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Point load determination of static elastic moduli using laser speckle interferometry
ARTICLE IN PRESS
Optics and Lasers in Engineering 42 (2004) 511–527
Point load determination of static elastic moduli using laser speckle interferometry Shahryar Shareef, Douglas R. Schmitt* Department of Physics, Institute for Geophysical Research, University of Alberta, Edmonton, AB T6G 2J1, Canada Received 1 September 2003; accepted 18 March 2004
Abstract Electronic speckle interferometry (ESPI) is used to determine the Young’s modulus E and Poisson ratio n of an isotropic material. Micron scale deformations of the surface of the block of polymethyl-methacrylate (PMMA) are induced by normal application of a known nearpoint force. These deformations are recorded in speckle interferometric fringe patterns. An iterative minimum error inversion technique is developed to obtain the elastic properties from the positions of fringe peaks and troughs observed in the fringe patterns. Sensitivity tests of the method on calculated fringe patterns using measured experimental uncertainties suggest the technique will provide measures of the elastic moduli to better than 5%. In an experimental test on a bloc of PMMA (acrylic) the technique gave values of E and n that differed from corresponding measures obtained using more conventional strain-gauge methods by less than 4%. r 2004 Published by Elsevier Ltd. Keywords: ESPI; Speckle interferometry; Elasticity; Young’s modulus; Poisson’s Ratio; Elastic properties
1. Introduction Prior knowledge of a material’s elastic properties is fundamentally important in engineering, physics and geophysics. Traditionally elastic properties are obtained by measuring strain as a function of stress, often to the plastic yield point of the material. These techniques are well understood [1,2]. Some disadvantages of such *Corresponding author. Tel.: +1-780-492-3985; fax: +1-780-492-0714. E-mail address: [email protected] (D.R. Schmitt). 0143-8166/$ - see front matter r 2004 Published by Elsevier Ltd. doi:10.1016/j.optlaseng.2004.03.004
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methods, however, are that they require specially machined samples, that strain gauges must often be applied, that testing machines themselves are expensive and bulky, and that such methods are not amenable to in situ moduli determinations. As such, there remains room for alternative and complementary methods. Optical interferometric techniques such as double exposure holography [3,4] and electronic speckle interferometry (ESPI) [5] have become popular for deformation analysis; and these methods have some potential to provide alternative and less intrusive methods for elastic modulus determination. The focus of this paper is on development of an ESPI-based method to measure elastic moduli. ESPI is a technique in which a micron scale displacement field produced by translation and deformation of a surface is encoded within a ‘fringe’ pattern. The raw fringe pattern is essentially a two-dimensional mapping of the variation in the phase of interfering coherent light on the surface of the object. The shape and density of the fringe pattern depends on the magnitude and the direction of the motion of the object under study. The advantage of such an optical technique is that samples often require little preparation, no sensors need be directly attached, and measurements are inherently calibrated. The surface displacements effected by application of the point force depend directly on the materials elastic coefficients. Consequently, a raw fringe pattern provides an indirect measure of the material’s elastic properties. In this paper a quantitative ESPI methodology to measure the static elastic properties of an isotropic material is developed and tested, this paper builds on an earlier contribution that employed less adaptable film-based holography [6] that was not amenable to accurate quantitative analysis. The method works by applying a known point load to the material surface and measuring the resulting displacement field with ESPI. Aside from the advantages inherent to the use of optical methods already mentioned, there are some particular benefits of the measurements here relative to more conventional strain gauge tests. First, the optical measurements may be non-destructively applied to the object or material of interest under in situ conditions without removal or sampling; this is useful in that it can be applied to not so well characterized materials such as wood, rock, or polymers whose deformation behaviour may be complicated by nonlinearity, anisotropy, or time-dependent anelasticity. Second, a full deformation field may be obtained allowing numerous measurements over the surface of the object that provides for greater statistical confidence in the final result in contrast to single point strain gauge determinations. The analytical treatment of the fringe formation and inversion is presented and a forward modeled fringe pattern is calculated using Bousinessesq’s well-known analytical solution [7] describing the surface displacements from a force on the boundary of a semi-infinite body, but more complex analyses that might, for example, include time-dependent responses are not precluded. The resulting fringe pattern is then inverted using the least-squares minimization method. Virtual experiments are first carried out to deduce what problems might be encountered in the analysis of such experimental fringe patterns and to assess level of experimental error. The experimental configuration is described and the method is applied to an acrylic block to obtain its Young’s modulus E and Poisson’s ratio n:
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2. Background In this section the mathematical background of the method including the relationship between induced displacements and final fringe patterns, and the inversion methodology are described. More details on the interferometric techniques may be found in the literature already cited [4,5] and only a brief overview is provided in order to prepare for presentation of the fringe inversion methodology [7]. The elastic modulus determination technique relies on measuring the displacements produced by applying a point force F on the flat surface of the object that is presumed here to be a semi-infinite solid or half space that fills the region zp0 (Fig. 1). The three components of the displacement vector at an arbitrary point D Uðx; yÞ ¼ ux ðx; yÞi þ uy ðx; yÞj þ uz ðx; yÞk
ð1Þ
are ux ¼
F ð1 2nÞð1 þ nÞx0 ; 2pEðx02 þ y02 Þ
ð2Þ
uy ¼
F ð1 2nÞð1 þ nÞy0 2pEðx02 þ y02 Þ
ð3Þ
and uz ¼
F ð1 n2 Þ pEðx02 þ y02 Þ1=2
ð4Þ
where x0 ¼ x x0 and y0 ¼ y y0 : Inspection of these relatively simple relationships indicates that the surface normal directed component uz of the displacement is greatest and that the magnitudes of all the components decay inversely proportional with the radial distance ðx02 þ y02 Þ1=2 from P: A deficiency of the above formulation is that it predicts infinite displacement at the point of force application that cannot, of course, be the case. In reality,
Fig. 1. Geometry of point load problem in Cartesian co-ordinate system x2y2z with origin at O. Surface of half-space coincides with x2y plane at z ¼ 0: Point force F applied normally to surface at point Pðx0 ; y0 Þ induces displacement U at point Dðx; yÞ:
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regardless of how small an indenter might be used, the force is actually distributed over a finite area. As such, Eqs. (2–4) are only valid at regions remote from P according to St. Venant’s principle [8]. This is not a problem for our application as the region near the indenter is difficult to illuminate properly and does not provide usable information. The ESPI method requires two coherent light sources split from one single laser source using a beam splitter (Fig. 2). As the surface under examination is displaced, the speckle pattern changes cyclically with respect to the magnitude and the direction of the displacement and the relative positions of the surface points Dðx; yÞ relative to the sources S1 and S2 as represented by unit directional vectors n1 and n2. The change in phase fðx; yÞ at a give surface point Dðx; yÞ due to the displacement is [5] fðx; yÞ ¼ kðx; yÞ Uðx; yÞ;
ð5Þ
where kðx; yÞ is called the sensitivity vector 2p ½n1 ðx; yÞ n2 ðx; yÞ : kðx; yÞ ¼ ð6Þ l The fringe pattern, however, is an image with intensities rðx; yÞ that vary over the range [0,1], these are directly related to the wrapped phase fðx; yÞ via [5] rðx; yÞ ¼ 12ð1 þ cos fðx; yÞÞ:
ð7Þ
This gives a maximum (1= white peak) and minimum (0= black trough) fringe intensities when f is an even and odd integral multiple of p; respectively. Shortly, it will be more convenient to express Eq. (5) in terms of a fringe order n where n is any real number 1 ð8Þ nðx; yÞ ¼ ðkðx; yÞ Uðx; yÞÞ: 2p In this situation fringe peaks and troughs correspond n being any integer and any odd integer divided by 2, respectively. It is important to note that the fringe grey scale convention described here follows the definition given by Eq. (7) but this is not
Fig. 2. Geometry of speckle interferometric experiment. Laser beam LB is split by beam splitter BS propagating to source points S1 and S2 where it is expanded by scattering to illuminate the surface of the object. Rays propagate from S1 and S2 to arbitrary surface point Dðx; yÞ with directions indicated by unit vectors n1 and n2, respectively.
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Fig. 3. Synthetic fringe pattern calculated for a point force F ¼ 1500 N applied at the centre of white cross at point P(x ¼ 9:5; y ¼ 3:2; z ¼ 0) to a material of E ¼ 3 GPa and n ¼ 0:45 created with light of wavelength l=829 nm. Dark and light lines delineate a few of the larger fringe peaks and troughs found by manual picking, the corresponding fringe orders are also shown. Bright fringe ‘A’ is reference fringe Fig. 4. Source positions used in calculations are S1(x ¼ 4:6; y ¼ 7:0; z ¼ 5:3) and S2(x ¼ 23:5; y ¼ 7:3; z ¼ 6:3).
necessarily used by other workers where different ESPI fringe determination methods relying on a variety of correlations are employed [4]. An example of a synthetic fringe pattern calculated using Eqs. (1)–(7) for a point load to the surface of an object with fringe orders given for a few peaks and troughs is shown in Fig. 3.
3. Fringe inversion method Extracting quantitative information from ESPI fringe patterns remains challenging. The direct determination of displacements at a point-by-point basis using specialized phase shifting techniques are perhaps the most popular way that quantitative information is extracted using ESPI (see Ref. [4] for an overview). However, if the expected induced displacement field has a well-defined analytic or numeric description, then a statistically based inversion methodology can be employed. That is, the parameters that are sought, which are in the present paper the elastic properties, may be directly obtained from the fringe pattern, or more precisely, knowledge of the spatial distribution of fringe order nðx; yÞ that the pattern maps. The technique here borrows from the geophysical literature in which large and often high-noise sets of observations are inversely modelled to provide some idea of the unseen internal structure of the earth. This method has recently been successfully applied to ESPI fringe patterns that recorded rigid-body translations [9] and to residual stress determination by blind-hole drilling [10].
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To determine the best values of Young’s modulus E and Poisson’s ratio n from the picked fringe positions of images, an iterative minimum error procedure has been employed. The method assumes that the character of the displacements due to a known load on the sample surface is described by Eqs. (2)–(4). Following through on substitution of these formulae into Eqs. (6) and (8) with algebraic rearrangement gives a form that will be amenable to least-squares inversion nðx; yÞ ¼ Aðx; yÞa þ Bðx; yÞb; where the knowns A and B rely on l; F ; x; and y ky kx Fx0 Fy0 A¼ 2 þ ; 4p ðx02 þ y02 Þ 4p2 ðx02 þ y02 Þ
kz F B¼ 2 2p ðx02 þ y02 Þ1=2
ð9Þ
ð10aÞ
! ð10bÞ
and a and b are the unknown constants containing the elastic properties a¼
ð1 2nÞð1 þ nÞ E
ð11aÞ
b¼
ð1 n2 Þ : E
ð11bÞ
and
In an experiment both A and B are easily calculated as the position and force are known, conversely a and b remain to be determined and they contain the elastic information n and E: Eq. (9) is useful as it allows nðxi ; yj Þ; the fringe order observed within the fringe pattern image at pixel ði; jÞ; to be expressed by matrix multiplication a ½nðxi ; yj Þ ¼ ½Aðxi ; yj Þ Bðxi ; yj Þ ; ð12Þ b which may be expanded for many fringe order observations at m pixels within the image 3 2 3 2 Aðx1 y1 Þ Bðx1 ; y1 Þ nðx1 ; y1 Þ 6 nðx ; y Þ 7 6 Aðx ; y Þ Bðx2 ; y2 Þ 7 2 2 2 2 7 6 7 6 7 6 7 6 6 nðx3 ; y3 Þ 7 6 Aðx3 ; y3 Þ Bðx3 ; y3 Þ 7 a 7¼6 7 6 ð13Þ 7 6 7 b ; 6 ^ ^ ^ 7 6 7 6 7 6 7 6 4 nðxm1 ; ym1 Þ 5 4 Aðxm1 ; ym1 Þ Bðxm1 ; ym1 Þ 5 nðxm ; ym Þ Aðxm ; ym Þ Bðxm ; ym Þ
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which may be represented as matrix algebra as N ¼ GM where N is the vector of observed fringe orders on the left had side of Eq. (13), G is the twocolumn matrix that holds the known information on the applied load and experimental geometry, and the unknown vector M ¼ ½a b T is to be determined. In this form, one can use the well known least squares minimization technique to determine M M ¼ ðG T GÞ1 GT N
ð14Þ
from which a and b may be derived. There is one major difficulty in the application of Eq. (14). The values in the fringe pattern itself range over [0 to 1] and hence are a wrapped fringe order only; the unwrapped true fringe order value is not necessarily known. An iterative procedure is employed to overcome this problem [11]. First, continuous fringe peaks and troughs are picked within the image. All of the pixels, each of which are mapped to a specific ðxi ; yj Þ position on the surface, along such a continuous feature must have constant fringe order n: For example, the fringe order at a peak must have an integral value, say n0 : The fringe order for the troughs adjacent to this peak must differ from n0 by 712: Similarly, the adjacent peaks must have values from n0 of 71. The fringe order values for subsequent farther peaks and troughs can also be assigned a fringe order relative to n0 as the difference in the fringe order between any two such continuous features must progressively increase or decrease in a well defined fashion. This is illustrated in Fig. 3 where the reference fringe A has a fringe order n0 ¼ 2 which is known absolutely because the fringe pattern is calculated. The fringe order for successive troughs and peaks that are chosen decreases in a clock-wise fashion around the force application point in Fig. 3. As the first step towards inversion, one feature of the fringe pattern is chosen as reference. An arbitrary but appropriate initial fringe order value n0 is assigned to this feature, for example a guessed integer value is assigned to the pixels associated with the reference fringe A in Fig. 3 because it is a fringe peak. Appropriately incremented fringe orders are then applied to the subsequent fringe peaks and troughs. Forward modelling of the anticipated fringe pattern can assist in assigning fringe order values. Once this procedure is carried out, a trial solution to M is found using Eq. (14). In most cases this initial ‘guess’ at n0 will not be correct, however, and this must result in an erroneous solution. There are a variety of ways to measure this least square error e but here e ¼ jN GMj2
ð15Þ
is employed. The entire procedure is then repeated by incrementing or decrementing the fringe order values by unity and again solving for a new trial M and corresponding fringe order error e by Eqs. (14) and (15), respectively. The procedure is repeated until a minimum error is detected. The fringe order assignment for this
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minimum error is most correct and will provide, in the least-square’s sense, the best value of M for determination of E and n: In the following sections this procedure is first tested on the simulated fringe pattern of Fig. 3 under a variety of assumptions and then applied to real experimental fringe patterns.
4. Test on simulated fringe pattern The calculated synthetic fringe pattern of Fig. 3 is inverted to get the E and n values by the fringe picking method described in the last section. Errors are purposefully introduced to the analysis in order to assess quantitatively the level of uncertainty that might be expected in the analysis. The fringe order and ðx; yÞ position are by default, as part of the calculation, absolutely known at every pixel in the forward modelled fringe pattern of Fig. 3. Consequently, one initial performance test is to directly invert these simulated values using Eq. (14) as this should provide the best solution possible. This was the case with the determined values of E and n differing from those input by less than one part in 10,000 (Table 1). This negligible difference is due to computer round off errors. Next, a series of 13 more realistic attempts were carried out in which the fringe peaks and troughs were manually picked (Table 1). Details of the fringe peak and trough picking procedure are described in Ref. [11] but the peaks and troughs picked in one of the trials within the series are superimposed on the fringe pattern of Fig. 3. The iterative procedure described in the previous section was applied to each of these
Table 1 Results from inversion of the synthetic image in Fig. 3 [Trial No.]
[Fringes picked]
No. of Points picked
E (GPa)
n
e
% Error in % Error in E n
1 2 3 4 5 6 7 8 9 10 11 12 13 Direct inv
8
106 121 126 121 116 109 121 117 123 121 126 121 116 —
3.0343 3.0136 3.0350 3.0347 3.0272 3.0082 3.0317 3.0025 2.9957 3.0290 3.0471 3.0284 3.0459 2.9999
0.4504 0.4510 0.4504 0.4505 0.4506 0.4511 0.4505 0.4513 0.4514 0.4499 0.4493 0.4499 0.4493 0.4500
0.0020 0.0017 0.0020 0.0034 0.0027 0.0025 0.0017 0.0024 0.0027 0.0314 0.0380 0.0332 0.0340 0.0088
1.14 0.45 1.16 1.15 0.90 0.27 1.05 0.08 0.14 0.96 1.57 0.94 1.53 0.003
—
0.08 0.22 0.08 0.11 0.13 0.24 0.11 0.28 0.31 0.02 0.15 0.02 0.15 0.0
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trials and are summarized in Table 1. For the most part, these results suggest that pixel picking errors are generally less than 1.5% for E and less than 0.4% for n: These results were encouraging especially given that only a limited number of points were employed in the inversion in each case; picking of more fringes should allow for even lower levels of error. Finally, an even more involved test sought to find the levels of error that could be introduced by uncertainties in the determination of the geometry of the experiment. These could arise from the uncertainty in finding the relative three-dimensional positions of the sources S1 and S2, the force application point P, and the many positions D(xi,yj) of the pixels in the image. In addition to direct measurement errors, distortion of the image due to camera placement and lens aberration adds to positioning problems. The force level F must also have some level of uncertainty that will propagate through to influence the final result. In the following, the input parameters for the inversion will be slightly changed from their exact values one at a time in order to assess the solution’s sensitivity to them. To do this, the calculated fringe pattern of Fig. 3 has been inverted once again first using the known input parameters to yield the results in the first row of Table 2. The co-ordinates of the picked points, the assigned fringe orders and the total number of picked points have been saved and subsequently used in the later inversions where only the input parameters are gradually changed. This keeps the consistency between different inversions. The percentage of error has been calculated from the first ‘error-free’ result relative to the later ones (Table 2). The degree of error that might be associated with a certain parameter is selected with some consideration of a real experimental arrangement, i.e., by estimating the amount of uncertainty that might exist in the experimental determination of those parameters. Accordingly, *
*
*
*
The co-ordinates of the laser sources could be uncertain by as much as 73 mm (cumulative from both the measuring ends) due to the complex positioning of the sources relative to the origin and difficulties associated with measuring points in 3-D space. This estimate is based on repeated attempts at determining the source positions at the lab bench. The viewing area of the sample surface may vary by as much as three millimetres due to the cumulative effect of the errors in measurement and in removal of lens distortion in the final image. The applied force is assigned an error of 730 N based on the 2% uncertainty specified for the load cell used in the later experiments. The co-ordinates of the force application point have been assigned an uncertainty of 72 mm which again arises from the measurement uncertainty.
It is evident from Table 2 that the co-ordinates of the point of force application (both x and y co-ordinates individually and combined as seen in rows 2, 3 and 4), x and z-coordinates of the first laser source S1, z-coordinate of the second laser source S2 and the length of the block are among the most sensitive parameters. These parameters individually have contributed more than 2% of error to each or both of
520
Trial no. Viewing area (cm) Force Point (cm)
19.4 14.3 19.4 14.3 19.4 14.3 19.4 14.3 19.4 14.3 19.4 14.3 19.4 14.3 19.4 14.3 19.4 14.3 19.4 14.3 19.4 14.3 19.4 14.3 19.4 14.3 19.4 14.3 19.4 14.3 19.4 14.3 19.4 14.3 19.1 14.3 19.7 14.3 19.4 14.0 19.4 14.6 19.7 14.6
X=8.53, X=8.35, X=8.53, X=8.35, X=8.53, X=8.53, X=8.53, X=8.53, X=8.53, X=8.53, X=8.53, X=8.53, X=8.53, X=8.53, X=8.53, X=8.53, X=8.53, X=8.53, X=8.53, X=8.53, X=8.53, X=8.35,
Y=2.70 Y=2.70 Y=2.52 Y=2.52 Y=2.70 Y=2.70 Y=2.70 Y=2.70 Y=2.70 Y=2.70 Y=2.70 Y=2.70 Y=2.70 Y=2.70 Y=2.70 Y=2.70 Y=2.70 Y=2.70 Y=2.70 Y=2.70 Y=2.70 Y=2.52
15 15 15 15 14.7 (70.3) 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 14.7
S1 cm x
y
z
S2 cm x y
z
4.6 4.6 4.6 4.6 4.6 4.3 4.9 4.6 4.6 4.6 4.6 4.6 4.6 4.6 4.6 4.6 4.6 4.6 4.6 4.6 4.6 4.3
7.0 7.0 7.0 7.0 7.0 7.0 7.0 6.7 7.3 7.0 7.0 7.0 7.0 7.0 7.0 7.0 7.0 7.0 7.0 7.0 7.0 6.7
5.3 5.3 5.3 5.3 5.3 5.3 5.3 5.3 5.3 5.0 5.6 5.3 5.3 5.3 5.3 5.3 5.3 5.3 5.3 5.3 5.3 5.6
23.5 23.5 23.5 23.5 23.5 23.5 23.5 23.5 23.5 23.5 23.5 23.2 23.8 23.5 23.5 23.5 23.5 23.5 23.5 23.5 23.5 23.2
6.3 6.3 6.3 6.3 6.3 6.3 6.3 6.3 6.3 6.3 6.3 6.3 6.3 6.3 6.3 6.0 6.6 6.3 6.3 6.3 6.3 6.0
7.3 7.3 7.3 7.3 7.3 7.3 7.3 7.3 7.3 7.3 7.3 7.3 7.3 7.0 7.6 7.3 7.3 7.3 7.3 7.3 7.3 7.6
E GPa % Error in E n
% Err. in n
3.02 3.14 2.86 2.77 2.96 3.11 2.97 3.05 2.99 3.01 3.28 3.06 3.01 3.02 3.03 3.17 3.05 3.14 2.72 3.04 2.96 3.07
0.0 1.5 0.9 0.9 0.0 1.1 0.8 0.3 0.3 1.6 3.2 0.4 0.5 0.2 0.2 2.4 1.2 1.3 2.5 0.2 0.5 4.8
0.00 3.77 5.34 8.27 2.00 2.79 1.70 0.96 0.90 0.58 8.58 1.27 0.52 0.02 .024 4.77 0.96 3.80 10.2 0.38 2.02 1.64
0.450 0.443 0.454 0.454 0.450 0.445 0.454 0.449 0.451 0.457 0.436 0.452 0.449 0.451 0.449 0.439 0.455 0.444 0.461 0.449 0.452 0.43
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F : (N) x100 Coordinates of the laser sources
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Table 2 Sensitivity tests using calculated fringe pattern
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Fig. 4. Least-squares error versus arbitrary fringe order assigned to fringe ‘A’ in Fig. 3.
the output values E and n: When all of them are considered together, the values of E and n resulted with 1.7% and 5% errors respectively (row 22, Table 2).
5. Experimental tests and results Experimental tests were conducted on an acrylic block and compared to corresponding strain-gauge type measurements carried out on a testing machine. The overall experimental configuration (Fig. 5a) is mounted on a specially prepared loading frame that holds the mechanical components. The experimental details are provided in the subsections below. Briefly, however, the recording process consists of capturing two images of the speckle patterns before and after the deformation, i.e. force application. A correlation procedure is then implemented to produce the fringe pattern rðx; yÞ from which is then taken the positions of the continuous fringe features as inputs to the inversion. 5.1. Sample preparation The laboratory tests were done using acrylic (polymethyl-methacrylate or PMMA) material. This material was selected as it remains elastic over the time periods of our test and it has a relatively low Young’s modulus for a solid; this greatly simplifies construction of the loading frame and selection of the load cells. The acrylic block was cut with dimensions of 25 cm 25 cm 5 cm. The block was machined square then annealed at low temperature in an oven in order to reduce residual stresses that may exist. This was qualitatively evaluated on the block by examining birefringent
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Voltmeter
Computer
CC D
Ram
S1 z n1
Load Cell
Indenter
S2 n2
y D
x
PMMA Block
(a)
CCD Camera Hydraulic Ram Beam Splitter
Indenter
S2 S1 PMMA Block
(b) Fig. 5. (a) Experimental configuration. CCD is the charge coupled camera with feed to the computerbased image capture and processing system. (b) Photograph of experimental equipment.
effects under cross-polarization, no fringes were observed near the centre of the annealed block. The surface of the acrylic block were first sand blasted with a fine grit then painted with a reflecting white paint to produce better scattering of laser light. This improves the image quality significantly; as the speckle technique requires an optically rough surface. This material is then placed on a solid steel surface (Fig. 5b) that is a part of the loading frame. As the goal of this study is to measure the elastic properties, it is important that alternative measures be available for comparison. A conventional strain gauge method, which is widely used in determining materials properties, has been used to measure the E and n values for the acrylic sample. A small piece of acrylic (12 cm 2.8 cm 0.69 cm) was cut from the sample used in the optical tests. The ends of the sample were fixed in the jaws of a universal testing machine. This machine is preset to apply 272.11 kg (600 lbs) of load over 5 s. The resultant strains (both axial and transverse) are measured by two strain gauges (1/16 in. grid length, 90 stacked gauges) attached on the sample. These strain gauges are connected to a signal conditioner and eventually to a digital scope that collects the data. The E is measured from the slope of the stress-strain curve [1] and the Poisson ratio is
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measured from the ratio of the transverse strain to the axial strain [2]. This procedure gave values of E ¼ 3:070:1 GPa and of n=0.3870.01. It is worthwhile noting that the manufacturer quotes a value of E ¼ 3:1 GPa for this material. 5.2. Mechanical system The mechanical system consists of the loading frame, sample, hydraulic piston, load cell, pump, hand-driven pressure accumulator, and associated recording equipment (Fig. 5). The force is applied to the sample via an indenter pushed onto the surface with a hydraulic ram. The hydraulic hand pump is connected to the cylinder and piston assembly and to an accumulator. The accumulator is used to achieve small changes in the line pressure. A load cell is attached between the piston and the indenter to yield the force exerted on the sample surface by the indenter. The signal from the load cell passes through a signal conditioner and the force is directly read in a calibrated voltmeter. The indenter tip is made approximately 0.5 cm in diameter to prevent the sample from cracking under the load. Note that care has been taken to avoid irreversible plastic deformation or fracturing on the material by keeping the loads relatively small (o1500 N). Reversibility was confirmed at the end of the optical tests as no fringes were seen in the live fringe pattern display when the applied force was released to its initial reference value. 5.3. Optical system The optical system consists of a coherent light source, a beam splitter, a CCD camera, source points, and computer-based high-speed image acquisition and processing system (Fig. 5). The optics are mounted on the floating optical bench but are not touching the loading frame in order to reduce any inadvertent relative motions between the sources and the object that would influence the fringe pattern. Coherent light was provided by a 35 mW stabilized laser diode emitting in the near infrared at l=829 nm. This wavelength range was chosen on the basis of the CCD camera sensitivity. The laser head module contains the laser, collimation optics, beam shaping optics and thermoelectric cooler, thermistor, and heat sink. Together these provide a highly stable, round collimated source. The beam propagates to a B50% splitter then onto the two source points S1 and S2 that scatter the light to illuminate the object’s surface. The rough surface produces a speckle pattern that is captured in a grey scale image (8 bit, 640 pixels horizontally and 480 pixels vertically) by a high-resolution Charged Coupled Device (CCD) camera. The output of this camera feeds to an image capture board on a 500 MHz computer (ca. 2001) where further processing occurs. The minimum experiment requires that two such speckle pattern images, one reference image captured immediately before deformation and one monitor image acquired once the object has been deformed. Two such speckle pattern raw images (Fig. 6) provide little useful information on their own. A fringe pattern, however, is calculated using a localized correlation procedure previously [7] that relies local
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Fig. 6. Raw speckle patterns captured (a) before and (b) after application of a 1300 N load at the point indicated in (a).
calculation of the Pearson’s correlation coefficient that has been shown to be equivalent to a mapping of rðx; yÞ: The top left corner of the image is chosen as the origin and the source positions are measured with respect to it. The co-ordinates of the point of force application are also given relative to this origin and all subsequent measurement such as positions of each pixel, are measured relative to the force application point. Examples of two such observed fringe patterns acquired in the series of experimental tests are shown in Figs. 7a and b for applied forces F of 1000 N and 1300 N, respectively. It is worth commenting that these fringe mappings are considerably noisier than the purely noise free patterns calculated (e.g. Fig. 3). A discussion of this speckle noise is beyond the scope of this paper but it has been found to be inherent to the correlation technique used here to calculate the fringe pattern [12].
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Fig. 7. Observed correlated fringe patterns for a (a) 1000 N force and (b) 1300 N force applied to the acrylic test block. Black and white lines superimposed on the fringes indicated locus of picked peaks and troughs, respectively. Fringe patterns calculated to compare using elastic properties obtained by the inversion of the fringes picked in (a) and (b) for the (c) 1000 N force and the (d) 1300 N force.
5.4. Fringe pattern analysis In the inversion, the experimental image is first corrected for perspective distortion [13]. The images captured suffer from perspective distortion mainly due to the angle of view of the camera, which captures the images of the undeformed and deformed states of the surface under investigation, with respect to the surface normal. Due to this oblique sight the inspected surface appears deformed in the recorded image and may not fill the full frame. A spatial transform maps the pixels of the recorded interferograms to new pixels in the output interferogram, such that identical points of the object surface, which are imaged to different pixels in the recorded images, are mapped to identical pixels in the output images. Then points are picked manually along the peak and trough of the bright and dark fringes respectively (Figs. 7a and b). The peak and trough are selected manually with the interpreter biases included. This method is not ideal but was used as it provides a larger level of error in the determination of the fringe extrema and as such is a good test of the method. It is worth nothing that it can be very difficult to remove the high frequency speckle noise in such fringe patterns without introducing further error.
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Table 3 Results of interferometric inversions Force (N) Trial No. No. Fringe picked
Points picked
E (GPa)
n
e
Average E Average n (GPa)
1000
1 2 3 4 5
8
280 391 386 484 597
3.31 3.11 3.05 3.15 3.16
0.37 0.39 0.39 0.39 0.38
0.56 0.58 0.74 0.67 0.48
3.170.3
0.3870.03
1300
1 2 3 4 5
6
297 330 436 308 503
3.35 3.16 3.19 2.9 2.81
0.37 0.37 0.39 0.39 0.38
4.4 6.8 5.35 4.5 5.0
3.170.3
0.3870.03
The results (Table 3) from a series of repeated tests were carried out on the fringe patterns of Figs. 7(a) and (b); the averages obtained are E ¼ 3:170:3 GPa and n=0.3870.03, in very good agreement with the conventionally determined values of E ¼ 3:070:1 GPa and of n=0.3870.01 and with the manufacturer’s quoted value of E ¼ 3:1 GPa. The optically derived elastic properties are used along with the input parameters to reproduce the experimental fringe patterns in Figs. 7(c) and (d) for purposes of comparison. Again, the character of the corresponding observed and calculated fringe patterns are similar indicating that the inversion procedure is working acceptably.
6. Concluding remarks The ESPI method employing fringe inversion by direct correlation resulted in values of Young’s modulus E and Poisson ratio n with errors of 2% and 0.3% respectively in the case of synthetic fringes. For experimental fringes the result of E for Plexiglas matched with the manufacturers specification of 3.1 GPa and is within 3% of the 3.0 GPa determined by the standard strain gauge method. The Poisson ratio n is found to be 0.38, which also matched with the value obtained by the strain gauge method (0.38). These results suggest that the point force method can be successfully used to obtain the elastic properties of isotropic materials. The major advantages of this technique are: that it is a non-destructive method and the sample can be considered for subsequent use. The sample needs little in the way of additional preparation in many cases. The full field image gives more information than the standard strain gauge method. Moreover, the inversion method is simple and straightforward. In the present paper, additional processing of the fringes was not employed but this may have added interpreter bias and error. As up
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to four separate fringe patterns can be calculated in near real time in our present system configuration, the technique can show some promise for time-lapse implementation to study progressive deformation for rheological studies. At present, we are able to acquire, calculate, and store four separate fringe patterns per second. Future work will focus on developing more automated methods of fringe inversion that may employ nearly all the data within the image eliminating the need for cumbersome and potentially error-prone manual fringe picking.
Acknowledgements The work was greatly assisted by technical data acquisition developments of F. Weichman, W. Engler, L. Tober, M.Diallo, M. Lazorek, and I. Rumzan. This work was made possible with funding provided by a NSERC Strategic Grant on ESPI.
References [1] ASTM Standard E111-97. Standard test method for Young’s Modulus, tangent modulus, and chord modulus. Annu Book ASTM Stand 2002;03(1):244–50. [2] ASTM Standard E132-97. Standard test method for poisson’s ratio at room temperature, Annu Book ASTM Stand 2002;03(1):275–77. [3] Vest CM. Holographic interferometry. New York: Wiley; 1979. p. 465. [4] Cloud GL. Optical methods of engineering analysis. New York: Cambridge University Press; 1995. p. 517. [5] Jones R, Wykes C. Holographic and speckle interferometry. Cambridge, UK: Cambridge University Press; 1983. p. 330. [6] Schmitt DR, Smither C, Ahrens TJ. In-situ holographic elastic moduli measurements from boreholes. Geophysics 1989;54(4):468–77. [7] Schmitt DR, Hunt RW. Optimization of fringe pattern calculation with direct correlations in speckle interferometry. Appl Opt 1997;36(34):8848–57. [8] Fung YC. Foundations of solid mechanics. Englewood Cliffs, NJ: Prentice-Hall, Inc.; 1965. p. 525. [9] Schmitt DR, Hunt RW. Model based inversion of speckle interferometer fringe patterns. Appl Opt 1998;37:2573–8. [10] Schmitt DR, Hunt RW. Inversion of speckle interferometer fringes for hole-drilling residual stress detrminations. Exp Mecha 2000;40:129–37. [11] Shareef S. Determination of elastic coefficients of materials by laser speckle interferometry. MSc thesis, Department of Physics, University of Alberta, Edmonton, Alberta; 2002. p. 123. [12] Engler WG. Laser speckle interferometry: a stochastic investigation. Department of Physics, University of Alberta, Edmonton, Alberta. 2002. p. 209. [13] Rastogi PK. Holographic interferometry: principles and methods. Berlin, New York: Springer; 1994. p. 170–3.
Optics and Lasers in Engineering 42 (2004) 511–527
Point load determination of static elastic moduli using laser speckle interferometry Shahryar Shareef, Douglas R. Schmitt* Department of Physics, Institute for Geophysical Research, University of Alberta, Edmonton, AB T6G 2J1, Canada Received 1 September 2003; accepted 18 March 2004
Abstract Electronic speckle interferometry (ESPI) is used to determine the Young’s modulus E and Poisson ratio n of an isotropic material. Micron scale deformations of the surface of the block of polymethyl-methacrylate (PMMA) are induced by normal application of a known nearpoint force. These deformations are recorded in speckle interferometric fringe patterns. An iterative minimum error inversion technique is developed to obtain the elastic properties from the positions of fringe peaks and troughs observed in the fringe patterns. Sensitivity tests of the method on calculated fringe patterns using measured experimental uncertainties suggest the technique will provide measures of the elastic moduli to better than 5%. In an experimental test on a bloc of PMMA (acrylic) the technique gave values of E and n that differed from corresponding measures obtained using more conventional strain-gauge methods by less than 4%. r 2004 Published by Elsevier Ltd. Keywords: ESPI; Speckle interferometry; Elasticity; Young’s modulus; Poisson’s Ratio; Elastic properties
1. Introduction Prior knowledge of a material’s elastic properties is fundamentally important in engineering, physics and geophysics. Traditionally elastic properties are obtained by measuring strain as a function of stress, often to the plastic yield point of the material. These techniques are well understood [1,2]. Some disadvantages of such *Corresponding author. Tel.: +1-780-492-3985; fax: +1-780-492-0714. E-mail address: [email protected] (D.R. Schmitt). 0143-8166/$ - see front matter r 2004 Published by Elsevier Ltd. doi:10.1016/j.optlaseng.2004.03.004
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methods, however, are that they require specially machined samples, that strain gauges must often be applied, that testing machines themselves are expensive and bulky, and that such methods are not amenable to in situ moduli determinations. As such, there remains room for alternative and complementary methods. Optical interferometric techniques such as double exposure holography [3,4] and electronic speckle interferometry (ESPI) [5] have become popular for deformation analysis; and these methods have some potential to provide alternative and less intrusive methods for elastic modulus determination. The focus of this paper is on development of an ESPI-based method to measure elastic moduli. ESPI is a technique in which a micron scale displacement field produced by translation and deformation of a surface is encoded within a ‘fringe’ pattern. The raw fringe pattern is essentially a two-dimensional mapping of the variation in the phase of interfering coherent light on the surface of the object. The shape and density of the fringe pattern depends on the magnitude and the direction of the motion of the object under study. The advantage of such an optical technique is that samples often require little preparation, no sensors need be directly attached, and measurements are inherently calibrated. The surface displacements effected by application of the point force depend directly on the materials elastic coefficients. Consequently, a raw fringe pattern provides an indirect measure of the material’s elastic properties. In this paper a quantitative ESPI methodology to measure the static elastic properties of an isotropic material is developed and tested, this paper builds on an earlier contribution that employed less adaptable film-based holography [6] that was not amenable to accurate quantitative analysis. The method works by applying a known point load to the material surface and measuring the resulting displacement field with ESPI. Aside from the advantages inherent to the use of optical methods already mentioned, there are some particular benefits of the measurements here relative to more conventional strain gauge tests. First, the optical measurements may be non-destructively applied to the object or material of interest under in situ conditions without removal or sampling; this is useful in that it can be applied to not so well characterized materials such as wood, rock, or polymers whose deformation behaviour may be complicated by nonlinearity, anisotropy, or time-dependent anelasticity. Second, a full deformation field may be obtained allowing numerous measurements over the surface of the object that provides for greater statistical confidence in the final result in contrast to single point strain gauge determinations. The analytical treatment of the fringe formation and inversion is presented and a forward modeled fringe pattern is calculated using Bousinessesq’s well-known analytical solution [7] describing the surface displacements from a force on the boundary of a semi-infinite body, but more complex analyses that might, for example, include time-dependent responses are not precluded. The resulting fringe pattern is then inverted using the least-squares minimization method. Virtual experiments are first carried out to deduce what problems might be encountered in the analysis of such experimental fringe patterns and to assess level of experimental error. The experimental configuration is described and the method is applied to an acrylic block to obtain its Young’s modulus E and Poisson’s ratio n:
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2. Background In this section the mathematical background of the method including the relationship between induced displacements and final fringe patterns, and the inversion methodology are described. More details on the interferometric techniques may be found in the literature already cited [4,5] and only a brief overview is provided in order to prepare for presentation of the fringe inversion methodology [7]. The elastic modulus determination technique relies on measuring the displacements produced by applying a point force F on the flat surface of the object that is presumed here to be a semi-infinite solid or half space that fills the region zp0 (Fig. 1). The three components of the displacement vector at an arbitrary point D Uðx; yÞ ¼ ux ðx; yÞi þ uy ðx; yÞj þ uz ðx; yÞk
ð1Þ
are ux ¼
F ð1 2nÞð1 þ nÞx0 ; 2pEðx02 þ y02 Þ
ð2Þ
uy ¼
F ð1 2nÞð1 þ nÞy0 2pEðx02 þ y02 Þ
ð3Þ
and uz ¼
F ð1 n2 Þ pEðx02 þ y02 Þ1=2
ð4Þ
where x0 ¼ x x0 and y0 ¼ y y0 : Inspection of these relatively simple relationships indicates that the surface normal directed component uz of the displacement is greatest and that the magnitudes of all the components decay inversely proportional with the radial distance ðx02 þ y02 Þ1=2 from P: A deficiency of the above formulation is that it predicts infinite displacement at the point of force application that cannot, of course, be the case. In reality,
Fig. 1. Geometry of point load problem in Cartesian co-ordinate system x2y2z with origin at O. Surface of half-space coincides with x2y plane at z ¼ 0: Point force F applied normally to surface at point Pðx0 ; y0 Þ induces displacement U at point Dðx; yÞ:
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regardless of how small an indenter might be used, the force is actually distributed over a finite area. As such, Eqs. (2–4) are only valid at regions remote from P according to St. Venant’s principle [8]. This is not a problem for our application as the region near the indenter is difficult to illuminate properly and does not provide usable information. The ESPI method requires two coherent light sources split from one single laser source using a beam splitter (Fig. 2). As the surface under examination is displaced, the speckle pattern changes cyclically with respect to the magnitude and the direction of the displacement and the relative positions of the surface points Dðx; yÞ relative to the sources S1 and S2 as represented by unit directional vectors n1 and n2. The change in phase fðx; yÞ at a give surface point Dðx; yÞ due to the displacement is [5] fðx; yÞ ¼ kðx; yÞ Uðx; yÞ;
ð5Þ
where kðx; yÞ is called the sensitivity vector 2p ½n1 ðx; yÞ n2 ðx; yÞ : kðx; yÞ ¼ ð6Þ l The fringe pattern, however, is an image with intensities rðx; yÞ that vary over the range [0,1], these are directly related to the wrapped phase fðx; yÞ via [5] rðx; yÞ ¼ 12ð1 þ cos fðx; yÞÞ:
ð7Þ
This gives a maximum (1= white peak) and minimum (0= black trough) fringe intensities when f is an even and odd integral multiple of p; respectively. Shortly, it will be more convenient to express Eq. (5) in terms of a fringe order n where n is any real number 1 ð8Þ nðx; yÞ ¼ ðkðx; yÞ Uðx; yÞÞ: 2p In this situation fringe peaks and troughs correspond n being any integer and any odd integer divided by 2, respectively. It is important to note that the fringe grey scale convention described here follows the definition given by Eq. (7) but this is not
Fig. 2. Geometry of speckle interferometric experiment. Laser beam LB is split by beam splitter BS propagating to source points S1 and S2 where it is expanded by scattering to illuminate the surface of the object. Rays propagate from S1 and S2 to arbitrary surface point Dðx; yÞ with directions indicated by unit vectors n1 and n2, respectively.
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Fig. 3. Synthetic fringe pattern calculated for a point force F ¼ 1500 N applied at the centre of white cross at point P(x ¼ 9:5; y ¼ 3:2; z ¼ 0) to a material of E ¼ 3 GPa and n ¼ 0:45 created with light of wavelength l=829 nm. Dark and light lines delineate a few of the larger fringe peaks and troughs found by manual picking, the corresponding fringe orders are also shown. Bright fringe ‘A’ is reference fringe Fig. 4. Source positions used in calculations are S1(x ¼ 4:6; y ¼ 7:0; z ¼ 5:3) and S2(x ¼ 23:5; y ¼ 7:3; z ¼ 6:3).
necessarily used by other workers where different ESPI fringe determination methods relying on a variety of correlations are employed [4]. An example of a synthetic fringe pattern calculated using Eqs. (1)–(7) for a point load to the surface of an object with fringe orders given for a few peaks and troughs is shown in Fig. 3.
3. Fringe inversion method Extracting quantitative information from ESPI fringe patterns remains challenging. The direct determination of displacements at a point-by-point basis using specialized phase shifting techniques are perhaps the most popular way that quantitative information is extracted using ESPI (see Ref. [4] for an overview). However, if the expected induced displacement field has a well-defined analytic or numeric description, then a statistically based inversion methodology can be employed. That is, the parameters that are sought, which are in the present paper the elastic properties, may be directly obtained from the fringe pattern, or more precisely, knowledge of the spatial distribution of fringe order nðx; yÞ that the pattern maps. The technique here borrows from the geophysical literature in which large and often high-noise sets of observations are inversely modelled to provide some idea of the unseen internal structure of the earth. This method has recently been successfully applied to ESPI fringe patterns that recorded rigid-body translations [9] and to residual stress determination by blind-hole drilling [10].
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To determine the best values of Young’s modulus E and Poisson’s ratio n from the picked fringe positions of images, an iterative minimum error procedure has been employed. The method assumes that the character of the displacements due to a known load on the sample surface is described by Eqs. (2)–(4). Following through on substitution of these formulae into Eqs. (6) and (8) with algebraic rearrangement gives a form that will be amenable to least-squares inversion nðx; yÞ ¼ Aðx; yÞa þ Bðx; yÞb; where the knowns A and B rely on l; F ; x; and y ky kx Fx0 Fy0 A¼ 2 þ ; 4p ðx02 þ y02 Þ 4p2 ðx02 þ y02 Þ
kz F B¼ 2 2p ðx02 þ y02 Þ1=2
ð9Þ
ð10aÞ
! ð10bÞ
and a and b are the unknown constants containing the elastic properties a¼
ð1 2nÞð1 þ nÞ E
ð11aÞ
b¼
ð1 n2 Þ : E
ð11bÞ
and
In an experiment both A and B are easily calculated as the position and force are known, conversely a and b remain to be determined and they contain the elastic information n and E: Eq. (9) is useful as it allows nðxi ; yj Þ; the fringe order observed within the fringe pattern image at pixel ði; jÞ; to be expressed by matrix multiplication a ½nðxi ; yj Þ ¼ ½Aðxi ; yj Þ Bðxi ; yj Þ ; ð12Þ b which may be expanded for many fringe order observations at m pixels within the image 3 2 3 2 Aðx1 y1 Þ Bðx1 ; y1 Þ nðx1 ; y1 Þ 6 nðx ; y Þ 7 6 Aðx ; y Þ Bðx2 ; y2 Þ 7 2 2 2 2 7 6 7 6 7 6 7 6 6 nðx3 ; y3 Þ 7 6 Aðx3 ; y3 Þ Bðx3 ; y3 Þ 7 a 7¼6 7 6 ð13Þ 7 6 7 b ; 6 ^ ^ ^ 7 6 7 6 7 6 7 6 4 nðxm1 ; ym1 Þ 5 4 Aðxm1 ; ym1 Þ Bðxm1 ; ym1 Þ 5 nðxm ; ym Þ Aðxm ; ym Þ Bðxm ; ym Þ
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which may be represented as matrix algebra as N ¼ GM where N is the vector of observed fringe orders on the left had side of Eq. (13), G is the twocolumn matrix that holds the known information on the applied load and experimental geometry, and the unknown vector M ¼ ½a b T is to be determined. In this form, one can use the well known least squares minimization technique to determine M M ¼ ðG T GÞ1 GT N
ð14Þ
from which a and b may be derived. There is one major difficulty in the application of Eq. (14). The values in the fringe pattern itself range over [0 to 1] and hence are a wrapped fringe order only; the unwrapped true fringe order value is not necessarily known. An iterative procedure is employed to overcome this problem [11]. First, continuous fringe peaks and troughs are picked within the image. All of the pixels, each of which are mapped to a specific ðxi ; yj Þ position on the surface, along such a continuous feature must have constant fringe order n: For example, the fringe order at a peak must have an integral value, say n0 : The fringe order for the troughs adjacent to this peak must differ from n0 by 712: Similarly, the adjacent peaks must have values from n0 of 71. The fringe order values for subsequent farther peaks and troughs can also be assigned a fringe order relative to n0 as the difference in the fringe order between any two such continuous features must progressively increase or decrease in a well defined fashion. This is illustrated in Fig. 3 where the reference fringe A has a fringe order n0 ¼ 2 which is known absolutely because the fringe pattern is calculated. The fringe order for successive troughs and peaks that are chosen decreases in a clock-wise fashion around the force application point in Fig. 3. As the first step towards inversion, one feature of the fringe pattern is chosen as reference. An arbitrary but appropriate initial fringe order value n0 is assigned to this feature, for example a guessed integer value is assigned to the pixels associated with the reference fringe A in Fig. 3 because it is a fringe peak. Appropriately incremented fringe orders are then applied to the subsequent fringe peaks and troughs. Forward modelling of the anticipated fringe pattern can assist in assigning fringe order values. Once this procedure is carried out, a trial solution to M is found using Eq. (14). In most cases this initial ‘guess’ at n0 will not be correct, however, and this must result in an erroneous solution. There are a variety of ways to measure this least square error e but here e ¼ jN GMj2
ð15Þ
is employed. The entire procedure is then repeated by incrementing or decrementing the fringe order values by unity and again solving for a new trial M and corresponding fringe order error e by Eqs. (14) and (15), respectively. The procedure is repeated until a minimum error is detected. The fringe order assignment for this
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minimum error is most correct and will provide, in the least-square’s sense, the best value of M for determination of E and n: In the following sections this procedure is first tested on the simulated fringe pattern of Fig. 3 under a variety of assumptions and then applied to real experimental fringe patterns.
4. Test on simulated fringe pattern The calculated synthetic fringe pattern of Fig. 3 is inverted to get the E and n values by the fringe picking method described in the last section. Errors are purposefully introduced to the analysis in order to assess quantitatively the level of uncertainty that might be expected in the analysis. The fringe order and ðx; yÞ position are by default, as part of the calculation, absolutely known at every pixel in the forward modelled fringe pattern of Fig. 3. Consequently, one initial performance test is to directly invert these simulated values using Eq. (14) as this should provide the best solution possible. This was the case with the determined values of E and n differing from those input by less than one part in 10,000 (Table 1). This negligible difference is due to computer round off errors. Next, a series of 13 more realistic attempts were carried out in which the fringe peaks and troughs were manually picked (Table 1). Details of the fringe peak and trough picking procedure are described in Ref. [11] but the peaks and troughs picked in one of the trials within the series are superimposed on the fringe pattern of Fig. 3. The iterative procedure described in the previous section was applied to each of these
Table 1 Results from inversion of the synthetic image in Fig. 3 [Trial No.]
[Fringes picked]
No. of Points picked
E (GPa)
n
e
% Error in % Error in E n
1 2 3 4 5 6 7 8 9 10 11 12 13 Direct inv
8
106 121 126 121 116 109 121 117 123 121 126 121 116 —
3.0343 3.0136 3.0350 3.0347 3.0272 3.0082 3.0317 3.0025 2.9957 3.0290 3.0471 3.0284 3.0459 2.9999
0.4504 0.4510 0.4504 0.4505 0.4506 0.4511 0.4505 0.4513 0.4514 0.4499 0.4493 0.4499 0.4493 0.4500
0.0020 0.0017 0.0020 0.0034 0.0027 0.0025 0.0017 0.0024 0.0027 0.0314 0.0380 0.0332 0.0340 0.0088
1.14 0.45 1.16 1.15 0.90 0.27 1.05 0.08 0.14 0.96 1.57 0.94 1.53 0.003
—
0.08 0.22 0.08 0.11 0.13 0.24 0.11 0.28 0.31 0.02 0.15 0.02 0.15 0.0
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trials and are summarized in Table 1. For the most part, these results suggest that pixel picking errors are generally less than 1.5% for E and less than 0.4% for n: These results were encouraging especially given that only a limited number of points were employed in the inversion in each case; picking of more fringes should allow for even lower levels of error. Finally, an even more involved test sought to find the levels of error that could be introduced by uncertainties in the determination of the geometry of the experiment. These could arise from the uncertainty in finding the relative three-dimensional positions of the sources S1 and S2, the force application point P, and the many positions D(xi,yj) of the pixels in the image. In addition to direct measurement errors, distortion of the image due to camera placement and lens aberration adds to positioning problems. The force level F must also have some level of uncertainty that will propagate through to influence the final result. In the following, the input parameters for the inversion will be slightly changed from their exact values one at a time in order to assess the solution’s sensitivity to them. To do this, the calculated fringe pattern of Fig. 3 has been inverted once again first using the known input parameters to yield the results in the first row of Table 2. The co-ordinates of the picked points, the assigned fringe orders and the total number of picked points have been saved and subsequently used in the later inversions where only the input parameters are gradually changed. This keeps the consistency between different inversions. The percentage of error has been calculated from the first ‘error-free’ result relative to the later ones (Table 2). The degree of error that might be associated with a certain parameter is selected with some consideration of a real experimental arrangement, i.e., by estimating the amount of uncertainty that might exist in the experimental determination of those parameters. Accordingly, *
*
*
*
The co-ordinates of the laser sources could be uncertain by as much as 73 mm (cumulative from both the measuring ends) due to the complex positioning of the sources relative to the origin and difficulties associated with measuring points in 3-D space. This estimate is based on repeated attempts at determining the source positions at the lab bench. The viewing area of the sample surface may vary by as much as three millimetres due to the cumulative effect of the errors in measurement and in removal of lens distortion in the final image. The applied force is assigned an error of 730 N based on the 2% uncertainty specified for the load cell used in the later experiments. The co-ordinates of the force application point have been assigned an uncertainty of 72 mm which again arises from the measurement uncertainty.
It is evident from Table 2 that the co-ordinates of the point of force application (both x and y co-ordinates individually and combined as seen in rows 2, 3 and 4), x and z-coordinates of the first laser source S1, z-coordinate of the second laser source S2 and the length of the block are among the most sensitive parameters. These parameters individually have contributed more than 2% of error to each or both of
520
Trial no. Viewing area (cm) Force Point (cm)
19.4 14.3 19.4 14.3 19.4 14.3 19.4 14.3 19.4 14.3 19.4 14.3 19.4 14.3 19.4 14.3 19.4 14.3 19.4 14.3 19.4 14.3 19.4 14.3 19.4 14.3 19.4 14.3 19.4 14.3 19.4 14.3 19.4 14.3 19.1 14.3 19.7 14.3 19.4 14.0 19.4 14.6 19.7 14.6
X=8.53, X=8.35, X=8.53, X=8.35, X=8.53, X=8.53, X=8.53, X=8.53, X=8.53, X=8.53, X=8.53, X=8.53, X=8.53, X=8.53, X=8.53, X=8.53, X=8.53, X=8.53, X=8.53, X=8.53, X=8.53, X=8.35,
Y=2.70 Y=2.70 Y=2.52 Y=2.52 Y=2.70 Y=2.70 Y=2.70 Y=2.70 Y=2.70 Y=2.70 Y=2.70 Y=2.70 Y=2.70 Y=2.70 Y=2.70 Y=2.70 Y=2.70 Y=2.70 Y=2.70 Y=2.70 Y=2.70 Y=2.52
15 15 15 15 14.7 (70.3) 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 14.7
S1 cm x
y
z
S2 cm x y
z
4.6 4.6 4.6 4.6 4.6 4.3 4.9 4.6 4.6 4.6 4.6 4.6 4.6 4.6 4.6 4.6 4.6 4.6 4.6 4.6 4.6 4.3
7.0 7.0 7.0 7.0 7.0 7.0 7.0 6.7 7.3 7.0 7.0 7.0 7.0 7.0 7.0 7.0 7.0 7.0 7.0 7.0 7.0 6.7
5.3 5.3 5.3 5.3 5.3 5.3 5.3 5.3 5.3 5.0 5.6 5.3 5.3 5.3 5.3 5.3 5.3 5.3 5.3 5.3 5.3 5.6
23.5 23.5 23.5 23.5 23.5 23.5 23.5 23.5 23.5 23.5 23.5 23.2 23.8 23.5 23.5 23.5 23.5 23.5 23.5 23.5 23.5 23.2
6.3 6.3 6.3 6.3 6.3 6.3 6.3 6.3 6.3 6.3 6.3 6.3 6.3 6.3 6.3 6.0 6.6 6.3 6.3 6.3 6.3 6.0
7.3 7.3 7.3 7.3 7.3 7.3 7.3 7.3 7.3 7.3 7.3 7.3 7.3 7.0 7.6 7.3 7.3 7.3 7.3 7.3 7.3 7.6
E GPa % Error in E n
% Err. in n
3.02 3.14 2.86 2.77 2.96 3.11 2.97 3.05 2.99 3.01 3.28 3.06 3.01 3.02 3.03 3.17 3.05 3.14 2.72 3.04 2.96 3.07
0.0 1.5 0.9 0.9 0.0 1.1 0.8 0.3 0.3 1.6 3.2 0.4 0.5 0.2 0.2 2.4 1.2 1.3 2.5 0.2 0.5 4.8
0.00 3.77 5.34 8.27 2.00 2.79 1.70 0.96 0.90 0.58 8.58 1.27 0.52 0.02 .024 4.77 0.96 3.80 10.2 0.38 2.02 1.64
0.450 0.443 0.454 0.454 0.450 0.445 0.454 0.449 0.451 0.457 0.436 0.452 0.449 0.451 0.449 0.439 0.455 0.444 0.461 0.449 0.452 0.43
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F : (N) x100 Coordinates of the laser sources
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Table 2 Sensitivity tests using calculated fringe pattern
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Fig. 4. Least-squares error versus arbitrary fringe order assigned to fringe ‘A’ in Fig. 3.
the output values E and n: When all of them are considered together, the values of E and n resulted with 1.7% and 5% errors respectively (row 22, Table 2).
5. Experimental tests and results Experimental tests were conducted on an acrylic block and compared to corresponding strain-gauge type measurements carried out on a testing machine. The overall experimental configuration (Fig. 5a) is mounted on a specially prepared loading frame that holds the mechanical components. The experimental details are provided in the subsections below. Briefly, however, the recording process consists of capturing two images of the speckle patterns before and after the deformation, i.e. force application. A correlation procedure is then implemented to produce the fringe pattern rðx; yÞ from which is then taken the positions of the continuous fringe features as inputs to the inversion. 5.1. Sample preparation The laboratory tests were done using acrylic (polymethyl-methacrylate or PMMA) material. This material was selected as it remains elastic over the time periods of our test and it has a relatively low Young’s modulus for a solid; this greatly simplifies construction of the loading frame and selection of the load cells. The acrylic block was cut with dimensions of 25 cm 25 cm 5 cm. The block was machined square then annealed at low temperature in an oven in order to reduce residual stresses that may exist. This was qualitatively evaluated on the block by examining birefringent
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Voltmeter
Computer
CC D
Ram
S1 z n1
Load Cell
Indenter
S2 n2
y D
x
PMMA Block
(a)
CCD Camera Hydraulic Ram Beam Splitter
Indenter
S2 S1 PMMA Block
(b) Fig. 5. (a) Experimental configuration. CCD is the charge coupled camera with feed to the computerbased image capture and processing system. (b) Photograph of experimental equipment.
effects under cross-polarization, no fringes were observed near the centre of the annealed block. The surface of the acrylic block were first sand blasted with a fine grit then painted with a reflecting white paint to produce better scattering of laser light. This improves the image quality significantly; as the speckle technique requires an optically rough surface. This material is then placed on a solid steel surface (Fig. 5b) that is a part of the loading frame. As the goal of this study is to measure the elastic properties, it is important that alternative measures be available for comparison. A conventional strain gauge method, which is widely used in determining materials properties, has been used to measure the E and n values for the acrylic sample. A small piece of acrylic (12 cm 2.8 cm 0.69 cm) was cut from the sample used in the optical tests. The ends of the sample were fixed in the jaws of a universal testing machine. This machine is preset to apply 272.11 kg (600 lbs) of load over 5 s. The resultant strains (both axial and transverse) are measured by two strain gauges (1/16 in. grid length, 90 stacked gauges) attached on the sample. These strain gauges are connected to a signal conditioner and eventually to a digital scope that collects the data. The E is measured from the slope of the stress-strain curve [1] and the Poisson ratio is
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measured from the ratio of the transverse strain to the axial strain [2]. This procedure gave values of E ¼ 3:070:1 GPa and of n=0.3870.01. It is worthwhile noting that the manufacturer quotes a value of E ¼ 3:1 GPa for this material. 5.2. Mechanical system The mechanical system consists of the loading frame, sample, hydraulic piston, load cell, pump, hand-driven pressure accumulator, and associated recording equipment (Fig. 5). The force is applied to the sample via an indenter pushed onto the surface with a hydraulic ram. The hydraulic hand pump is connected to the cylinder and piston assembly and to an accumulator. The accumulator is used to achieve small changes in the line pressure. A load cell is attached between the piston and the indenter to yield the force exerted on the sample surface by the indenter. The signal from the load cell passes through a signal conditioner and the force is directly read in a calibrated voltmeter. The indenter tip is made approximately 0.5 cm in diameter to prevent the sample from cracking under the load. Note that care has been taken to avoid irreversible plastic deformation or fracturing on the material by keeping the loads relatively small (o1500 N). Reversibility was confirmed at the end of the optical tests as no fringes were seen in the live fringe pattern display when the applied force was released to its initial reference value. 5.3. Optical system The optical system consists of a coherent light source, a beam splitter, a CCD camera, source points, and computer-based high-speed image acquisition and processing system (Fig. 5). The optics are mounted on the floating optical bench but are not touching the loading frame in order to reduce any inadvertent relative motions between the sources and the object that would influence the fringe pattern. Coherent light was provided by a 35 mW stabilized laser diode emitting in the near infrared at l=829 nm. This wavelength range was chosen on the basis of the CCD camera sensitivity. The laser head module contains the laser, collimation optics, beam shaping optics and thermoelectric cooler, thermistor, and heat sink. Together these provide a highly stable, round collimated source. The beam propagates to a B50% splitter then onto the two source points S1 and S2 that scatter the light to illuminate the object’s surface. The rough surface produces a speckle pattern that is captured in a grey scale image (8 bit, 640 pixels horizontally and 480 pixels vertically) by a high-resolution Charged Coupled Device (CCD) camera. The output of this camera feeds to an image capture board on a 500 MHz computer (ca. 2001) where further processing occurs. The minimum experiment requires that two such speckle pattern images, one reference image captured immediately before deformation and one monitor image acquired once the object has been deformed. Two such speckle pattern raw images (Fig. 6) provide little useful information on their own. A fringe pattern, however, is calculated using a localized correlation procedure previously [7] that relies local
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Fig. 6. Raw speckle patterns captured (a) before and (b) after application of a 1300 N load at the point indicated in (a).
calculation of the Pearson’s correlation coefficient that has been shown to be equivalent to a mapping of rðx; yÞ: The top left corner of the image is chosen as the origin and the source positions are measured with respect to it. The co-ordinates of the point of force application are also given relative to this origin and all subsequent measurement such as positions of each pixel, are measured relative to the force application point. Examples of two such observed fringe patterns acquired in the series of experimental tests are shown in Figs. 7a and b for applied forces F of 1000 N and 1300 N, respectively. It is worth commenting that these fringe mappings are considerably noisier than the purely noise free patterns calculated (e.g. Fig. 3). A discussion of this speckle noise is beyond the scope of this paper but it has been found to be inherent to the correlation technique used here to calculate the fringe pattern [12].
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Fig. 7. Observed correlated fringe patterns for a (a) 1000 N force and (b) 1300 N force applied to the acrylic test block. Black and white lines superimposed on the fringes indicated locus of picked peaks and troughs, respectively. Fringe patterns calculated to compare using elastic properties obtained by the inversion of the fringes picked in (a) and (b) for the (c) 1000 N force and the (d) 1300 N force.
5.4. Fringe pattern analysis In the inversion, the experimental image is first corrected for perspective distortion [13]. The images captured suffer from perspective distortion mainly due to the angle of view of the camera, which captures the images of the undeformed and deformed states of the surface under investigation, with respect to the surface normal. Due to this oblique sight the inspected surface appears deformed in the recorded image and may not fill the full frame. A spatial transform maps the pixels of the recorded interferograms to new pixels in the output interferogram, such that identical points of the object surface, which are imaged to different pixels in the recorded images, are mapped to identical pixels in the output images. Then points are picked manually along the peak and trough of the bright and dark fringes respectively (Figs. 7a and b). The peak and trough are selected manually with the interpreter biases included. This method is not ideal but was used as it provides a larger level of error in the determination of the fringe extrema and as such is a good test of the method. It is worth nothing that it can be very difficult to remove the high frequency speckle noise in such fringe patterns without introducing further error.
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Table 3 Results of interferometric inversions Force (N) Trial No. No. Fringe picked
Points picked
E (GPa)
n
e
Average E Average n (GPa)
1000
1 2 3 4 5
8
280 391 386 484 597
3.31 3.11 3.05 3.15 3.16
0.37 0.39 0.39 0.39 0.38
0.56 0.58 0.74 0.67 0.48
3.170.3
0.3870.03
1300
1 2 3 4 5
6
297 330 436 308 503
3.35 3.16 3.19 2.9 2.81
0.37 0.37 0.39 0.39 0.38
4.4 6.8 5.35 4.5 5.0
3.170.3
0.3870.03
The results (Table 3) from a series of repeated tests were carried out on the fringe patterns of Figs. 7(a) and (b); the averages obtained are E ¼ 3:170:3 GPa and n=0.3870.03, in very good agreement with the conventionally determined values of E ¼ 3:070:1 GPa and of n=0.3870.01 and with the manufacturer’s quoted value of E ¼ 3:1 GPa. The optically derived elastic properties are used along with the input parameters to reproduce the experimental fringe patterns in Figs. 7(c) and (d) for purposes of comparison. Again, the character of the corresponding observed and calculated fringe patterns are similar indicating that the inversion procedure is working acceptably.
6. Concluding remarks The ESPI method employing fringe inversion by direct correlation resulted in values of Young’s modulus E and Poisson ratio n with errors of 2% and 0.3% respectively in the case of synthetic fringes. For experimental fringes the result of E for Plexiglas matched with the manufacturers specification of 3.1 GPa and is within 3% of the 3.0 GPa determined by the standard strain gauge method. The Poisson ratio n is found to be 0.38, which also matched with the value obtained by the strain gauge method (0.38). These results suggest that the point force method can be successfully used to obtain the elastic properties of isotropic materials. The major advantages of this technique are: that it is a non-destructive method and the sample can be considered for subsequent use. The sample needs little in the way of additional preparation in many cases. The full field image gives more information than the standard strain gauge method. Moreover, the inversion method is simple and straightforward. In the present paper, additional processing of the fringes was not employed but this may have added interpreter bias and error. As up
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to four separate fringe patterns can be calculated in near real time in our present system configuration, the technique can show some promise for time-lapse implementation to study progressive deformation for rheological studies. At present, we are able to acquire, calculate, and store four separate fringe patterns per second. Future work will focus on developing more automated methods of fringe inversion that may employ nearly all the data within the image eliminating the need for cumbersome and potentially error-prone manual fringe picking.
Acknowledgements The work was greatly assisted by technical data acquisition developments of F. Weichman, W. Engler, L. Tober, M.Diallo, M. Lazorek, and I. Rumzan. This work was made possible with funding provided by a NSERC Strategic Grant on ESPI.
References [1] ASTM Standard E111-97. Standard test method for Young’s Modulus, tangent modulus, and chord modulus. Annu Book ASTM Stand 2002;03(1):244–50. [2] ASTM Standard E132-97. Standard test method for poisson’s ratio at room temperature, Annu Book ASTM Stand 2002;03(1):275–77. [3] Vest CM. Holographic interferometry. New York: Wiley; 1979. p. 465. [4] Cloud GL. Optical methods of engineering analysis. New York: Cambridge University Press; 1995. p. 517. [5] Jones R, Wykes C. Holographic and speckle interferometry. Cambridge, UK: Cambridge University Press; 1983. p. 330. [6] Schmitt DR, Smither C, Ahrens TJ. In-situ holographic elastic moduli measurements from boreholes. Geophysics 1989;54(4):468–77. [7] Schmitt DR, Hunt RW. Optimization of fringe pattern calculation with direct correlations in speckle interferometry. Appl Opt 1997;36(34):8848–57. [8] Fung YC. Foundations of solid mechanics. Englewood Cliffs, NJ: Prentice-Hall, Inc.; 1965. p. 525. [9] Schmitt DR, Hunt RW. Model based inversion of speckle interferometer fringe patterns. Appl Opt 1998;37:2573–8. [10] Schmitt DR, Hunt RW. Inversion of speckle interferometer fringes for hole-drilling residual stress detrminations. Exp Mecha 2000;40:129–37. [11] Shareef S. Determination of elastic coefficients of materials by laser speckle interferometry. MSc thesis, Department of Physics, University of Alberta, Edmonton, Alberta; 2002. p. 123. [12] Engler WG. Laser speckle interferometry: a stochastic investigation. Department of Physics, University of Alberta, Edmonton, Alberta. 2002. p. 209. [13] Rastogi PK. Holographic interferometry: principles and methods. Berlin, New York: Springer; 1994. p. 170–3.