Exploiting measurement-based validation for a high-fidelity model of dynamic indentation of a hyperelastic material

Luis Felipe-Sesé,1* Francisco A. Díaz,1 and Eann A. Patterson2

*lfelipe@ujaen.es

1 Departamento de Ingeniería Mecánica y Minera, Campus las Lagunillas, Universidad de Jaén, 23071, Jaén, Spain

2School of Engineering, University of Liverpool, The Quadrangle, Brownlow Hill, Liverpool L69 3GH, U.K.

Abstract

The high-speed (up to 1m/s) loading and unloading of a rubber block by a rigid wedge has been simulated using a finite element model. The model has been refined and validated using measured data obtained from combined fringe projection and two-dimensional digital image correlation during the real-time loading and unloading of a rubber block by an aluminium wedge. Comprehensive data was obtained from the experiment in the spatial and temporal domains at resolutions of 0.075mm and 0.00125 seconds respectively. Image decomposition techniques were used to represent the predicted and measured displacement fields in order to allow a number of quantitative measures to be employed to assess the fidelity of the model. It was demonstrated that it was essential to refine the Mooney-Rivlin material model to provide an accurate representation of the material behaviour for the speeds of the loading and unloading. The refined model was validated using the European Committee for Standardization (CEN) procedure and found to exhibit differences relative to the experiment of less than 5%. Together the predicted and measured data fields probably represent the best description yet of a soft material being deformed by a rigid indenter.

1.INTRODUCTION

Nowadays, the investigation of engineering problems that involves materials which experience large deformation, such as rubber and biological materials, has focused the attention of many researchers [1]. An important aspect of research on such materials is the analysis of their behaviour when subject to indentation for which it is important to be able to describe their mechanical behaviour using theoretical models. Analytical [2 - 8] and numerical [9 - 11] studies are available in the literature which all include significant assumptions. For instance, that strains are small and they are below the elastic limit [5, 7 & 10], that an elastic half-plane can be considered [6, 8, 10 & 11], that the contact area is much smaller than the radius of the indenter, that there is frictionless contact [7, 10 & 11] or that the external angle of the indenter is very small [8]. Some of these assumptions are not always well supported by experiments [12 - 15]. Although data from experiments is essential for the validation of analytical and numerical models, no substantial work can be found in the literature relating to integrated analysis involving modelling and experiments of high-speed, large indentation of soft materials. In this study, a simple low-cost approach for the measurement of three-dimensional displacements is applied to the analysis of the high-speed indentation of a block of hyperelastic material by a rigid indenter and the resultant data is used to refine and validate a computational model of the event.

One of the most extensively used techniques for full-field three-dimensional displacement measurements is digital image correlation [16], which for high speed events, as in case of this work, requires the use of two high-speed cameras. In this study, to avoid the need for two expensive high-speed cameras, an alternative technique has been used which combines two well-known methods namely two-dimensional digital image correlation [17] and fringe projection [18]. Since the two methods employ similar experimental set-ups, i.e. a camera perpendicular to the measured surface, it is possible to measure simultaneously in-plane and out-of-plane displacements using only one camera and a fringe projector [19, 20]. Two-dimensional Digital Image Correlation (2D-DIC) has been used to measure in-plane displacements and Fringe Projection (FP) to measure out-of-plane displacements. The resultant data was used to refine a finite element model of the hyperelastic model undergoing high-speed indentation by the rigid wedge using innovative, quantitative comparison methods [21]. An initial computational model was found to give relatively poor results and so additional material compression tests were performed at different load rates to refine the calibration of the material model. Tchebichef polynomials (of the first kind) [21] were used to decompose maps of surface strain from both experiment and simulation to permit detailed quantitative comparisons. The variation of the surface strain fields as a function of time increased the dimensionality of the comparison process compared to some prior work [12, 21] and hence a number of quantitative measures have been explored as means of evaluating the degree to which the model was a surrogate for the experiment and these are believed to represent an advance compared to earlier comparison work for dynamic cases [22].

2.NUMERICAL MODELLING

An explicit numerical analysis was conducted using the commercially-available software, Abaqus 6.11 [23] for a silicone rubber block of dimensions 60x60x20mm subject to indentation at speeds varying from 400 to 1000mm/s by an aluminium wedge of thickness 20mm with a tip radius of 1.68mm and an included angle of 73.45 degrees as shown in figure 1. The indenter was modelled as a non-deformable body using 520 bi-dimensional rigid solid elements (type R3D3). The unfilled silicone block was modelled using 150,000 four-noded reduced integration elements (type C3D8I). The size of these elements varied according to their position in the block, as shown in figure 2. The minimum element size controlled the minimum time-step in the explicit solution and so the values were chosen to give an appropriate temporal and spatial resolution. The time step Δt is related to the element size by the Courant–Friedrichs–Lewy (CFL) condition [24] which states that for stability the time step must be sufficiently small to ensure that information has time to propagate through the spatial discretization, i.e.

(1)

where is the minimun element length and cd is the dilatational wave speed in the material, which assumed from the literature to be 1050m/s [25].

The silicone rubber is known to exhibit a non-linear elastic behaviour that can be modelled using the Mooney-Rivlin relationship [26, 27] in which the strain energy function is described by [28]

(2)

where U is the strain energy per unit of a reference volume, and are the first and second stress invariants and C10 and C01 are the specific material parameters. In the first instance, the material parameters were taken as C10 =0.17 and C01=0.1 based on the Shore A Hardness of the material (42 from experiments) and the Batterman& Köhler relationship [29]. Subsequently, the material model was refined by providing empirical data from compression tests which were used by the Abaqus subroutine to evaluate the response of the material. The uniaxial compression tests were performed on two cubes of the material with a side of length 30mm which were loaded between platens at 400m/s and 1000m/s respectively using the same machine as described below for the indentation experiments. The results from these tests are shown in figure 3. The applied load was obtained from the load cell of the test machine and used to compute engineering stress by dividing the applied load by the original cross-section area of the specimen; while the displacement of the cross-head of the test machine was used to evaluate engineering strain by dividing the measured displacement by the original height of the specimen. The data in figure 3 were used to evaluate the material parameters, C10 and C01 using the EDIT MATERIAL tool from Abaqus [23], and were found to be 1.5 and 1.1 respectively at 400m/s and 2.1 and 1.0 respectively at 1000m/s.

The movement of the wedge into the block and the release of the wedge after indentation were both modelled using wedge speeds obtained from the experiments. The results from the initial and refined models are shown in figures 4 and 5.

3.EXPERIMENTS

3.1MEASUREMENT TECHNIQUE

Indentation and contact of soft materials, such as silicone rubber, usually generates large in-plane displacements and significant out-of-plane displacements. This implies the need to measure surface displacements in all three dimensions simultaneously during the dynamic loading. In this study a combined fringe projection and two-dimensional digital image correlation method has been utilised based on earlier work [19, 20]. The method employs a conventional colour LCD projector and a colour RGB high-speed camera, as shown schematically in figure 6. The projector is used to create a blue & white fringe pattern with a sinusoidal intensity profile in one direction on the measured surface, as shown in figure 1, which permits fringe projection analysis to be performed in the blue spectrum. In addition, a red speckle pattern on a white background is painted onto the measured surface to permit two-dimensional digital image correlation to be performed in the red spectrum.

Images of the measured surface were captured before and during loading using the high-speed RGB camera and the images separated into the Red, Green and Blue components. The Red data were processed using a commercial digital image correlation algorithm (VIC 2D from Correlated Solutions Inc.). The Blue data containing the fringe projection information were processed using a five-step phase-shifting algorithm [18] combined with a quality unwrapping algorithm [30].

Since a telecentric lens was not employed on the camera, the measured in-plane displacements from the digital image correlation were distorted by the out-of-plane displacement of the block. The measured in-plane displacements were corrected using the out-of-plane displacements measured using the fringe projection system. The correction methodology had been developed previously [19] and is based on a pin-hole model with the assumption that the relatively small out-of-plane displacement does not affect the image quality due to any defocusing effect. A calibration procedure was used to align the optical axis of the camera perpendicular to a plane reference surface and to obtain the distance from the reference surface to the camera [19]. The measured x- and y- displacements were then obtained in the plane of the reference surface and the z-displacements along the optical axis of the camera, i.e. perpendicular to the reference surface, with the origin at the point of initial contact between the block and the wedge such that y-displacements are negative in the direction of indentation.

3.2EXPERIMENTAL SET-UP

The experimental set-up is shown in figure 6 and consists of a rubber block (60x60x20mm) mounted on a rigid, stationary platen in a servo-hydraulic test-machine and which was indented by an aluminium wedge driven by the hydraulic actuator of the test machine. A high-speed RGB camera recorded images of the red-speckled block on which a LCD projector displayed a blue-white fringe pattern.

The preparation of the rubber block followed the procedure established previously [12]. In more detail, it was cast in a mould using a bi-component silicone, QSil 226 (ACC Silicones UK). The two components were mixed together and then the gas generated by the mixing process was removed using a vacuum pump for 15 minutes after which the material was cured in a mould for 24 hours using the temperature cycle recommended by the manufacturer. The resultant block was transparent and so a very small amount of white paint was sprayed onto one square face to facilitate the optical measurements. The white paint did not form a coating and consisted of fine speckles which had a texture and appearance similar to dusting the surface with a fine white talcum. The discontinuous paint layer ensured the paint could not be subject to any strain and hence would not be susceptible to cracking or peeling during loading of the block. An even less dense speckle of red paint was sprayed onto the same face to provide a pattern for the two-dimensional digital image correlation.

The rubber block was located in the centre of the rigid, fixed platen of a hydraulic impact test-machine (ESH frame and load actuator controlled by instrumentation from Phoenix Calibration & Services). A fine layer of talcum powder was applied to the top surface of the block to prevent adhesion of the indenter. The indenter consisted of a machined aluminium (Al 2024) wedge of thickness 20mm with tip radius and included angle of 1.68mm and 73.45 degrees respectively, as shown in figure 1. The indenter was attached to the actuator of the hydraulic test-machine which had a maximum capacity of 25kN. However, the actuator was operated in displacement control and was driven into the block at 400mm/s to a depth of nominally 6mm from an initial position 3mm above the top surface of the block, and then it was pulled out at a rate of 1000mm/s. The actual position and speed of the wedge tip is shown in figure 7.

During a test, images were captured at 800 frames per second using a high-speed RGB camera (IDT N4-S1) with a resolution of 1024 x 1024 pixels which was fitted with a 105mm lens (Sigma) that gave a magnification of 13.4 pixels/mm. In addition, an LCD projector (Epson EB-x11) was employed to project a blue and white fringe pattern with 1.8mm pitch with the lines nominally parallel to the motion of the wedge. No extra lighting was employed for the digital image correlation because it affected the contrast of the projected fringes. The data was processed as described above using a facet size and overlap that were both 21 and 21 pixels for the two-dimensional digital image correlation. Typical maps of displacement obtained from the experiments are shown in figures 4 and 5.

4.RESULTS

Typical results from the computational model and experiments are shown in figure 5 for 0.0225s into the test which corresponds approximately to the maximum indentation. About thirty images were collected during the period of a test but five images were selected for more detailed analysis at the times shown by the filled symbols in figure 7. Even at this reduced temporal density it is still difficult to make detailed quantitative comparisons of the displacement fields from experiment and numerical model because of the quantity of spatial data, the different densities of the datasets and the different coordinate systems of the datasets. To facilitate a comparison of the results, the strain maps were treated as images and Tchebichef polynomials were used for image decomposition following the procedure recommended by CEN [31]. In this process, a displacement field can be treated as an image I(i, j) where the intensity of the image corresponds to the magnitude of the displacement. The image can be described by a set of coefficients or shape descriptors, sk such that

(3)

where Tk(i, j) are the Tchebichef polynomials. Consequently, a matrix of data can be represented by a feature vector containing the coefficients, sk. The number of polynomials and hence coefficients is dependent on the level of correlation required between the original dataset and its reconstruction from the feature vector. In this study a correlation coefficient of 0.95 was assumed to be acceptable in common with prior work [21, 22] and the number of polynomials required to achieve this level of correlation was explored for the x, y and z displacement fields, as shown in figure 8. On the basis of the trends shown in figure 8, twenty shape descriptors were used to describe the x and y displacement maps and one hundred for the z-displacement maps. A higher number of shape descriptors was required to represent z- displacement due to the presence of high frequency noise and the shape of the displacement map.

The set of shape descriptors describing each displacement field were treated as a feature vector. The vectors obtained from experiments and simulations were compared using two metrics, i.e. the Euclidean distance between them and their Concordance correlation coefficient. The Euclidean distance is simply the straight-line distance between the locations represented by the vectors in multi-dimensional space, so that two coincident vectors would have a Euclidean distance of zero. The evolution of the Euclidean distance between the feature vectors representing the displacement maps from the experiment and simulations are shown in figure 9 during the indentation and subsequent release. The Concordance correlation coefficient, c [32] provides an indication of the extent to which the components of the feature vector fall on a straight line of gradient unity when plotted against one another. It is a combination of the Pearson's correlation coefficient r, which measures the deviation from a best-fit regression line, and a bias correction factor Cb, which measures the deviation of the best-fit line from a gradient of unity through the origin. Hence, the Concordance correlation coefficient was taken as

(4)

where the Pearson correlation coefficient is

(5)

and the bias correction is

(6)

where e, m, and em are the standard deviations and co-variance of two sets of shape descriptors representing the data fields from experiment and model, that have means, e and m respectively such that

(7)

The Concordance coefficients for the shape descriptors representing the data from experiment and simulation are shown in figure 10 for the x, y and z-displacements during the indentation and subsequent release.

5. DISCUSSION

The tests performed in this study present some challenges in the experimental methodology because the high frame rates imply short exposure times for the image capture, and this in turn requires a high level of illumination. It was also important to achieve a high level of contrast in both the speckle and projected fringes. A higher level of general illumination would improve the former but be detrimental to the latter. Hence, to overcome these challenges a number of measures were taken. A projector with high brightness was employed to provide good contrast for the fringes and a high level of illumination for the digital image correlation. In addition, the lens aperture (f=2.8) was adjusted to maximise the quantity of light that reached the camera sensor and the exposure time (621 s) was optimised to avoid blurred images and provide maximum time for recording images. The experimental set-up was very similar to that used previously by the authors to measure three-dimensional shape changes in a rubber sheet [20] and with this arrangement the minimum measurement uncertainties were assessed and found to be 8.32.4µm and 23.8.6.8µm for in-plane and out-of-plane measurements respectively. A similar level of relative uncertainty, i.e. 0.83% and 2.4% for in-plane and out-of-plane displacements respectively, could be expected in the current study because the optical arrangements and data processing algorithms are effectively the same.

Figure 4 shows a qualitative comparison of the data fields obtained by experiment at t=0.005s (2mm indentation), 0,01125s (4mm indentation), t=0.0275 (4mm indentation in release) and t=0.03s (2mm indentation in release) and those predicted using the initial models and the refined model. While, Figure 5 shows a qualitative comparison of the data fields obtained by experiment at the maximum indentation and those predicted using the initial and refined models. The in-plane data fields from the experiment are effectively free of random noise, especially for the y-displacements. The out-of-plane or z-displacement map exhibits a relatively high level of noise which is in part due to the low levels of displacement being measured, i.e. about one-fifth of the in-plane values, and in part because the paint speckle pattern has not been completely removed from the fringe projection data by the image separation process and, hence, influences the processing. This has been observed previously and is not a feature of these experiments but a generic weakness of the combined fringe projection and two-dimensional digital image correlation technique, particular for low levels of out-of-plane displacement. It should be noted that no smoothing has been applied to any of the displacement maps in figures 4 and 5.

The overall level of qualitative agreement between the data from the experiment and initial simulation in figure 4 is good; however there are a number of distinct differences particularly in the detailed shapes of the contours. In particular, the x-displacement fields in figure 4 exhibit substantial differences in shape for 2mm of displacement loading and unloading. However, at 4mm of displacement measured and predicted shapes seem closer although the magnitudes are somewhat different. For the maximum level of indentation, data in figure 5 shows that shape of the x, y and z-displacement fields measured and predicted by the initial model are similar but different in magnitude.

In this initial model the values for the material parameters in the Mooney-Rivlin model were based on the static behaviour of the material and a Shore hardness value. As a consequence of the differences observed between the predictions from the initial model and the results from the experiments, it was decided to calibrate the Mooney-Rivlin material model using test speeds more appropriate to the indentation experiment. Uniaxial compression tests were performed using a cube of side length 30mm which was cast from the same material as the block indented in the experiments shown in figure 6. The compression tests were performed at 400mm/s and 1000mm/s which corresponded to the nominal loading rates in the indentation experiments. The results from these calibration tests are shown in figure 3 and were used to refine the Mooney-Rivlin material model. The value of the material parameters, C10 and C01 in equation (1) obtained at 400mm/s and 1000mm/s were used to create models known as A and B. The maps of displacement for the refined model in figure 5 were obtained from model A because the indentation was performed at nominally 400mm/s and hence model A is likely to provide a better representation. The indentation was released at nominally 1000mm/s and hence the release of displacements is likely to be better represented by model B. However, in practice the loading machine was unable to maintain the loading rate throughout the indentation event so that the wedge speed had slowed down to about 200mm/s when the wedge tip was indented about 5.5mm or about 90% of its travel into the block, as shown in figure 7. On release the wedge speed only reached 1000mm/s when the wedge tip had already withdrawn about 1.5mm or 25% of its outward journey. The softer response of the loading system than the nominal response used in the model is likely to influence the displacement maps during the indentation and release events and this is illustrated in figures 9 to 10.

The behaviour of the Euclidean distances between the feature vectors representing the displacement fields from the models and experiment are shown in figure 9. The overall trend is that the Euclidean differences are less than 0.25mm with the values varying as a function of both the indentation depth and the measured displacements, i.e. the Euclidean distances are largest for the y-displacement fields and smallest for the x-displacement fields, where corresponds to the direction of travel of the wedge. At maximum indentation, there are no differences between model A and B for the x and z- displacements and a very small difference for the y-displacements; however, there are important differences during the indentation and release stages. For all of the displacement components, model A tends to exhibit a lower Euclidean distance during indentation than during release, i.e. the model A data loops anti-clockwise during the event, which is consistent with the material parameters having been obtained at the nominal indentation rate and not the faster release rate. Conversely, the model B data tends to loop clockwise with lower Euclidean distances, i.e. better agreement between the model and experiment, during the release stage of the event which nominally occurred at the rate represented by the material parameters in model B. In the case of the z-displacements, the plots of Euclidean distance execute figures of eight with both models exhibiting better agreement with experiment during the earlier stages of loading than the corresponding levels of indentation during release. This is not unexpected because in the final stages of release and immediately post-release there was a substantial recoil from the test machine and some particles of talc fell from the top surface of the block, as can be seen in figure 11. The recoil caused a substantial rigid body motion and the falling talc acted extraneous speckle which made it difficult to remove the rigid body motion and to obtain high-quality data. Hence, the final data value in the release event is unreliable and should probably be ignored.

The Euclidean distance between the feature vectors representing the displacement fields measured in the experiment and predicted by the models are a function of the overall magnitude of the displacement fields, because the largest shape descriptor is always indicative of the average value in the image. However, the concordance correlation coefficient is normalised and hence potentially provides a better basis for comparison. In figure 10, it can be seen that the concordance correlation coefficients for the x and y displacement fields at the maximum indentation are almost the same (0.975 and 0.987 respectively) and close to unity, implying close agreement between the model and experiment. However, it is apparent from the qualitative comparison in figure 5 that the displacement fields from the model are good but not excellent representations of the fields from the experiment. Hence, the Concordance correlation coefficient may not be such a useful measure of the comparison between shape descriptors, perhaps because the statistics underpinning it are dominated by a large number of zero-valued descriptors that do not contribute to the representation of the displacement fields. In figure 12 the shape descriptors from the feature vectors representing the displacement fields from the experiment and simulation have been plotted against one another for the duration of the event using model A for the indentation and model B during the release. The shape descriptors with a magnitude of less than 10% of the largest shape descriptor in each feature vector have been excluded following the approach of Labeas et al [33]. Each of the remaining shape descriptor forms a trace during the event which if the model is an excellent representation of the experiment would track along the straight-line of gradient unity. There will always be some uncertainty in the data so that this perfect correlation will not occur and engineers might find the model acceptable if all of the shape descriptors stayed within a zone defined by lines of relative error equal to 5%, where relative error is defined as the uncertainty divided by the maximum value of the measurand. This zone is shown in figure 12 and it can be seen that all of the shape descriptors remain within the zone throughout the indentation and release event implying that the model is a good representation of the experiment. The shape descriptors for the x and z-displacement fields are within a 2% error band implying excellent agreement between measured and predicted results. For y-displacements differences are larger and the first, second and sixth shape descriptors exhibit the largest excursions from the line of equality. The first shape descriptor in the Tchebichef polynomial representation describes the overall magnitude of the shape and its locus traces a figure of eight in a similar manner to the Euclidean distance plots in figure 9; implying that the magnitude during indentation is less well-represented by the model, which tends to underestimate it, and during unloading the magnitude is initially well-represented but then returns to the previous under-represented state. The second shape descriptor for the z-displacement field shows only small changes during the loading and unloading event which implies a consistent difference between the measured and predicted shapes. The sixth shape descriptor traces out a locus parallel to the line of equality implying this component of the shape changes with loading but the difference between the measured and predicted shapes does not change with loading and unloading. In the Tchebichef polynomial representation, the second shape descriptor describes a tilt about a central in-plane axis and the sixth shape descriptor a curvature about the perpendicular in-plane axis. The presence of the loci for these shape descriptors below the line of gradient equal one for the z-displacements implies that the model is underestimating these components of the deformed out-of-plane shape. It can be observed in figure 5 that the z-displacement field predicted by the simulation is symmetric about the y-axis whereas the measured field is slightly eccentric with larger values on the right edge than the left edge. This difference in the symmetry is probably responsible for the differences in the second shape descriptor seen in figure 10. Along the horizontal centreline of the measured z-displacement field in figure 5 there is almost a 'ridge' of 0.5mm displacement, which is not present in the corresponding predicted field, and is probably responsible for the differences in the sixth shape descriptor observed in figure 10.

6.CONCLUSIONS

A finite element model of a rubber block (60x60x20) subject to high-speed indentation and subsequent release by a rigid wedge has been developed using experimental data to refine and validate the model. The experiments were conducted using a high-speed hydraulic test-machine which displaced the wedge with a tip radius of 1.68mm, 6mm into the block at a nominal speed of 400mm/s and then released it at a rate of 1000mm/s. Combined fringe projection and two-dimensional digital image correlation was used with a single high-speed colour camera to obtain images at 800 frames/second which were processed to yield the three-dimensional displacement fields over the 60mm square face of the block that was parallel to the direction of indentation. This experimental dataset is believed to be the first published for the dynamic indentation of a hyperelastic material.

Qualitative comparisons of the in-plane and out-of-plane displacement fields predicted by the initial model and measured in the experiment implied that the model was not a good representation of the reality. The initial model, which utilised a Mooney-Rivlin material model defined via the Shore A hardness of the rubber material, was refined by using empirical data from uniaxial compression tests on a cube of rubber material performed at 400mm/s and 1000mm/s. Qualitative comparisons indicated the refined models were more accurate representations of reality. More detailed analysis of the predicted and measured displacement fields was performed using image decomposition based on Tchebichef polynomials to reduce the dimensionality of the displacement fields. The resultant feature vectors were used to make detailed quantitative comparisons of the datasets using Euclidean distances and the Concordance correlation coefficient. These quantitative approaches to comparing data-rich fields could provide a route to the automated refinement and updating of models [34]. It has also been demonstrated that they allow exploration of the nature of the differences between predicted and measured data fields.

The work represents a case study in the advanced use of information-rich datasets from experimental measurements to refine and validate a complex computational mechanics model through quantitative comparison of the predicted and measured datasets as recommended recently by CEN [31]. The updated model was successfully validated using the displacement fields measured in the indentation experiments; and hence perhaps represents the most reliable model of dynamic indentation of a soft material yet reported thus demonstrating the value of such an approach.

ACKNOWLEDGEMENTS

This work has been conducted with the financial support of Junta de Andalucía through the programme ‘Programa para la formación de PDI pre-doctoral en Universidades Públicas Andaluzas en áreas de conocimiento consideradas deficitarias por necesidades docentes’. The authors also acknowledge the generosity of University of Liverpool where the experimental was conducted. EAP is a grateful recipient of a Royal Society Wolfson Research Merit Award.

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[29] W. Battermannm E. Köhler, 1982, Elastomere Federung, elastische Lagerungen: Grundlagen ingenieurmässiger Berechnung und Konstruktion. Ed: Ernst, 1982

[30]A Dorfmann, A Muhr. Constitutive models for rubber. 1999 CRC Press Taylor & Francis Group London.

[31]CEN 2014, Validation of computational solid mechanics models, Comité Européen de Normalisation, Brussels, CWA 16799.

[32] L. I-Kuei Lin. A Concordance Correlation Coefficient to Evaluate Reproducibility. BiometricsVol. 45, No. 1 (Mar., 1989), 255-268

[33] G. Labeas, V.P. Pasialis, X. Lin, X. & E.A. Patterson, 2015. On the validation of solid mechanics models using optical measurements and data decomposition, Simulation Modelling Practice & Theory, 52, 92-107.

[34]W. Wang, J.E. Mottershead, C.M. Sebastian, E.A.Patterson, 2011, Shape features and finite element model updating from full-field strain data, Int. J. Solids Struct. 48(11-12), 1644-1657.

Figure captation:

· Figure 1: A photograph (left) of the silicone block during contact by the indenter whose dimensions are given in the diagram (right).

· Figure 2: Detail of the finite element mesh employed for the 60x60x20 block and the wedge.

· Figure 3: Stress-strain response at 400 mm/s and 1000 mm/s from uniaxial compression tests on a cube of side length 30mm. The data is based on the mean of three tests performed at each speed.

· Figure 4: Displacement maps for x-displacements for experimental data (top) and predicted by the initial finite element model (middle) and refined model (bottom) obtained at 2mm (t =0.005s), 4mm (t =0.01125s) during indentation and 4mm (t =0.0275s) and 2mm (t =0.03s) during release or unloading.

· Figure 5: Displacement maps for x-displacements (top), y-displacements (middle) and z-displacements (bottom) obtained at 6mm indentation (t=0.0225s) using FP+2D-DIC (left) and predicted by the initial (centre) and refined model A (right) finite element models.

· Figure 6: Photograph of the experimental set-up (left) and a schematic diagram of the optical layout (right).

· Figure 7: Typical experimental data showing the position and speed of the wedge tip as a function of time with the filled symbols indicating the times at which images were subject to more detailed analysis.

· Figure 8: Correlation coefficient between original and reconstructed displacement fields as a function of the number of Tchebichef coefficients or shape descriptors employed in the image decomposition of the experimental displacement fields.

· Figure 9: Evolution of the Euclidean distance between the feature vectors representing the x-displacement (top), y-displacement (middle) and z-displacement (bottom) maps from the experiment and simulation for models A (green line & square symbols) and B (red line & diamond symbols) during the indentation and subsequent release.

· Figure 10: Concordance correlation coefficient, c for the shape descriptors representing the x-, y-, and z- displacement maps from experiment and simulation during the indentation event (model A) and subsequent release (model B).

· Figure 11: Photograph of the wedge and rubber block just after zero indentation in the release vent showing the recoil of the test machine and some particles of talc falling from the top surface of the block. These events caused the final datasets obtained in the test to be unreliable. A video of this phase of the test is available as supplementary material.

· Figure 12: Comparison of the largest shape descriptors (SD) representing the x- (top), y- (middle) and z- (bottom) displacement fields obtained from experiment (abscissa) and model A (ordinate) during the indentation and model B during the release.

FIGURES

Figure 1: A photograph (left) of the silicon block during contact by the indenter whose dimensions are given in the diagram (right).

Figure 2: Detail of the finite element mesh employed for the 60x60x20 block and the wedge.

Figure 3: Material response in terms of engineering stress and strain at 400 mm/s and 1000 mm/s from uniaxial compression tests on a cube of side length 30mm.

Figure 4: Displacement maps for x-displacements for experimental data (top) and predicted by the initial finite element model (bottom) obtained at 2mm (t =0.005s), 4mm (t =0.01125s) during indentation and 4mm (t =0.0275s) and 2mm (t =0.03s) during release or unloading.

Figure 5: Displacement maps for x-displacements (top), y-displacements (middle) and z-displacements (bottom) obtained at 6mm indentation (t=0.0225s) using FP+2D-DIC (left) and predicted by the initial (centre) and refined model A (right) finite element models.

Figure 6: Photograph of the experimental set-up (left) and a schematic diagram of the optical layout (right).

Figure 7: Typical experimental data showing the position and speed of the wedge tip as a function of time with the filled symbols indicating the times at which images were subject to more detailed analysis.

Figure 8: Correlation coefficient between original and reconstructed displacement fields as a function of the number of Tchebichef coefficients or shape descriptors employed in the image decomposition of the experimental displacement fields.

Figure 9: Evolution of the Euclidean distance between the feature vectors representing the x-displacement (top), y-displacement (middle) and z-displacement (bottom) maps from the experiment and simulation for models A (green line & square symbols) and B (red line & diamond symbols) during the indentation and subsequent release.

Figure 10: Concordance correlation coefficient, c for the shape descriptors representing the x-, y-, and z- displacement maps from experiment and simulation during the indentation event (model A) and subsequent release (model B).

Figure 11: Photograph of the wedge and rubber block just after zero indentation in the release event showing the recoil of the test machine and some particles of talc falling from the top surface of the block. These events caused the final datasets obtained in the test to be unreliable. A video of this phase of the test is available as supplementary material.

Figure 12: Comparison of the largest shape descriptors (SD) representing the x- (top), y- (middle) and z- (bottom) displacement fields obtained from experiment (abscissa) and model A (ordinate) during the indentation and model B during the release.

24

-12-10-8-6-4-20-0,4-0,3-0,2-0,10Strss (MPa)StrainTest 400 mm/sTest 1000 mm/s

mmExperimental data Initial model

200300400500600700800

200

300

400

500

600

700

800

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

100200300400500600

100

200

300

400

500

600

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

200300400500600700800

200

300

400

500

600

700

800

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

100150200250300350400450500

50

100

150

200

250

300

350

400

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

100150200250300350400450500

50

100

150

200

250

300

350

400

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

200300400500600700800

200

300

400

500

600

700

800

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

100150200250300350400450

50

100

150

200

250

300

350

400

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

200300400500600700800

200

300

400

500

600

700

800

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0,200,20,60,2-0,6-0,20,60,2-0,6-0,20,30,15-0,3-0,15mmmmmmt=0.005s t=0.01125s t=0.0275s t=0.03s

100200300400500600700800900

100

200

300

400

500

600

700

800

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

100200300400500

50

100

150

200

250

300

350

400

450

500

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

200300400500600700800

200

300

400

500

600

700

800

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

200400600800

200

300

400

500

600

700

800

0

0.2

0.4

0.6

0.8

1

0100200300

50

100

150

200

250

300

350

0

0.2

0.4

0.6

0.8

1

1.2

0100200300

50

100

150

200

250

300

350

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0100200300

50

100

150

200

250

300

350

-5

-4

-3

-2

-1

0

mmmmmm

50100150200250300350

50

100

150

200

250

300

350

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

100200300400500600

100

200

300

400

500

600

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

100200300400500600

100

200

300

400

500

600

-5.5

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

100200300400500600

100

200

300

400

500

600

0

0.2

0.4

0.6

0.8

1

1.2

Experimental ResultsFEM initial modelFEM refined model

42131234

ProjectorCameraSilicone blockHydraulic actuator

715mm450mm

312

B)A)

-600

-400

-200

0

200

400

600

800

1000

1200

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

0

0,005

0,01

0,015

0,02

0,025

0,03

0,035

w

e

d

g

e

s

p

e

e

d

(

m

m

/

s

)

W

e

d

g

e

P

o

s

i

t

i

o

n

(

m

m

)

Test time (s)

Wedge position

Wedge Speed

00,20,40,60,811,2020406080100Correlation coefficient Number of shape descriptors x- displacement field y- displacement field z- displacement field

00,050,10,150,20,250,30,3501234567Euclidean distance (mm)Indentation (mm)00,050,10,150,20,250,30,3501234567Euclidean distance (mm)Indentation (mm)00,050,10,150,20,250,30,3501234567Euclidean distance (mm)Indentation (mm)

0,10,20,30,40,50,60,70,80,911,101234567Correlation CoefficientIndentation (mm)x- displacementy- displacementz- displacement

-1.5-1-0.500.51

-1.5

-1

-0.5

0

0.5

1

Experiment shape descriptors

Model shape descriptors

-0.2-0.100.10.20.30.4

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Experiment shape descriptors

Model shape descriptors

SD 1

SD 2

SD 6

SD 9

SD 15

SD 20

SD 26

data8

data9

data10

-1.5-1-0.500.51

-1.5

-1

-0.5

0

0.5

1

Experiment shape descriptors

Model shape descriptors

-1.5-1-0.500.51

-1.5

-1

-0.5

0

0.5

1

Experiment shape descriptors

Model shape descriptors

-1.5-1-0.500.51

-1.5

-1

-0.5

0

0.5

1

Experiment shape descriptors

Model shape descriptors

SD 3

SD 5

SD 8

SD 10

SD 12

data6

data7

data8

-1.5-1-0.500.51

-1.5

-1

-0.5

0

0.5

1

Experiment shape descriptors

Model shape descriptors

SD 1

SD 2

SD 6

SD 11

SD 15

data6

data7

data8

A)

B)

18,42 mm

73,45

o

R 1,68 mm

yxz

5 mm5 mm8,25 mm30 mm20 mm5 mm5 mm2 mmABCDEHGF

ElementtypeElementSize(mm)XYZA0,250,250,25B10,50,25C0,50,50,25D120,25E0,510,25F122G10,51H10,52I0,250,52

I15 mm