# Characterization of Necking Phenomena in High-Speed Experiments by Using a Single Camera

- Gilles Besnard
^{1, 2}, - Jean-Michel Lagrange
^{1}, - François Hild
^{2}Email author, - Stéphane Roux
^{2}and - Christophe Voltz
^{3}

**2010**:215956

https://doi.org/10.1155/2010/215956

© Gilles Besnard et al. 2010

**Received: **6 January 2010

**Accepted: **24 June 2010

**Published: **12 July 2010

## Abstract

The purpose of the experiment described herein is the study of material deformation (here a cylinder) induced by explosives. During its expansion, the cylinder (initially 3 mm thick) is thinning until fracture appears. Some tens of microseconds before destruction, strain localizations occur and induce mechanical necking. To characterize the time of first localizations, 25 stereoscopic acquisitions at about 500,000 frames per second are used by resorting to a single ultra-high speed camera. The 3D reconstruction from stereoscopic movies is described. A special calibration procedure is followed, namely, the calibration target is imaged during the experiment itself. To characterize the performance of the present procedure, resolution and optical distortions are estimated. The principle of stereoscopic reconstruction of an object subjected to a high-speed experiment is then developed. This reconstruction is achieved by using a global image correlation code that exploits random markings on the object outer surface. The spatial resolution of the estimated surface is evaluated thanks to a realistic image pair synthesis. Last, the time evolution of surface roughness is estimated. It gives access to the onset of necking.

## 1. Introduction

For detonics applications, objects subjected to very high deformations (about 50% to 100% strains) are to be observed in very short times (i.e., less than 100 s). To characterize the phenomenon of necking and to compare experimental results with hydrodynamic computations, ultra-fast cinematography is very useful. This diagnostic, which is resolved in space and time, is used to monitor external surfaces of expanding objects. For the present applications, dedicated cameras are used [1]. Thanks to a stereoscopic setup, 3D reconstructions are possible.

The stereovision technique is used for mechanical observations. Numerous applications exist for quasistatic experiments [2–5] where stereovision is coupled with digital image correlation [6]. The latter is a nonintrusive measurement technique that provides a large density of measurement points. Thanks to the generalization of digital cameras (with CCD or CMOS sensors), the use of stereo-correlation tends to develop in the field of fast dynamics such as, for example, torsion and tensile tests on Hopkinson bars [7, 8]. However, the use of stereovision for quantitative purposes for high-speed experiments is marginal. Recently it was shown that the use of stereovision to monitor detonics tests [9, 10] is possible with CCD cameras. However, the lack of resolution of these sensors (typically, pixels at 10 fps) is a strong limitation. In the present study, film cameras with a revolving mirror are used. They have a very high resolution (e.g., pixels at 10 fps). However, additional treatments are necessary because of the digitization of the developed film and the specific technology of these cameras.

The objective of the present paper is to provide a characterization of the surface quality of the object, and the time of inception of localized phenomena. First, the experimental setup is presented. Because of the use of a specific optical chain, the implemented techniques are introduced and characterized (resolution and distortions). Then, the stereovision coupled with digital image correlation is presented. A synthetic case is analyzed to determine the detection resolution (i.e., the minimum defect size). Last, in order to improve the quality of the 3D reconstruction, a correction method which allows for large displacements is presented. The whole procedure is finally illustrated to analyze a true experiment.

## 2. Stereovision Principle

This first part deals with the reconstruction of an object based upon stereoscopic observations. The principle of formation of images via mirrors and a rotating mirror framing camera and its calibration are introduced. The specific global digital image correlation algorithm used to perform stereomatching is finally presented.

### 2.1. Formation of Images via Mirrors

*same*camera are possible. In this part, theoretical expressions of the transformation matrix, that is, relating the 3D coordinates of a point of the scene and its projection in the image plane are recalled. This is performed within the framework of an orthographical model with mirror, which is an appropriate model for the measurements performed herein since the object size is very small (height: 100 mm, diameter: 100 mm, see Figure 1(a)) in comparison with the distance between the camera and the object (ca. 16 m).

Because of the large optical path between the camera and the object (about 16 m), image formation is split into two stages, illustrated in two dimensions in Figure 1(a). This process is identical for the two mirrors and only the generic case is considered in the sequel. Let be a point in the image scene. It is imaged at point by mirror . The orthogonal projection of onto the image plane is denoted by .

where the parameters denote the coefficients of the transformation matrix , whose expression is simplified compared with those obtained in the case of pinhole model [6]. In the present case, for and [14].

### 2.2. Calibration and Stereoscopic Reconstruction

enforcing the condition [14].

The expected conditions for can be checked as a self-consistent validation of the calibration. It is worth noting that with the model used herein, the coefficients are vanishingly small.

*in situ*calibration target) is implemented since lighting is obtained by pyrotechnic flashes that are only used during the experiment itself. The latter further requires that the explosive be introduced only at the very end of the experiment preparation, for safety reasons. Prior to that, the position of various objects (the mirrors in particular) may change slightly because of operator manipulations. Therefore, the calibration target has to remain in the field of view (and hence will be destroyed during the explosion). Therefore, the proposed camera calibration procedure is not "optimal" in the sense that all the field of view is not calibrated. However, it will be shown in Section 4.2 that the distortions remain small, thereby only having a small impact on the quality of the reconstruction.

where (resp., ) are the coefficients of the left (resp., right) transformation matrix.

### 2.3. Registration by Digital Image Correlation

## 3. Experimental Setup

The experiment reported herein aims at studying the mechanical behavior of copper under high-speed loading conditions. The material is a high-purity copper (UNS C10100 - ISO Cu-OFE grade). The studied object is a cylinder (length: 100 mm, internal diameter: 100 mm, wall thickness: 3 mm). Different forming steps are needed to obtain the final sample. First, a thick blank is deep drawn to a cup form. Second, the cylinder is turned by flospinning. Last, the hemispherical top of the piece is cut out and the cylindrical part is kept. The final microstructure is obtained by a heat-treatment to trigger recrystallization and stress relaxation. The average grain size is 25 m (this value is constant in both directions). The external surface is polished to ensure good reflectivity for laser velocimetry measurements.

## 4. Characterization of the Optical Chain

Rotating mirror framing film cameras are used for quantifications of local (necking) phenomena. The latter ones are observed via a random pattern that must be characterized. If the random pattern is too fine, there is not enough contrast (i.e., small gradients) and the resolution of the correlation procedure is not sufficient. Conversely, if the speckle is too coarse, large element sizes are needed so that the number of measurement points decreases. The optimal size of the pattern is used in the synthetic case described in Section 5.1. Moreover, the cameras used herein are complex since they are made of a principal lens, 25 secondary lenses, and many mirrors are used to form an image. These optical devices may generate distortions that are to be characterized.

### 4.1. Resolution

A resolution calibration target, similar to a Foucault pattern, is put in place of the object. The former consists of 6 small plates joined together to form a mm plate. The pattern consists of a succession of horizontal and vertical lines with varying thicknesses ranging from 0.6 mm to 1.4 mm with a 0.1 mm step. The center of the plate is then put in place of the observed object (located at a distance of 16 m from the camera) and at an angle of with respect to the optical axis to estimate the depth of field.

The resolution of the optical chain is sought to assess the minimum size that can be observed, and the size of the random pattern to be deposited for an optimal observation. To quantify this size, contrast of each line is analyzed to deduce the cut-off frequency corresponding to 50% of the dynamic range. The latter is obtained by analyzing a zone close to the edges of the plate and by rescaling the amplitudes between 0 and 1. Local contrast is obtained in an identical way for horizontal and vertical lines.

In Figure 7(b), variations of contrast that would be obtained if the calibration target was seen through a linear optical system of Gaussian transfer function of Full Width at Middle Height (FWMH) ranging from 0.5 to 3.9 mm are plotted in solid lines. These values are given for magnifications of about 20. The experimental curve lies between the lines corresponding to an FWMH of 1.1 and 1.3 mm, meaning that it will be difficult to distinguish correctly elements smaller than these two sizes. Thus, the speckle deposited onto the cylinder must have at least a diameter of 1.2 mm. This characterization is useful for the realistic synthesis of images discussed in Section 5.1.

### 4.2. Lens Distortions

Because of the use of a rotating mirror framing camera and many mirrors between the object and the camera, the estimation of optical distortions is an important step in the experiments. Techniques utilized to determine the distortions of the *whole* optical system require the acquisition of one image per secondary lens. It is different from procedures followed to analyze quasistatic experiments [17] or even for the case of a dynamic test [18]. Moreover, because of the complexity of the optical chain, it is not possible to project the distortions found onto a polynomial basis as generally performed [6, 17]. In the present case, the frame-to-frame distortions were found negligible with respect to that of the whole optical chain.

## 5. Application

In this part, the stereovision technique is applied to analyze a cylinder expansion caused by blast loading. It is worth noting that other types of loading configurations have been used in the literature [13, 20]. First, a synthetic case representative of the experiment is studied to estimate the performances of the technique and in particular the resolution of the reconstruction. This enables for the evaluation of the minimum size of observable and quantifiable defects. In the present experiments, the observed surface undergoes important deformations (beyond 100% strain). This is the reason why the computation is not carried out with the initial reference but rather with an updated reference that causes a cumulation of measurement errors. A reduction in the size of the reconstructed surface is observed since the points that leave the initial region of interest are not taken into account. To improve the performances of the approach, a precorrection for large displacements is performed. It consists in seeking a uniform translation to apply to the images so that, on average, the region of interest is motionless. Then the DIC algorithm is run using the prior translation as an initialization of the displacement field. This procedure makes the computation faster, more stable and more accurate. Finally, the stereovision technique is applied to the experiment itself [21–24].

### 5.1. Detection Level

Before applying the stereo-correlation procedure to an experimental case, it is important to evaluate the size and amplitude of defects that can be detected. The hydrodynamic code HESIONE [25] predicts the shape of the specimen at different stages of evolution. For any instant of time, , the predicted surface is projected onto the actual surface by least squares minimization.

A DIC analysis was performed on those artificial images, based on the same choice of parameters as the one used in the experiment, namely, pixel elements are selected based on the signal-to-noise ratio. A comparison between the measured and prescribed displacements for each bump allows for the evaluation of the resolution. To carry out this analysis, the measured and imposed shapes are unfolded onto a plane as suggested by Luo and Riou [26].

It is concluded that for very severe experimental conditions (rotating mirror high-speed camera at 16 m optical distance from the specimen, magnification of 22, small radius of curvature, and poorly contrasted surface texture), the limit of detection of such bumps is of the order of 5 mm, and a minimum size of about 10 mm is needed to allow for a reliable quantification of the perturbation. Moreover, 250 m amplitudes are resolved reliably for those conditions. These conclusions hold for a fixed element size of 16 pixels. Smaller elements lead to too noisy measurements to secure the determination, whereas larger elements are too coarse. This level is to be compared with the 3D reconstruction uncertainty achieved herein. A level of m is estimated by randomly perturbing the position of the calibration points, and reconstructed points with realistic values [19].

Possible improvements involve drastic changes in the experimental set-up. CCD camera could offer images in digital format directly, thus limiting the digitation step in the analysis. However, access to similar pixel sizes still represents a technical challenge. A better resolution could also be obtained through a higher magnification, at the expense of a smaller frame.

### 5.2. Large Displacement Handling

In the context of detonics, very high strain levels between consecutive images have to be captured. This fact is a major difficulty for DIC. A specific procedure has been designed to allow for a much more robust analysis in this context. As a side benefit, displacement fields appear to be less subjected to noise.

The principle of the method is simply to initialize the DIC analysis, which is in the present case an iterative procedure, by a prior determination of the displacement field obtained via a simulation of the experiment. This allows for a convergence of the displacement determination into the deepest minimum, and avoids trappings into secondary minima. Let us note that a multiscale strategy is adopted in the DIC analysis for the same purpose of limiting secondary minima trappings [16]. However, at the largest scales, the contrast of the images is significantly reduced and hence some nodes or zones may be polluted by such artifacts. In contrast, a fair prior estimate of the displacement field, which may still be inaccurate, requires DIC to address only the remaining corrections displacements and hence can be tackled with less coarsened images. As this procedure only affects the initial displacement field, it does not affect the final one at convergence as can be checked by perturbing this prior determination, and checking that the final determination is unaffected. This robustness allows for some tolerance on the quality of this first displacement field, and hence, small effects such as the motion of the cylinder axis are neglected.

### 5.3. Experimental Reconstruction

After the experiment, the film composed of 25 acquisitions is developed and the 25 images are digitized independently from each other. From the fixed elements of the scene (yellow paper in Figure 1(a) or the calibration target shown in Figure 14), repositioning of images is performed by adjusting a translation correction so that the elements remain motionless [19].

In order to investigate on a quantitative basis the onset of necking, it is proposed to base the analysis on the standard deviation of the normal surface displacements. This standard deviation can be seen as a measure of the roughness of the expanding surface, which is expected to remain small for a uniform strain of the surface, and to display a sudden increase when necking (at least necking that can be captured by the DIC analysis, i.e., at a large enough scale). The standard deviation is expected to be sensitive to the sampled surface as the lower part of the specimen is subjected to a larger strain. Thus the standard deviation is estimated over three regions, namely, first globally over the entire field of analysis, second over a central zone (where edge effects are avoided), and third on a zone located at the bottom of the specimen where necking is seen to occur first.

## 6. Conclusion

A successful attempt is reported to reconstruct complex 3D displacement fields for high-speed blast experiments under very severe experimental conditions using a rotating mirror high-speed camera. Both large-scale shape changes as well as local features associated with necking can be captured. The techniques were also set up to estimate the resolution and distortions of the optical chain allowing for the analysis of a synthetic case representative of the experimental test.

Specific procedures useful to enhance the performance of such stereo-correlation analyses have been developed. In particular, the combination of precorrections and global digital image correlation algorithm provides both robust and accurate 2D displacement fields that are suitable to stereo-correlation to get three-dimensional displacement fields. Other directions for improving the experiment itself are currently being investigated, namely, surface marking, which can sustain the very strain rates of the specimen, submicrosecond lighting, and digital cameras [27, 28] that will in a near future replace the film camera used in the present study. In the same spirit, combining more than two views of the same scene is a stimulating direction for increasing the accuracy of the reconstruction even with images having a low contrast or specular reflections that may always occur unexpectedly [14, 29, 30]. Stereovision is thus a very powerful tool to analyze high-speed experiments.

Last, let us note that the fact that the observed object is initially cylindrical makes the DIC procedures difficult due to the perspective issues. An a priori knowledge on the initial shape of the object would make the reconstruction step easier and faster. It would also allow to further increase the reconstructed surface area.

## Authors’ Affiliations

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