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Journal of Composite

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 DOI: 10.1177/002199837801200404

1978 12: 390Journal of Composite MaterialsJames T.S. Wang and John N. Dickson

Interlaminar Stresses in Symmetric Composite Laminates  

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390

Interlaminar Stresses in

Symmetric Composite Laminates1

JAMES T. S. WANG2

School of Engineering Science and MechanicsGeorgia Institute of Technology

Atlanta, Georgia 30332

AND

JOHN N. DICKSON

Advanced Structures DepartmentLockheed-Georgia CompanyMarietta, Georgia 30063

(Received March 30, 1978)

ABSTRACT

Interlaminar stresses and displacements of each layer satisfying geo-metrical boundary conditions are represented in series of Legendre poly-nomials. The extended Galerkin procedure is used for each layer, and thefinal solutions are obtained by requiring the satisfaction of continuityconditions at each interface and the stress boundary conditions at exteriorplanes. A computer program has been prepared, and numerical results arepresented and compared with existing data.

INTRODUCTION

f NTERLAMINAK STRESSES PLAY an important role in causing failures of struc-Itural components made of composite materials. The peculiar phenomenon of

high interlaminar normal and shear stresses occurring in a region near the free edgeof composite laminates subjected even to in-plane loading has been experimentallyobserved by Pagano and Pipes [ 1 ] . Rybicki [2] studied the problem using the finiteelement method. Pipes et al. [3, 4] have computed the interlaminar stresses using afinite difference approach. Since high stress gradients occur near the free edge,

1 Results ot the paper have been accepted for presentation at the Eighth U.S. National

Congress of Applied Mechanics, Los Angeles, California.2 Also, Consultant, Lockheed-Georgia Company.

J. CCAfP~/T~ AM 77~/~7~, Vol. 12 (Oct. 1978),

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finite difference as well as finite element techniques become difficult in estimatingstresses in this area, and certain artificial manipulations were made. Hsu andHerakovich [5, 6] analyzed the problem by dividing each layer into an inner regionwhere classical laminated plate theory is assumed valid and a small region near thefree edge. They used a perturbation method, and concluded that their results pro-vide better insight than the finite difference solutions by Pipes [3]. Recent finiteelement results by Wang and Crossman [7] reveal some new interesting phenomena.The free edge effects treated as boundary layer problems have been studied by Tanget al. [8, 9]. The present paper applied the extended Galerkin’s method with

displacements and interlaminar stresses represented by complete sets of Legendrepolynomials. The method leads to a systematic procedure for computing inter-laminar stresses directly. The procedure has literally no limitations to the numberof plies involved in a laminate.

FORMULATION AND ANALYSIS

Figure 1. Geometry and coordcnates

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The geometry, coordinates and some symbols for a rectangular laminate sub-jected to uniform longitudinal strain, e, are shown in Figure 1. For symmetriccomposite laminates, only one quarter of a laminate (X > 0, Y > 0 and Z < 0) willbe considered in the analysis. Following the linear theory of anisotropic elasticity,the equilibrium condition for each layer based on the principle of minimum poten-tial energy may be represented by the following general equations in terms ofdisplacements with i, j ranging from I to 3:

where V and A are volume and boundary surface area respectively, L ii and Bi¡ arelinear differential operators, j~~ is the surface traction, x~ and ui are coordinates anddisplacements normalized with respect to their respective coordinate dimensions ofthe region under study. Displacement components in x, y and z directions corre-sponding to uniform axial strain are represented as follows:

For laminates consisting of orthotropic layers with elasticity axes along y and zdirections, the warpmg function u(y,z) should be zero. On the other hand, if angleplies are involved, the warping function will exist. However, many practical lay-upsresult in rather minimal warping and the effect of u(y,z) on the stresses should besmall. In this study, the warping function is taken to be zero. Consequently, onlytwo Euler’s equations and their corresponding boundary conditions involved inequation ( 1 ) will be considered. While cross-ply laminates will be used for examin-mg the proposed procedure of analysis, a numerical example involvmg angle-plylaminates is also presented later for comparison of existing results.

Guided by the symmetry of deformation, displacement functions u2 and U3satisfying geometncal boundary conditions along y = 0 are lepresented by doublesenes as follows:

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where Pa are the Legendre polynomials and Pm are the shifted Legendre poly-nomials. The interlaminar normal and shearing stresses treated as unknown loadingsat a general kth interface is also represented in terms of Legendre polynomials asfollows:

Since Legendre polynomials are complete, convergence of solutions are ex-

pected. Applymg the Galerkin procedure on equation (1) for i,j = 2 and 3 in

conjunction with equations (2) through (8), one obtains a system of mL X 2aLsimulations algebraic equations relating V ma’ Wmal Qml ~ TM , Tm ~ am , and e. Ithas been verified that the coefficient matrix for Y,,ta and Wma is symmetric. It isfurther found that all coefficients associated with W~ ~ vanish. As a result, one

equation only involves the coefficients of normal stress of adjacent interfaces. Fromthis, one can subsequently conclude that the first term of the series expansionrepresenting the average interlaminar normal stress vanishes along every interface.This is clearly the consequence of the loading condition considered, under whichthe resultant interlaminar normal force must be zero along every interface. Fromthe remainmg part of the system of equations, V,n~ and W ma are solved in terms ofadjacent mterlaminar stress coefficients and the applied strain C. By satisfying con-tinuity conditions along each interface, one obtains recurrence relations relating thecoefficients of interlaminar stresses at adjacent interfaces and Woo of adjacent plies.Finally, by satisfying the free of stress condition along the lower boundary planeand zero shearing stress and displacement w along the mid-plane of the laminate,one can calculate mid-plane normal stresses. Interlaminar stresses along other inter-faces are then calculated through the recurrence relations.

NUMERICAL EXAMPLES

The elasticity constants with respect to principal matenal axes used in [5] are

considered for all numerical examples in this study:

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The constants E1 1 = 20 X 106 psi and E2 =2.1 X 106 psi which are slightlydifferent from the above numbers were used in [3]. Using the computer programgiven in [10], elasticity constants with reference to x, y and z coordinates arecomputed for use in equation (1). The applied uniform axial strain exx = 6 = 1 willbe considered throughout this study.

The laminates consisting of four identical plies symmetrically stacked in

[0/90/90/0] and [90/0/0/90] sequence are first considered. The number of termsused in y and z directions for displacement functions will be represented by mL XaL . To examme the convergence of solutions, a [0/90/90/0] laminate with b = 0.2in. and t = 0.005 in., or b = 40t, is investigated by considering various number ofterms in the series. The effect of the number of terms in the z direction is firstexamined. For aL = 2,3 and 4 with mL = 12, the average transverse displacement,Wo o , and coefficients corresponding to m = 10 for interlaminar normal and shear-ing stresses are given in Table 1. Discrepancies for m < 10 are all smaller as

expected.

Table 1.

For mL = 16, discrepancies corresponding to various o~ are smaller than thosestress coefficients shown in Table 1. It is found for this case that al o ~ 1 ~ = 0.043 X106, T10~ ~ ~ = 0.026 X 106 and 0, (2) = 0.069 X 106 which differ appreciablyfrom those listed in Table 1. It is clear from this examination that the number ofterms in the z direction represented by aL has small effects. The effect of thenumber of terms in the y direction will be examined by considering aL = 2. Thevalue of m~ varying from 12 to 30 has been investigated. The coefficients ofinterlammar stresses have negligible changes for mL beyond 20. It is now evidentthat solutions are convergent following the present procedure. The maximum valueof ML equal to 30 has been considered in this study. As m increases, the inter-laminar normal stress coefficients first increases and then sharply decreases to asmaller value. However, the interlaminar shearing stress coefficients first increases,then decreases to a smaller value, and increase, subsequently again. It indicates thatthe interlaminar normal stresses converge bettei than the interlaminar shear stresses.

Interlaminar Normal Stresses

Stresses for [0/90/90/0] laminates with b = 40t have been computed for several

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Figure 2. a along Z = 0 for b = 40t

Interlaminar normal stress dis-tributions along interface at z = 0for various b to t ratios are plottedin Figure 4 for [0/90/90/0] lami-

nates. Although the nondimension-alized boundary layer width de-creases as the width of the laminateicreases as being reported in [5],the actual boundary layer widthequals essentially one ply thicknessfor all cases as shown in Figure 4. Itis also notcd that based on 30 X 2

approximation, the interlaminarnormal stress at Y = b decreases asthe width of the laminate increases.

Slight waviness of the stress distri-bution corresponding to smallervalues of b shown in Figure 4 indi-cates that to use more terms than

30 X 2 would be desirable. Similarstress distributions with lowei stresslevels are found along Z = -t inter-face.

mL X aL cases. Results on inter-

laminar normal stresses shown in

Figures 2 and 3 exhibit very steepgradient for a near the free edge.The magnitude of a at the free edgecontinues to increase as more terms

in the y direction are used, indicat-ing the possibility of the free edgepoints being singular. The presentresults are compared with results byHsu and Herakovich [5]. Curves

shown in Figures 5-8 of [5] haveno reference to specific interface.

They are assumed for the interfacebetween the first and second layers,and the curve for b = 40t case is

shown in Figure 3 with present re-sults. At any rate, the peak stressesat the edge given in [5] are sub-

stantially lower, and the boundarylayer zones are smaller than the

present and the finite difference re-sults [3].

H’tgure 3. a a/on?, /. = -1 for h = 401 t

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I

Figure 4. a alonx Z = 0

Interlaminar normal stresses

along Z = 0 and Z = -t for

[0/90/90/0] and [90/0/0/90]laminates are plotted in Figure5. Along Z = 0, the normal

stress vanes in similar manner

for both cases with their signs re-versed. However, the magnitudeat Y = b for [90/0/0/90] case issubstantially higher than the

[0/90/90/0] case. It is also

noted that a exhibits totally dif-ferent variations near the free

edge for those two cases along Z=-t. These results contradict the

earlier findings given in [3-6]but t confirm the observation

made in [7] that these two

laminates behave fundamentallydifferent.

For the case b = 15.04 t, the

perturbation and finite differ-ence solutions given in Figure 8

of [5] are shown in Figure 6. The peak stiesses at Y = b according to the presentanalysis are higher than the finite difference solution and are noticeably higher thanthe perturbation solution. The boundary layer widths based on the present analysislies between those of perturbation and finite difference solutions.

For [0/90/90/0] and [90/0/0/90] laminates with b = 8t, present results shownin Figures 7 and 8, generally support recent finite element solutions given in [7]where very fine element sizes were used. Results shown in Figure 8 also confirm thefindings presented in [7] that the interlaminar normal stresses for [0/90] and[90/0] laminates are not merely a reverse of sign.

Interlammar normal stiesses foi a symmetric laminate consisting of 8 layers of

[±45/0/90]S with b = 40t are presented in Figure 9. The maximum stress occurringat Y = b on the mid-plane of the eight layer laminate is several times higher thanthat in the four layer laminate of [0/90] S, and the sign reversed at Z = -2t and -3t. Normal stresses for 190/0/-45/45], case have also been computed. The maxi-mum stiess OCCUII1J1g at Y = b on the plane Z = -t and the mode of stressdistribution along all interfaces aie consistent with the results shown m Figure 13of [7 J . This confirms the shaip diffeieiice of these two lammates discussed m [7] .

Interlaminar Shearing Stresses

For the lom ply Idl1llnate of [01’>0] with b = 40t, the distribution ot shearing

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Figure 5. a for b = 40t.

H’tgure 6. a /or h = l5 041

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Figure 7. a along Z = 0 for b = 8t.

Figure 7. (Coutrnuod)

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Figure 8. a along Z = -t for b = 8t. Figure 8. (Continued)

stress between 0° and 90° layers are computed based on various values of m LandaL. The convergence is seemingly not as rapid as the convergence for the inter-laminar normal stress as being indicated earlier. The stress becomes negligible atpoints away from the free edge. Results for 0.9b < Y < b for 28 X 2 and 30 X 2

approximations are plotted in Figure 10. The stress increases to a peak value at Yequals approximately 0.97b and decreases toward 0 as Y approaches b. However, atY equals approximately 0.99b, the stress increases agam to a finite value at Y = bwhere the stress should be zero. The point having the lowest value moves closer tothe free edge as the number of m~ mcreases. Same phenomena occurring in eightply [± 45/0/90] and [90/0/+45] laminates are shown in Figures 1 1 and 12 basedon 30 X 2 approximation. The mathematical difficulty may be due either to theGibb’s phenomenon for generalized Fourier series representation of a function, orto the possibility of the free edge points bemg singular. If this phenomenon is

ignored, the peak stress occurred at y < 0.99b should be considered for design. Itshould also be reported that the convergence based on 30 X 2 approximationseemingly worsens as the b to t ratio decreases. The maximum interlaminar sheanngstress in the eight ply laminates is considerably higher than that occurred m thefour ply laminate of 10/901,. As shown in Figure 10, the peak stress for the fourply laminate based on the present analysis is comparable but lower than the per-turbation solution given in [5].

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Figure 9. a for h = 40t Figure 9. (Coiitiyziied)

Hgure 10. T alollg / -i (01 h -1UI fïgure I I. T for h = 40t

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Figure 12. T for b = 40t

CONCLUDING REMARKS

A systematic and straightforwardprocedure for computing interlaminarstresses directly is presented. Numeri-cal results indicate the possibility offree edge points at interfaces beingsingular. The computer storage and

computation time are governed mainlyby the number of different ply orien-tations. Therefore, the present pro-cedure has literally no restriction onthe number of plies involved in a lami-nate. Convergence of solutions appearsto improve for laminates having largerply width to thickness ratio. This is a

favorable feature as the number of

plies in a practical laminate will be

substantially more than four or eight.Consequently, realistic laminates will

have much larger b to t ratios than

those considered in the numerical

examples.

ACKNOWLEDGMENT

A number of discussions concerning mathematical aspects of generalized Fourierseries and boundary layer problems generously provided by Dr. D. V. Ho at GeorgiaInstitute of Technology is greatly appreciated. Some computational assistance pro-vided by Dr. Luther S. Long at Lockheed-Georgia Company is acknowledged.

REFERENCES

1. N. J. Pagano and R. B. Pipes, "Some Observations on the Interlaminar Strength of Com-posite Laminates," International J Mechanical Sciences, Vol. 15, (1973), p. 679.

2. E. F. Rybicki, "Approximate Three-Dimensional Solutions for Symmetric Laminates underInplane Loading," J Composite Materials, Vol 5, (1971), p. 354

3. R. B. Pipe, "Interlaminar Stresses m Composite Laminates," Al ML-TR-72-18, May 19724. R. B. Pipes and N. J. Pagano, "Interlaminar Stresses in Composite Laminates under Uni-

form Axial Extension," J Composite Materials, Vol 4, (1970), p 538.5. P. W. Hsu and C. T. Herakovich, "A Perturbation Solution for Interlaminar Stresses in

Composite Laminates," ASTM 4th Conference on Composite Materials Testing and Dc-sign, Valley Forge, PA, May 1976.

6. P. W. Hsu and C. T. Herakovich, "Edge Effects in Angle-Ply Composite Laminates," J

Composite Materials, Vol 11, (1977), p 422.7. A. S. D. Wang and F. W. Crossman, "Some New Results on I dge I flect in Symmetric

Composite Laminates," J Composite Materials, Vol 11, (1977), p 92

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8. S. Tang, "A Boundary Layer Theory - Part I: Laminated Composites in Plane Stress," J.Composite Materials, Vol. 9, (1975), p. 33.

9. S. Tang and A. Levy, "A Boundary Layer Theory - Part II: Extension of Laminated FiniteStrip," J. Composite Materials, Vol. 9, (1975), p. 42.

10. J. N. Dickson, "Computer Programs for the Determination of Equivalent Stiffness orSection Properties of Laminated Composite Plates, Beams, and Skin-Stringer Panels,"SMN327, Lockheed-Georgia Co., February 28, 1974.