Thermal Residual Stresses in Thick Graphite/Epoxy Composite Laminates--Uniaxial Approach

by D. Joh, K.Y. Byun, and J. Ha

ABSTRACT--Thermal residual stresses have been known to be very large in laminates of continuous-fiber-reinforced polymer composites. When the thickness of the laminate is large, however, the measurement of the residual stresses raises questions on the accuracy of the conventional methods. A novel concept of layer separation is developed to measure quantitatively and precisely the tensile residual stresses in thick plates with layered distribution of residual stresses. It is applied to thick [02/904113s, AS/3501-6 graphite/epoxy laminates. The test specimens were mechanically modeled into the thin strips for the application of the new concept of layer separation. The tensile residual stresses measured in the 90-deg layers of these laminates are nonuniform throughout the specimen, and vary from 55.6 MPa to 71.4 MPa. It is very inter- esting to compare these values with the transverse strength #7 of AS/3501-6 unidirectional composites, which is 65.4 MPa.

Introduction The adverse effects of the residual stresses on the fatigue life

and strength of structural components are known to be sig- nificantly large, and they must be considered in the design of structures. In particular, thermal residual stresses which are com- mon in hot-rolled steel plates or polymer-matrix composite laminates have been recognized to be difficult to characterize due to the various factors involved, such as nonlinear or viscoplastic material properties at high temperatures and cooling conditions. Therefore, pro'ely analytical attempts to assess the effect of thermal residual stresses cannot be successful without measuring the residual stresses in the plates.

The unique characteristic of the residual stresses is that the stresses are in a self-equilibrium state. To measure the latent strains caused by these residual stresses, many investigators have developed various experimental methods, either destructive or nondestructiveJ 4 Nondestructive methods, such as acoustical techniques s, 6 and X-ray methods, 7 are not very successful with residual stresses in a thick member especially when the material is highly nonhomogeneous or anisotropic, because the sensitivity is limited. In destructive methods, such as hole-drilling 8~~ and

D. Joh is Assistant Professor, and K.E Byun and J. Ha are Graduate Students, University of Missouri-Rolla, Department of Mechanical Engineering, Rolla, MO 65401. Original manuscript submitted: June 15,1991. Final manuscript received: October 3, 1992.

surface-layer removal, the measurements are conducted indirect- ly, and calibrations are needed. The accuracy of the data in these methods is relatively low when the distribution of the residual stresses is complicated or discontinuous, or when the thickness of the member is very large. Thus, the direct and precise meas- urement of the residual stresses whichvary through the thickness of thick plates has been challenging.

This paper presents a novel optical technique to measure quantitatively and precisely the residual strains which correspond to thermal residual stresses varying through the thickness of thick layered plates.

Concept of Layer Separation Thermal residual stresses are developed in each layer because

the free thermal expansion or contraction is confined by other layers which have different coefficients of thermal expansion. In this novel concept of layer separation, each layer is separated from the other layers in the laminate at room temperature. Then free expansion or contraction will occur in each layer, as the cohesive constraints are removed when the layers are separated. The residual stress can be determined by measuring the released strain in the layer, because the relationship between the released strain and the residual stress is linear at room temperature.

/ 1"

/

Fig. 1 .__~jR. T- e curve of a 90-deg layer in cross-ply composite laminates

70 * March 1993

~,% B . . y ~ To M(~rk Sysiem Spectrn+n

t:p;,o

/ / , ~B'\\

Fig. 2.--A typical optical setup of moire interferometry

A 90-deg layer in a cross-ply polymer composite laminate is considered for the description of the concept of layer separation. Generally, the coefficient of thermal expansion of 90-deg layers is much larger than the global coefficient of thermal expansion of the laminate. The 90-deg layers are subjected to residual tensile stress in cooling. As schematically shown by using ~R - T - ~ curve in Fig. 1, the history of the tensile residual stress developed in a 90-deg layer is generally considered to be nonlinear when the laminate is cooled down from the curing temperature of the polymer matrix, Tc, to room temperature T~m along path I. The objective of the concept of layer separation is to measure the residual stress oR at T,m. The residual stress in this layer cannot be measured directly, however, because the residual strain, aR, of the layer is latent in the lmninate. When this layer is separated from other layers at room temperature, T,~, the tensile residual stress is relieved elastically along path 1I, accom- panied by the compressive released strain e,. The slope of this curve of linear relaxation is the elastic modulus of the 90-deg layer, E2, at room temperature.

To measure the released strain, the gages must be attached to this layer before the layer is released by separation. When the thickness of this layer is very small, however, it is practically impossible to use electrical strain gages. In the method presented in this paper, high-sensitivity molt6 interferometry is used to measure the deformation of each layer over extremely small areas. Provided the residual strain eR is equal to the negative released strain -e, the residual stress at point A in Fig. 1, can be determined by using linear-elastic constitutive relationships of the material through path II.

For cross-ply composite laminates, the state of residual stresses in each layer is biaxial, and the plane-stress constitutive relation- ship must generally be employed to determine the residual stres- ses by using the residual strains measured in two directions. In this paper, however, a simplified relationship is used to determine the tensile residual stress in 90-deg layers by measuring residual strains in one direction only.

The basic ideas of high-sensitivity moir6 interferometry are explained in detail in the literature.'. ~2 In moir6 interferometry a high-frequency cross-line diffraction grating is replicated on the specimen. When the specimen is deformed, the diffraction grat- ing deforms with the specimen. By illuminating the deformed specimen grating with well-collimated laser beams, interference patterns are obtained which represent the deformation of the underlying specimen. The resulting fringe patterns correspond to those of moir6 with 2,400/?/mm (60,960/?/in.). In-plane U and V displacement fields are determined, with sensitivities of 0.208

A

(a) '"

0 . $

Z Z

(b) (c )

Fig. 3.--(a) A [0m/g0m]s laminate, in which the fiber orientation in 0-deg layers coincides with the x direction. (b) A sketch of residual stresses distributed through the thickness of a cross-ply laminate at an arbitrary cross-section A-A. (c) The stress state which is superimposed on the residual stresses of Fig. 3b then cross-section A-A is cut

gm (8.20 gin.) per half-fringe order. This sensitivity is as high as that of the electric strain-gage method. The resulting data are the whole-field contour map of deformations, while the strain-gage method produces discontinuous data averaged over each gage area. Consequently, the spatial resolution of moir6 interferometry is incomparably higher than that of the electric strain gage methods. The most typical optical setup of moir6 interferometry is shown in Fig. 2.

When the ratio of the length or width of the plate to the thickness is relatively large, it can be assumed that the stresses are the function of the position in the thickness direction only at the region away from free edges. At this region, all stresses except for the in-plane stresses must vanish for laminates of symmetric stacking sequences. These assumptions can be brought from the analogy to the analysis of free-edge stresses in composite laminates ~3 and expressed using the coordinate system in Fig. 3(a) as

c~j = ~j(z) i , j=x,y (1)

~Gij = Ooij = 0 i , j=x,y Ox 3y (2)

o~j = ~jz= 0 j=x,y, andz

at positions away from free edges.

(3)

Experimental Mechanics * 71

[021904 ]13s

L ,J

Fig. 4.--Thin-strip specimen cut from a free edge surface of a laminated plate

Modeling of Thin-strip Specimens In Fig. 3(h), residual-stress distribution is schematically shown

at a cross-section A-A of a [0m/90m]~ laminate, in which the fiber orientation of the 0-deg layers coincides with the x direction. When the laminate is sectioned along this cross section, the state of stress on this surface can be embodied by superimposing the stress field of Fig. 3(c) on that of Fig. 3(b). The two stress states are exactly the same with opposite signs. The resulting stress distribution represents the null stresses which satisfy the traction- free condition of free surfaces. It must be noted that the stresses of Fig. 3(c) also satisfy the self-equilibrium condition of forces on this free surface as the initial residual stresses of Fig. 3(b). The superimposing stresses in the y direction develop interlaminar stresses, 6z and %, near the free-edge surface, which causes corresponding deformations in the y-z plane. However, when the width and the length of the plate are sufficiently large compared to the thickness of the plate, these deformations cannot affect the x dimension of the free-edge surface, since the deformations are of the plane-strain state confined in the y-z plane by the large undeformed part of the plate. This argument has been verified experimentally by using moir6 interferometry.

For practical convenience in the final procedure of layer separation, however, a thin strip which is cut from this free-edge surface, as depicted in Fig. 4, is used as the test specimen. The aforementioned argument on the plane-strain state of deforma- tions near the free edge cannot be applied to this thin strip specimen which has two free-edge surfaces, because the length of the strip may change when it is sectioned from the large constraining plate.

Nonzero Effects of Strip Sectioning on Specimen Length

To show the effect of strip sectioning on the longitudinal dimension of the specimen, a representative volume element is used with the stacking sequence [0m/90~m],, as shown in Fig. 5(a). When this element is in the original plate, the two faces are subjected to the residual stresses in the y direction, as schemati- cally shown in Fig. 5 (b). When this element is sectioned from the laminated plate, the stresses in Fig. 5(c) are superimposed on those in Fig. 5(b) to satisfy the traction-free condition on the two exposed surfaces. In order to simplify the analysis further, the stresses of Fig. 5(c) will be considered as the sum of the two decomposed parts, as shown in Figs. 5(d) and 5(e). With the

(a)

i

Om �9 o - ~ ~ . ~ , , ' , o . . . . . , ,

Om

(b) (c) (or)

Fig. 5.--(a) A representative volume element in the [0m/90~m]s laminate. (b) Residual stresses in the y direction. (c) Stresses superimposed on the residual stresses when the element is separated from the laminate. (d) Stresses superimposed on the 0-deg layers. (e) Stresses superimposed on the 90-deg layers

72 ~ March 1993

F :2::: : : : : : : : : : :.::::::: :::: ; L-':~I~ : L'- 7:

~"~- Diffraction Grating (a)

1 (b)

Fig. 6.--(a) Replication of diffraction grating on a thin-strip specimen. (b) Layers of the specimen separated into a comb shape by using a low-speed diamond cutter

thickness ratio, ~, between the layers of two different fiber orientations in the representative volume element, the compres- sive residual stress in the 0-deg layers must be { times as large as the tensile residual stress in the 90-deg layers to satisfy the zero-resultant force condition in the y direction. The 90-deg layers subjected to the tensile stress, ony, will contract in the x direction due to the effect of Poisson' s ratio, and the 0-deg layers will expand in the same direction with the Poisson's effect of the tensile stress, ~CYny. However, if the interlaminar bonding is perfect, the effect of shear stress, Zzy, must be considered at the interfaces between layers, that is, {ony on the 0-deg layer will cause the adjoining 90-deg layers to deform in the y ~lirection. Defining g~ as the coefficient of mutual influence on the strains in the y direction between layers, the tensile strain it} the 0-deg layers can be expressed as gtEny, in which ~y is the strain caused by Oqy in the 90-deg layers. By using the rule-of-mixture concept for the stress in Fig. 5(d), the total force P in the y direction is expressed in terms of stress and strains as

+ W 1~ 1 + ~ E2glEqy (4)

where E,, and E2 are the Young's moduli in the material principal directions.

From eq (4), eny is determined as

•~ ~qly = ~.~E 1 ..~ E 2 ~ l ( 5 )

Assuming the layers are not mutually constrained in the x direction, the compressive strains caused by the effect of Poisson's ratio are determined in the x direction as

gl~OnyV21 ~ " = - g'eaYv2' - ~E, + g,E2 (6)

90 ~(~]yV 12 ei'leX = - -e .~yVl2 = ~E, + [.t,E2 (7)

where v12 and v2, are the Poisson's ratios defined as v12 = - E2/~I when the load is in fiber direction, and v2, = - e,/e2 when the load is in transverse direction in the unidirectional composite.

By applying the same procedure to the compressive stress, ~onY, on the 0-deg layers, as depicted in Fig. 5(e), the noncon- strained tensile strains in the x direction are given by

0 V21 ~(~y e"~ - g ~ E ~ (8)

r go_ v,21a,{o.~ ~ = (9)

By superimposing the strains obtained in eqs (6)-(9) for each layer, respectively, the total nonconstrained strains in each layer, ~y and en~, are given by

2 2 4 , = _2 v2,~(_, - ~ , )E,o~

(10)

90 - - - - V 121~( 1 - - ~ t~ )E2Ol l y

I~qx - ,u, IE~ - -]- ~ ( 1 Jr- g2,)E,E2 + g,{2 E (11)

However, since the layers are bonded together, they are forced to have the same strain, ~], in the x direction. The stresses developed in each layer by the mutual constraint in the x direction are

(12)

o7 = E~(~: - 4 ~ = E~{< + ~I A J (13)

where A = ~ [ iE~ ]- ~ ( 1 -[" ~.~21)E1E2 -}- ~.~i~2~11 , and B = v2,~(1 - I.tZl)E, =v,2~ ( 1 - g~)E2, since EIV2, = E2V,2 by the symmetry of the compliance matrix for orthotropic materials. These stresses must also satisfy the zero-resultant force condition in the x direction, because there is no net external load in this

Experimental Mechanics �9 73

direction. Thus, the compressive stress in the 0-deg layer must be ~ times as large as the tensile stress in the 90-deg layer, that is,

G~ + ~(ygx~ = 0 (14)

By substituting eqs (12) and (13) into eq (14), the x strain, ~, induced by the thin-strip cutting, is determined to be

~ _ B~(E~ - E2)l~qy A(E, + ~E2) (15)

The coefficient of mutual influence, g~. is a function of the interlaminar shear moduli of each layer, G~3 and Gl2, and the thickness of the 0-deg layer which is more compliant to shear stress, "czy, than the 90-deg layer. This coefficient, g~, is also a function of the specimen thickness, t. When the thickness, t, of the specimen is very large compared to that of 0-deg layer, ~.~--~ 0, and from eq (15), e~ = 0. When t = 0, g~ ~-- 0, and the resulting strain is reduced to

E: = ~VI2(EI - Ez)(~ny El(El-1- ~f2) (16)

For the plates ofisotropic materials, such as hot-rolled steel plates in which the residual stresses are distributed varying through the thickness only, E~ = Ezin eq (15), and ~2 = 0.

Layer Separation and Release Strain The next step of the experiment is to determine the released

strain, ~,. A diffraction grating is replicated on one of the two faces of the thin-strip specimen in the region away from the two free ends, as shown in Fig. 6(a). After the specimen is cut perpen- dicular to the longitudinal axis of the specimen along the dashed line in the region of diffraction grating, each layer is separated by using a low-speed diamond cutter, in the shape of a comb, as shown in Fig. 6(b), leaving a large area of the specimen grating, which will be used to align the optical system with the specimen, intact. Being separated, each layer will deform elastically in accordance with the residual stress that the layer has before being separated. Using the fringe pattern obtained from the deformed specimen grating on each separated layer, eris determined quan- titatively with high precision. In this process of separation, the residual stresses in the y direction are also released. These stresses are smaller than those stresses in the same direction that the layers had in the unsectioned plate, because they are relieved significantly when the thin-strip specimen is sectioned from the plate with the two stress-free faces close to each other. However, it must be recalled here that what is measured by moire inter- ferometry is the difference between the two geometrical dimen- sions before and after the specimen deforms. Therefore, regardless of the consecutive changes of three-dimensional stress states which are induced by each of the cuttings involved, the final fi'inge pattern will show the total deformations in the x direction in each layer, except for the initial strain, ~.

A question may arise about the validity of this method, how- ever, because when the layers are separated using a low-speed diamond cutter, a boundary layer of high machining residual stresses may be formed near the cut surface and affect the true deformation fields induced by hygrothermal residual stresses only. In order to measure the undesirable effect of these machin- ing residual stresses on the final result, a unidirectional com- posite plate of the same material, which is free of residual stresses in the inter layer scale, is formed into the comb shape using the same cutter. The rake angle of the cutter to the specimen-grating

Fig. 7.--Deformation field at the free edge formed when the specimen is cut perpendicular to the layers

surface is maintained at 90 deg. When the thickness of the teeth in the comb-shaped specimen is large compared with the thick- ness of the boundary layer of machining residual stresses, the effect of the cutting stresses on the deformation of each tooth is also small compared with that of the initial residual stresses. As the thickness of each separated layer is reduced, the effect of machining residual stresses on the deformations increases. When this machining-stress effect is not negligible, a compensation is needed for these effects to determine the correct values of defor- mations caused by thermal residual stresses only.

Implementation of the Technique A thick AS/3501-6 composite laminate was fabricated with the

stacking sequence of [0J904]~3~. Thin slices were cut from the laminate by using a low-speed diamond cutter, as shown in Fig. 4. The length L and the thickness t of the strip were 300 mm and 2.54 ram, respectively, and the width was 19.8 ram, which is the same as the thickness of the laminated plate. The faces of the strip specimens were ground smooth and flat. After the specimens were dried in the oven, the cross-line diffraction gratings with a frequency of 1,200 ~/mm were replicated on each specimen at the middle section away from the two ends. The replication was done in a vacuum chamber to avoid moisture absorption.

The thin-strip specimens with diffraction gratings were cut perpendicular to the layers using a low-speed diamond cutter. At the new free edges thus formed by cutting, the residual stresses were released, satisfying the stress-free boundary conditions. The fringe pattern in Fig. 7 demonstrates the deformation at the free edge. The 90-deg layers have more fringes than the 0-deg layers, which exhibits that the compressive released strains in the 90-deg layers are larger than the tensile released strains in the 0-deg layers. However, the thickest 90-deg layer at A does not have as many fringes as predicted. This anomalous behavior may be conelated with the initial cracks observed only in this layer.

Then the layers were separated in the shape of a comb using the same cutter. A 0.15-ram thick diamond wafering blade of 76-mm diameter was used with 10 rpm. By balancing the weight of the specimen and the grip of the cutter, the normal force applied to the specimen by the blade was kept at about 25 grams. A high-power magnifying lens was used to place the cutting blade precisely between the layers of the specimen. The separated length of each layer was 5 mm. After the separation, a mild

74 ~ March 1993

mine this machining effect on the deformations. The cutting was done perpendicular to the fibers with the same speed of the cutting blade and the same pressure as were employed for the cross-ply specimen. The moir6 fringe pattern obtained from this unidirec- tional specimen showed virtually a null field, which indicates that the machining residual stress for AS/3501-6 is zero when it is cut with a low-speed diamond cutter.

T e s t R e s u l t s

Fig. 8.--A moire fringe pattern of released strains developed when the layers of the [02/90411as AS/3501-6 specimen are separated

degreaser was used to remove the cutting oil from the surface of the specimen. The comb-shaped specimens were put in the oven to expel the moisture absorbed after the specimen grating was replicated.

In the moir6 fringe pattern of Fig. 8, x-displacement fields are shown for each of the separated layers. The fringe patterns on the separated teeth show the compressive deformations linear-elas- tically released from the tensile residual stresses in the 90-deg layers. The angular deviation of the fringes from the y direction indicated the rigid-body rotations of the teeth, which was caused by the nonsymmetric deformations of the material at the roots of the teeth. However, the rotation of the fringes does not affect the measurement of the normal strain in the x direction.

To measure the strain e: induced in the strip-cutting process, the specimen grating was applied on the free-edge surface of the laminated plate prior to sectioning of the strip specimens from the plate. When the strip specimens were sectioned from the large plate, the diffraction gratings deformed with the underlying strip specimen, and the strain e* was measured quantitatively in the moir6 interferometer.

An anomalous phenomenon which needs a special attention is the apparently small residual deformation in the thick 90-deg middle layer. Cracks were initially observed in this layer when the specimens were prepared for the replication of diffraction gratings. It can be concluded from these cracks that, in the 90-deg layers, the tensile residual stress is larger than the transverse strength/7'~. As the tensile residual stress is significantly released when the cracks are formed, the measured released strain appears smaller in this layer than in other thinner 90-deg layers. However, the tensile residual stresses in all the 90-deg layers must be the same, since there cannot be any effect of layer thickness on the magnitude of the residual stresses. The reason that the cracks are not formed in the other 90-deg layers may be explained by the deformation-regulating effect of the adjacent 0-deg layers. In a cross-ply laminate in which the thickness of 90-deg layers is large, the deformation is not effectively regulated by the 0-deg layers from being locally concentrated. As the thickness of each layer is reduced, the deformation-regulating effect increases be- cause the spacing between the 0-deg layers decreases.

Since the deformations caused by machining residual stresses could be incorporated into the fringe pattern of Fig. 8 when the layers are separated, a unidirectional composite plate of AS/3501-6 was prepared and cut into the comb shape to deter-

Strain Analysis The released strains in the 90-deg layers were determined from

the displacement contours of moir6 fringes using the equations which are used for small-strain fields, as given by

~ = OU 1 {AN,,'~ F (19)

where ANx is the fringe-order difference between a set of two fringes, Ax is the distance between the two fringes, and MF is the magnification factor of the fringe pattern.

The residual strain, e R, was obtained by reversing the sign of the released strain and compensating for the strains developed in the procedure of machining. The resulting thermal residual strain is calculated as

e" = _ (~, - e* _ e~) (20)

where m is the residual strain caused by machining for the layer separation, which is negligibly small for AS/3501-6 composites, andC is 32 p-rn/m. The determined residual strains in the 90-deg layers were randomly nonuniform microscopically throughout the thickness and length of the laminate, which shows the non- uniform fiber volume fraction in the layers. The residual strain R measured by using the fringe pattern of Fig. 8 showed a variation between 4,250 p-m/m and 5,460 p,m/m, the average value of which was 4,860 p-m/m.

Stress Analysis Residual stresses in each layer of cross-ply laminates are

two-dimensional, in general, at regions away from free edges. To determine the biaxial residual stresses from the measured strains, the plane-stress constitutive equations must be used. What is presented in this paper, however, is a simplified one-dimensional approach to measure the tensile residual stress in the 90-deg layers.

As there are no in-plane shear stresses in the principal direc- tions of each layer in cross-ply laminates, the constitutive relationship is reduced to

yo: \ 90 '~" [~'R2J ~QI2 Q22J [~2.J

(21)

where the subscripts 1 and 2 represent the fiber and the transverse directions in each layer, respectively, and the Q~, the so-called reduced stiffnesses, are approximated in terms of the engineering constants ofAS/3501-6 composites as follows, with errors of less than one percent.

Et E~ Q " - 1 -v~2vz~ - 1 - (0.30)(0.027) = E~

Experimental Mechanics ~ 75

VuE2 V2~E, - - - - - ~ 0.30E2

OJ2= 1 -V,2V2, 1 -V,~V2,

E2 Q22 - ~E2

1 - v~2v2L (22)

The 90-deg layers of the [021904]13s laminate are compressed in the y direction of fibers, but the compressive strain e9o was

�9 90 measured to be less than three percent of the tensile strain ER2, which shows a good correlation with the prediction made by the classical lamination theory. Then, from eq (21) the tensile residual stress in the 90-deg layers is determined as

90 E90 90 cruz = O~ Rt + Q22ga2 -~ 0.35E2(0.03e~) + Eze~ ~ --~ EaSR~ (23)

The tensile residual stresses in the 90-deg layers, calculated by using the measured strain values and eq (23), are very non- uniform throughout the specimen. The average value of these tensile residual stresses is 63.5 MPa which is 97 percent of the transverse strength f~ of AS/3501-6 unidirectional composites.

Discussion and Conclusions

The method of layer separation was implemented to measure the tensile residual strain and stress in a [02/904] t3~ AS/3501-6 laminate at room temperature. Due to the difficulty involved in obtaining the biaxial strain values at a point, a simplified uniaxial approach was employed with the thin-strip specimens to deter- mine the residual strains and stresses in one direction. With the high sensitivity and high spatial resolution of moir6 inter- ferometry, the residual deformations in the thin separated layers were measured quantitatively. The whole-field contour map of the released deformation field was obtained. The effect of machining residual stresses on the result was zero for the tested material. The distribution of the tensile thermal residual stress was nonuniform microscopically, which may be an indication of the nonuniform fiber volume fraction in the material. It varies from 55.6 MPa to 71.4 MPa. However, the residual stress was macroscopically uniform throughout the thickness of the com- posite laminate, which indicates that the residual stress is not a function of layer thicknesses. It is very interesting to compare these values with the transverse strength F2~ of AS/3501-6

unidirectional composites, which is 65.4 MPa. The cracks ob- served in the thick middle layer also show that the residual stresses are at the level of failure stress. However, cracks are not observed in all the thin 90-deg layers in spite of the same magnitude of the residual stress, because the localized stress concentration near failure can be more effectively regulated by the 0-deg layers when the 90-deg layers are thinner.

Acknowledgments

This research was sponsored by the National Science Founda- tion Science and Technology Center at Virginia Tech through contract No. CR-4374-427203 and the Manufacturing Research and Training Center of the State of Missouri. The authors grate- fully acknowledge their support. The authors also would like to express special thanks to Mr. R.E. Bohlmann at McDonnell Aircraft Company for his support with specimen manufacturing.

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Stresses in Review, "Report at the SAE Golden Anniversary Meeting, Detroit (Jan. 1955).

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3. Tebedge, N., Alpsten, G.A., and Tall, L., Measurements of Residual Stresses - - A Study of Methods, Fritz Eng. Lab. Rep. No. 337.8, Lehigh Univ. (Feb. 1971).

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5. Hildebrand, t3.P. and Brenden, 8. 8, "'An Introduction to Acoustical Holog- raphy," Plenum Press, New York (1972).

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7. Ruud, C.O., "A Review of Nondestructive Methods for Residual Stress Measurement," J. Metals, 33 (7), 3 5 4 0 (July 1981).

8. Bert, C. W. and Thompson, G.L., "A Method for Measuring Planar Residual Stresses in Rectangularly Orthotropic Materials, " J. Comp. Mat., 2 (4), 244-253 (1968).

9. Ajovalasit, A., "'Measurement of Residual Stresses by the Hole-Drilling Method: Influence of Hole Eccentricity, "J. Strain Analysis, 14 (4), 171-178 (1979).

10. "Determining Residual Stresses by the Hole-Drilling Strain-Gage Method" ASTM, 183781 (1981).

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7 6 �9 March 1993