Plate bending investigations by comparative holographic moiré interferometry. Evaluation by the finite-element method

Optics and Lasers in Engineering 11 (1989) 15—25

Plate Bending Investigations by Comparative Holographic Moire Interferometry. Evaluation by the Finite-element Method V. Sainov, E. Simova Central Laboratory of Optical Storage and Processing of Information, Bulgarian Academy of Sciences, Sofia 1113, P.O. Box 95, Bulgaria

& E. Manoah Laboratory of Precision Mechanics, Bulgarian Academy of Sciences, Sofia 1113, G. Bontchev Str., bl.4, Bulgaria (Received 25 July 1988; accepted 12 October 1988)

ABSTRACT Bending of clamped defect-containing and defect-free circular plates, subjected to equal uniformly distributed loading is investigated. Using the finite-element method (FEM), a numerical model is produced and the results from the itumerical computations are compared with those from a holographic experiment, based on comparative holographic moire interferometry (CHMI). The theoretical and experimental approach to solving the problem is discussed and the measurement accuracy is analyzed. Good agreement between the numerical and experimental results is observed. The possibility of applying the techniques developed to a wide scope of mechanical problems in non-destructive testing (detection of hidden defects, cracks and debonds, as well as investigations on material fatigue and plastic deformations) is pointed out.

1 INTRODUCTION Comparative holographic moire interferometry (CHMI) is a contactiess method which does not require preliminary preparation of the tested 15

Optics and Lasers in Engineering O143-8166/89I$O3~5O© 1989 Elsevier Science Publishers Ltd, England. Printed in Northern Ireland

V. Sainov, E. Simova, E. Manoah

16

objects and is suitable for non-destructive testing. It allows the comparison of two macroscopically identical objects. The moire fringes, arising from the superimposition of the compared specimens’ interference patterns, visualize the differences in the mechanical responses at equal loading conditions.’-3 This makes possible the detection of hidden defects and the analysis of mechanical strains and stresses, wear, plasticity and material fatigue. In the present work, CHMI is used to investigate the displacement field in bending of circular clamped plates which are subjected to uniformly distributed loading. The objective is to demonstrate the advantage of combining the experimental techniques with numerical methods for the purpose of solving mechanical problems. The experimentally obtained results are compared with those obtained by computations using the finite-elements method. The peculairities of the experiment are discussed and the measurement accuracy is assessed. Good agreement between the experimental and numerical results is observed.

2 PRINCIPLES OF CHMI In CHMI the interferograms of the master and test specimens are superimposed so that their identical points coincide. Moire fringes are formed as a beat between the two interference patterns and they represent lines of equal difference in the mechanical responses at equal loading of the compared specimens. If there is a defect, the fringes are localized around it.4’5 The basic equation of CHMI is I~1+cos{~(ko_ki).(Zm+Zt)}cos{~(ko—ki).(Zm—Zt)}

(1)

assuming that t~k 0 kom k0~=0 and Ak1 = k,,,, k~,=0, where Lm and L, are the displacement vectors of the master and test object with components u, v and w along the x-, y- and z-axes, respectively, k1 and k0 are the unit vectors of illumination and observation, and ~Xis the —

—

wavelength of the coherent light source used. The moire fringes are described with the second cosinusoidal term in eqn (1), which modulates the fast-oscillating first term—the so-called carrier frequency. As the moire fringes represent a beat between two interference patterns, they can be observed as long as the interference fringes have sufficient contrast and density and are localized in or close to the plane of the investigated object. The moire fringes are the areas where the carrier

Comparative holographic moire interferometry and plate bending

17

frequency is nullified, i.e. (ko”kj).(Lm

Lf)(2N+ 1)A12

(2)

The presence of a defect in elastic deformations is expressed by the difference in the displacement vectors of the defect-containing and defect-free specimens:6 L,(x, y, z) = Lm(X, y, z) + AL(x, y, z) (3) where AL(x, y, z) is the additional displacement caused by the defect. The term AL(x, y, z)I is much less than ILm(X, y, z)I. The thin elastic plates subjected to a uniformly distributed loading obey the classical theory of plates, so IUm(:)l, lt’,n(t)I <
i

i + cos

{~.2 cos

{~.cos2 (.~)Aw(x)}

(4)

The quantitative assessment of the differences in the mechanical responses of the defect-containing and defect-free plate is carried out by eqns (2) and (4).

3 EXPERIMENTAL The test objects are circular duralumin plates 1~5mm thick with a diameter of 46 mm, cut from one sheet. Young’s modulus E and Poisson’s ratio ~t of the material are 706 Pa and 034, respectively. On the rear non-illuminated surface, at a distance a = 7~4mm from the plate center, a circular defect (R, = 3~9mm) is inflicted by thinning the plate to a thickness h 1 = 0~8mm (Fig. 1). This defect does not produce

Fig. 1.

Investigated object.

V. Sainov, E. Simova, E. Manoah

18

visible changes in the investigated surface. The plates are clamped by a ring along the periphery to a vacuum chamber and this reduces the illuminated area to one with a radius R 16 mm. Loading of the plate is achieved by uniformly distributed pressure of 0~1MPa. 3.1

Experimental set-up

The optical configuration for experimental investigations by CHMI is shown in Fig. 2. The light source is a He—Ne laser with A = 632~8nm. This set-up provides image-plane hologram recording with unity magnification and localization of the interference pattern in the recording plane. The light-sensitive recording material is placed in an immersion tank which serves also for in situ development. The silver halide HP-650 holographic plates with a gelatin—polyacrylamide carrying matrix, the immersion liquid and the modified GP-3 developer provide a high signal-to-noise ratio and wide dynamic range. The latter, together with the properly selected ratio between the energies of the exposures, ensures equal intensities of the reconstructed wavefronts and high contrast of the interference fringes.7 A double-exposure hologram of the defect-free plate is recorded in the light-sensitive material, the plate being subjected to loading between exposures. The master plate is replaced by the defect-containing one and a second double-exposure hologram is recorded in the same light-sensitive material, subjecting the plate to the same loading. The four wavefronts are reconstructed simultaneously and photographed or viewed by a TV camera. Illuminating bam

/

Pressure Chambre Fig. 2.

2f Reference beam

Immersion tank

/2f

PhOtO

or TV

Holographic plate

Optical configuration for investigation by CHMI.

Comparative holographic moire interferometry and plate bending

19

A disadvantage of this method of CHMI is the repeated loading of the master plate, which can eventually result in wear or fatigue of the material and erroneous results. This can be avoided by recording a separate double-exposure hologram of the master plate which is aligned with the hologram of each tested plate during reconstruction. Doubleexposure holograms have higher diffraction efficiency than fourexposure holograms, but the absorption at the reconstruction wavelength should be kept at a minimum. Usually, this is achieved through bleaching. However, it leads to deterioration of the signal-to-noise ratio. The silver halide superfine light-sensitive HP-650 holographic plates and the GP-3 developer provide minimal absorption in the red spectral region and do not require bleaching. 3.2

Numerical analysis

Let us consider the deformation of a circular isotropic plate with radius R in which there is a circle with a radius R, where the thickness h1 is smaller than that of the rest of the plate. The geometric scheme of the problem, as well as the coordinates used, are shown in Fig. 1. We assume that the plate as a whole is symmetrical about its mid-plane. Such an assumption is feasible and (according to the San Venant principle) it could lead to erroneous results only in the close vicinity of the 8thin area contours. The quantity D, or flexural rigidity, has the form D = Eh/12(1 ,i2); X’~e Q/Q, —

D

2); 1=Eh,/12(1—,u

X~EQ 1

where E is Young’s modulus of elasticity of the plate, IL is Poisson’s ratio and Q, is the region in the plate with a thickness h, (Q = ~ U Q~). In order to investigate the bending of a plate with a step-wise change in thickness, the finite-element method (FEM) is used. The solution is obtained with the universal program system STRUDL. The finiteelements mesh of the region occupied by the plate contains 126 elements with 80 nodes, the thin region being modelled by six triangular finite elements, as shown in Fig. 3. Triangular and quadrangular shallow shell finite elements SSHQ2 and AHS1 with five degrees of freedom at the nodes are used (three translations u, v and w along the x-, y- and z-axis and two rotations aw/ax and 3w/3y). The use of shallow shell finite elements to solve the problem aims at improving the accuracy of the obtained results. As is evident, the finite-elements mesh becomes denser in the vicinity of the plate’s center and defect region. Although the scheme is not too dense, comparison of the numerical

V. Sainov, E. Simova, E. Manoah

20

55

1

~

~ 3

57V~I\~TTh-~Z\/\~L\! 1.9 n

704

1 59

Fig. 3.

Finite element mesh of the investigated object.

analysis results with the exact solution for a master (defect-free) plate, as well as the stability of the results on changing the scheme, is evidence of its convergence9 and of the possibility for the mesh model to describe with sufficient accuracy the plate deformation. 3.3 Results and discussion The experimentally obtained holographic interferograms of clamped circular plates, subjected to uniformly distributed loading, are presented in Fig. 4. In Fig. 4b a strongly pronounced asymmetry of the central fringes in the direction of the defect thinned part of the plate is observed, but it is insufficient to localize the defect and to assess its influence. The moirégram in Fig. 4c visualizes the difference in the displacement field between the master and the defect-containing plate and allows qualitative and quantitative assessment of the influence of such types of defects on the plate’s bending for each point of the investigated surface. Figure 5 displays the plots describing the bending of the master and defect plates, obtained from the quantitative interpretation of the interferograms. The calculations on the master plate are accomplished by the formula (1+1/\/2)w(x)=NA,

N=1, 2,3,...

The number of fringes N is determined visually, neglecting fractional parts. Thus, the accuracy is not higher than ow = A/(1 + 1/V2). Figure 6 shows the bending curves of the master and defect plate, obtained by

Comparative holographic moire interferometry and plate bending

(a)

~

21

~

Fig. 4. Interferograms. (a) Double-exposure interferogram of defect-free plate; (b) double-exposure interferogram of a defect-containing plate; (c) comparative holographic moirégram.

V. Sainov, E. Simova, E. Manoah

22 0 —2 -1.

-6

! -‘a -12 C .~

-11.

V 0

-16

.,~

~

~

..~

~

a ~

~

diltonce from cnter to fixed erxi olong x-oxis(rnm)

Fig. 5.

Plate bending, determined by CHMI. plate.

•,

Master plate; x, defect-containing

numerical calculations by the FEM. It should be noted that a numerical model which would account for all experimental peculiarities and strip the experiments of effects which cannot be taken into account, is too complex to be constructed. In our case, the ring fixing the plate to the vacuum chamger reduces the usable illuminated area of the plate to a 0 -2

-6

E

E -~40 •0

-1L

1.16 .18

~p

-~

-~

-~

-~

~

i~

,~

distance from center to fixed end along x oxis (mm)

FIg. 6.

Plate bending, determined by numerical computations by FEM. plate; x, defect-containing plate.

•,

Master

Comparative holographic moire interferometry and plate bending

23

radius of 16 mm. The ring also affects the bending, changing slightly the form of the bent plate and the boundary conditions of the problem which differ from those of the original one. The differences observed between the plots in Figs 5 and 6 increase from the center to the periphery of the plate, where they are largest. A substantial advantage of CHMI is that in the comparative measurements the master and defect plates are put under equal loading conditions and only the differences in the displacement field of the two plates are taken into account. For this reason, when analyzing and assessing quantitatively the influence of the defect on the plate’s bending, the inconsistencies considered above will be reduced to a minimum. This is demonstrated in Fig. 7, which shows the differences in the displacement fields of the defect-containing and defect-free plates. Good agreement between the experimental data from the moirégram and numerical model is observed. The experimental data are obtained from the number of moire fringes along the x-axis, passing through the center of the defect, in accordance with formulae (2) and (4). Therefore, the accuracy of measurement is OAw 0.2 ~m. The midpoints of the moire fringes are sampled with an accuracy of 0.1 mm along the x-axis, and the error depends on the fringes’ width and on the magnification of the viewing system. The difference between the two curves is within the error in measuring the x-coordinate and the calculated experimental values of w(x). The systematic error from measuring the illumination angle ci,, the error from loading, and the

,

1.6

.

*

~ 1,2 V .9

~l2-8-1.O

~

~

1~

l~

2b

distance from center to fixed end along x-axis (mm)

Fig. 7. Numerically and experimentally determined differences in bending of master and defect plates.

+,

Expenmenal;•, numerical.

24

V. Sainov, E. Simova, E. Manoah

error in Young’s modulus of elasticity and Poisson’s ratio can be neglected. The systematic error caused by the manner of fixing the plate to the vacuum chamber has the strongest influence on the difference between the experiments and the numerical model and on the reproducibility of the experimental results. The reproducibility of fixing and loading of the plates must also be taken into account, as it leads to the appearance of moire fringes which can be interpreted as being caused by the defect, or as a difference in the mechanical response. It should be noted that the maxima of both curves coincide and this indicates the center of the defect thinned part of the plate. 4 CONCLUSIONS The method, using CHMI, is contactless, relatively accessible and suitable for qualitative and quantitative analysis of various mechanical problems. It provides full visualization of the differences in the displacement field of the compared specimens over the whole surface. The present work uses CHMI to analyze the bending of clamped circular plates, subjected to uniformly distributed loading. The experimentally determined difference in bending a defect-free and a defect-containing plate is compared with the results from numerical computations by FEM. Good agreement between the experimental and numerical results is observed. CHMI is suitable for detecting flaws, cracks and debonds, as well as for investigating fatigue, wear and plasticity of the materials. The method does not impose more rigorous requirements on the optical configuration and stability during recording than the conventional holographic interferometry. Prior to proceeding to the solution of a specific mechanical problem for comparative testing, the proper optical configuration, type and level of loading should be selected. This guarantees more reliable detection, localization and assessment of the investigated defects.

ACKNOWLEDGEMENTS

The present work was accomplished as part of a contract, supported by the Committee of Science. The authors appreciate the assistance of thier colleagues from CLOSPI and the encouraging advice of Prof. Dzhupanov from the Institute of Mechanics and Biomechanics.

Comparative holographic moire interferonwtry and plate bending

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REFERENCES 1. Kozachok, A. G., Holographic Methods of Investigations in Experimental Mechanic. Mashinostroenie, Moscow, 1984, pp. 121—44. 2. Youren, Xu, Vest, C. M. & DeIp, E. J., Digital and optical moire detection of flaws applied to holographic non-destructive testing, Opt. Lett., 8(8) (1983), 452—4. 3. Rastogi, P., Comparative holographic moire interferometry in real-time. Appl. Opt., 23(6) (1984) 924—7. 4. Rastogi, P., Comparative holographic interferometry: a non-destructive inspection system for detection of flaws. Exp. Mech., 25(4) (1985), 325—37. 5. Simova, E. & Sainov, V., Comparative holographic moire interferometry for non-destructive testing; comparison with the conventional holographic interferometry. Opt. Eng. (to be published). 6. Lurie, A. I., Theory of Elasticity. Nauha, Moscow, 1970.

7. Sainov, V. & Simova, E., Report of the National Conference, Optika ‘87, Varna, 18—20 May, Vol. 2, p. 411, p. 463 (in Bulgarian). 8. Timoshenko, S. P. & Woinovsky-Krieger, S., Theory of Plates and Shells,

McGraw—Hill, New York, 1959. 9. Zienkiewicz, 0. C., The Finite Element Method in Engineering Science.

McGraw—Hill, New York, 1971.