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Dynamic material parameters in an anisotropic plate estimated by phase-stepped holographic interferometry
Oprio ond Laser.$ in Engineering 24 (1996) 429-454 Cop@ht 0 1996 Elsevier Science Limited Printed m Northern Ireland. All rights reserved 014X-R166/Y6/$1500 0143-8166(95)00100-x
ELSEVIER
Dynamic Material Parameters in an Anisotropic Plate Estimated by Phase-stepped Holographic Interferometry Karl-Evert
FallstrBm,
Kenneth Staffan
Olofsson, Schedin
Henrik
0. Saldner
&
Division of Experimental Mechanics, Lule5 University of Technology, S-97187 Luleii. Sweden. Tel: + 46-920 91000. Fax: + 46-920 91047. E-mail: [email protected] (Received 3 October 1994; revised version received 18 April 1995; accepted 24 April 1995) ABSTRACT Material parameters in an anisotropic plate are determined using two non-destructive measuring techniques: real-time phase-stepped electronic speckle pattern interferometry and dual-reference-beam pulsed holographic interferometry. The first technique is used to measure the lower modes of vibration of the plate with free-free boundary conditions. Finite element analysis is then used to determine two effective Young’s moduli and the in-plane shear modulus. The second technique is used to detect transient bending waves propagating in the plate and acoustic waves propagating in the surrounding air. A double-pulsed laser is used both to generate the waves and to make holographic recordings of the wave fields. The stiffness of the plate is estimated using the measured deformation field and an analytical solution to the plate impact problem.
1 INTRODUCTION Non-destructive testing (NDT) is an important tool to improve the quality of a product. In order to guarantee that the properties of a component match the given requirements, a non-destructive test can be used to find material defects and elastic constants. Knowledge of the elastic constants of fibre composites is important in optimum design and quality control of a product. The stiffness of standard materials such as metals and homogeneous plastics can easily be tested with conventional methods, such as ultrasonic and tensile 429
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testing. Defects and elastic constants in materials like fibre-reinforced plastics can hardly be found with standard methods. In this case an additional or alternative method has to be applied. One way to achieve this is to use optical techniques like holographic interferometry, speckle interferometry, shearography, etc. Sometimes such experimental methods are combined with finite element simulations into a hybrid method’ to become a powerful tool in NDT. One advantage of these nondestructive methods is that they give a whole-field visual display. These techniques have been known since the mid-1960s but have been much improved in recent years. One of the advances is that interferograms can now be transformed into 3-D displacement maps, which simplifies the interpretation of the measured results. In this paper speckle interferometry* and pulsed holographic interferometry” are used to estimate the elastic constants of a composite plate. In 1987, Fallstrom & Molin4 reported a method to estimate the stiffness of orthotropic plates from the lower modes of vibration. They used traditional electronic speckle pattern interferometry (ESPI) with poor quality interferograms. An up-to-date electronic holography technique, real-time phase-stepped ESPI, was used in the present study. In 1990, Fallstrom et ~1.~7~used holographic interferometry to study the propagation of transient bending waves in isotropic plates. A closed solution to the Euler plate equation was given with the initial conditions modelled as a Dirac-pulse in space and time. In this paper, a solution for an orthotropic plate is given with the same initial conditions. An important advantage is that these experiments can be performed not only on free plates but also on plates with any boundary conditions, because only the initial part of the propagating wave is studied, i.e. the bending waves have not reached the boundaries for the time periods which are investigated. In Refs 5 and 6 a pendulum was used to impact the plates. A disadvantage of using a pendulum is that the impact time can be quite long (several tens of microseconds) compared with the wave propagation time. In this paper, however, the double-pulsed laser is used both to generate and to record the bending waves, thereby no triggering problems arise and the impact can be regarded almost as a Dirac-pulse both in space and time. A disadvantage is that the transferred impulse to the plate is difficult to determine because knowledge of the mass and the velocity of the evaporated material from the plate can barely be measured. It is possible to estimate the impulse from the interference pattern, for example, but in this experiment this parameter is of no interest.
Dynamic material parameters in an anisotropic plate
2 SPECIMEN
DESCRIPTION
A glass-fibre reinforced composite plate is examined in this paper. plate was wound on a flat tool in a filament winding machine and pressed between two steel blocks to achieve a constant thickness. plate was cured for 4 h in a convection oven, The properties of the are shown in Table 1.
Material
431
The then The plate
TABLE 1 Properties of the Test Plate
Density Laminate sequence Dimension
2027 kg/m’ (2W - 24),* 7.7 X 17.3 X 0.315 cm
* The plate consists of three layers oricnlcd at 2” relative to the longer side of the plate and three layers at -2” relative to the same side. Furthormore. the plate is symmetric with respecl to the neutral line of the plate.
For orthotropic plates there are three main material parameters which describe the stiffness of the plate: the Young’s module in the xand y-directions (E, and E,.) and the in-plane shear modulus (~5,). These parameters can be determined from the first three vibration modes. 3 ELECTRONIC
HOLOGRAPHY
3.1 Experimental arrangement A real-time phase-stepped speckle interferometric technique called electronic (electro-optic) holography has been used in its time-average mode. The system used was developed in the USA by United Technologies Research Center (UTRC),7.X see Fig. 1. The plate is illuminated by a 500 mW frequency-doubled Nd:YAG laser (wavelength 532 nm). The image of the plate interferes with a smooth reference beam from an optical fibre onto a CCD camera. The piezo-mounted mirror (PZT) introduces optical phase shifts of 90” between subsequent images. The four latest images are treated by an image processor and presented as interferograms on the monitor in real-time (30 frames per second). A host computer controls the image processor and the phase-stepping mirror, PZT. To reduce. the speckle noise and improve the quality of the interferograms, speckle averaging is performed. A
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OPTICAL HEAD
PZT bhib.
(a) PZT (phase step)
CCD
Video recorder (b)
Fig. 1. Optical arrangement for electronic holography. (a) The optical head of the electronic holography system and the analysed plate: PZT, piezo-mounted mirror; BE, beam expander; BS, beam-splitter; R, relay lens. (b) The electronic components of the system.
number of interferograms are averaged to produce the final picture. Each interferogram is given a slightly different illumination direction to give a different speckle noise. The speckle average is not performed in real-time. A time-average interferogram, as presented by the system after speckle averaging, is shown in Fig. 2a. A loudspeaker is used as an exciter to vibrate the plate.
Dynamic material purameters in an unisotropic plate
43.1
The second PZT mirror, in the illumination beam, is used both to find the phase relation between anti-nodes and for the quantitative evaluation of the mode shape. This mirror is set to vibrate at equal frequency, amplitude and phase as parts of the object, as a bias vibration. These parts of the object will then act as new nodal lines. The real nodal lines will be given a number of fringes that correspond to the amplitude of the bias vibration of the PZT. By switching the bias vibration on and off, the phase of the anti-nodes can be determined from the increase or decrease of the number of fringes in each anti-node. This facility is needed to understand complicated modes of vibration of the object, especially when several sides of the structure are imaged simultaneously. Jansson ef al.’ have studied modes of vibration of the violin. Three sides of the violin were studied simultaneously via mirrors. The bias facility made it possible to decide if anti-nodes are in phase or not. Numerical evaluation of the vibration amplitude distribution can also be performed by recording three interferograms, the first without bias, the second with positive bias and the third with negative bias together with a post-processing routine; see the details in Ref. 8. The interferogram in Fig. 2a is one of the three interferograms that have been used to obtain Fig. 2b. The vibration mode in Figs 2a and b is -the second mode of the free-free plate at 200 Hz and is called the (0,2)-mode. The label (0,2) means zero nodal line parallel to the longer side of the plate (x-direction) and two nodal lines parallel to the shorter side (ydirection). To obtain the free-free boundary conditions of the plate, the supports have to be moved to the location of the nodal lines. We have used small pieces of rubber as supports. By tuning the excitation frequency through a resonance frequency, the nodal lines will not move if the supports are placed correctly. A real-time-field technique is favourable to visualize the modes of vibration and the supports. High vibration amplitudes, giving a large number of fringes, can be used with the present technique, which makes it easier to place the supports correctly. Interferograms of the other two modes, (1,l) at 128 Hz and (2,O) at 304 Hz, are shown in Figs 2c and d.
3.2 Experimental
results and discussion
A finite element (FE) model of the plate is used with free-free boundary conditions. The FE model of one-quarter of the plate has 160
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64
y-axis (mm)
0-o
x-axis (mm) W
Fig. 2. First three modes of vibration for the plate with free-free boundary conditions. (a) Interferogram of the (0,2)-mode and (b) a plot of the measured mode shape of the (0,2)-mode. Interferograms of (c) the (l,l)-mode and (d) the (2,0)-mode.
Dynamic
material
parameters
in an anisotropic
(4 Fig. 2.
(Continued.)
plate
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triangular, three node, flat, shell elements. The geometry and the density of the plate are known. A parameter study is performed by changing the two Young’s moduli E, and Ey and the in-plane shear modulus Er. From the FE analysis it was shown that each one of the three resonance frequencies in Fig. 2 was strongly coupled to only one of the Young’s moduli or the in-plane shear modulus. The resonance frequency of the (l,l)-mode is most dependent on the shear modulus and only slightly dependent on the Young’s moduli. The frequencies of modes (0,2) and (2,0) depend most on the Young’s moduli in the Xand y-directions, respectively. The results of the FE analysis are shown in Figs 3a-c, one graph for each mode. The experiments give the three resonance frequencies. These frequencies are used together with the FE results to decide which E,, E, and E,$are looked for.
4 DUAL-REFERENCE-BEAM PULSED INTERFEROMETRY
HOLOGRAPHIC
One problem with traditional holographic interferometry is that there is a sign ambiguity for the fringes. It is difficult to know if the fringe pattern shows a valley or a hill, therefore it would be better if the interferograms could be directly transformed into a 3D displacement map. This problem has been solved by Crawforth et al.,” amongst others. They used a double-pulsed dual-reference-beam holographic measurement technique to achieve this. 4.1 Experimental arrangement The holographic set-up is a standard one (Ref. 3, pp. 57-64), arranged to be most sensitive to the out-of-plane displacement of the object. A beam splitter divides the pulses from a double-pulsed ruby laser into two beams; see Fig. 4a. One beam is used to create the bending waves and the other to record the bending waves. About 70% of the laser pulse energy is used to generate the bending waves. The object is the glass-fibre reinforced epoxy plate described in Section 2. The plate is held vertically using a clamp at one edge. The light used to record the hologram is deflected via two mirrors and becomes divergent after passing through a negative lens. The divergent beam from the negative lens is used to form an object beam and a reference beam which are recorded by photographic film as a hologram.
Dynamic
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plate
431
(4
Relative change of parameters
0.95
1
1.05
Relative change of parameters
Fig. 3. Frequency change versus parameter change for the three fundamental modes. Each mode is strongly related to only one parameter: (a) the (I,l)-mode and the shear modulus E,; (b) the (0,2)-mode and the Young’s modulus in the x-direction E,; and (c) the (2,0)-mode and the Young’s modulus in the ),-direction E, (on the next page).
The plate is recorded in an undeformed state, since the plate is impacted by the ruby laser pulse simultaneously with the first exposure. The next laser pulse is deliberately delayed with respect to the first pulse by between 1 and 800 ps. As the second pulse also impacts the plate simultaneously with the second exposure, the impact caused by
438
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Relative change of parameters Fig. 3.
(Continued. )
this pulse does not affect the interferometric data. During the delay, the rotating mirror (RM) has shifted the reference beam by about 0.01”. The second pulse records the plate in a deformed state caused by the first laser beam impact, but with a slightly shifted reference beam. The pulse length or exposure time is as short as 25 ns, so even propagating bending waves are recorded frozen in the hologram. In this manner, a series of two-reference-beam holograms can be recorded by repeating the experiment with increasing time delay. 4.2 Evaluation
of dual-reference-beam
interferograms
In the reconstructed image of this dual-reference-beam interferogram” the plate is covered with a set of cosinusoidal fringes. These interference fringes display contours of equal, out-of-plane displacement. The reconstruction arrangement is shown in Fig. 4b. A helium-neon laser (HeNe) illuminates a Michelson interferometer (M, BS, PZT) to produce two similar beams of light as output. A mirror (PZT) can be moved by a piezo-electric crystal to shift the phase in one beam relative to the other. The mirror (M) is adjusted to give an angular difference between reference wave 1 (Ref. 1) and reference wave 2 (Ref. 2) equal to the one used in the experimental arrangement in Fig. 4a. Reference waves 1 and 2 illuminate the hologram film plate (H) and reconstruct the object in its undeformed and deformed state simultaneously. The
Dynamic
(4
material
parameters
MIRROI P
in an unisotropic
plate
439
MIRROR
4 --+
HOLOGRAM
RY
(/
SPLITTER
for generating and recording bending wave propagaFig. 4. (a) Optical arrangement tion in a plate using double-pulsed dual-reference-beam holographic interferometry. and evaluation of dual-reference-beam (b) Optical arrangement for reconstruction interferograms.
440
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al.
undeformed and the deformed object interfere, giving the cosinusoidal fringes. These fringes can now be controlled by moving the (PZT) mirror. To determine the phase in the reconstructed interferogram, a phase-stepping technique is used.” The interferogram is reconstructed and recorded by a CCD camera (CCD) four times. The mutual phase difference of the two reference beams is changed each time between two reconstructions by the phase shift introduced by the piezo-mounted mirror (PZT). The local intensities &(x, y) in the interferogram of reconstructions k = 1, 2, 3, 4 are given by I,&, I’) = U(X,_Y){l+ rn(X, y> c0.s (+(X, y) + (k - 1)7r/2)},
(1)
where a(x, y) denotes the local mean intensity, m(x, y) the fringe contrast and 4(x, F) the interference phase. The phase 4(x, _v) at each point in the detector plane is
Ldx, y) - L(4 Y)
4J(xp y)=tan ‘&xv) Id
I&,
y
)I
(2)
The phase at each point, given by eqn (2), is evaluated by a computer program. To remove the 27r phase ambiguities a noise-immune phase unwrapping algorithm” is used. The coupling between the phase and the deformation of the object is found in eqn (3),
where A is the ruby laser wavelength, k, is the illumination vector and k, is the observation vector, both of unit length. w is the displacement vector. The out-of-plane displacement of the plate is calculated from the phase using eqn (3). 4.3 Laser-generated
bending waves
About 70% of the first laser pulse, which has an energy of 0.4-0.6 J, from the ruby laser is used to excite the bending waves (see Fig. 4a). The laser beam is deflected via a mirror to a positive lens, which focuses the beam onto the surface of the plate. The focused spot is typically 0*4mm in diameter. Most materials will melt and evaporate under such intense irradiation. The vapour leaves the surface at a very high velocity according to the high temperature created. This transfers a mechanical impulse to the target, causing it to recoil, similar to an impact of very short duration.
Dynamic material parameters in an anisotropic
plate
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The surface temperature is of the order of 104 K. A sharp bang can be heard and the generated blast will create an evaporated crater, about 0.5 mm in diameter, in the steel plate surface. Figures 5a and b show the propagating bending waves in a steel plate, 100 ps after the impact. Figure Sa is an interferogram and Fig. 5b is obtained using the phase-stepping and dual-reference-beam techniques. Figure 5b shows a deep valley in the vicinity of the impact point. The explanation of this is that the local high temperature spot gives rise to local surface tensions which force the plate to deform.14 These thermal stresses disturb the created bending waves, and much of the laser pulse energy is used to generate these stresses. However, if a spot of gelatine (water and gelatine powder) is placed at the impact centre most of the evaporating process can take place outside the structure, inside the gelatine. As the gelatine has a low evaporating point most of the energy is used for evaporation. In Ref. 14 it is shown that the impulse transferred to the plate can be many times larger with gelatine at the impact point than without, and the thermal stresses can almost be removed. Visualization impacted plate
4.4
of and effects of acoustic
waves outside the
When double-exposure or time-average hologram interferometry experiments are interpreted, the influence of pressure variations in the air surrounding the object are normally neglected. When rapid events are studied, like transient bending waves in plates, it is found that the influence of these spatial variations in the surrounding air cannot always be neglected. Bending waves with supersonic speed produce a sound pressure field localized quite close to the object. The probing laser light in the holographic recording is affected by changes in optical path length (which is the integrated product of the index of refraction and the geometrical path length) in the air as well as the changes which are introduced by object deformation or object motion itself. Figure 6 displays an unwrapped phase map showing the 2D projection of the transient acoustic near field around an impacted cantilever steel plate. It is the integrated values of the acoustic pressure changes from normal atmospheric pressure that appear as darker and brighter regions in the picture. Brighter regions indicate an increase in pressure of about 1% from the undisturbed conditions and darker regions a decrease in pressure with the same amount.‘” The data for the experiment are: plate dimension, 300 X 30 X 1 mm;
(b) 0.8 0.6
Fig. 5. (a) Interfcrogram of a propagating bending wave in a steel plate 100~s after the start of impact. (b) The corresponding 3D image obtained using dual-referencebeam recording.
Dynamic material parameters in an anisotropic plate
Fig. 6. outside
443
An unwrapped phase map of the 2D projected transient acoustic pressure field an impacted steel plate. The edge of the steel plate is seen as the vertical line in the centre of the picture.
444
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impactor, 0.5 g air gun; velocity of the gun, about lOOm/s; impulse transferred to the plate, about O-05 Ns. The impacting point is at the middle of the plate and the plate is impacted at right angles. The figure shows the event about 140 ps after the start of the impact. Acoustic waves can be seen moving both up and down the plate. They are known as trace matched acoustic waves and are generated by the travelling flexural bending waves moving with supersonic speed along the cantilever plate. Note that along the plate an increase in pressure on one side corresponds to a decrease on the other side. The transferred momentum from an impact with a laser pulse is much smaller than the impact with an air gun bullet. The experiments show that the surrounding acoustic field generated by the laser pulse affects the measured phase so little that this contribution can be ignored. 4.5 Theory of bending wave propagation
in anisotropic
plates
The theory of propagating bending waves will be treated briefly. A more detailed description is given in Ref. 16. The equation of motion for the out-of-plane displacement w of an orthotropic plate is”
01 YXXXX+ 2(D1* + 2 * &kxqy
+ &yvyyy
+ @w,n
= PC& Y, 0,
(4)
where Oij is the flexural moduli, h the thickness of the plate, the volume density of the plate, p(x, y, t) the load per unit area and the subscript comma indicates partial differentiation. This equation is based on classical plate theory and it is assumed that the principal material directions coincide with the X- and y-axes. Consider the transformations
g=x a
(5)
where a and b are constants. Then the differential equation becomes DI, %+2.&i - W,,-,,+ 2 a4
a*. b*
&Z wzFF + - w, b4
- -
+ phw,,, = p(x, y)
(6)
The impact force is modelled as a Dirac-pulse in time and space. Choose a and b in such a way that D1,/a4 = D2,1b4. A suitable choice is that a is, for example, the distance between the impact point to a
Dynamic material parameters in an anisotropic plate
445
maximum along the x-axis and b is the distance between the impact point to the same maximum along the y-axis.‘R*‘9Equation (6) can then be solved using Laplace transformation with respect to time variables and Fourier transformation with respect to X and 7. In polar coordinates (in the transformed system) the solution to eqn (6) will be
X
sin v(Dy,/ph)
th2dl + &[(sin (2p))2]/4
Ad1 + c[(sin (2p))2]/4
(7) where
,p,tD”
@a)
a4 Dp
=
02
12
2&,
(7) can be converted
(Ref. 20, p.464) to
+ (Fresnel C($)r}) where I = impulse transferred
@b)
a2b2
WV2 - XJ Wl
&=
Equation
+
ddp’
to the plate. P(cos (e - e))’
’ = 4th&k 30:
k, =
=
0
J 1
+ k,)lph + Of2 4
D:: -DE 4
k=
h
(9)
_
2k(sinc2’P))’ k+k,
(lOa> (lob) (1Oc) (104
K.-E. Ftillstriim et al.
446
The disadvantage with eqn (9) is that it is necessary to use numerical integration to estimate a numerical value of w. An approximate equation which gives a result which is very close to the exact solution if -0.15 5 k/(k + k,) 5 0.15 (valid for almost all plates) is W(T, 8, t) = ?r
” sin (x) du + 3 --cos c, z ( Cl x
1I
x2-,
cl =4&k, Equations (9) and system setting
+ k cos 48
(11) can be converted
8 = arctan %JM
(
ttan
cp 1
sin c, d
1
cos
48
I
(11)
(12) to the untransformed
W) W)
where 50 and r are the angle and the distance from the impact point measured in the untransformed system. 4.6 Estimating the parameters From the experiments the spatial location of the extremes of the transient bending wave can be found. If the density (p) and the thickness (h) for a plate are known, it is possible to estimate D,, and D,2 + 2D,, from eqn (9) or (11). Furthermore, D22 can be calculated using the relation D,,/a4 = D2,/b4. For estimating D,, it is necessary to know D,2. Plates are usually made out of identical layers. Normally the layers can be considered as identical and the stacking sequence of the layers is known. This being the case, it is possible to determine the D,2 modulus of the plate if the Poisson’s ratio for a single layer is known, using for example the laminate theory.2’ The term D,, + 2D,, influences the propagating bending waves very little, outside the first minimum, in the x-direction. Therefore, it is possible to estimate an approximate value of the D,, modulus by changing this modulus in eqn (11) so that the extremes measured from the interferogram or the amplitude diagram coincide with those calculated from eqn (11). To get a starting value for this iteration
Dynamic
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in an anisotropic
plate
447
process one can omit the last term in eqn (11). Then an approximate location of the extremes, in the x-direction, can be estimated by
where n = 1,2, . . . (odd value for minimum and even for maximum) and r is the distance from the impact point to a maximum or a minimum. Choosing another angle (e.g. q = 45”) it is then possible to estimate the term Qz + 2D,, by repeating the above procedure. The iterative process is then continued to the point where the results from eqn (11) coincide with the results from the interferogram. If the flexural moduli are knowr. it is then possible to estimate the effective material parameters (E,, E,, E,r) of plates.2’ 4.7 Experimental results and discussion Figures 7a and b show the phase map and the 3D plot of the orthotropic plate 70 ps after the impact start, respectively. Gelatine has been applied on the plate surface at the impact point. In the phase map one can observe that the bending waves have been reflected at the longer side of the plate. As the theory described in the foregoing part is valid only for infinite plates, that is, before the bending waves have reached the boundary, one must use a delay less then 70 ps for this plate, when estimating the flexural moduli for the plate. In the 3D plot, one can observe the indentation in the vicinity of the impact point. The interpretation is not obvious, but the disturbance is either a minor effect of thermal disturbance or that a small part of the laser light that has gone through the plate. Figures 8a and b show the displacement map and the 3D plot of the same plate but 50 ps after the impact start. No reflected waves can be found. From the sampled values one can now determine a (the distance from the impact point to, for example, the first maximum along the x-axes) and b (the corresponding distance along the y-axes). Then, finding the distances from the impact point to the maxima and the minima for the bending waves in different directions, it is possible to estimate the D,, , D22 and D,* + 2D,, as described in Section 4.6. To increase the accuracy in the estimations it is better to make the above measurements in many directions, e.g. every lo” around the plate and at different times. These measurements can also be performed from an interferogram.
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Phase map and corresponding bending wave in an orthotropic
et al.
3D image of the propagating laser-generated plate, 70 ps after the start of impact.
Figure 9 shows an interferogram of the same plate 60~s after impact start. The problem with the evaluation of such an interferogram is the sign ambiguity of the fringes. It is difficult to know if the fringe pattern shows a valley or a hill. It is also difficult to determine the exact location of maxima and minima of the bending waves.
Dynamic material parameters in an anisotropic plate
Fig.
8.
Amplitude map and corresponding 3D image of the propagating generated bending wave in an orthotropic plate. SO ~CLS after impact start.
449
laser-
Figures 10a and b show a 3D map respective contour map, 60 ps after impact start, obtained from eqn (11). Another possibility to estimate the parameters could be to adjust the parameters until the contour map obtained by the theory coincides with the experimental amplitude map or interferogram.
K.-E.
Fig. 9.
A traditional
FiilktrBm
et al.
interferogram of the same event as shown 60 ps after the start of laser impact.
in Fig. 8 but now at
Dynamic material parameters in an anisotropic plate
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(4
1.
C
(b)
0.05 0.025 0 -0.025 -0.05 -0.075
-0.1 Fig. 10.
-0.05
(a) and (b) 3D map respective obtained
0
0.05
contour map 60~s from eqn (11).
0.1 after the start of impact,
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TABLE 2 Comparison between Young’s Moduli and Shear Modulus obtained with Different Methods* Method
FE BW Ra St DY
46.2 39.9 44.7 49 41
16.6 14.9 16.2 16 14
6.9 6.1 6.8
*FE, the method described in Section 3; BW, the method described in Section 4; Ra, Rayleigh’s method;“2 St, static test; Dy, dynamic test.
5 CONCLUDING
REMARKS
In Table 2 the effective stiffness parameters obtained with the two methods proposed in this paper are compared with three other methods. Rayleigh’s method is described in Ref. 22 and the other methods are a static tensile test of bars cut out of the test-plate and a dynamic vibration test of the same bars. Static and dynamic measurements have not been performed to determine the in-plane shear modulus. Using the theory, in Ref. 21 the Djjestimated results in this paper are converted to the engineering material parameters (E,, Ey, E,). To get the in-plane shear modulus (E,) it is necessary first to estimate D,,.This is possible if Poisson’s ratio is known for each layer of the plate. A realistic value for this parameter for this plate is V = 0.26. Table 2 shows that the parameters agree quite well between tg different methods. The static and dynamic values are uncertain because when bars are cut from the plate cracks often appear. In the tensile test, a strain gauge was glued to the bar and this gives only local static Young’s moduli. The differences in stiffness parameters measured by holographic interferometry arise from phase errors, uncertainty in wave position, etc. Several experiments have been performed with different time delays between the two pulses and with different quantities of gelatine at the impact point. The experimental error in the stiffness parameters is &5%. Another uncertainty in the result from the experiments with bending waves is that Kirchhoff’s plate theory is valid only for low frequencies, and therefore the results are founded on the extremes near the impact point.
Dynamic material parameters in an anisotropic
plate
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The material parameters estimated by studying the bending waves are smaller compared with the two other methods performed on a plate. One reason for this could be that the impact time cannot be regarded as a Dirac function in time, because it takes time for the energy to evaporate the gelatine or the material in the plate. No difference was found in the propagation velocity of the bending waves, comparing experiments with or without gelatine at the impact point.
REFERENCES 1. Cardon, A. H., Mixed numerical-experimental techniques; recent developments and potentials for experimental mechanics. Proc. IOrh Int. Conf: on Experimental Mechanics, Lisboa, Portugal, 1994. 2. Sirohi, R. S. (ed.), Speckle Metrology. Marcel Dekker, New York, 1993. 3. Vest, C. M., Holographic Interferometry. John Wiley, New York, 1979. 4. F;illstrom, K.-E. & Molin, N.-E., A non-destructive method to determine properties in orthotropic plates. Polymer Composites, 8 (1987), 103-108. 5. Fallstrom, K.-E., Gustavsson, H., Molin, N.-E. & Wghlin, A., Transient bending waves in plates studied by hologram interferometry. Exp. Mech., 29 (1989) 378-387. 6. Fallstrom, K.-E., Lindgren, L.E., Molin, N.-E. & Wdhlin, A., Transient bending waves in anisotropic plates studied by hologram interferometry. Exp. Mech., 29 (1989), 409-413. 7. Stetson, K. A., Brohinsky, W.
8.
9.
10.
11. 12.
13.
R., Wahid, J. & Bushman, T., An electro-optic holography system with real-time arithmetic processing. .I.Non Destructive Eval., 8 (1989), 69-76. Pryputniewicz, R. J. & Stetson, K. A., Measurement of vibration patterns using electro-optic holography. SPIE, 1162 (1989) 456-467. Jansson, E. V., Molin, N.-E. & Saldner, H. O., On eigenmodes of the violin-electronic holography and admittance measurements. J Acoust. Sot. Am., 95 (1994), 1100-1105. Crawforth, L., Lee, C.-K. & Munce, A. C., Application of pulsed laser holographic interferometry to the study of magnetic disk drive component motions, hologram interferometry and speckle metrology. Proc. 1990 Fall Conference of The Society for Experimental Mechanics, pp. 404-409. SEM, Publications, Bethel, 1990. Dandliker, R., Thalmann, R. & Willemin, J.-F., Fringe interpolation by two-reference-beam holographic interferometry: reducing sensitivity to hologram misalignment. Opt. Commun ., 42 (1982), 301-306. Creath, K., Phase-measurement interferometry techniques. In Progress in Optics, ed. E. Wolf, Vol. XXVI, pp. 349-393. Elsivier, Amsterdam, 1988. Huntley, J. M., Noise-immune phase unwrapping algorithm. Appl. Opt., 28
(1989), 3268-3270. 14. Fallstrom, K.-E., Molin, N.-E., Olofsson,
K., Palagyi, P. & W8hlin, A., A study of the deformation of a steel plate when impacted by a focused laser pulse. NDT&E In?., submitted.
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15. Schedin, S., Gren, P. & Whhlin, A., Visualisation of transient acoustic field in air surrounding an impacted plate using pulsed holographic interferometry. J. Acoust. Sot. Am., submitted. 16. Lindblom, O., Naslund, R. & Persson, L.-E., A study of vibrations in infinite plates. Appl. Math., submitted. 17. Jones, R. M., Mechanics of Composite Materials. Scripta, 1975. 18. Fallstrom, K.-E. & Jonsson, M., A non destructive method to determine material properties in anisotropic plates. Polymer Composites, 12 (1991), 293-305. 19. McIntyre,
M. E. & Woodhouse, J., On measuring wood properties, Parts 1 and 2. J. Catgut Acoust. Sot., 42 (1984); 43 (1985). 20. Gradshteyn, I. S. & Ryzhik, I. M., Table of Integrals, Series and Products. Academic Press, New York, 1965. 21. Tsai, S. W. & Hahn, H. T., introduction to Composite Material. Technomic Publishing, Westport, Connectitut, 1980. 22. Fallstrom, K.-E., Determining material properties in anisotropic plates using Rayleigh’s method. Polymer Composites, 12 (1991), 306-314.
ELSEVIER
Dynamic Material Parameters in an Anisotropic Plate Estimated by Phase-stepped Holographic Interferometry Karl-Evert
FallstrBm,
Kenneth Staffan
Olofsson, Schedin
Henrik
0. Saldner
&
Division of Experimental Mechanics, Lule5 University of Technology, S-97187 Luleii. Sweden. Tel: + 46-920 91000. Fax: + 46-920 91047. E-mail: [email protected] (Received 3 October 1994; revised version received 18 April 1995; accepted 24 April 1995) ABSTRACT Material parameters in an anisotropic plate are determined using two non-destructive measuring techniques: real-time phase-stepped electronic speckle pattern interferometry and dual-reference-beam pulsed holographic interferometry. The first technique is used to measure the lower modes of vibration of the plate with free-free boundary conditions. Finite element analysis is then used to determine two effective Young’s moduli and the in-plane shear modulus. The second technique is used to detect transient bending waves propagating in the plate and acoustic waves propagating in the surrounding air. A double-pulsed laser is used both to generate the waves and to make holographic recordings of the wave fields. The stiffness of the plate is estimated using the measured deformation field and an analytical solution to the plate impact problem.
1 INTRODUCTION Non-destructive testing (NDT) is an important tool to improve the quality of a product. In order to guarantee that the properties of a component match the given requirements, a non-destructive test can be used to find material defects and elastic constants. Knowledge of the elastic constants of fibre composites is important in optimum design and quality control of a product. The stiffness of standard materials such as metals and homogeneous plastics can easily be tested with conventional methods, such as ultrasonic and tensile 429
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testing. Defects and elastic constants in materials like fibre-reinforced plastics can hardly be found with standard methods. In this case an additional or alternative method has to be applied. One way to achieve this is to use optical techniques like holographic interferometry, speckle interferometry, shearography, etc. Sometimes such experimental methods are combined with finite element simulations into a hybrid method’ to become a powerful tool in NDT. One advantage of these nondestructive methods is that they give a whole-field visual display. These techniques have been known since the mid-1960s but have been much improved in recent years. One of the advances is that interferograms can now be transformed into 3-D displacement maps, which simplifies the interpretation of the measured results. In this paper speckle interferometry* and pulsed holographic interferometry” are used to estimate the elastic constants of a composite plate. In 1987, Fallstrom & Molin4 reported a method to estimate the stiffness of orthotropic plates from the lower modes of vibration. They used traditional electronic speckle pattern interferometry (ESPI) with poor quality interferograms. An up-to-date electronic holography technique, real-time phase-stepped ESPI, was used in the present study. In 1990, Fallstrom et ~1.~7~used holographic interferometry to study the propagation of transient bending waves in isotropic plates. A closed solution to the Euler plate equation was given with the initial conditions modelled as a Dirac-pulse in space and time. In this paper, a solution for an orthotropic plate is given with the same initial conditions. An important advantage is that these experiments can be performed not only on free plates but also on plates with any boundary conditions, because only the initial part of the propagating wave is studied, i.e. the bending waves have not reached the boundaries for the time periods which are investigated. In Refs 5 and 6 a pendulum was used to impact the plates. A disadvantage of using a pendulum is that the impact time can be quite long (several tens of microseconds) compared with the wave propagation time. In this paper, however, the double-pulsed laser is used both to generate and to record the bending waves, thereby no triggering problems arise and the impact can be regarded almost as a Dirac-pulse both in space and time. A disadvantage is that the transferred impulse to the plate is difficult to determine because knowledge of the mass and the velocity of the evaporated material from the plate can barely be measured. It is possible to estimate the impulse from the interference pattern, for example, but in this experiment this parameter is of no interest.
Dynamic material parameters in an anisotropic plate
2 SPECIMEN
DESCRIPTION
A glass-fibre reinforced composite plate is examined in this paper. plate was wound on a flat tool in a filament winding machine and pressed between two steel blocks to achieve a constant thickness. plate was cured for 4 h in a convection oven, The properties of the are shown in Table 1.
Material
431
The then The plate
TABLE 1 Properties of the Test Plate
Density Laminate sequence Dimension
2027 kg/m’ (2W - 24),* 7.7 X 17.3 X 0.315 cm
* The plate consists of three layers oricnlcd at 2” relative to the longer side of the plate and three layers at -2” relative to the same side. Furthormore. the plate is symmetric with respecl to the neutral line of the plate.
For orthotropic plates there are three main material parameters which describe the stiffness of the plate: the Young’s module in the xand y-directions (E, and E,.) and the in-plane shear modulus (~5,). These parameters can be determined from the first three vibration modes. 3 ELECTRONIC
HOLOGRAPHY
3.1 Experimental arrangement A real-time phase-stepped speckle interferometric technique called electronic (electro-optic) holography has been used in its time-average mode. The system used was developed in the USA by United Technologies Research Center (UTRC),7.X see Fig. 1. The plate is illuminated by a 500 mW frequency-doubled Nd:YAG laser (wavelength 532 nm). The image of the plate interferes with a smooth reference beam from an optical fibre onto a CCD camera. The piezo-mounted mirror (PZT) introduces optical phase shifts of 90” between subsequent images. The four latest images are treated by an image processor and presented as interferograms on the monitor in real-time (30 frames per second). A host computer controls the image processor and the phase-stepping mirror, PZT. To reduce. the speckle noise and improve the quality of the interferograms, speckle averaging is performed. A
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OPTICAL HEAD
PZT bhib.
(a) PZT (phase step)
CCD
Video recorder (b)
Fig. 1. Optical arrangement for electronic holography. (a) The optical head of the electronic holography system and the analysed plate: PZT, piezo-mounted mirror; BE, beam expander; BS, beam-splitter; R, relay lens. (b) The electronic components of the system.
number of interferograms are averaged to produce the final picture. Each interferogram is given a slightly different illumination direction to give a different speckle noise. The speckle average is not performed in real-time. A time-average interferogram, as presented by the system after speckle averaging, is shown in Fig. 2a. A loudspeaker is used as an exciter to vibrate the plate.
Dynamic material purameters in an unisotropic plate
43.1
The second PZT mirror, in the illumination beam, is used both to find the phase relation between anti-nodes and for the quantitative evaluation of the mode shape. This mirror is set to vibrate at equal frequency, amplitude and phase as parts of the object, as a bias vibration. These parts of the object will then act as new nodal lines. The real nodal lines will be given a number of fringes that correspond to the amplitude of the bias vibration of the PZT. By switching the bias vibration on and off, the phase of the anti-nodes can be determined from the increase or decrease of the number of fringes in each anti-node. This facility is needed to understand complicated modes of vibration of the object, especially when several sides of the structure are imaged simultaneously. Jansson ef al.’ have studied modes of vibration of the violin. Three sides of the violin were studied simultaneously via mirrors. The bias facility made it possible to decide if anti-nodes are in phase or not. Numerical evaluation of the vibration amplitude distribution can also be performed by recording three interferograms, the first without bias, the second with positive bias and the third with negative bias together with a post-processing routine; see the details in Ref. 8. The interferogram in Fig. 2a is one of the three interferograms that have been used to obtain Fig. 2b. The vibration mode in Figs 2a and b is -the second mode of the free-free plate at 200 Hz and is called the (0,2)-mode. The label (0,2) means zero nodal line parallel to the longer side of the plate (x-direction) and two nodal lines parallel to the shorter side (ydirection). To obtain the free-free boundary conditions of the plate, the supports have to be moved to the location of the nodal lines. We have used small pieces of rubber as supports. By tuning the excitation frequency through a resonance frequency, the nodal lines will not move if the supports are placed correctly. A real-time-field technique is favourable to visualize the modes of vibration and the supports. High vibration amplitudes, giving a large number of fringes, can be used with the present technique, which makes it easier to place the supports correctly. Interferograms of the other two modes, (1,l) at 128 Hz and (2,O) at 304 Hz, are shown in Figs 2c and d.
3.2 Experimental
results and discussion
A finite element (FE) model of the plate is used with free-free boundary conditions. The FE model of one-quarter of the plate has 160
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64
y-axis (mm)
0-o
x-axis (mm) W
Fig. 2. First three modes of vibration for the plate with free-free boundary conditions. (a) Interferogram of the (0,2)-mode and (b) a plot of the measured mode shape of the (0,2)-mode. Interferograms of (c) the (l,l)-mode and (d) the (2,0)-mode.
Dynamic
material
parameters
in an anisotropic
(4 Fig. 2.
(Continued.)
plate
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triangular, three node, flat, shell elements. The geometry and the density of the plate are known. A parameter study is performed by changing the two Young’s moduli E, and Ey and the in-plane shear modulus Er. From the FE analysis it was shown that each one of the three resonance frequencies in Fig. 2 was strongly coupled to only one of the Young’s moduli or the in-plane shear modulus. The resonance frequency of the (l,l)-mode is most dependent on the shear modulus and only slightly dependent on the Young’s moduli. The frequencies of modes (0,2) and (2,0) depend most on the Young’s moduli in the Xand y-directions, respectively. The results of the FE analysis are shown in Figs 3a-c, one graph for each mode. The experiments give the three resonance frequencies. These frequencies are used together with the FE results to decide which E,, E, and E,$are looked for.
4 DUAL-REFERENCE-BEAM PULSED INTERFEROMETRY
HOLOGRAPHIC
One problem with traditional holographic interferometry is that there is a sign ambiguity for the fringes. It is difficult to know if the fringe pattern shows a valley or a hill, therefore it would be better if the interferograms could be directly transformed into a 3D displacement map. This problem has been solved by Crawforth et al.,” amongst others. They used a double-pulsed dual-reference-beam holographic measurement technique to achieve this. 4.1 Experimental arrangement The holographic set-up is a standard one (Ref. 3, pp. 57-64), arranged to be most sensitive to the out-of-plane displacement of the object. A beam splitter divides the pulses from a double-pulsed ruby laser into two beams; see Fig. 4a. One beam is used to create the bending waves and the other to record the bending waves. About 70% of the laser pulse energy is used to generate the bending waves. The object is the glass-fibre reinforced epoxy plate described in Section 2. The plate is held vertically using a clamp at one edge. The light used to record the hologram is deflected via two mirrors and becomes divergent after passing through a negative lens. The divergent beam from the negative lens is used to form an object beam and a reference beam which are recorded by photographic film as a hologram.
Dynamic
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parameters
in an anisotropic
plate
431
(4
Relative change of parameters
0.95
1
1.05
Relative change of parameters
Fig. 3. Frequency change versus parameter change for the three fundamental modes. Each mode is strongly related to only one parameter: (a) the (I,l)-mode and the shear modulus E,; (b) the (0,2)-mode and the Young’s modulus in the x-direction E,; and (c) the (2,0)-mode and the Young’s modulus in the ),-direction E, (on the next page).
The plate is recorded in an undeformed state, since the plate is impacted by the ruby laser pulse simultaneously with the first exposure. The next laser pulse is deliberately delayed with respect to the first pulse by between 1 and 800 ps. As the second pulse also impacts the plate simultaneously with the second exposure, the impact caused by
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Relative change of parameters Fig. 3.
(Continued. )
this pulse does not affect the interferometric data. During the delay, the rotating mirror (RM) has shifted the reference beam by about 0.01”. The second pulse records the plate in a deformed state caused by the first laser beam impact, but with a slightly shifted reference beam. The pulse length or exposure time is as short as 25 ns, so even propagating bending waves are recorded frozen in the hologram. In this manner, a series of two-reference-beam holograms can be recorded by repeating the experiment with increasing time delay. 4.2 Evaluation
of dual-reference-beam
interferograms
In the reconstructed image of this dual-reference-beam interferogram” the plate is covered with a set of cosinusoidal fringes. These interference fringes display contours of equal, out-of-plane displacement. The reconstruction arrangement is shown in Fig. 4b. A helium-neon laser (HeNe) illuminates a Michelson interferometer (M, BS, PZT) to produce two similar beams of light as output. A mirror (PZT) can be moved by a piezo-electric crystal to shift the phase in one beam relative to the other. The mirror (M) is adjusted to give an angular difference between reference wave 1 (Ref. 1) and reference wave 2 (Ref. 2) equal to the one used in the experimental arrangement in Fig. 4a. Reference waves 1 and 2 illuminate the hologram film plate (H) and reconstruct the object in its undeformed and deformed state simultaneously. The
Dynamic
(4
material
parameters
MIRROI P
in an unisotropic
plate
439
MIRROR
4 --+
HOLOGRAM
RY
(/
SPLITTER
for generating and recording bending wave propagaFig. 4. (a) Optical arrangement tion in a plate using double-pulsed dual-reference-beam holographic interferometry. and evaluation of dual-reference-beam (b) Optical arrangement for reconstruction interferograms.
440
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al.
undeformed and the deformed object interfere, giving the cosinusoidal fringes. These fringes can now be controlled by moving the (PZT) mirror. To determine the phase in the reconstructed interferogram, a phase-stepping technique is used.” The interferogram is reconstructed and recorded by a CCD camera (CCD) four times. The mutual phase difference of the two reference beams is changed each time between two reconstructions by the phase shift introduced by the piezo-mounted mirror (PZT). The local intensities &(x, y) in the interferogram of reconstructions k = 1, 2, 3, 4 are given by I,&, I’) = U(X,_Y){l+ rn(X, y> c0.s (+(X, y) + (k - 1)7r/2)},
(1)
where a(x, y) denotes the local mean intensity, m(x, y) the fringe contrast and 4(x, F) the interference phase. The phase 4(x, _v) at each point in the detector plane is
Ldx, y) - L(4 Y)
4J(xp y)=tan ‘&xv) Id
I&,
y
)I
(2)
The phase at each point, given by eqn (2), is evaluated by a computer program. To remove the 27r phase ambiguities a noise-immune phase unwrapping algorithm” is used. The coupling between the phase and the deformation of the object is found in eqn (3),
where A is the ruby laser wavelength, k, is the illumination vector and k, is the observation vector, both of unit length. w is the displacement vector. The out-of-plane displacement of the plate is calculated from the phase using eqn (3). 4.3 Laser-generated
bending waves
About 70% of the first laser pulse, which has an energy of 0.4-0.6 J, from the ruby laser is used to excite the bending waves (see Fig. 4a). The laser beam is deflected via a mirror to a positive lens, which focuses the beam onto the surface of the plate. The focused spot is typically 0*4mm in diameter. Most materials will melt and evaporate under such intense irradiation. The vapour leaves the surface at a very high velocity according to the high temperature created. This transfers a mechanical impulse to the target, causing it to recoil, similar to an impact of very short duration.
Dynamic material parameters in an anisotropic
plate
441
The surface temperature is of the order of 104 K. A sharp bang can be heard and the generated blast will create an evaporated crater, about 0.5 mm in diameter, in the steel plate surface. Figures 5a and b show the propagating bending waves in a steel plate, 100 ps after the impact. Figure Sa is an interferogram and Fig. 5b is obtained using the phase-stepping and dual-reference-beam techniques. Figure 5b shows a deep valley in the vicinity of the impact point. The explanation of this is that the local high temperature spot gives rise to local surface tensions which force the plate to deform.14 These thermal stresses disturb the created bending waves, and much of the laser pulse energy is used to generate these stresses. However, if a spot of gelatine (water and gelatine powder) is placed at the impact centre most of the evaporating process can take place outside the structure, inside the gelatine. As the gelatine has a low evaporating point most of the energy is used for evaporation. In Ref. 14 it is shown that the impulse transferred to the plate can be many times larger with gelatine at the impact point than without, and the thermal stresses can almost be removed. Visualization impacted plate
4.4
of and effects of acoustic
waves outside the
When double-exposure or time-average hologram interferometry experiments are interpreted, the influence of pressure variations in the air surrounding the object are normally neglected. When rapid events are studied, like transient bending waves in plates, it is found that the influence of these spatial variations in the surrounding air cannot always be neglected. Bending waves with supersonic speed produce a sound pressure field localized quite close to the object. The probing laser light in the holographic recording is affected by changes in optical path length (which is the integrated product of the index of refraction and the geometrical path length) in the air as well as the changes which are introduced by object deformation or object motion itself. Figure 6 displays an unwrapped phase map showing the 2D projection of the transient acoustic near field around an impacted cantilever steel plate. It is the integrated values of the acoustic pressure changes from normal atmospheric pressure that appear as darker and brighter regions in the picture. Brighter regions indicate an increase in pressure of about 1% from the undisturbed conditions and darker regions a decrease in pressure with the same amount.‘” The data for the experiment are: plate dimension, 300 X 30 X 1 mm;
(b) 0.8 0.6
Fig. 5. (a) Interfcrogram of a propagating bending wave in a steel plate 100~s after the start of impact. (b) The corresponding 3D image obtained using dual-referencebeam recording.
Dynamic material parameters in an anisotropic plate
Fig. 6. outside
443
An unwrapped phase map of the 2D projected transient acoustic pressure field an impacted steel plate. The edge of the steel plate is seen as the vertical line in the centre of the picture.
444
et al.
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impactor, 0.5 g air gun; velocity of the gun, about lOOm/s; impulse transferred to the plate, about O-05 Ns. The impacting point is at the middle of the plate and the plate is impacted at right angles. The figure shows the event about 140 ps after the start of the impact. Acoustic waves can be seen moving both up and down the plate. They are known as trace matched acoustic waves and are generated by the travelling flexural bending waves moving with supersonic speed along the cantilever plate. Note that along the plate an increase in pressure on one side corresponds to a decrease on the other side. The transferred momentum from an impact with a laser pulse is much smaller than the impact with an air gun bullet. The experiments show that the surrounding acoustic field generated by the laser pulse affects the measured phase so little that this contribution can be ignored. 4.5 Theory of bending wave propagation
in anisotropic
plates
The theory of propagating bending waves will be treated briefly. A more detailed description is given in Ref. 16. The equation of motion for the out-of-plane displacement w of an orthotropic plate is”
01 YXXXX+ 2(D1* + 2 * &kxqy
+ &yvyyy
+ @w,n
= PC& Y, 0,
(4)
where Oij is the flexural moduli, h the thickness of the plate, the volume density of the plate, p(x, y, t) the load per unit area and the subscript comma indicates partial differentiation. This equation is based on classical plate theory and it is assumed that the principal material directions coincide with the X- and y-axes. Consider the transformations
g=x a
(5)
where a and b are constants. Then the differential equation becomes DI, %+2.&i - W,,-,,+ 2 a4
a*. b*
&Z wzFF + - w, b4
- -
+ phw,,, = p(x, y)
(6)
The impact force is modelled as a Dirac-pulse in time and space. Choose a and b in such a way that D1,/a4 = D2,1b4. A suitable choice is that a is, for example, the distance between the impact point to a
Dynamic material parameters in an anisotropic plate
445
maximum along the x-axis and b is the distance between the impact point to the same maximum along the y-axis.‘R*‘9Equation (6) can then be solved using Laplace transformation with respect to time variables and Fourier transformation with respect to X and 7. In polar coordinates (in the transformed system) the solution to eqn (6) will be
X
sin v(Dy,/ph)
th2dl + &[(sin (2p))2]/4
Ad1 + c[(sin (2p))2]/4
(7) where
,p,tD”
@a)
a4 Dp
=
02
12
2&,
(7) can be converted
(Ref. 20, p.464) to
+ (Fresnel C($)r}) where I = impulse transferred
@b)
a2b2
WV2 - XJ Wl
&=
Equation
+
ddp’
to the plate. P(cos (e - e))’
’ = 4th&k 30:
k, =
=
0
J 1
+ k,)lph + Of2 4
D:: -DE 4
k=
h
(9)
_
2k(sinc2’P))’ k+k,
(lOa> (lob) (1Oc) (104
K.-E. Ftillstriim et al.
446
The disadvantage with eqn (9) is that it is necessary to use numerical integration to estimate a numerical value of w. An approximate equation which gives a result which is very close to the exact solution if -0.15 5 k/(k + k,) 5 0.15 (valid for almost all plates) is W(T, 8, t) = ?r
” sin (x) du + 3 --cos c, z ( Cl x
1I
x2-,
cl =4&k, Equations (9) and system setting
+ k cos 48
(11) can be converted
8 = arctan %JM
(
ttan
cp 1
sin c, d
1
cos
48
I
(11)
(12) to the untransformed
W) W)
where 50 and r are the angle and the distance from the impact point measured in the untransformed system. 4.6 Estimating the parameters From the experiments the spatial location of the extremes of the transient bending wave can be found. If the density (p) and the thickness (h) for a plate are known, it is possible to estimate D,, and D,2 + 2D,, from eqn (9) or (11). Furthermore, D22 can be calculated using the relation D,,/a4 = D2,/b4. For estimating D,, it is necessary to know D,2. Plates are usually made out of identical layers. Normally the layers can be considered as identical and the stacking sequence of the layers is known. This being the case, it is possible to determine the D,2 modulus of the plate if the Poisson’s ratio for a single layer is known, using for example the laminate theory.2’ The term D,, + 2D,, influences the propagating bending waves very little, outside the first minimum, in the x-direction. Therefore, it is possible to estimate an approximate value of the D,, modulus by changing this modulus in eqn (11) so that the extremes measured from the interferogram or the amplitude diagram coincide with those calculated from eqn (11). To get a starting value for this iteration
Dynamic
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in an anisotropic
plate
447
process one can omit the last term in eqn (11). Then an approximate location of the extremes, in the x-direction, can be estimated by
where n = 1,2, . . . (odd value for minimum and even for maximum) and r is the distance from the impact point to a maximum or a minimum. Choosing another angle (e.g. q = 45”) it is then possible to estimate the term Qz + 2D,, by repeating the above procedure. The iterative process is then continued to the point where the results from eqn (11) coincide with the results from the interferogram. If the flexural moduli are knowr. it is then possible to estimate the effective material parameters (E,, E,, E,r) of plates.2’ 4.7 Experimental results and discussion Figures 7a and b show the phase map and the 3D plot of the orthotropic plate 70 ps after the impact start, respectively. Gelatine has been applied on the plate surface at the impact point. In the phase map one can observe that the bending waves have been reflected at the longer side of the plate. As the theory described in the foregoing part is valid only for infinite plates, that is, before the bending waves have reached the boundary, one must use a delay less then 70 ps for this plate, when estimating the flexural moduli for the plate. In the 3D plot, one can observe the indentation in the vicinity of the impact point. The interpretation is not obvious, but the disturbance is either a minor effect of thermal disturbance or that a small part of the laser light that has gone through the plate. Figures 8a and b show the displacement map and the 3D plot of the same plate but 50 ps after the impact start. No reflected waves can be found. From the sampled values one can now determine a (the distance from the impact point to, for example, the first maximum along the x-axes) and b (the corresponding distance along the y-axes). Then, finding the distances from the impact point to the maxima and the minima for the bending waves in different directions, it is possible to estimate the D,, , D22 and D,* + 2D,, as described in Section 4.6. To increase the accuracy in the estimations it is better to make the above measurements in many directions, e.g. every lo” around the plate and at different times. These measurements can also be performed from an interferogram.
448
Fig. 7.
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Phase map and corresponding bending wave in an orthotropic
et al.
3D image of the propagating laser-generated plate, 70 ps after the start of impact.
Figure 9 shows an interferogram of the same plate 60~s after impact start. The problem with the evaluation of such an interferogram is the sign ambiguity of the fringes. It is difficult to know if the fringe pattern shows a valley or a hill. It is also difficult to determine the exact location of maxima and minima of the bending waves.
Dynamic material parameters in an anisotropic plate
Fig.
8.
Amplitude map and corresponding 3D image of the propagating generated bending wave in an orthotropic plate. SO ~CLS after impact start.
449
laser-
Figures 10a and b show a 3D map respective contour map, 60 ps after impact start, obtained from eqn (11). Another possibility to estimate the parameters could be to adjust the parameters until the contour map obtained by the theory coincides with the experimental amplitude map or interferogram.
K.-E.
Fig. 9.
A traditional
FiilktrBm
et al.
interferogram of the same event as shown 60 ps after the start of laser impact.
in Fig. 8 but now at
Dynamic material parameters in an anisotropic plate
451
(4
1.
C
(b)
0.05 0.025 0 -0.025 -0.05 -0.075
-0.1 Fig. 10.
-0.05
(a) and (b) 3D map respective obtained
0
0.05
contour map 60~s from eqn (11).
0.1 after the start of impact,
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452
TABLE 2 Comparison between Young’s Moduli and Shear Modulus obtained with Different Methods* Method
FE BW Ra St DY
46.2 39.9 44.7 49 41
16.6 14.9 16.2 16 14
6.9 6.1 6.8
*FE, the method described in Section 3; BW, the method described in Section 4; Ra, Rayleigh’s method;“2 St, static test; Dy, dynamic test.
5 CONCLUDING
REMARKS
In Table 2 the effective stiffness parameters obtained with the two methods proposed in this paper are compared with three other methods. Rayleigh’s method is described in Ref. 22 and the other methods are a static tensile test of bars cut out of the test-plate and a dynamic vibration test of the same bars. Static and dynamic measurements have not been performed to determine the in-plane shear modulus. Using the theory, in Ref. 21 the Djjestimated results in this paper are converted to the engineering material parameters (E,, Ey, E,). To get the in-plane shear modulus (E,) it is necessary first to estimate D,,.This is possible if Poisson’s ratio is known for each layer of the plate. A realistic value for this parameter for this plate is V = 0.26. Table 2 shows that the parameters agree quite well between tg different methods. The static and dynamic values are uncertain because when bars are cut from the plate cracks often appear. In the tensile test, a strain gauge was glued to the bar and this gives only local static Young’s moduli. The differences in stiffness parameters measured by holographic interferometry arise from phase errors, uncertainty in wave position, etc. Several experiments have been performed with different time delays between the two pulses and with different quantities of gelatine at the impact point. The experimental error in the stiffness parameters is &5%. Another uncertainty in the result from the experiments with bending waves is that Kirchhoff’s plate theory is valid only for low frequencies, and therefore the results are founded on the extremes near the impact point.
Dynamic material parameters in an anisotropic
plate
453
The material parameters estimated by studying the bending waves are smaller compared with the two other methods performed on a plate. One reason for this could be that the impact time cannot be regarded as a Dirac function in time, because it takes time for the energy to evaporate the gelatine or the material in the plate. No difference was found in the propagation velocity of the bending waves, comparing experiments with or without gelatine at the impact point.
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