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A modified digital phase-shift moiré technique for impact deformation measurements
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Optics and Lasers in Engineering 42 (2004) 653–671
A modified digital phase-shift moire! technique for impact deformation measurements Patricia Verleysen*, Joris Degrieck Laboratory Soete, Department of Mechanical Construction and Production, Gent University, Sint-Pietersnieuwstraat 41, 9000 Gent, Belgium Received 1 September 2003; received in revised form 9 May 2004; accepted 21 May 2004 Available online 28 July 2004
Abstract Due to the short loading times and high deformation rates inherent to impact experiments, measurement of the occurring deformations is not straightforward. Presented in this paper is a technique to obtain the displacements and deformations of a specimen subjected to a uni-axial impact load. During the experiment the deformation of a line grating attached to the specimen is captured using a streak camera. From the recorded deforming grating the specimen displacements are automatically derived using an advanced numerical algorithm, based on the interference between the specimen grating and a virtual reference grating. Numerical interference is considered because it allows that the pitch of the reference grating is adapted to the changing amplitude of the deformation. Indeed, at each moment of the deformation process the pitch of the reference grating is chosen such that the highest possible accuracy and sensitivity is guaranteed. Because of this, large changes in deformation amplitude are allowed, and the technique is applicable to a wide range of materials. Eventual imperfections of the specimen grating and temperature effects are taken into account. Specimen displacements are extracted automatically by means of a phase-shifting technique. The non-contact measurement technique yields high resolution, quantitative information on the specimen deformation, along the entire length of the specimen and during the full duration of the experiment. Interaction by the operator is excluded. Results are presented of a high strain rate tensile test on a steel sheet specimen showing local deformations up to about 170%. r 2004 Elsevier Ltd. All rights reserved. Keywords: Digital phase-shift; Displacement measurement; High strain rate
*Corresponding author. Tel.: +32-9-264-34-35; fax: +32-9-264-35-87. E-mail address: [email protected] (P. Verleysen). 0143-8166/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2004.05.001
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1. Introduction Several test techniques exist during which a material sample is subjected to a uniaxial, impact load: Hopkinson (tensile or compressive) bar tests [1], falling weight tests and fly wheel experiments [2]. Due to the short duration of these experiments, typically of the order of one to a few milliseconds, as well as the short rise time of the load and the corresponding high deformation rates, measurement of the evolution of the specimen deformation is not straightforward. In addition, wave propagation phenomena often accompany impact experiments. Interaction of the waves with the measurement devices must be avoided both to generate reliable measurements and to avoid disturbing the experiment. Non-contact measurement techniques are therefore preferable. Most existing techniques only give limited information: the time history of the displacement of one or a few points, or the wholefield specimen deformation at certain moments in time. Additional difficulties arise from the fact that during an experiment the local deformation can vary over several orders of magnitude. When testing ductile materials, local strains of more than 100% are not exceptional. To monitor damage mechanisms and strain localisation effects in the specimen as well as the influence of the specimen geometry, high-resolution information on the evolution of the deformation along the length of the specimen is desirable. A non-contact technique is developed which gives quantitative information on the time evolution of the axial displacement field in a specimen subjected to an impact load. A line grating of 2 lines/mm is applied to the specimen. During the experiment the deformation of the grating is recorded by means of a rotating drum streak camera. Since a coarse grating is used, the technique used to extract the displacements from the recorded deforming grating must be carefully selected. For example, fringe centre positioning techniques give insufficient information, and are thus excluded. A more advanced numerical technique is developed to extract the displacements from the recorded picture. The phase of the digital interferogram, which arises from the interference between the digitised recorded specimen grid and a well-chosen virtual reference grid is related to the specimen displacement. The phase is determined by a phase-shifting technique [3,4]; by shifting the reference grating over well-determined distances, four interferograms are created, and these are subsequently used to calculate the phase and hence the displacements. The digital implementation of the geometrical interference between the reference and deformed grating, and the phase-shifting technique implies numerous practical advantages. More important however, is that it also allows the use of a reference grating with a variable pitch. A procedure was established to adapt the pitch of the reference grating automatically to the changing specimen deformation. As a result, the highest possible accuracy and sensitivity are obtained at each moment. Limitations of the amplitude of the specimen deformation are also overcome. The algorithm takes into account eventual imperfections of the original specimen grating. Results of a high strain rate (750/s) tensile experiment on a ductile steel sheet are presented. As will be shown, the combination of the non-contact measurement setup with the numerical processing technique yields highly accurate, quasi-continuous
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(both in time and space) information on the specimen displacements. Data on the value and accuracy of the measurement technique are given.
2. Deformation measurement set-up The set-up for the displacement measurement is schematically represented in Fig. 1. A line grid is attached to the surface of the specimen perpendicular to the loading direction. The image of the line grid is projected on a photographic film that is laid inside the drum of a rotating drum camera, in this case used as a streak camera. Prior to the actual experiment, the drum of the camera is brought to a welldetermined rotational speed. As the specimen is subjected to a high strain rate uniaxial load, a flash lightens the specimen and the image of the deforming grid is recorded on the moving film. The time evolution of the specimen deformation is thus recorded on the film. Although the measurement principle is rather simple, practical implementation, is more cumbersome. Finding an optimal combination of test parameters, such as the density of the line grating, the rotational speed of the camera drum, the spatial and light sensitivity of the photographic film, and the image magnification is not straightforward. Indeed, a compromise must be found between the opposing demands on the actual recording of the deforming specimen grid and those posed by the numerical processing technique. The first mainly concern the light exposure of the photographic film, the latter are related to the inherent accuracy of the technique. A high density of the line grating, a high velocity of the drum, a large magnification of the image, and a photographic film with a high spatial sensitivity all have a positive influence on the accuracy of the measurement technique, but have an adverse effect on the light exposure of the film. The velocity of the specimen particles also has an important influence on the choice of the parameters. To avoid motion blurring the rotational speed of the camera must be high enough and the width of the slit of the camera (see Fig. 1) must be sufficiently small.
Fig. 1. Schematical representation of the setup for the displacement measurement.
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The optimal set of parameters was established experimentally. A grating density of 2 lines/mm, combined with Kodak Tri-X pan 220 film, no image amplification and a rotational speed of 100 rps proved to give good results. The selected rotational speed yields a maximum recording time of 10 ms on the film; since the duration of an impact experiment lies typically in the interval [0.2 ms; 2 ms] this is largely sufficient. Furthermore, the use of a rather coarse grating has numerous practical advantages. The production of a highly accurate, coarse grating is straightforward, no sophisticated techniques are required to attach the grating onto the specimen, a strong but common light source can be used to illuminate the specimen, and the image of the deforming grating can be captured by commercially available photographic film. Since the technique should be able to manage large specimen deformations, the grid and the way it is fixed on the specimen should be able to follow the specimen deformation. A gelatine grid attached to the specimen surface using photographic stripping film, on which a high quality reprint from a master grating was made, proved to fulfil the requirements. The gelatine has a sufficiently large deformability and is firmly attached to the specimen surface. The grating can be applied to different types of materials, as long as their surface is smooth or can be made smooth.
3. Experimental results 3.1. Impact experiment The way the high strain rate deformation is applied to the specimen is of no importance for the deformation measurement technique; as long as the specimen surface is accessible, the technique can be used to measure axial displacements during high strain rate experiments using various methods such as a high speed hydraulic testing machine, a falling weight device or a fly wheel. Here application of the deformation measurement technique is illustrated by means of a split Hopkinson tensile bar experiment [5,6]. A material sample is fixed between two long bars, the input bar and the output bar. A tensile wave with a certain length is generated at the free end of the input bar. This wave propagates towards the specimen and applies a high strain rate load to the specimen. From strain gage measurements on the bar, the history of the strain, the strain rate and the stress in the specimen can be determined. However, at each moment only one value for the strain, the strain rate, and the stress are obtained [7]. These values are mean values along the length of the specimen. The measurement technique described in this paper was used to monitor the specimen displacement and deformation distribution during the experiment. For the experiment described here a stainless steel sheet, type 301LN, is tested because of its deformation capacity. Its geometry is presented in Fig. 2. Since the specimen is glued between the input bar and the output bar, glue zones are provided on both sides of the test section. The actual section that will be subjected to the high
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Fig. 2. Specimen geometry used for the impact experiment. Here, a radius of 5 mm, a length of 6 mm and a width of 5 mm are considered.
strain rate tensile load can be divided into three parts: a 6 mm long, 5 mm wide central section, and two transition zones in which the width spans from 5 to 15 mm. The specimen thickness is 1.2 mm. 3.2. Recorded picture Fig. 3 represents the recorded picture. Each horizontal line represents the image of the line grid at one moment during the experiment. Before the beginning of the loading, the lines do not change in width. Once the specimen is subjected to the tensile load, the lines are stretched. The lower in the picture, the wider become the lines. At a certain moment a crack appears near the middle of the specimen. Afterwards, one part of the specimen moves away from the other part. The duration of the experiment, from the start of the loading to the nucleation of the crack, is about 0.75 ms. At the moment of cracking, the mean strain in the specimen was a bit higher than 50%, but as will be shown, local strain peaks of 150% occur.
4. Treatment of the recorded image—digital phase-shift technique 4.1. Introduction A procedure was derived to calculate the specimen displacements from the deforming grid of Fig. 3 automatically. The following demands are posed to the processing technique: *
(Quasi-) continuous information, both in time and space, should be obtained. During an impact experiment the distribution of the deformation along the length
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Fig. 3. Deforming specimen grid recorded by means of a rotating drum camera used as streak camera.
*
of a specimen changes extremely fast; mostly the transition from more or less homogeneous deformations to severe strain localisation takes less than a few microseconds. Also the zone in which the strain localises is very small for most materials. High spatial and time resolution is thus required. Since a rather coarse grating is used, the spatial resolution of fringe centre positioning techniques is not sufficient, The technique should be valid for different amplitudes of deformation. It should be applicable for a wide range of materials, and also from the start towards the end of the experiment, so small to eventually high deformations should be allowed.
As will be shown, straightforward use of geometric moire! is not possible. Nonetheless, the geometric interference between two sets of lines, the recorded deforming grid and a reference grating, is used to calculate the displacement field. In Section 4.2, a relation is established between the displacement and the phase of the interference pattern. The value of the phase is determined by a digital phase-shifting technique: four virtual interference patterns are formed; each time the reference grating is shifted over a known distance. To account for the changing amplitude of the specimen deformation, the pitch of the reference grating is adapted in such a way that it follows the changing specimen deformation. In Sections 4.3 and 4.4 details are given on the implementation of the technique.
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4.2. Principle An equation valid for the superposition of two parallel gratings, a reference grating and a deformed grating, will be established. The equation will be used to calculate the in-plane displacements, perpendicular to the lines. The reference grating used is cosinusiodal with a pitch p: The brightness at position X of this grating is then given by the equation 2pX Iref ¼ A0 þ A1 cos : ð1Þ p A0 and A1 are, respectively, the mean value and amplitude of the brightness, and both are constant. Fig. 4 represents the normalised brightness of a part of the recorded grating before the start of the actual deformation. This corresponds with a part of the horizontal line at the top of Fig. 3. As can be seen in Fig. 4, the signal can be represented, with a very good approximation, by a cosine function. Consequently, if the specimen grating deforms axially and the point with coordinate x moves to point X after a displacement U; perpendicular to the lines, the brightness of the deformed grating is given by 2pðX U Þ Ispec ¼ B0 þ B1 cos : ð2Þ p B0 and B1 are, respectively, the mean value and amplitude of the brightness. However, due to uneven illumination and optical aberrations, B0 and B1 vary from point to point. The displacement U can be a real displacement due to loads, but it can also take into account the initial imperfections of the specimen grating. When the deformed grating given by Eq. (2) is superposed with the reference grating given by Eq. (1), the following brightness distribution is obtained: IðX Þ ¼ Ispec Iref
ð3Þ
1.2
before loading cosine
normalised signal
1 0.8 0.6 0.4 0.2 0 7
7.5
8 X (mm)
8.5
9
Fig. 4. Detail of the brightness of the specimen grating recorded before the actual deformation of the specimen and a cosine fitted to the signal.
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or:
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A1 B1 2pU cos IðX Þ ¼ p 2 A1 B1 2pð2X UÞ 2pðX UÞ 2pX þ A1 B0 cos þ cos þ A0 B1 cos p p p 2 þ A0 B0 :
ð4Þ
When small deformations are considered, the frequencies, with respect to X ; of the first cosine-term of the right- hand term of Eq. (4) are substantially lower than those of the other three cosine-terms. This term represents the actual moire! fringes. The three higher frequency cosine-terms in Eq. (4) can be filtered away using a low-pass filter function. The brightness distribution thus obtained can be written as IðX Þ ¼ I0 ðX Þ þ I1 ðX Þ cos fðX Þ;
ð5Þ
where I0 is the average brightness, I1 the amplitude of the brightness, and f the phase, given by fðX Þ ¼
2pUðX Þ : p
ð6Þ
Eqs. (5) and (6) are in agreement with those given in [8], valid for in-plane geometric moire! . In classical moire! applications, the higher frequency components of Eq. (4) are most often filtered away by the optics used to record the moire! pattern. Therefore, from a moire! fringe pattern, formed by interference of a reference grid and a deformed grid, the displacement U at each point of the image can be obtained by measuring the phase f of the cosinusoidal wave at that point p UðX Þ ¼ fðX Þ : ð7Þ 2p Since, I0 and I1 in Eq. (5) have no constant, known value, calculation of the phases is not straightforward. In such cases, the phases can be obtained with a high accuracy and sensitivity using a phase-shifting technique [9]. The phases are determined by recording (or calculating) the moire! fringe patterns three times or more by shifting the phase of the reference grid over known amounts, namely 2p divided by the number of phase-shifts. So, if four shifts are considered, the reference grid is superposed on the deformed line grid and this results in an image with brightness Ia¼0 : Then the reference grid is shifted over p/2, superposed on the deformed grid, this results in an image with brightness Ia¼p=2 : Subsequently, the reference grid is shifted over another p/2, so Ia¼p is obtained. From the obtained values of the brightness Ia¼0 ; Ia¼p=2 ; Ia¼p and Ia¼3p=2 ; the phases can be calculated by the Eq. [3] Ia¼3p=2 Ia¼p=2 f ¼ arctan : ð8Þ Ia¼0 Ia¼p As can be seen, the phase-shift method has the advantage that the phase in each point of the image is obtained independently from the phases in other points, and
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thus no interpolation between the fringes is required. Thus, a much higher resolution is obtained. 4.3. Deformation compatible reference grating 4.3.1. Principle Using the principle described in Section 4.2, four interference patterns are needed to calculate the displacements. The interferograms can be obtained by multiplying the brightness values of the original recorded grating by a reference grating (see Eq. (4)). Subsequently, the higher frequency components of the patterns have to be filtered out to obtain moire! patterns corresponding with Eq. (5). A correct transition of Eq. (4) to (5) can only be guaranteed if the frequencies corresponding with the first cosine-term and the frequencies of the other cosine-terms do not overlap. Fig. 5a represents the brightness along the axis of the specimen recorded before the actual loading of the specimen (see also Fig. 4). Fig. 5b represents the corresponding normalised spatial spectrum. As expected, a sharp peak appears at a spatial frequency of 2000/m, corresponding with a wavelength of 0.5 mm, which equals the spatial period of the specimen grating. A peak shift to the left is noticed when the specimen is stretched homogeneously. Due to non-homogeneous deformation, the frequency spectrum broadens. Fig. 6a represents the recorded brightness along the axis of the specimen just before the crack appears (at tE0.75 ms). The corresponding normalised spatial spectrum is shown in Fig. 6b. The sharp peak disappeared and spread over a wider zone of spatial frequencies. Compared to the peak in Fig. 5b a peak shift to the left is indeed observed. The signal even contains frequencies more or less around HALF the frequency of the original grating (1000/m). This is in agreement with Eq. (2) for the deformed grid. The normalised spatial spectra, obtained by multiplying the brightness values of Figs. 5a and 6a by a reference grating, are shown in Fig 7a and b, respectively. The pitch of the reference grating is equal to the pitch of the undeformed specimen grating. In Fig. 7a, two distinct peaks can be distinguished: one at 2000/s and the other at 4000/m, corresponding to the third and fourth terms, and the second righthand term of expression (4), respectively. The low frequencies of the actual moire! pattern (first term) are separated from the higher frequency components. As illustrated in Fig. 7b, the above is not valid when the specimen deformations become larger. In this case the peaks shown in Fig. 7a are smeared out and shifted to the left. The higher the displacements, the lower the frequencies corresponding with the second to fourth term of Eq. (4), whereas the frequencies corresponding with the moire! fringes become higher. As a result, when a certain deformation is reached, the frequencies corresponding to the moire! fringes interfere with the higher frequency components of Eq. (4). Filtering out the higher frequency terms, to obtain the desired moire! pattern, becomes more and more precarious, and finally impossible. On the other hand, if the pitch of the reference grating ‘follows’ the deformation, useful results are obtained. Fig. 8 shows the normalised spatial spectrum corresponding with the specimen grating of Fig. 6a multiplied by a reference grating
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1.2 before loading
normalised signal
1 0.8 0.6 0.4 0.2 0 0
5
(a)
10 X (mm)
15
20
1 Normalised spectrum (-)
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 (b)
1000
2000 3000 4000 Spatial frequency (1/m)
5000
6000
Fig. 5. (a) Brightness of the specimen grating, along the full length of the specimen, recorded before the actual deformation of the specimen. (b) Normalised spectrum of the brightness of Fig. 5a.
having a pitch equal to the mean pitch of the deformed specimen grating. No overlap between the moire! term and the other terms occurs. Fig. 9a and b represent the normalised spectra obtained by filtering out the higher frequency components of Figs. 7a and 8, respectively. Eq. (7) assumes that the pitch of the reference grating is equal to the pitch of the undeformed specimen grating. In that case, the obtained deformations can be interpreted as absolute deformations. This is not the case, if at time t a reference grating with pitch pðtÞ; different from the pitch of the undeformed grating p(t ¼ 0Þ; is used. Then, the deformations obtained using Eq. (7) are the specimen deformations relative to the reference grating, noted as Urel ðX ; tÞ: The absolute deformations can be derived as follows. Once the relative displacements Urel ðX ; tÞ are calculated (see Fig. 10), the original position can be
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1.2 just before rupture normalised signal
1 0.8 0.6 0.4 0.2 0 0
5
10
15
20
25
X (mm)
(a) 1 Normalised spectrum (-)
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 (b)
1000
2000 3000 4000 Spatial frequency (1/m)
5000
6000
Fig. 6. (a) Brightness of the specimen grating, along the full length of the specimen, recorded just before rupture of the specimen. (b) Normalised spectrum of the brightness of Fig. 6a.
obtained by the following equation pðt ¼ 0Þ : xðX ; tÞ ¼ ðX Urel ðX ; tÞÞ pðtÞ The absolute displacements UðX ; tÞ are given by pðt ¼ 0Þ pðt ¼ 0Þ UðX ; tÞ ¼ X xðX ; tÞ ¼ X 1 : þ Urel ðX ; tÞ pðtÞ pðtÞ
ð9Þ
ð10Þ
The displacements UðX ; tÞ are expressed using the coordinates X of the deformed grating. However, it is more common to recalculate the displacements to the coordinate x of the undeformed specimen. The displacement Uðx; tÞ corresponds with the displacement of a specific point, originally located at a distance x from the beginning of the specimen, at time t:
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0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
1000
2000
3000
4000
5000
6000
5000
6000
Spatial frequency (1/m)
(a)
Normalised spectrum (-)
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
(b)
1000
2000 3000 4000 Spatial frequency (1/m)
Fig. 7. Normalised spectrum of the superposition of the original recorded grating with a reference grating with pitch p ¼ 0:5 mm at t ¼ 0 ms (a) and just before rupture (b).
If the algorithm is applied to a part of the specimen grating recorded, before the actual start of the loading, small displacements are noticed. They are due to imperfections of the specimen grating, an eventual misfit between the reference grating and specimen grating, or temperature effects. Although most often these effects have a negligible influence on the final displacements, a correction is implemented. The displacements of the undeformed grating Uðx; to0Þ are calculated, and subsequently subtracted from the displacements Uðx; tÞ: Ucor ðx; tÞ ¼ Uðx; tÞ Uðx; to0Þ:
ð11Þ
4.3.2. Practical implementation As illustrated in Section 4.3.1, if a reference grating with a pitch varying in time is used, the algorithm can also be used for materials showing large changes in
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Normalised spectrum (-)
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
1000
2000 3000 4000 Spatial frequency (1/m)
5000
6000
Fig. 8. Normalised spectrum of the superposition of the original recorded grating with a reference grating with a pitch adapted to the deformation before rupture.
1 Normalised spectrum (-)
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
1000
2000 3000 4000 Spatial frequency (1/m)
5000
6000
0
1000
2000 3000 4000 Spatial frequency (1/m)
5000
6000
(a) 1 Normalised spectrum (-)
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 (b)
Fig. 9. Normalised spectrum of the moir!e pattern at t ¼ 0 ms and just before rupture.
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Fig. 10. Schematic representation of the brightness of the reference grating at t ¼ 0 ms (start of loading), of the reference grating at time t and of the deformed grating at time t:
deformation amplitude during the experiment. At t ¼ 0; a pitch of the reference grating equal to the pitch of the undeformed specimen is used. The moire! patterns I1 ; I2 ; I3 and I4 ; and the corresponding phases are calculated using Eq. (8). Since, the arctangent function is confined to the range of p to p, even when the actual phase is out of this range, the occurring phase jumps need to be eliminated. To connect the phase values smoothly, a phase unwrapping procedure is used. After the unwrapping, the displacements relative to the reference grating are calculated using Eq. (7). Then, the absolute displacements are calculated using Eqs. (9) and (10). From the obtained displacements the mean value of the specimen pitch is derived, and this value is used as pitch of the reference grating for the next time step. If the reference pitch pðtÞ for the next time step is taken equal to the mean pitch of the previous time step, at each moment, the displacements relative to the reference grating are small. This guarantees that filtering out of the higher frequency components is not critical. Once the Uðx; tÞ-values are calculated, a correction is implemented using Eq. (12). 4.4. Fulfilment of the continuity requirement An important advantage of using a phase-shifting technique is that the information contained in each point of the recorded specimen grating can be used. Depending on the spatial resolution used to digitise the recorded (Fig. 3) deforming grating, more or less continuous information on the displacements, both in time and space, is obtained. The brightness of the recorded grating is digitised using 256 greyscale levels. The resolution, of the scanning device is chosen such that 28 (in the beginning of the experiment) to 40 (towards the end) pixels per pitch length are obtained, corresponding with, on average, respectively 56 to 80 pixels/mm on the
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specimen. In time, 800 horizontal lines, i.e. time steps of 0.8 ms/800=1 ms are considered. Scanning resolutions up to 6 times higher could be used, because the grain size of the photographic film is very small. The scanning resolution used here was chosen on the basis of considerations related with the processing time of the algorithm on one hand, and continuity of the obtained displacements on the other. As will be shown in Section 5, with the scanning resolution used here, quasicontinuous displacement data are obtained. 5. Results 5.1. Specimen displacements The specimen displacements corresponding with Fig. 3 are presented in Fig. 11. The brightness of the used greyscale is proportional to the amplitude of the displacement. Some irregularities, due to an inferior quality of the recorded grating in that particular zone, are noticed (right corner below). Each horizontal line gives the displacements along the axis of the specimen at a certain moment in time. As expected, fracture of the specimen results in a discontinuity of the displacement field. Fig. 12 shows the displacements along the axis of the specimen at different time intervals, using time steps of 100 ms. Each vertical line in Fig. 11 gives the displacement-time history of a specific point of the specimen. The time histories of the displacement in 7 points of the specimen are illustrated in Fig. 13. The following points are selected: the middle of the specimen (point 1), 2 points located at a distance of 1.5 mm on both side of the middle (point 2 and 3), 2 points at a distance of 3 mm (point 4 and 5) and 2 points at a distance of 5.5 mm (point 6 and 7). Points 4 and 5 delimit the central zone of the specimen, the zone with a constant cross-section. Points 6 and 7 are located in the middle of the transition zones. The slope of lines tangent to the curves depicted in Fig. 13 yields the velocity of the considered point. In this experiment, it was noticed that the particle velocities differ from point to point. The velocity of each single point, however, is relatively constant during the experiment. Values varying from 1.2 to 9 m/s, respectively, correspond to the point far left (Fig. 3) and the point far right of the specimen. A line of picture 3 recorded just before the actual loading of the specimen is used to estimate the resolution of the measurement technique. Fig. 14 represents the calculated displacements, which should be zero. However, displacements with amplitudes less than 74 mm are found, except towards the right end, where, as mentioned earlier the quality of the recorded grating is of lesser quality. 5.2. Specimen deformations The derivative of the displacement curves represented in Fig. 12 yields the local deformation. The strain is given by: qU ðx; tÞ eðx; tÞ ¼ : ð12Þ qx
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Fig. 11. Greyscale picture of the time evolution of the specimen displacements calculated from the recorded deforming grating represented in Fig. 3.
To minimise the influence of local perturbations in the displacement signal, the strain value in point x is calculated as the slope of the x-displacement curves in the interval [x0.25 mm; x+0.25 mm]. Fig. 15 shows the strain along the length of the specimen, calculated from the displacement profiles represented in Fig. 12. From this picture, it can be concluded that the strain is not at all constant along the length of the specimen. As expected from the specimen geometry, the smallest strain values are obtained at the ends of the specimen. In the transition zones, the strain gradually increases towards the central section of the specimen. However, the strain is clearly not constant in the zone where the cross-section is constant (x in [5 mm, 11 mm]). This is in contrast with what is generally assumed. Furthermore, the homogeneity clearly decreases
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7
Displacement (mm)
6 5 4 3 2 1 0 0
2
4
6
8 10 x (mm)
12
14
16
Fig. 12. Displacement profiles along the axis of the specimen at different moment in time, using timesteps of 100 ms.
Point 7 Point 5
Displacement (mm)
6 5
Point 3
4 Point 1
3
Point 2 2
Point 4 Point 6
1 0 0
0.1
0.2
0.3 0.4 0.5 Time (ms)
0.6
0.7
0.8
Fig. 13. Time history of the displacement of 7 points of the specimen: the middle (point 1), 2 points located at a distance of 1.5 mm on both side of the middle of the specimen (point 2 and 3), 2 points at a distance of 3 mm (point 4 and 5) and 2 points at a distance of 5.5 mm (point 6 and 7).
in time. If time progresses, the strain localises in a small zone of the specimen. Just before fracture, peak values of more than 170% are reached. Before strain localisation, the strain profiles are approximately symmetrical around the middle of the specimen. In Fig. 16 the time evolution of the strain in some points of the specimen is presented. The points are the same as those given in Fig. 13, however, because of the symmetry of the strain field, only 4 points (1, 2, 4 and 6) are considered. The slope of the tangent lines yields the strain rate. Notice that the strain rate is not at all uniform in the specimen.
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6
8
10
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4
6
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Fig. 15. Strain profiles along the axis of the specimen at different moment in time, using timesteps of 100 ms.
6. Conclusions In this contribution, a combined experimental-numerical technique is presented to measure the history of the axial displacements and deformations along the length of a specimen subjected to a high strain rate uni-axial load. The technique can be applied to a wide range of materials and high strain rate experiments. The results of the technique applied to a split Hopkinson tensile bar experiment are presented. They prove that highly accurate and valuable information is obtained. The comprehensive, quantitative information on the specimen deformation cannot be obtained by the classical measurement techniques, but it is essential for all studies
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Point 2 Point 4 Point 6 0
0.1
0.2
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Fig. 16. Time history of the strain in 4 points of the specimen: the middle (point 1), a point located at a distance of 1.5 mm of the middle of the specimen (point 2), a point at a distance of 3 mm (point 4) and a point at a distance of 5.5 mm (point 6).
aiming at an in-depth understanding of high strain rate experiments, or of the high strain rate behaviour of materials.
References [1] Hopkinson B. A method of measuring the pressure produced in the detonation of high explosives or by the impact of bullets. Philos Trans R Soc London Series A. 1914;213:437–52. [2] Manjoine M, Nadai A. High-speed tension test at elevated temperatures—Parts II and III. J Appl Mech 1942;A:77–91. [3] Kreis T. Quantitative evaluation of the interference phase. Holographic interferometry: principles and methods. Berlin; Akademic Verlag; 1996. p. 101–70 [chapter 4]. [4] Degrieck J, Van Paepegem W. Application of digital phase-shift shadow moir!e to micro deformation measurements of curved surfaces. Opt Laser Eng 2001;36:29–40. [5] Kolsky H. An investigation of the mechanical properties of materials at very high rates of loading. Proc Phys Soc London Sec. B 1949;62:676–700. [6] Verleysen P, Degrieck J, Taerwe L. Experimental investigation of the impact behaviour of cementious composites. Mag Concr Res 2002;54(4):257–62. [7] Verleysen P, Degrieck J. Improved signal processing for split Hopkinson bar tests on (quasi-) brittle materials. Exp Tech 2000;24(6):31–3. [8] Kobayashi A, editor. Handbook on experimental mechanics. SEM, 2nd revised edition, 1993, p. 997–1007. [9] Kujawinska M, Osten W. Fringe pattern analysis methods: up-to-date review. Proc SPIE 1998;3407:56–66.
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A modified digital phase-shift moire! technique for impact deformation measurements Patricia Verleysen*, Joris Degrieck Laboratory Soete, Department of Mechanical Construction and Production, Gent University, Sint-Pietersnieuwstraat 41, 9000 Gent, Belgium Received 1 September 2003; received in revised form 9 May 2004; accepted 21 May 2004 Available online 28 July 2004
Abstract Due to the short loading times and high deformation rates inherent to impact experiments, measurement of the occurring deformations is not straightforward. Presented in this paper is a technique to obtain the displacements and deformations of a specimen subjected to a uni-axial impact load. During the experiment the deformation of a line grating attached to the specimen is captured using a streak camera. From the recorded deforming grating the specimen displacements are automatically derived using an advanced numerical algorithm, based on the interference between the specimen grating and a virtual reference grating. Numerical interference is considered because it allows that the pitch of the reference grating is adapted to the changing amplitude of the deformation. Indeed, at each moment of the deformation process the pitch of the reference grating is chosen such that the highest possible accuracy and sensitivity is guaranteed. Because of this, large changes in deformation amplitude are allowed, and the technique is applicable to a wide range of materials. Eventual imperfections of the specimen grating and temperature effects are taken into account. Specimen displacements are extracted automatically by means of a phase-shifting technique. The non-contact measurement technique yields high resolution, quantitative information on the specimen deformation, along the entire length of the specimen and during the full duration of the experiment. Interaction by the operator is excluded. Results are presented of a high strain rate tensile test on a steel sheet specimen showing local deformations up to about 170%. r 2004 Elsevier Ltd. All rights reserved. Keywords: Digital phase-shift; Displacement measurement; High strain rate
*Corresponding author. Tel.: +32-9-264-34-35; fax: +32-9-264-35-87. E-mail address: [email protected] (P. Verleysen). 0143-8166/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2004.05.001
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1. Introduction Several test techniques exist during which a material sample is subjected to a uniaxial, impact load: Hopkinson (tensile or compressive) bar tests [1], falling weight tests and fly wheel experiments [2]. Due to the short duration of these experiments, typically of the order of one to a few milliseconds, as well as the short rise time of the load and the corresponding high deformation rates, measurement of the evolution of the specimen deformation is not straightforward. In addition, wave propagation phenomena often accompany impact experiments. Interaction of the waves with the measurement devices must be avoided both to generate reliable measurements and to avoid disturbing the experiment. Non-contact measurement techniques are therefore preferable. Most existing techniques only give limited information: the time history of the displacement of one or a few points, or the wholefield specimen deformation at certain moments in time. Additional difficulties arise from the fact that during an experiment the local deformation can vary over several orders of magnitude. When testing ductile materials, local strains of more than 100% are not exceptional. To monitor damage mechanisms and strain localisation effects in the specimen as well as the influence of the specimen geometry, high-resolution information on the evolution of the deformation along the length of the specimen is desirable. A non-contact technique is developed which gives quantitative information on the time evolution of the axial displacement field in a specimen subjected to an impact load. A line grating of 2 lines/mm is applied to the specimen. During the experiment the deformation of the grating is recorded by means of a rotating drum streak camera. Since a coarse grating is used, the technique used to extract the displacements from the recorded deforming grating must be carefully selected. For example, fringe centre positioning techniques give insufficient information, and are thus excluded. A more advanced numerical technique is developed to extract the displacements from the recorded picture. The phase of the digital interferogram, which arises from the interference between the digitised recorded specimen grid and a well-chosen virtual reference grid is related to the specimen displacement. The phase is determined by a phase-shifting technique [3,4]; by shifting the reference grating over well-determined distances, four interferograms are created, and these are subsequently used to calculate the phase and hence the displacements. The digital implementation of the geometrical interference between the reference and deformed grating, and the phase-shifting technique implies numerous practical advantages. More important however, is that it also allows the use of a reference grating with a variable pitch. A procedure was established to adapt the pitch of the reference grating automatically to the changing specimen deformation. As a result, the highest possible accuracy and sensitivity are obtained at each moment. Limitations of the amplitude of the specimen deformation are also overcome. The algorithm takes into account eventual imperfections of the original specimen grating. Results of a high strain rate (750/s) tensile experiment on a ductile steel sheet are presented. As will be shown, the combination of the non-contact measurement setup with the numerical processing technique yields highly accurate, quasi-continuous
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(both in time and space) information on the specimen displacements. Data on the value and accuracy of the measurement technique are given.
2. Deformation measurement set-up The set-up for the displacement measurement is schematically represented in Fig. 1. A line grid is attached to the surface of the specimen perpendicular to the loading direction. The image of the line grid is projected on a photographic film that is laid inside the drum of a rotating drum camera, in this case used as a streak camera. Prior to the actual experiment, the drum of the camera is brought to a welldetermined rotational speed. As the specimen is subjected to a high strain rate uniaxial load, a flash lightens the specimen and the image of the deforming grid is recorded on the moving film. The time evolution of the specimen deformation is thus recorded on the film. Although the measurement principle is rather simple, practical implementation, is more cumbersome. Finding an optimal combination of test parameters, such as the density of the line grating, the rotational speed of the camera drum, the spatial and light sensitivity of the photographic film, and the image magnification is not straightforward. Indeed, a compromise must be found between the opposing demands on the actual recording of the deforming specimen grid and those posed by the numerical processing technique. The first mainly concern the light exposure of the photographic film, the latter are related to the inherent accuracy of the technique. A high density of the line grating, a high velocity of the drum, a large magnification of the image, and a photographic film with a high spatial sensitivity all have a positive influence on the accuracy of the measurement technique, but have an adverse effect on the light exposure of the film. The velocity of the specimen particles also has an important influence on the choice of the parameters. To avoid motion blurring the rotational speed of the camera must be high enough and the width of the slit of the camera (see Fig. 1) must be sufficiently small.
Fig. 1. Schematical representation of the setup for the displacement measurement.
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The optimal set of parameters was established experimentally. A grating density of 2 lines/mm, combined with Kodak Tri-X pan 220 film, no image amplification and a rotational speed of 100 rps proved to give good results. The selected rotational speed yields a maximum recording time of 10 ms on the film; since the duration of an impact experiment lies typically in the interval [0.2 ms; 2 ms] this is largely sufficient. Furthermore, the use of a rather coarse grating has numerous practical advantages. The production of a highly accurate, coarse grating is straightforward, no sophisticated techniques are required to attach the grating onto the specimen, a strong but common light source can be used to illuminate the specimen, and the image of the deforming grating can be captured by commercially available photographic film. Since the technique should be able to manage large specimen deformations, the grid and the way it is fixed on the specimen should be able to follow the specimen deformation. A gelatine grid attached to the specimen surface using photographic stripping film, on which a high quality reprint from a master grating was made, proved to fulfil the requirements. The gelatine has a sufficiently large deformability and is firmly attached to the specimen surface. The grating can be applied to different types of materials, as long as their surface is smooth or can be made smooth.
3. Experimental results 3.1. Impact experiment The way the high strain rate deformation is applied to the specimen is of no importance for the deformation measurement technique; as long as the specimen surface is accessible, the technique can be used to measure axial displacements during high strain rate experiments using various methods such as a high speed hydraulic testing machine, a falling weight device or a fly wheel. Here application of the deformation measurement technique is illustrated by means of a split Hopkinson tensile bar experiment [5,6]. A material sample is fixed between two long bars, the input bar and the output bar. A tensile wave with a certain length is generated at the free end of the input bar. This wave propagates towards the specimen and applies a high strain rate load to the specimen. From strain gage measurements on the bar, the history of the strain, the strain rate and the stress in the specimen can be determined. However, at each moment only one value for the strain, the strain rate, and the stress are obtained [7]. These values are mean values along the length of the specimen. The measurement technique described in this paper was used to monitor the specimen displacement and deformation distribution during the experiment. For the experiment described here a stainless steel sheet, type 301LN, is tested because of its deformation capacity. Its geometry is presented in Fig. 2. Since the specimen is glued between the input bar and the output bar, glue zones are provided on both sides of the test section. The actual section that will be subjected to the high
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Fig. 2. Specimen geometry used for the impact experiment. Here, a radius of 5 mm, a length of 6 mm and a width of 5 mm are considered.
strain rate tensile load can be divided into three parts: a 6 mm long, 5 mm wide central section, and two transition zones in which the width spans from 5 to 15 mm. The specimen thickness is 1.2 mm. 3.2. Recorded picture Fig. 3 represents the recorded picture. Each horizontal line represents the image of the line grid at one moment during the experiment. Before the beginning of the loading, the lines do not change in width. Once the specimen is subjected to the tensile load, the lines are stretched. The lower in the picture, the wider become the lines. At a certain moment a crack appears near the middle of the specimen. Afterwards, one part of the specimen moves away from the other part. The duration of the experiment, from the start of the loading to the nucleation of the crack, is about 0.75 ms. At the moment of cracking, the mean strain in the specimen was a bit higher than 50%, but as will be shown, local strain peaks of 150% occur.
4. Treatment of the recorded image—digital phase-shift technique 4.1. Introduction A procedure was derived to calculate the specimen displacements from the deforming grid of Fig. 3 automatically. The following demands are posed to the processing technique: *
(Quasi-) continuous information, both in time and space, should be obtained. During an impact experiment the distribution of the deformation along the length
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Fig. 3. Deforming specimen grid recorded by means of a rotating drum camera used as streak camera.
*
of a specimen changes extremely fast; mostly the transition from more or less homogeneous deformations to severe strain localisation takes less than a few microseconds. Also the zone in which the strain localises is very small for most materials. High spatial and time resolution is thus required. Since a rather coarse grating is used, the spatial resolution of fringe centre positioning techniques is not sufficient, The technique should be valid for different amplitudes of deformation. It should be applicable for a wide range of materials, and also from the start towards the end of the experiment, so small to eventually high deformations should be allowed.
As will be shown, straightforward use of geometric moire! is not possible. Nonetheless, the geometric interference between two sets of lines, the recorded deforming grid and a reference grating, is used to calculate the displacement field. In Section 4.2, a relation is established between the displacement and the phase of the interference pattern. The value of the phase is determined by a digital phase-shifting technique: four virtual interference patterns are formed; each time the reference grating is shifted over a known distance. To account for the changing amplitude of the specimen deformation, the pitch of the reference grating is adapted in such a way that it follows the changing specimen deformation. In Sections 4.3 and 4.4 details are given on the implementation of the technique.
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4.2. Principle An equation valid for the superposition of two parallel gratings, a reference grating and a deformed grating, will be established. The equation will be used to calculate the in-plane displacements, perpendicular to the lines. The reference grating used is cosinusiodal with a pitch p: The brightness at position X of this grating is then given by the equation 2pX Iref ¼ A0 þ A1 cos : ð1Þ p A0 and A1 are, respectively, the mean value and amplitude of the brightness, and both are constant. Fig. 4 represents the normalised brightness of a part of the recorded grating before the start of the actual deformation. This corresponds with a part of the horizontal line at the top of Fig. 3. As can be seen in Fig. 4, the signal can be represented, with a very good approximation, by a cosine function. Consequently, if the specimen grating deforms axially and the point with coordinate x moves to point X after a displacement U; perpendicular to the lines, the brightness of the deformed grating is given by 2pðX U Þ Ispec ¼ B0 þ B1 cos : ð2Þ p B0 and B1 are, respectively, the mean value and amplitude of the brightness. However, due to uneven illumination and optical aberrations, B0 and B1 vary from point to point. The displacement U can be a real displacement due to loads, but it can also take into account the initial imperfections of the specimen grating. When the deformed grating given by Eq. (2) is superposed with the reference grating given by Eq. (1), the following brightness distribution is obtained: IðX Þ ¼ Ispec Iref
ð3Þ
1.2
before loading cosine
normalised signal
1 0.8 0.6 0.4 0.2 0 7
7.5
8 X (mm)
8.5
9
Fig. 4. Detail of the brightness of the specimen grating recorded before the actual deformation of the specimen and a cosine fitted to the signal.
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or:
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A1 B1 2pU cos IðX Þ ¼ p 2 A1 B1 2pð2X UÞ 2pðX UÞ 2pX þ A1 B0 cos þ cos þ A0 B1 cos p p p 2 þ A0 B0 :
ð4Þ
When small deformations are considered, the frequencies, with respect to X ; of the first cosine-term of the right- hand term of Eq. (4) are substantially lower than those of the other three cosine-terms. This term represents the actual moire! fringes. The three higher frequency cosine-terms in Eq. (4) can be filtered away using a low-pass filter function. The brightness distribution thus obtained can be written as IðX Þ ¼ I0 ðX Þ þ I1 ðX Þ cos fðX Þ;
ð5Þ
where I0 is the average brightness, I1 the amplitude of the brightness, and f the phase, given by fðX Þ ¼
2pUðX Þ : p
ð6Þ
Eqs. (5) and (6) are in agreement with those given in [8], valid for in-plane geometric moire! . In classical moire! applications, the higher frequency components of Eq. (4) are most often filtered away by the optics used to record the moire! pattern. Therefore, from a moire! fringe pattern, formed by interference of a reference grid and a deformed grid, the displacement U at each point of the image can be obtained by measuring the phase f of the cosinusoidal wave at that point p UðX Þ ¼ fðX Þ : ð7Þ 2p Since, I0 and I1 in Eq. (5) have no constant, known value, calculation of the phases is not straightforward. In such cases, the phases can be obtained with a high accuracy and sensitivity using a phase-shifting technique [9]. The phases are determined by recording (or calculating) the moire! fringe patterns three times or more by shifting the phase of the reference grid over known amounts, namely 2p divided by the number of phase-shifts. So, if four shifts are considered, the reference grid is superposed on the deformed line grid and this results in an image with brightness Ia¼0 : Then the reference grid is shifted over p/2, superposed on the deformed grid, this results in an image with brightness Ia¼p=2 : Subsequently, the reference grid is shifted over another p/2, so Ia¼p is obtained. From the obtained values of the brightness Ia¼0 ; Ia¼p=2 ; Ia¼p and Ia¼3p=2 ; the phases can be calculated by the Eq. [3] Ia¼3p=2 Ia¼p=2 f ¼ arctan : ð8Þ Ia¼0 Ia¼p As can be seen, the phase-shift method has the advantage that the phase in each point of the image is obtained independently from the phases in other points, and
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thus no interpolation between the fringes is required. Thus, a much higher resolution is obtained. 4.3. Deformation compatible reference grating 4.3.1. Principle Using the principle described in Section 4.2, four interference patterns are needed to calculate the displacements. The interferograms can be obtained by multiplying the brightness values of the original recorded grating by a reference grating (see Eq. (4)). Subsequently, the higher frequency components of the patterns have to be filtered out to obtain moire! patterns corresponding with Eq. (5). A correct transition of Eq. (4) to (5) can only be guaranteed if the frequencies corresponding with the first cosine-term and the frequencies of the other cosine-terms do not overlap. Fig. 5a represents the brightness along the axis of the specimen recorded before the actual loading of the specimen (see also Fig. 4). Fig. 5b represents the corresponding normalised spatial spectrum. As expected, a sharp peak appears at a spatial frequency of 2000/m, corresponding with a wavelength of 0.5 mm, which equals the spatial period of the specimen grating. A peak shift to the left is noticed when the specimen is stretched homogeneously. Due to non-homogeneous deformation, the frequency spectrum broadens. Fig. 6a represents the recorded brightness along the axis of the specimen just before the crack appears (at tE0.75 ms). The corresponding normalised spatial spectrum is shown in Fig. 6b. The sharp peak disappeared and spread over a wider zone of spatial frequencies. Compared to the peak in Fig. 5b a peak shift to the left is indeed observed. The signal even contains frequencies more or less around HALF the frequency of the original grating (1000/m). This is in agreement with Eq. (2) for the deformed grid. The normalised spatial spectra, obtained by multiplying the brightness values of Figs. 5a and 6a by a reference grating, are shown in Fig 7a and b, respectively. The pitch of the reference grating is equal to the pitch of the undeformed specimen grating. In Fig. 7a, two distinct peaks can be distinguished: one at 2000/s and the other at 4000/m, corresponding to the third and fourth terms, and the second righthand term of expression (4), respectively. The low frequencies of the actual moire! pattern (first term) are separated from the higher frequency components. As illustrated in Fig. 7b, the above is not valid when the specimen deformations become larger. In this case the peaks shown in Fig. 7a are smeared out and shifted to the left. The higher the displacements, the lower the frequencies corresponding with the second to fourth term of Eq. (4), whereas the frequencies corresponding with the moire! fringes become higher. As a result, when a certain deformation is reached, the frequencies corresponding to the moire! fringes interfere with the higher frequency components of Eq. (4). Filtering out the higher frequency terms, to obtain the desired moire! pattern, becomes more and more precarious, and finally impossible. On the other hand, if the pitch of the reference grating ‘follows’ the deformation, useful results are obtained. Fig. 8 shows the normalised spatial spectrum corresponding with the specimen grating of Fig. 6a multiplied by a reference grating
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1.2 before loading
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Fig. 5. (a) Brightness of the specimen grating, along the full length of the specimen, recorded before the actual deformation of the specimen. (b) Normalised spectrum of the brightness of Fig. 5a.
having a pitch equal to the mean pitch of the deformed specimen grating. No overlap between the moire! term and the other terms occurs. Fig. 9a and b represent the normalised spectra obtained by filtering out the higher frequency components of Figs. 7a and 8, respectively. Eq. (7) assumes that the pitch of the reference grating is equal to the pitch of the undeformed specimen grating. In that case, the obtained deformations can be interpreted as absolute deformations. This is not the case, if at time t a reference grating with pitch pðtÞ; different from the pitch of the undeformed grating p(t ¼ 0Þ; is used. Then, the deformations obtained using Eq. (7) are the specimen deformations relative to the reference grating, noted as Urel ðX ; tÞ: The absolute deformations can be derived as follows. Once the relative displacements Urel ðX ; tÞ are calculated (see Fig. 10), the original position can be
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1.2 just before rupture normalised signal
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25
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6000
Fig. 6. (a) Brightness of the specimen grating, along the full length of the specimen, recorded just before rupture of the specimen. (b) Normalised spectrum of the brightness of Fig. 6a.
obtained by the following equation pðt ¼ 0Þ : xðX ; tÞ ¼ ðX Urel ðX ; tÞÞ pðtÞ The absolute displacements UðX ; tÞ are given by pðt ¼ 0Þ pðt ¼ 0Þ UðX ; tÞ ¼ X xðX ; tÞ ¼ X 1 : þ Urel ðX ; tÞ pðtÞ pðtÞ
ð9Þ
ð10Þ
The displacements UðX ; tÞ are expressed using the coordinates X of the deformed grating. However, it is more common to recalculate the displacements to the coordinate x of the undeformed specimen. The displacement Uðx; tÞ corresponds with the displacement of a specific point, originally located at a distance x from the beginning of the specimen, at time t:
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2000
3000
4000
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5000
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Normalised spectrum (-)
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(b)
1000
2000 3000 4000 Spatial frequency (1/m)
Fig. 7. Normalised spectrum of the superposition of the original recorded grating with a reference grating with pitch p ¼ 0:5 mm at t ¼ 0 ms (a) and just before rupture (b).
If the algorithm is applied to a part of the specimen grating recorded, before the actual start of the loading, small displacements are noticed. They are due to imperfections of the specimen grating, an eventual misfit between the reference grating and specimen grating, or temperature effects. Although most often these effects have a negligible influence on the final displacements, a correction is implemented. The displacements of the undeformed grating Uðx; to0Þ are calculated, and subsequently subtracted from the displacements Uðx; tÞ: Ucor ðx; tÞ ¼ Uðx; tÞ Uðx; to0Þ:
ð11Þ
4.3.2. Practical implementation As illustrated in Section 4.3.1, if a reference grating with a pitch varying in time is used, the algorithm can also be used for materials showing large changes in
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Normalised spectrum (-)
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
1000
2000 3000 4000 Spatial frequency (1/m)
5000
6000
Fig. 8. Normalised spectrum of the superposition of the original recorded grating with a reference grating with a pitch adapted to the deformation before rupture.
1 Normalised spectrum (-)
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1000
2000 3000 4000 Spatial frequency (1/m)
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6000
0
1000
2000 3000 4000 Spatial frequency (1/m)
5000
6000
(a) 1 Normalised spectrum (-)
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 (b)
Fig. 9. Normalised spectrum of the moir!e pattern at t ¼ 0 ms and just before rupture.
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Fig. 10. Schematic representation of the brightness of the reference grating at t ¼ 0 ms (start of loading), of the reference grating at time t and of the deformed grating at time t:
deformation amplitude during the experiment. At t ¼ 0; a pitch of the reference grating equal to the pitch of the undeformed specimen is used. The moire! patterns I1 ; I2 ; I3 and I4 ; and the corresponding phases are calculated using Eq. (8). Since, the arctangent function is confined to the range of p to p, even when the actual phase is out of this range, the occurring phase jumps need to be eliminated. To connect the phase values smoothly, a phase unwrapping procedure is used. After the unwrapping, the displacements relative to the reference grating are calculated using Eq. (7). Then, the absolute displacements are calculated using Eqs. (9) and (10). From the obtained displacements the mean value of the specimen pitch is derived, and this value is used as pitch of the reference grating for the next time step. If the reference pitch pðtÞ for the next time step is taken equal to the mean pitch of the previous time step, at each moment, the displacements relative to the reference grating are small. This guarantees that filtering out of the higher frequency components is not critical. Once the Uðx; tÞ-values are calculated, a correction is implemented using Eq. (12). 4.4. Fulfilment of the continuity requirement An important advantage of using a phase-shifting technique is that the information contained in each point of the recorded specimen grating can be used. Depending on the spatial resolution used to digitise the recorded (Fig. 3) deforming grating, more or less continuous information on the displacements, both in time and space, is obtained. The brightness of the recorded grating is digitised using 256 greyscale levels. The resolution, of the scanning device is chosen such that 28 (in the beginning of the experiment) to 40 (towards the end) pixels per pitch length are obtained, corresponding with, on average, respectively 56 to 80 pixels/mm on the
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specimen. In time, 800 horizontal lines, i.e. time steps of 0.8 ms/800=1 ms are considered. Scanning resolutions up to 6 times higher could be used, because the grain size of the photographic film is very small. The scanning resolution used here was chosen on the basis of considerations related with the processing time of the algorithm on one hand, and continuity of the obtained displacements on the other. As will be shown in Section 5, with the scanning resolution used here, quasicontinuous displacement data are obtained. 5. Results 5.1. Specimen displacements The specimen displacements corresponding with Fig. 3 are presented in Fig. 11. The brightness of the used greyscale is proportional to the amplitude of the displacement. Some irregularities, due to an inferior quality of the recorded grating in that particular zone, are noticed (right corner below). Each horizontal line gives the displacements along the axis of the specimen at a certain moment in time. As expected, fracture of the specimen results in a discontinuity of the displacement field. Fig. 12 shows the displacements along the axis of the specimen at different time intervals, using time steps of 100 ms. Each vertical line in Fig. 11 gives the displacement-time history of a specific point of the specimen. The time histories of the displacement in 7 points of the specimen are illustrated in Fig. 13. The following points are selected: the middle of the specimen (point 1), 2 points located at a distance of 1.5 mm on both side of the middle (point 2 and 3), 2 points at a distance of 3 mm (point 4 and 5) and 2 points at a distance of 5.5 mm (point 6 and 7). Points 4 and 5 delimit the central zone of the specimen, the zone with a constant cross-section. Points 6 and 7 are located in the middle of the transition zones. The slope of lines tangent to the curves depicted in Fig. 13 yields the velocity of the considered point. In this experiment, it was noticed that the particle velocities differ from point to point. The velocity of each single point, however, is relatively constant during the experiment. Values varying from 1.2 to 9 m/s, respectively, correspond to the point far left (Fig. 3) and the point far right of the specimen. A line of picture 3 recorded just before the actual loading of the specimen is used to estimate the resolution of the measurement technique. Fig. 14 represents the calculated displacements, which should be zero. However, displacements with amplitudes less than 74 mm are found, except towards the right end, where, as mentioned earlier the quality of the recorded grating is of lesser quality. 5.2. Specimen deformations The derivative of the displacement curves represented in Fig. 12 yields the local deformation. The strain is given by: qU ðx; tÞ eðx; tÞ ¼ : ð12Þ qx
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Fig. 11. Greyscale picture of the time evolution of the specimen displacements calculated from the recorded deforming grating represented in Fig. 3.
To minimise the influence of local perturbations in the displacement signal, the strain value in point x is calculated as the slope of the x-displacement curves in the interval [x0.25 mm; x+0.25 mm]. Fig. 15 shows the strain along the length of the specimen, calculated from the displacement profiles represented in Fig. 12. From this picture, it can be concluded that the strain is not at all constant along the length of the specimen. As expected from the specimen geometry, the smallest strain values are obtained at the ends of the specimen. In the transition zones, the strain gradually increases towards the central section of the specimen. However, the strain is clearly not constant in the zone where the cross-section is constant (x in [5 mm, 11 mm]). This is in contrast with what is generally assumed. Furthermore, the homogeneity clearly decreases
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7
Displacement (mm)
6 5 4 3 2 1 0 0
2
4
6
8 10 x (mm)
12
14
16
Fig. 12. Displacement profiles along the axis of the specimen at different moment in time, using timesteps of 100 ms.
Point 7 Point 5
Displacement (mm)
6 5
Point 3
4 Point 1
3
Point 2 2
Point 4 Point 6
1 0 0
0.1
0.2
0.3 0.4 0.5 Time (ms)
0.6
0.7
0.8
Fig. 13. Time history of the displacement of 7 points of the specimen: the middle (point 1), 2 points located at a distance of 1.5 mm on both side of the middle of the specimen (point 2 and 3), 2 points at a distance of 3 mm (point 4 and 5) and 2 points at a distance of 5.5 mm (point 6 and 7).
in time. If time progresses, the strain localises in a small zone of the specimen. Just before fracture, peak values of more than 170% are reached. Before strain localisation, the strain profiles are approximately symmetrical around the middle of the specimen. In Fig. 16 the time evolution of the strain in some points of the specimen is presented. The points are the same as those given in Fig. 13, however, because of the symmetry of the strain field, only 4 points (1, 2, 4 and 6) are considered. The slope of the tangent lines yields the strain rate. Notice that the strain rate is not at all uniform in the specimen.
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Displacement (mm)
0.004 0.002 0 -0.002
0
2
4
6
8
10
12
14
16
-0.004 -0.006 -0.008 -0.01 -0.012
x (mm)
Fig. 14. Displacements calculated at a time to0 give an indication on the resolution of the measurement technique.
2 1.8 1.6 Strain (-)
1.4 1.2 1 0.8 0.6 0.4 0.2 0 0
2
4
6
8 10 x (mm)
12
14
16
Fig. 15. Strain profiles along the axis of the specimen at different moment in time, using timesteps of 100 ms.
6. Conclusions In this contribution, a combined experimental-numerical technique is presented to measure the history of the axial displacements and deformations along the length of a specimen subjected to a high strain rate uni-axial load. The technique can be applied to a wide range of materials and high strain rate experiments. The results of the technique applied to a split Hopkinson tensile bar experiment are presented. They prove that highly accurate and valuable information is obtained. The comprehensive, quantitative information on the specimen deformation cannot be obtained by the classical measurement techniques, but it is essential for all studies
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2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0
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Point 1
Point 2 Point 4 Point 6 0
0.1
0.2
0.3
0.4 0.5 Time (ms)
0.6
0.7
0.8
Fig. 16. Time history of the strain in 4 points of the specimen: the middle (point 1), a point located at a distance of 1.5 mm of the middle of the specimen (point 2), a point at a distance of 3 mm (point 4) and a point at a distance of 5.5 mm (point 6).
aiming at an in-depth understanding of high strain rate experiments, or of the high strain rate behaviour of materials.
References [1] Hopkinson B. A method of measuring the pressure produced in the detonation of high explosives or by the impact of bullets. Philos Trans R Soc London Series A. 1914;213:437–52. [2] Manjoine M, Nadai A. High-speed tension test at elevated temperatures—Parts II and III. J Appl Mech 1942;A:77–91. [3] Kreis T. Quantitative evaluation of the interference phase. Holographic interferometry: principles and methods. Berlin; Akademic Verlag; 1996. p. 101–70 [chapter 4]. [4] Degrieck J, Van Paepegem W. Application of digital phase-shift shadow moir!e to micro deformation measurements of curved surfaces. Opt Laser Eng 2001;36:29–40. [5] Kolsky H. An investigation of the mechanical properties of materials at very high rates of loading. Proc Phys Soc London Sec. B 1949;62:676–700. [6] Verleysen P, Degrieck J, Taerwe L. Experimental investigation of the impact behaviour of cementious composites. Mag Concr Res 2002;54(4):257–62. [7] Verleysen P, Degrieck J. Improved signal processing for split Hopkinson bar tests on (quasi-) brittle materials. Exp Tech 2000;24(6):31–3. [8] Kobayashi A, editor. Handbook on experimental mechanics. SEM, 2nd revised edition, 1993, p. 997–1007. [9] Kujawinska M, Osten W. Fringe pattern analysis methods: up-to-date review. Proc SPIE 1998;3407:56–66.