ACTA MECHANtCA SINICA, Vol. 4. No. 4, November, 1988

ISSN 0567--7718

Science Press, Beijing, China Allerton Press, INC., New York, U. S. A.

USING MOIRI~ INTERFEROMETRY

AND BOUNDARY

INTEGRAL

HYBRID

M E T H O D TO D E T E R M I N E M I X E D - M O D E S T R E S S I N T E N S I T Y FACTOR Jia Youquan

Xie Jinan

Wang Yinyan

(Departmentof Mechanics, Tianjin University) ABSTRACT: The paper describes a hybrid experimentaland numerical method of Moir6 Interferometry and the boundary-integral-elementmethod. The interference patterns used for the evaluation of the displacement vector are obtained by Moir6 Interferometry. The boundary displacements obtained experimentally are conveniently used for the calculation of the stress intensity factor in the body by the boundary-integral-method. Some examples bear witness to the effectiveness and accuracy of the hybrid technique. KEY WORDS: Moir6interferometrymethod, boundary integral element metho.d, hybrid method, stress intensity factor. I. INTRODUCTION Experimental methods can be used to determine stress components or displacement vector on the boundary and onthe surface of solids. It is, however, not always possible to obtain the stress field by experiments alone. It is sometimes necessary to utilize numerical method to supplement experimental methods. The boundary integral element (BIE) method is very convenient to be used along with a number of experimental methods. The BIE method used for the stress analysis is based on the numerical solution of integral equations. The method is very well suited for solving two and three dimensional problems as it reduces them to boundary solutions, i.e., only the boundary values need to be determined. The hybrid numerical and experimental method for the analysis of three-dimensional problems permits the determination of stress components at interior points nondestructively. The presented method has another advantage in comparison with a purely numerical or a purely experimental approach, that is, the initial complicated problem is reduced to a problem with simple boundary conditions.

H. THE BASIC PRINCIPLE OF THE HYBRID METHOD 1. Basic Equation of Contour Integral Before going on to the determination of the stress intensity factor, we first describe the computation of a contour integral tx], The equation may be derived by Betti's reciprocal work theorem for plane elastic state with vanishing body force.

fc(U. ~-- ~,. t) ds

0

Received 20 March 1987, Revised 13 March1988; Project is supported by the Science Fundation of the State Education Commissionof China.

(1)

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where u, t are the displacement field and the traction vector on the boundary C of a simple connected and bounded region R, which corresponds to an arbitrary specified equilibrium problem and tl, t are the displacement and the traction corresponding to any other such problem. In order to apply the reciprocal work theorem to a body with edge crack, we first remove from the body points within a circle of radius e centered at the crack tip as "indicated in Fig.1. The boundary of the remaining body is decomposed into two parts, the circular boundary of the removed region denoted by C,, and the remaining'boundary denoted by C,. Assuming that the crack surface is traction free, the following may be derived from the Equation (1):

(u" ~- u" t)ds = f

I,=-~

(u" t- u" t)ds

(2)

C'~

Ce

Fig. 1

The.se integrals can be evaluated with infinitely small ~ and an "appropriately" selected auxiliary equilibrium state. The stress and displacement near the crack tip expressed by the natural polar coordinates as shown Jn Fig. 1 has the following form:

o__,, ,,,

,_co#]~,_[,2,,,

,.,#]~,,}

u,_ u:__~( ~ )', {[_,2,,:_ ,,,,,,.,_;..,_,,i,.,~_],:-,_ [,2~_ ,,co,;_ ,,co#],,:.,,} + ~[,-,I~)

o,.__~,,.,,,., [ (.~co,,;_oos_~),:.,_(~s,,,~_3,.,,,~_),c,,] _,_,~r-,.,'",

o, ~[(,,~o,,~_,_ co,,~),c,_(3,,,.,;_,_~,i,,~),c,,]-,-,~r-,,'.,, ' -" J'",c,-'+~-jj =

1

(3)

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Jia Youquan et al.: Mixed-Mode Stress Intensity Factor

321

o and u~ are the radial and tangential components of the displacement u ~ at the crack tip and

Ur

K t = lira (2~zr)1/2 ae In = 0

(4)

r~O

K. = lim (2rtr) '/2 a,010=o

(5)

r~O

are the stress intensity factors. The auxiliary elastic state used in the reciprocal equation is:

u, = 2(2~r)l/z(

k)

1

U~

k-){[ _ (2k _l)sin302 +3sin~lCx+[(Zk_l)cos302 ' 0

_

_cos~lC2~

G

~'=

2(2rtr3)1/2 ( l + k ) [ ( 7 c ~

g0 =

2(2rcr3)'/2(1 + k)

3 c o s ~ ) C 1 + (7sin~,0 -

cos

+ 3cos~ C , +

sin

+ sin

[( 3o 0) ( 3 sin~-+sin~

C1 -

sinO)C2 ]

3cos

(6)

C2

0) ]

+cos~

C2

where C1 and C2 are arbitrary constants, G is shear modulus of elasticity, k = 3 - 4st for the plane strain state; k = (3 - St)/(1 + St) for the plane stress state. Now on the inner circular boundary the contour "integral in terms r 'he traction and displacement (relative to that of the crack tip) components takes the form {. I, =

-

E(,, -

,,o).

t -

las

t ~

=/" d-

[ # , ( u , - u ~ + #,o(uo- u ~ - a , a , - a,oao]r'dO

(7)

/t

With (3) (6) substituted into Equation (7), it follows t h a t I, = C1K, - C2K.. + O(1)

(8)

In the above equation, the remainder term goes to zero with a --* 0. It follows from Equation (2) that

C,K~ - C2K n = ~ [(u - u ~ JC

" - ~s't]ds

(9)

It is important to note that the contour C (or F) involves only the boundary since both t and t" necessarilyvanish on the crack surface, i.e., the crack surface is traction free. In the meantime, the surrounding curve may be an arbitrary curve containing the crack tip. Furthermore, the rigid body displacement u ~ may be discarded in the evaluation of equation (9) since the distribution of u ~ t'on any closed contour vanishes even when the origin is contained in the interior. It remains only to obtain u and t on the outer boundary from the prescribed data, so that the contour integral may be evaluated as a linear combination of arbitrary constants C1 and C2, the coefficients of which are the desired stress intensity factors Ki and K n. Let C1 = l, C2 = 0 and C1 ='0, C2 = - 1 , we may find

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1988

out both K I and K n from the Equation (9). As shown in Fig. 2, a region containing the crack tip may be removed from the model, the radius of which is r'. Using the Moir6 Interferometry to determine the displacement u (u x and %) on the boundary F x or F2, and then applying the integral Equation (9), it is very easy to calculate Ki and K..

Fig. 2

Although the above discussion is about boundary cracks, it is very easy to extend the method to the case with center cracks through the division of the model into districts.

2. Numerical Evaluation of the Contour Integral Equation From Equation (9) we can get Cl KI - C2Kll = ~ 6" [(lg - uo) t - d" t]d$

i=l Jri

(10)

Since u ~ is equal to zero on any closed contour and the rigid displacement u ~ may be neglected, Equation (10) may be simplified.

c, ~, - c2K,, = E j~r ( " ~ - ~.,)as i=1

i

= ~ f (ux'~x + u,'~ - ux't~- uS't,)ds

(11)

i=1 Jr i

In Eq. (11) we can take Ci = 1, C2 = 0 or C2 = - 1 , C1 = O. Then

fr i ~

lm m 1 (12)

....

l~

x

2

a

f Zt~tds= ~t~'l ~ Nl(~):u~(r162

Jri

r' [

+~trz

j-1

.. 1c% ~" , d s = !2t, ifx ;) r i

N2(r162162

. -1

Nl(')'h'~r")

ar + t . t-~ 2

" fl

N2")'u~r") d'

All integrals are calculated by using four points Gaussian integration method.

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dia Youquan et al.: Mixed-Mode Stress Intensity Factor

323

IH. THE EXPERIMENT OF MOIR]~ INTERFEROMETRY 1. The Principle of Moir6 Interferometry The specimen grating is formed by two beams of coherent light. The optical path of grating production is illustrated in Fig. 3. The pitch of grating can be obtained by geometrical relations.

p = 2/(sini A + sin/B)

or

p = 2/2sin~cos/~

(13)

where iA and i B are the incident angle of the two beams of light respectively, as shown in Fig.4. A

!Lase specimen

--

x

.V

Fig. 3

Fig. 4

According to the basic theory of diffraction, the diffraction beams obey the following equation: p' = n2 / ( s i n / - sin0,)

(14)

in which n is the order number of the diffraction waves; 0, is the angle of the nth diffraction wave; i is the angle of the incident; p' is the pitch of the grating. Another virtual grating is added in front of the specimen grating. When the specimen grating deformed, these two gratings interact to form fringes of a Moir6 interferometry pattern, which is viewed and photographed with a camera and from which we can get the displacements. Ordinarily the pitch of grating is very small. Suppose the frequency isf, f = 1/p = 1200 - 2400 lines/mm, which depends on the incident angle of light, The pitch of the interference fringe depends on the rigid movement and the deformation of the specimen surface. If 6x stands for the pitch of the Moir6 fringes (see Fig.5), according to reference

N

\N'

Fig. 5 [3], [4], we have fix = )~/[~r(cosiA -- cosiB) + ~x(siniA + siniB)-]

(15)

in which ex is the strain of the specimen along X axis; ay is the rigid rotating angle of the specimen

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around the Yaxis, if~y < < 1. Equation (15) makes it clear that more attention shouht be paid to the influence of %, when this method is used. If ~y is very small the angle iA is equal to i B, and Equation (15) becomes: 6~ = )~/[ex(siniA + sin/a) ]

(16)

Substituting (13) into (16) we get

~Jx'Sx = p where 6x" ex = U~. Therefore, the amount of the deformation expressed by one order of fringe isp. Nth order of fringe stands for the deformation u~ = N~p U~ = U,p

I17)

where U~, Ur are the components of displacement in the x and y directions, respectively. N,, N r are the fringe orders when lines of the reference grating are perpendicular to the x and y directions, respectivelyl 2. Method of Experiment (l) There are two methods to measure Ux and Ur. In one method, use is made of a special loading frame, in which the specimen can be loaded from both vertical and horizontal directions. In another method one uses a special optical system, which can produce the interference of the light waves along x-axis and y-axis. The former method is used by the author. A special loading frame is designed and made in our laboratory, as shown in Fig. 6.

Fig 6 (2) First, the holographic film is pasted on the surface of the specimen and then it is exposed two times separately before and after loading. After tearing the film carefully from the specimen, the film is developed and fixed. The clear Moir6 fringe pattern is obtained by reproducing in the special reproducing optical system [5]. (3) The reproducing light source can be either laser light or white light. If laser light is used, the

Vol. 4, No. 4

dia Youquan et al,: Mixed-Mode Stress Intensity Factor

325

reproducing pattern will be influenced much by speckle and noise. The white light is much better, The fringe pattern illustrated in Figs. 8 and 9 is that of the displacement Ux and Ur of a simple supported beam. The pattern is reproduced using white light.

IV. EXAMPLES OF CALCULATION Example 1 A plexiglass beam, its size being shown in Fig. 7. is loaded at the center in which P = 44.42N; E = 3334.38MPa;; ~t = 0.37. Us, Uy are measured by the method of Moir6 Interferometry. The contour of displacement is shown in Fig. 8 and Fig. 9. The model of calculation is from the contour of Ur (refer to Fig. 9). The mathematical model is shown in Fig. 10. The hybrid method suggested in this paper is used to calculate K 1 along F 2, K 1 = 0.433 MNm-Z/2. Compared with Boundary Collocation Method, the relative error of the method is about 1%.

Fig. 7

Fig.9

Fig.l/

Example 2

Fig.8

Fig.t0

Fig.12

A plexiglass beam with an inclined crack, its size being shown in Figoll is loaded of two points. It is the case of pure bending. P = 111.2N; E = 2687MPa; # = 0.33. The contour of displacement of Ux and Uy is shown in Fig. 12 and Fig.13. The model of the calculation is shown in

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1988

Fig. 14. The mixed-mode S I F K 1 and K 2 are calculated along F2. The resuh is Ki = 2.686MNm - 3/2, Kn = l : 1 2 7 M N m - 3 / 2 . Compared with Boundary Collocation Method, the relative errors of the KI and Kn are 1% and 1.6% respectiw~ly.

r2

C

Fig.14

Fig.13

V. CONCLUSION The hybrid numerical and experimental method, especially the boundar~ element and the Moir6 Interferometry combined together, for the determination of stress intensity factor is very effective and highly accurate. For this reason, it is expected that the further development of this method will make it an effective tool for stress analysis.

REFERENCES [1] Stern, M.Becker, E.B. and Dunham R.S. Contour Integral Computation of Mixed-modeStress Intensity Fact~ Int. Soum. of Fracture 12, (1976) 35%-368. [2] Qronch B.L. and Bthrified, A.M. Boundary Element Method in Solid Mechanics (198,3). [3] Wang Yinyan, ReleaseFreezingStress Moir6Interferometry Method and It's Application in Determination of the SIF., Thesis, Tianjin University, China (1984), (In Chinese). [4] Lu Hua, The Application of Moir6 Interferometry Method in the Fracture Mechanics, Thesis, Tianjin University, Tian]in, China (1984), (In Chinese). [5] Luo Zhishan, Yuan Fuxiang, Zhang Guiqin, Xaio Wei, Moirb Interferometry of Sticking Film and It's Application. Journal of Experiment Mechanics 1, 3, (1986), (In Chinese). [6] Post D. Developments of Moir~ Interferometry Opt. Engineering, 21, 3, (1982).