates. One can notice that the convention in DMTA is to approximate the strainrate by \(\dot {\varepsilon }=4f\varepsilon _{max}\) [24] whereas one uses here \(\dot {\varepsilon }_{max}\), i.e. ≈ 2π f ε _{ m a x } which would lead to an horizontal shift of about 0.2 s^{−1}. This is not enough to explain the significant differences observed in Fig. 10(b).

The strain amplitude sensitivity. The strain varies along the sample length and during the test from 0.01 to 0.35 %, which means that the following test is close to DMTA in term of deformation amplitude (0.1 %) but somewhere between hydraulic machine tests and SHPB tests in term of strainrates. The potential impact of the strain amplitude on the storage modulus variation, especially at low strains, is not clear and could introduce some variation of storage modulus simply due to strain amplitude variations. This point can affect the global shape of the master curve.

Limitation of a 1D approach. It has been observed that the temperature was not perfectly uniform over a material section (see supplementary data) due to both higher heat losses at the edges and unsymmetrical strain localization. It seems important to investigate how such an approach could be extended to a 2D case which would not require this assumption. The present authors are currently working on a strategy based on the use of the Virtual Fields Method and a subsetbased equilibrium to overcome such an issue.
 1.
Fracture of brittle materials at high strainrate. Brittle materials like glass or concrete are notoriously difficult to test at high rates of strain. A preliminary numerical simulation on glass with a ± 60 μ m excitation amplitude (limit of the current ultrasonic excitation system), and considering a very low damping coefficient of β ≈ 10^{−9} leads to a maximum stress of σ = ±95 M P a, i.e. about the fracture stress of glass. As the material is not damped, the actuator can work over its whole displacement range to reach the failure stress of glass. Combining such test with a 2D generalization of the method would allow characterizing glass failure at high strainrates. An initial experimental proof of principle has been conducted and a glass specimen has been successfully fractured.
 2.
Transverse fracture of composites. Because of the very low failure stress of a unidirectional (UD) composite in the direction transverse to the fibres, its tensile fracture stress is very difficult to obtain using SHPB experiments. A preliminary finite element study showed that an amplitude of ± 100 μ m would be required to reach σ = 50 MPa with E = 10 GPa and β ≈ 10^{−8}. While this is larger than what the current system used here can provide, it is not impossible to reach. An alternative could also be to fix a mass at the free end, this would however require a significant numerical test design campaign. The situation is even more difficult for the through thickness tensile properties where very small specimens have to be used. In this case, a thin laminate could be sandwiched between two steel blocks and displacement measurements just performed on the steel blocks with a direct reading to the transverse stress from the free end steel block.
 3.
Adhesives. The same idea as for the throughthickness composites test could be employed for adhesives which are notoriously difficult to test at high rates [39].
 4.
Yielding of engineering alloys. From initial finite element simulations, the actuator would also need an amplitude of ± 160 μ m to yield an aluminium sample (with σ _{ y } = 280 MPa, E = 72 GPa and β ≈ 10^{−10}). Again, this is too large for the current setup but a specific horn could be designed to boost the excitation amplitude further.
 5.
Extended strainrate range. The current strainrate range is imposed by the excitation frequency, the strain concentration within the sample and the covered range of temperatures (considering timetemperature superposition). It is possible to reach higher apparent strainrates by cooling down the sample prior to the start of the test, e.g. building a temperature controlled enclosure. According to preliminary calculations and tests, cooling the specimen down to 10^{∘}C only would allow reaching the equivalent of 10^{3} s^{−1} at room temperature, while cooling down to −10^{∘}C would allow reaching 10^{4} s^{−1} at room temperature for PMMA samples. Working on sample geometry (using notches for instance) would lead to strain concentrations and thus higher local strainrates. Nevertheless, the suitability of the low spatial resolution camera would need to be verified, and the 1D approach would most probably not be enough.
 6.
Heterogeneous materials. Although the present paper uses a simple homogeneous PMMA specimen for the sake of validation of the test concept, it is clear that the main asset of this imagebased DMTA is to investigate heterogeneous materials. Indeed, while a technique such as DMTA is now well mastered and allows capturing the behaviour of rheologically simple materials over a very large range of strainrates (see Fig. 10(b)), it cannot provide spatial maps of properties and therefore, cannot derive detailed properties for heterogeneous materials. An interesting case for instance could be injected polymeric sheets which have different properties in the skin than in the core. The present method has the potential to provide spatiallyresolved properties.
Conclusion
The present work demonstrates the feasibility of a multiparametric identification on a single sample and falls within an effort to invent new highstrain test methodologies based on fullfield imaging and inverse identification, to both overcome the limits of standard experimental strategies and take advantage of the deformation heterogeneities to achieve a fullcharacterization of a material from a “oneshot” test. It has been demonstrated that using a 20 kHz high power ultrasonic excitation combined with infrared thermography and ultrahigh speed imaging, a PMMA sample could be subjected to apparent strainrates varying from less than 10^{0} s^{−1} to 10^{2} s^{−1} and temperatures varying from ambient to its T _{ g }. Within the studied strainrate range, a significant increase in storage modulus has been evidenced that was not visible in the DMTA results. This interesting observation justifies the interest in developing new measurement techniques to capture the material behaviour at intermediate and high strainrates. The validity of the approach has been checked using Finite Element modelling combined with synthetic grid image deformation. It demonstrates, under simple behaviour assumptions, that the error on identification is expected to remain below 10 % over the sample length. This has been partially confirmed by the experimental data scatter. Finally, the potential of this new test technique has been underlined and it is expected that it will evolve to join the toolkit of researchers and engineers in mechanics of materials.
Footnotes
Notes
Acknowledgments
This material is based on research sponsored by the Air Force Research Laboratory, under agreement number FA95501510293. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Air Force Research Laboratory or the U.S. Government. The authors would like to thank Dr Jason Foley (AFOSR), Dr Matt Snyder (EOARD) and Dr David Garner (EOARD) for supporting this work.
Prof. Pierron acknowledges funding from EPSRC through grant EP/L026910/1, as well as from the Wolfson Foundation through a Royal Society Wolfson Research Merit Award.
The authors are also grateful to Prof. Clive Siviour from the Department of Engineering Science at the University of Oxford for providing the DMTA results and for helpful discussions.
The authors are also grateful to Dr. Frances Davis from the Engineering and the Environment research group from the University of Southampton for performing the quasistatic tests, density measurements and DSC analysis on PMMA samples, as well as for helpful discussions.
All data supporting this study are openly available from the University of Southampton repository at http://doi.org/10.5258/SOTON/D0207.
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