Rs In Laminated Fiber Epoxy Composites

Residual Surface Stresses in Laminated Cross-ply Fiber-epoxy Composite Materials by Harold E. Gascoigne

ABSTRACT--Residual (curing) stresses in a cross-ply laminated plate are related to the strains released when individual plies are separated. Released displacements are determined using high-sensitivity moir6 interferometry and linearized strain-displacement equations are used to determine residual strains. Elastic orthotropic stress-strain relations are used to calculate residual stresses remote from free-edges of a [902o/02o/902o] graphite-epoxy cross-ply panel. The measured strains compare favorably with those predicted by laminated plate theory. In a second example, the circumferential and radial residual strains and stresses at the end-section of a thick-walled cross-ply graphite-epoxy cylinder are determined.

Introduction Residual stresses in composite laminates result from various sources during manufacture but particularly from differences in thermal contraction of adjacent plies when a laminate is cooled from its curing temperature.These residual stresses depend on laminate construction and material properties as well as the fabrication process. Residual stresses are especially important at the free edge of a laminate where high values can create matrix cracking or delamination initiation. Cut-outs (holes) in laminated structures create free edges where the basic in-plane residual stresses generate strong interlaminar shear and normal stresses9 Organic, metal, and ceramic matrix laminates are likely to possess residual stresses after fabrication9 Since fabrication residual stresses may be large, it is essential that they be understood so that designers can estimate their influence. Two recent studies involving composite material have used moir6 interferometry to measure deformations due to released residual stresses; the first in a thick cross-ply ring I

and the second in a flat cross-ply panel. 2 Neither of these studies presents a unified experimental procedure which results in determination of the residual stresses at preselected locations. It is the goal of this paper to present a significant step toward such a unified procedure9 Only cross-ply laminates are considered in this paper in which representative residual stresses are determined in a flat panel and in a cylinder9

Approach During cooling from the curing temperature, certain plies contract more than their neighbors, leading to lockedin deformations at the ambient (operating) temperature9 When free-edges of a laminate are present, interlaminar as well as in-plane stresses are created. This 'free-edge problem' has been studied extensively in cases of mechanical loading. Since interracial forces (stresses) acting between plies that differentially contract during cooling from the curing temperature create deformations in the plies, elimination of these interfacial forces (stresses) will relieve the residual deformations in the plies9 Assuming elastic response, if all of the interface stresses are removed, each ply returns to a stress-free state throughout. In the present approach, plies are separated at interfaces using diamond cutting tools after replicating a cross-line diffraction grating on the surface where the residual strains and stresses are to be determined. The relieved deformations (strains) are the negative of the deformations (strains) produced by the residual stresses resulting from cool-down9 The relieved displacements are determined using moir6 interferometry and the corresponding relieved strains are determined from the linearized strain-displacement relations 4 9

3

.

.

.

.

OU 1 ON~ HaroM E. Gascoigne is Professor of Mechanical Engineering, California Polytechnic State University, San Luis Obispo, CA 9340Z Paper was presented at the 1992 SEM Spring Conference on Experimental Mechanics held in Las Vegas NV on June 8-11. Original manuscript submitted: April 31, 1992. Final manuscript received." May 27, 1993

OV 1 aNy

~ X - - 0 X - - f 0X

~Y----0y f Oy

l(aN~

(1)

~NyI (2)

Experimental Mechanics 9

27

where U, V are components of displacement in the x and y directions, respectively; N~ and Ny are fringe orders corresponding to U and V, respectively; a n d f i s the frequency of the reference grating which was 2400 g/mm (60, 960 g/in.) throughout. The corresponding residual stresses are determined from the general classical thermoelastic equations written for the cross-ply laminate as 5 cy!

-Ql! Qlz Q13 0

0

e~ - o ~ I A T "

0

0'2

Q12 Q22 Q23 0 0 0 9 (Y3 ---- Q13 Q23 Q33 0 0 0 "~23 0 0 0 Q44 0 0 "1~31[ 0 0 0 0 Q550 0 0 0 0 0 Q66 "[12 J

e~ - ~2AT E~ - o~3AT 723

71z

(3)

where the coordinate axes coincide with symmetry axes of the material. Because the grating is applied on a surface after cool-down, the residual strains measured are the terms (e~- ~IAT), etc. Since the surfaces on which the gratings are replicated are free of surface tractions, eq (3) simplifies to

~12

L0

0

Q66

[~12J

(4)

with the superscript R denoting residual strain or stress with Ez

Q,, -

1022 --

'1.)12 'O21

E2

QI~ ---

with the principal material directions: x perpendicular to the plane of the laminate, y parallel to the 0-deg fibers and z parallel to the 90-deg fibers. Residual strains and stresses at locations affected by and at locations not affected by free-edges are sought. To determine the strains in the plane of a free-edge, strips are isolated from an edge as shown in Fig. 1. If the lengths of the sides of the plate are sufficiently large, the edge effects will not be felt sufficiently far from the edge and the strains and stresses will be independent of y and z. Near the edges, interlaminar shear stresses are required to equilibrate the in-plane stresses created by differential thermal contraction. In turn, to satisfy equilibrium, the interlaminar shear stresses require interlaminar normal stresses. These interlaminar stresses have signs depending on the coefficients of thermal expansion, c~, of the plies. Figure 1 shows the sense of the stresses developed for c~90 > a0. Relieving the interlaminar stresses will relieve the locked-in residual strains in all plies9 To demonstrate, in a particular case, the release of deformations that were locked-in during cooling from the curing temperature, a 38 x 41-mm (1.50 x 1.61-in.) coupon was cut from a [902d02d9020] ]M7/8551-7A (Hercules Materials Co.) graphite/epoxy flat laminate having a thickness of 8.38 mm (0.330 in.). Each lamina was one-third of the total laminate thickness, i.e., each lamina had a thickness of 2.79 mm (0.110 in.). The panel had been stored in the laboratory in which subsequent measurements were made for several weeks to minimize hygroscopic effects. A 1200 g/ram (30 480 g/in.) cross-line reflective diffrac-

1)12 E2 l -- 1)12 1)21

Q66 = G12

1 - ~12 'D21

(5)

The normal strain perpendicular to the surface may be determined from '1-)13 1)23 ~3 = E~ - ~3AT = - 7 G1 - - - (Y2 E2

(6)

The 'tension value' of Poisson's ratio, a)~l, is defined as the decrease in length of an element in the one-direction due to a tensile force in the two-direction divided by the extension in the two-direction. The compressive value of Poisson's ratio, v~l, is correspondingly defined as the increase in length of an element in the one-direction due to a compressive force in the two-direction divided by the contraction in the two-direction. Differences between the Poisson's ratios in tension and compression in composites has been observed previously. 68

(a) I

,

l

o- o -oo -oo

~ _ _--

I-I

o o- ~o 1o7 o6 1 7 6.~,..r'l,..~ oo9o9o9 o o

r,

-ooo - o -'v ~ u 1 7 6 ooooo oo o o onono n

..~.--n,.-~

~v

,F

90

90

!i

/i

J

(~90~ ~0 Examples

Flat Laminate

As a first example, consider a [90d0d90n] flat laminate shown in Fig. 1. The x, y, z, coordinate system is aligned

28 9 March 1994

(b)

Fig, 1 - - G r o s s - p l y laminate. (a) Laminate construction and coordinate system. (b) Formation of residual stresses upon cooling from curing t e m p e r a t u r e

Ny

Nz

Fig. 2-Ny and Nz displacement fringes for surface ply of [902o/0zd90~0]cross-ply laminate. (a) Initial 2.16-mm (0.085-in.) layer thickness removed. (b) 0.50-mm (0.020-in.) thickness layer removed from original 2.16-ram layer

tion grating was replicated on one of the lateral surfaces using PC-10 adhesive (Measurements Group Inc.). The thickness of the adhesive layer in the replication process is typically less than 25 p-m (0.001 in.). The coupon with the replicated diffraction grating was placed into the field of the mterferometer and the null-field obtained in the projection plane for both the y- and z-direction displacement fields, V and W, respectively. The specimen was removed from the interferometer. The mold from which the grating was taken was then placed in the interferometer to verify that the same null-field was obtained. Thereafter, the interferometer could be brought to null-field immediately prior to determining the residual displacements in the plane of the grating using the grating mold as the reference for released deformation. This was a critical step in the procedure since several hours could elapse between ply separation and obtaining the final displacement interferograms. The outer (90-deg) ply was separated from the center (O-deg) ply using a diamond-wafering saw cutting -

4

parallel to the interface between the 90-deg and 0-deg plies. The thickness of the layer removed was 2.16 mm (0.085 in.). The relieved displacements in the plane of the surface are shown in Fig, 2. The effect of the edges is clearly evident extending inward from each edge approximately one-third of the length of the side of the coupon. Thus, approximately the central one-third of the length of the edge is free of edge-effect. The representative strain released perpendicular to the fibers was -0.37 percent (residual strain +0.37 percent) and representative strain released parallel to the fibers was +0.014 percent (residual strain -0.014 percent). Figure 2 also shows the corresponding displacement fields after a 0.5 l-ram (0.020-in.) surface layer was removed from the 2.16-mm thick original layer. No significant change in response was observed indicating that the locked-in deformations were fully released with the first cut at the interface and, thus, substantiated the premise that removal of stresses at the interface returns the lamina to a stress-free state.

Experimental Mechanics 9

29

TABLE 1--MATERIAL PROPERTIES* ii

viously described was replicated on one lateral surface. A 0.5 l-ram (0.020-in.) surface layer was removed using a diamond-wafering saw. Thus, the interlaminar tractions acting at the plane of the cut were eliminated and the surface slice relaxed to its stress-relieved state. The resulting Vand Wdisplacement fringes are shown in Fig. 3. The sign of the fringe gradient was determined using carrier fringes of extension. 9 The excellent fringe clarity and high fringe density permit reliable determination of the fringe gradients and strain. The residual (tensile) strain, ey, perpendicular to the fiber direction was +0.45 percent. The corresponding residual tensile stress was ~y = 35 MPa (5.1 ksi). At other locations on the boundary of the hole, residual shear stresses may reach significant levels which, in combination with normal stresses, may cause in-plane matrix cracking. The residual deformations in the plane of the free-edge are determined in a similar way. A cross-line grating was replicated on the edge of the coupon perpendicular to the z axis. Two cuts were made nearly parallel to the interface between the 0-deg and 90-deg plies using a 0.35-mm (0.014-in.) diamond slitting saw denoted as Cut I and Cut II in Fig. 4. These cuts were made at a slight angle with respect to the ply interfaces--the interfaces are indicated by the symbol 0 . . . . . . 0. The purpose of these cuts was to separate the central 0-deg lamina from the 90-deg lamina, thereby removing the interfacial stresses. The purpose of the slight angle of the cut relative to the interface was to reveal the slope of both the U and V field displacement fringes near and at the interface as seen in Fig. 4. Next, Cut III (Fig. 3) was made by parting off a layer of 0.80 mm (0.032 in.) parallel to the grating. Three separate strips were thus created; one from the central lamina and one each for the two outer laminae. Each strip was heldin place on a flat glass plate using two-sided adhesive tape, and the glass plate was inserted into the interferometer after taking great care to carefully align the interferometer for nullfield using the specimen grating mold. The U (x-direction) and V (y-direction) displacement fringes are Shown in Fig. 4. One entire interface is visible along 0 . . . . . . 0 adjacent to the chevron shaped (v or ^) fringes in the U-field. These chevron fringes indicate a change in sign of the throughthe-thickness normal residual-strain component ~ crossing the interface from positive in the central 0-deg lamina to negative in the outer 90-deg lamina. This is similar to the reversal in sign of the through-the-thickness normal stress very near the interface of a mechanically loaded cross-ply laminate described in Ref. 3 (page 209). The residual strains in the uniformly responding center onethird of the length of the edge are: ex= +0.50 percent in the center lamina and -0.22 percent in the outer lamina, ey =

I

Propertyt

IM7/8551-7A

AS4/3501-6

El(ten.)

1 6 5 x 1 0 S ( 2 4 x 1 0 a)

145 x103 ( 2 1 x 1 0 a)

E1 (com.)

148 x 103 (21.5 x 103) 143 x 103 (20.7 x 103)

E2(ten.)

7 . 8 x 1 0 a ( 1 . 1 4 x 1 0 a) 9.3 x103 (1.35 x103)

E3*(ten.) Su~ (ten.) Su~ (com.) Su2 (ten.) v12

7.8 x10a (1.14 x103) 9 . 3 x 1 0 a ( 1 . 3 5 x 1 0 a) 2480 (360) U 1650 (240) 1720 (250) 53 (7.7) U 0.3 0.3

v2("

0.014

v2a

U

U

c~r

+21.6 x 106/~

+21.6 x 108/~

~_

-0.4 x 10~/~

-0.1 x 108/~

0.020

*Hercules Materials Co. data base tStiffness and strength in MPa (ksi) at 20~ T= Transverse to fibers L = Along fibers **Calculated from reciproicity at 20~ U = Unknown *Property not available--assumed to be approximately E2

It is interesting to compare the strains released in the experiment with the residual strains predicted by laminated plate theory (LPT). The appendix gives the LPT analysis corresponding to the present example. With uncertainty in material properties, principally the values of the coefficients of thermal expansion, the measured values of the in-plane strains on the lateral surfaces away from the edges and those predicted by the LPT agree quite favorably. Table 2 gives these results. Using the material properties given in Table 1, residual stresses in the 90-deg layer are (~y= 28 MPa (4.1 ksi), which is approximately 50 percent of the transverse tensile strength, and (~z= -20 MPa (-2.9 ksi) or about one percent of the longitudinal compressive strength. To portray the influence of cut-outs (holes) on the state of residual strain (stress) near the intersection of the cut-out and the surface of the laminate, a 9.38-mm (0.375-in.) diameter hole was machined through the thickness of a coupon taken from the same panel. A diamond-core drill was used with liberal amounts of coolant and the cutting speed reduced to minimize cutting damage and introduction of machining stresses. A cross-line grating as pre-

TABLE 2--COMPARISON OF RESIDUAL STRAINS DETERMINED BY EXPERIMENT AND LAMINATED PLATE THEORY III

Measured Theory

30 9 March 1994

I

I

III

I

II

0-deg Lamina ] I to Fibers (percent) _L.to Fibers (percent) Not Measured Not Measured -0.037 +0.34

I

I

90-deg Lamina [ I to Fibers (percent) .1_to Fibers (percent) -0.014 +0.36 -0.016 +0.32

-0.04 percent in the center lamina and +0.33 percent in the outer lamina. An elemental strip shown in Fig. 4 at the intersection of the free-edge and the lateral surface of the laminate experiences uniaxial loading; a special case of plane stress. The values of residual strains at this location

are ex = -0.22 percent, and ey = +0.33 percent. This meant that when the residual stresses were released, the elemental strip responded as if a compressive stress, Cry,acted alone resulting in a Poisson's ratio v~ = 0.22/0.33 = 0.67. This seemingly large value of Poisson's ratio in compression has been observed in similar laminae. 8 Thick-walled Cylinder

9

Cut III

D

0.8010.032)

'

V

i.i

I ,,,,,u

41(1.61)

i /~0.35(0.014) Dimensions: ram(in] 3301

O)

Thick-walled cross-ply laminated cylinders are of current interest for use in submersible structures. The influence of curing stresses has been studied theoretically under the assumptions of linear-elastic response and temperature-independent properties. 1~The same methodology as described for the flat laminate can be used to determine the residual strains on surfaces of a cross-ply cylinder. A portion of a cylinder near its end section is shown in Fig. 5. At an end section of the cylinder, there will be a free-edge effect with the strains and stresses varying in the axial (z) direction. The r-0 plane is similar to the x-y plane at the free-edge of the laminate as shown in Fig. 4. Likewise, there will be residual strains and stresses in the r-0 plane. The representative strains released in the plane of the free surface at the end of the cylinder due to the removal of the circumferential and radial stresses, co and crrrespectively, (at the free surface 6z = 0) will be independent of 0

V - Field

W - Field

(b) Fig. 3-Released residual displacement fields for cross-ply laminate with through-the-thickness hole. (a) Coupon dimensions. (b) V and Wdisplacement fields (perpendicular and parallel to fiber direction at surface)

Fig. 4--U and V residual displacement fringes on the edge of a [902o/020/902o]cross-ply laminate

Experimental Mechanics 9 31

Fig. 5--Surface slice removal for thick-walled crossply cylinder

because of axial symmetry. Anomalies in material properties and fabrication will, of course, give local variations from average (representative) values. However, these strains and stresses may depend on axial (z) position. The following procedural steps are suggested to reveal the important characteristics of the residual stresses at surface points of the cylinder. (1) To determine the axial dependence of e0 and ez on, say, the outer surface, a small flat surface can be produced by carefully machining or grinding using diamond tools. The depth of this flat can be

made, say, one-third of the thickness of the outer ply without causing an appreciable force release in the outer ply. This fiat is desirable since replication on curved surfaces is very difficult. Next, a grating G~ is replicated on this flat and, subsequently, a thin layer containing the grating is removed with the thickness of the layer confined to the outer ply. The released strains will indicate the nature of the axial dependence in the outer ply. The same procedure could be used to determine the axial dependence of the surface strains on the inside surface but with greatly increased difficulty due to the concave curvature of the surface. To determine the released strains in the r-0 plane, a grating G2 is replicated on the end section of the cylinder and a thin slice parallel to the surface is removed. This piece can be cut into two portions from which information can be extracted for the circumferential plies from one portion and for the axial plies from the other portion (see Fig. 6). Stresses ~0 are removed by radial cuts and stresses (~r are removed by making short, straight cuts approximately parallel to the plies. As an example of these steps, a wafer containing G2 was taken from a AS4/3501-6 graphite/epoxy cylinder with ([903/0]20/903) stacking sequence designated 89-1A which was used in a previous study. 11 The cured plies had a thickness of 0.18 mm (0.0072 in.) giving a representative thickness of the three-ply circumferential layers of 0.55 mm (0.0217 in.). The cylinder had an inside diameter of 17.78 cm (7.00 in.) and a nominal wail-thickness of 1.58 cm (0.62 in.). Figure 6 shows the displacement fringes in the r-0 plane near the outside diameter resulting from the release of both 60 and ~YrThe circumferential and radial stresses for the outer four circumferential plies which are given in Fig. 6 were determined from eq (4) with the principal material directions for the 903 plies taken as 1 = 0, 2 = z, 3 = r. The exact locations of calculated stresses are shown for each ply in Fig 6. The use of carder fringes

Fig. 6--Circumferential and radial residual displacement fringes at end section of a thick-walled cross-ply cylinder. (a) Microphotograph showing cuts that release ~o and ~r. (b), (c), (d) are Vdisplacement fringes with different carrier fringes of rotation. (e), (f), (g) are U displacement fringes with different carrier fringes of rotation

32 9 March 1994

of rotation were indispensable in accurately determining the released strains. 12 Averaging of fringe spacings and fringe angles over the areas shown together with standard deviations are given. Since the outer circumferential plies had compressive circumferential residual strains, the value of v+21 was used in calculating the residual stresses using eq (4). Some small uncertainty exists in the residual strains determined at the boundary due to minor distortion of the grating caused by epoxy shrinkage. This uncertainty is discussed in the following secton. The value of ~0 = -140 MPa (-20.3 ksi) in the outer ply is in fair agreement with the value of -15 ksi determined using a layer-removal method for rings cut from the same cylinder.13The values of ~r calculated in the outer four circumferential plies is considerably larger (approximately ten times larger) than those determined by experiment in Ref. 13 or calculated for a similar cylinder in Ref. 10. It is interesting to note that the radial residual stress, or, increases very quickly from zero at the free surface to +14.9 MPa (2.2 ksi) near the inner surface of the outer ply as shown in Fig. 6. Experimental Error Sources

As in any experiment, Uncertainties occur that may deteriorate the accuracy of the results. The following uncertainties and sources of error are considered. (1) Uncertainty of the specimen-grating frequency. The method by which the master grating (from which specimen replicate copies used in the present work were made) was 14 made precludes an uncertainty of more than 1 fringe/ram in the frequency. This uncertainty would lead to a negligible error of 0.04 percent in the strains using eq (1) and eq (2). (2) Out-of-plane warpage of the specimen grating due to release of residual stresses. If the out-of-plane warpage causes a rotation about an axis perpendicular to the reference grating lines, no error is introduced. If warpage causes a rotation of the grating about a line parallel to the reference grating lines, an extraneous moir6 frequency Fe = -f W2/2 is caused where W is the rotation angle (assumed to be small) a n d f i s the frequency of the reference grating (see Ref. 4) For a rotation W = 0.01 rad, the extraneous strain a, = -50 p,m/m or -0.005 percent. It is highly unlikely that the deformations of the removed surface layers in the present study would be this large. Thus, the error due to out-of-plane warpage is judged to be insignificant. (3) Introduction of carrier fringes of rotation introduce extraneous fringes with frequency Fe = -f 02/2 where 0 is the angle between the initial specimen grating lines and the reference grating lines andfis the frequency of the reference grating (see Ref. 12). The carrier fringes are related to 0 by Fc = f O. The largest cartier frequency used in the present study was 35 fringes/ram for the outer circumferential ply shown in Fig. 6. The extraneous strain in this case is - 106 gm/m or approximately -0.01 percent, making the apparent value of 8rat this location 0.028 percent which is approximately 28-percent low. At other locations in Fig. 6 this error is small enough to be neglected. (4) Rigid-body rotations of portions of layers occur after release of residual stresses. Since a four-beam optical system was used, rigid-body motions of the grating do not

affect calculation of shear strains (Ref. 4, page 345). Normal strain calculations are not affected by rigid-body rotations regardless of the type of optical system used. (5) Shear lag in the specimen grating. Because the diffraction grating is separated from the composite surface by a thin layer of epoxy approximately 25-gm (0.001-in.) thick, abrupt changes in strain are not transmitted precisely to the external surface of the grating. High-strain gradient zones on the composite surface are enlarged on the grating surface and peak strains are reduced. Values of strain determined at ply interfaces where resin-rich zones may exist will therefore be less accurate than at locations away from such high-strain gradient regions. Since all numerical results presented in this paper are for locations away from ply interfaces, shear-lag error is judged to be unimportant. (6) Distortion of the grating at saw cut or free edge. If extreme care is used in cutting with diamond wafering blades, no damage to the grating occurs. Slow speeds are recommended with the use of liberal amounts of cutting fluid to keep the edge being cut cool and free of debris which can damage the cut edge. The blade removes material which is not available for analysis. The width of the saw blade can often be selected to closely match the thickness of a ply so that complete plies may be removed with a single cut. Wafering blades may be obtained which remove as little as 0.20 mm (0.008-in.). A more serious error may be introduced near a boundary due to shrinkage in the epoxy used to replicate the grating. This shrinkage near the boundary is manifested by distortion of the grating and the generation of 'parasitic' fringes similar to those in photoelastic coatings. This causes error principally in normal strains perpendicular to the boundary. These parasitic fringes can be readily identified by introducing carrier fringes of rotation and observing any curvature or 'hooking' of the fringes near the boundary. These fringes can be minimized, and in some cases completely eliminated, by carefully cleaning any excess uncured epoxy from the intersection of the grating plane and the boundary during replication. (7) Uncertainty in material properties. Except at resinrich or resin-starved locations in the material, the representative values given in Table 1 are felt to be accurate within +2-3 percent for modulii and within +3-5 percent for the coefficients of thermal expansion. Thus, in most cases, the greatest errors in the calculated values of strain and stress are caused by uncertainties in material properties.

Conclusions (1) The determination of residual surface strains and stresses for cross-ply laminates as described in this paper is a straightforward but tedious process. (2) Extreme care must be taken, in the method described, in releasing inter-ply tractions using diamond-cutting tools so as not to damage the excised section or the diffraction grating. (3) For the flat [9020/020/9020] graphite-epoxy laminate, good agreement was found between the experimentally determined in-plane residual surface strains and corre-

ExperimentaI Mechanics 9 33

sponding residual surface strains predicted by laminated plate theory, (4) Residual surface strains and stresses at the end section near the outer surface of a thick-walled ([903/0]2o/903) graphite-epoxy cylinder were determined. The introduction of carrier fringes in moir6 interferometry is an important tool in increasing the accuracy of the results where the load-induced fringes are irregular or sparse. (5) Material properties used in constitutive equations should be accurately determined to insure that the calculated residual stresses are realistic.

Acknowledgments Appreciation is expressed to the Photomechanics Group in the Department of Engineering Science and Mechanics at Virginia Polytechnic Institute and State University for their assistance during 1989-1990 when the work described herein was initiated. Special thanks are extended to Professor Daniel Post for his advice and encouragement. The author wishes to thank the Hercules Materials Company for supplying the specimens used in the study.

APPENDIX Laminated Plate Analysis In classical laminated plate theory (LPT)5 the following assumptions are made. (1) Displacements are continuous across the interface between two lamina. (2) The in-plane displacements at any point are related to the displacements of the geometric midplane. (3) Plane sections remain plane. (4) In-plane strains are related to displacement gradients at the geometric midplane. (5) Strains vary linearly (at most) across laminate thickness. (6) Through-thethickness normal strains are neglected. Figure 7 shows the coordinate system and deformations in the x-y plane. From geometry (A1)

v = Vo - xo~

with ~Uo

OY

(A2)

Also

References W=Wo--X 1. Lee, J., Czarnek, R. and Guo, Y., "Interferometric Study of Residual Strains in Thick Composites," Proc. 1989 SEM Spring Conf. on Exp. Mech., 356-364 (1989). 2. Lee, J. and Czarnek, R., "Measuring Residual Strains in Composite Panels Using Moird Interferometry, '" Proc. 1991 SEM Spring Conf. on Exp. Mech., 405-415 (1991). 3. Herakovich, C.T., "Free Edge Effects in Laminated Composites," Handbook of Composites, ed. C.T. Herakovich and EM. Tarnopolskii, Elsevier Science Publishers, 2, Chap. 4 (1989). 4. Post, D., "Moir~ lnterferometry," Handbook of Experimental Mechanics, ed. A. S. Kobayashi, Prentice Hall, Englewood Cliffs, NJ, Chap. 7 (1987). 5. Agarwal, B.D. and Broutman, L.J., Analysis and Performance of Fiber Composites, John Wiley and Sons, Inc., 148-155 (1990). 6. Halpin, J.C., "Primer on Composite Materials," Technomic Publishing Co., Lancaster, PA, 168-172 (1984). 7. Bert, C. W., "Modelsfor Fiber Composites with Different Properties in Tension and Compression," J. Eng. Mat. Tech., 344-349 (1977). 8. Gascoigne, H.E. and Abdallah, M. G., "Strain Analysis of a Bonded, Dissimilar, Composite Material T-Joint Using Moir~ lnterferometry," Opt. and Lasers in Eng., 13, 155-165 (1990). 9. Han, B. and Post, D., "The Tilted Plate Method for Introducing Carrier Fringes of Extension in Moirg Interferometry," EXPERIMENTAL TEC~Qt;ES, 13 (7), 25-29 (1989). 10. Hyer, M. W., "Hydrostatic Response of Thick Laminated Composite Cylinders, " J. Reinforced PIastics and Composites, 7, 321-340 (1988). 11. Abdallah, M. G., Cairns, D.S. and Gascoigne, H.E., "Experimental Investigation of Thick-Walled Graphite~Epoxy Composite Ring Under External Hydrostatic Compressive Loading," Proc. 1991 SEM Spring Conf. on Exp. Mech., Milwaukee, 626-631 (1991). 12. Guo, Y., Post, D., and Czarnek, R., "The Magic of Carrier Fringes in Moir( lnterferometry," EXPERtMEN'rALMECHANtCS, 29, 169-173 (1989). 13. Abdallah, M.G., "Residual Stresses in Thick-Walled Composite Rings," Proc. 1992 SEM Spring Conf. on Exp. Mech., Las Vegas, 1063-1070 (1992). 14. Post, D., MeKelvie, J., Tu, M. and Dai, F., "Fabrication of Holographic Gratings Using a Moving Source," Appl. Opt., 28 (15), 3494-3497 (1989).

34 9

March 1994

~z

(A3)

The linearized strain-displacement relations give

Ov OVo

- ~y

Ow ~ Z - ~z

3y

3Wo

O2Uo x Oy2

b2Uo

~z - X ~z ~

by 3w bVo 3Wo ~(y~= -q--+ --V-= - - + - - - 2 x oz oy Oz 3y

32Uo

(A4)

~yOz

Equation (A4) can be written in matrix form as

,,z

LvyzJ o

L~yzJ

(A5)

o

where e y, and ~ z, and "~~ z are the midplane strains and ~:y, ~c~, and ~Cyzare the plate curvatures. The stress-strain relations for the kth lamina are

yF f ~176 1 lfyt LQ,6Q2006JLyJ

(same)

,,,

(A6)

.

"p

v,

f!ot; r _A""_ _

I P8"6"6"6"8"6"8"~ ooooooooooo ,oggg~ )oooooooooor ooooooooooo ioooooooooo( )oooooooooo

LB, ID'J

7dOg,

(All)

where, for the case of cross-ply laminate symmetric with respect to the midplane,

oooooooooo( oooooooooo

1

oo x oooooo( )or ~ 3)00000000000( O0 000000 )0000000000r 00000000000

~oooooooonor

X

[A'] = [A]-'

[B'] =

[0]

{M T} = {0}

(A12)

with

)lot n

Deformed

[A]~ = [Au]~= ~ (Qij)k ( hi: - hk-, )

Undeformed

k=l

Fig. 7--Definition of the undeformed and deformed cross section of a laminated plate

where ~0 are the transformed stiffnesses which are related to the stiffness in the principal material directions, Qu, by eq 5.61 of Ref. 5. In' the case of thermal strains now considered, the mechanical strains are given by

where hk is the thickness of the kth lamina. Hence, as expected, the plate curvatures are zero for the symmetric cross-ply laminate. The thermal force resultant for the present case is

INIV!I

[Q22[ff'T'4;-Q120~L] [Qlll~Lnt'Q120[,T 1 (A13)

"~y~J tYyzJ [~yzj

(A7)

From eq (A12) the inverse of [A], [A]-~, gives the elements of [A'] as

where {e} are the total lamina strains given by

A'I 1--'-

A22 AuA22 - A22

Iy1,1 f](]yzyt

['YyzJ [~yzJ

-Alz Atl2 _-AuA22 - A~2 Au A'22 -A uA22 - A~2

(A8)

with the thermal strains given by

where Ali = ( 2Q22 + Qu ) h

V;yzJ L~y~ArJ

(A9)

where ~y, o~ and ~yz are the (transformed) coefficients of thermal expansion. The thermal stresses for the kth lamina are

~Ez-Ol,zAT I[ rzJk [O,6Qz6066J,[yyz-~zATJ O'y/~ oTI ,.[TT

__

__

A22 = ( 2Qu + Q2z ) h with

EL

I OllO] 2 #161 I~y -- 0(,yAT ] =1 a120%2 QG6| --

Am = ( 3Qm ) h

O l I --

(AlO)

The midplane strains and curvatures are related to the thermal force (Nr) and thermal moment ( M r) resultants by

1 - 19LragrL ~Lr EL Q12 --1 - ~Lra~rc Er Q22 --'1 - ~Lra&L

(A14)

Experimental Mechanics 9

35

Computing the components of the midplane strains gives

4= (2Qll+Q22)

[(Qn+2Q,2)o~L+(2Q22+Q12)o~r] - 3Q12 [(2Qn+Q,2)o~L+(Q22+2Q,2)o~r]

(2Q22+Qll) (2Qll+Q22)

-

9Q22

AT

(2Q22+Q11)[(2Q11+Q12)o~L+(Q22+2Q12)otr] 3Q12[(Q1,+2Q12)o~L+(2Q22+Q12)o~r] -

AT

(2Q22+Qll) (2Qll+Q22) - 9Q~2

(A15)

Since v12v21 is very small compared to unity, it will be neglected so that Qll -~ El, Q22 Defining m

= EffE1,

=

E2, Q12 -~ v12E2

r = V12m and rewriting (A15) gives e~ = (2+m) [(l+2r)aL+(2m+r)O~r] - 3r [(2+r)o~L+(m+2r)o~r] A T (2m+l) (2+m) - 9r 2 = (-3r[(l+2r)aL+(2m+r)O~r]+(2m+l)[(2+r)otL+(m+2r)~r] A T

(2m+l) (2+m)

-

9r z

(A16)

Numerical Example Using average values over the temperature range of AT = -160~ to be (XL= -0.4 X 10-6/~ C~r= -21.6 x'10-6/~ gives ey~ = -304 x 10 .6 m/m. Hence, in the 0-deg layer, the mechanical strain parallel to the fiber is ~Mt0deg]~--- ~ y

--

~L A T = -368

x 10 -6

mJm or -0.037 percent

and, in the 90-deg layer, the mechanical strain perpendicular to the fiber is ~[90 deg]= ey - o~r AT = +3152

36

9

March 1994

x 10 -6

m/m or +0.32 percent

The corresponding total midplane strain perpendicular to the fiber is ez~ = -93 x 10 -6 m/m and the mechanical strain in the 0-deg layer perpendicular to the fiber is

~M[0deg]=

~o --

[~TAT = +3363

x 10 -6

m/m or +0.34 percent

The mechanical strain parallel to the fiber in the 90-deg layer is

~zM[90deg]~-- -157

• 10 -6

m/m or -0.016 percent