ANGLE PLY LAMINATES

MICHAEL R. PIGGOTTUniversity of Toronto,

Toronto, Ontario, Canada

INTRODUCTION

These laminates have been neglected from the earliesttimes, despite the success of filament-wound pressurevessels. Thus, it was known since the early 1970s thatthese vessels could be very strong, even though they wereeffectively [±φ]s angle ply laminates, with φ ∼= 35◦ forideal strength and stiffness [1]. This was because suffi-ciently good experimental techniques to test flat couponshad yet to be developed. The classical (1960s era) exper-iments with long, narrow test coupons did not break anyof the fibers (see Tsai [2] and Jackson and Cratchley[3]). Later attempts to remedy this were unsuccessful;the specimens broke prematurely and did not achieve thepotential strength. Recently, though, a way to demon-strate the effective strength, without the need to constructa pressure vessel and test it to destruction, has beendiscovered.

NEW APPROACH

This is to mold a short, flat, wide coupon and test that inthe normal way or with notches. The laminate needs to besymmetrical (or balanced) so that it does not twist whenstressed in tension. Figure 1 shows a broken laminatesubjected to such a test [4].

The notched tests do indeed show that you can achievethe high strengths of filament-wound vessels; see Fig. 2.Naturally, unnotched tests have to be used to measurethe stiffnesses, but these confirm the viability of the testmethod for approaching the potential strengths of angleply laminates.

Figure 3 shows that the grips do not constrain thecoupons too much, or else the results would followthe upper (100% restrained) curve [5]. It also shows thatthe American Society for Testing and Materials (ASTM)coupon fails to measure the innate stiffnesses of theselaminates, as noted in the most recent versions of ASTM D3039. Figure 4 demonstrates one of the more remarkablepotential properties of balanced (symmetrical) angle plylaminates.

In fact, Poisson’s ratio can increase without apparentlimit when the matrix has a sufficiently low Young’smodulus compared with the fibers. Figure 4 shows theeffect of modulus ratio, E1/E2, on the maximum Poisson’sratio achievable by having a soft (low Young’s modulus)matrix coupled with stiff enough fibers. E1, the Young’s

Wiley Encyclopedia of Composites, Second Edition. Edited by Luigi Nicolais and Assunta Borzacchiello.© 2012 John Wiley & Sons, Inc. Published 2012 by John Wiley & Sons, Inc.

modulus of the constituent lamina in the fiber direction,and E2, the Young’s modulus of the lamina in the directionnormal to the fibers, are approximately equal to Vf Efand Em/Vm, respectively, so that E1/E2 ∼= VmVf Ef/Em, Vfbeing the fiber volume fraction and Vm being that of thepolymer matrix. The laminate angle has to be judiciouslychosen for this: at the top of the figure are the laminateangles selected.

The line in Fig. 4 is given by νxymax = ν12 + 0.45√

E1/E2and is independent of G12. The points are thecalculations from lamination theory. Carbon-reinforcedsilicone rubbers can be made with E1/E2 ∼ 19, 000,so a [±5]s laminate made from that should have aPoisson’s ratio of about 70. However, it is very difficult todemonstrate experimentally that Poisson’s ratio can begreater than 10.

Fracture Toughness

The fracture resistance of quasi-isotropic [0, ±45, 90]nslaminates, when tested across the fibers (the in-plane frac-ture toughness), is acceptable, but angle ply laminates canbe better. The ASTM E 1922 fracture toughness test, whilesuitable for quasi-isotropic laminates, cannot be used forthe angle ply laminates. This is because the coupon is notwide enough. So, angle ply laminates have to be testedwith a coupon similar to that shown in Fig. 1. Someresults with this configuration are shown in Fig. 5. Resultsfor carbon–epoxy and glass–epoxy laminates are shown,compared with those for a carbon–epoxy quasi-isotropiclaminate [6].

Design with Angle Ply Laminates

The almost universal use of pseudo-isotropic laminates isthought to be the safest option for the current applicationsof high-performance fiber composites in aircraft wings andso on. But supposing we were to be more adventurous, wemight chose to leave out the 90◦ layers that can cause pre-mature cracking and add weight. If we also leave out the0◦ layers, we have the option of the angle ply laminate. Inthe axial direction, we can make it as stiff as needed, withconfidence also in its strength and toughness. Moreover,by choice of angle, we can have the stiffness needed in theoff-axis direction as well.

Designing for stiffness with angle ply laminates alsotakes care of the strength. If we have our main laminatedirection, x, aligned with the maximum stress, then asFig. 6 shows, the ratios of stresses and stiffnesses are closeto each other for [±φ]ns angle ply laminates with φ between15◦ and 45◦. Moreover, having biaxial tensile stresses, thatis, σx > 0 and σy > 0, is beneficial. The bursting pressureof pressure vessels, filament wound at an angle of 55.7◦,

1

2 ANGLE PLY LAMINATES

Figure 1. Fractured [±15]2s carbon–epoxy laminate tested inthe wide coupon notched tensile test.

1.6

1.2

0.8

0.4

0

Str

engt

h (G

Pa)

30 60 90

Angle (degrees)

LaminateHoopAxial tube

Figure 2. Results for glass–epoxy from wide coupon tests com-pared with values from tube tests.

where σx = 2σy (Fig. 6), is greater than when the sametube is tested uniaxially, (σx > 0 with σy = 0) [7].

We will make the process as general as possible byassuming biaxial stresses. Further, we will let the majordesign criterion to be to limit the strains, as is customary.Let the minimum stress be a multiple, α, of the maximumstress. We set the principal axis, x, in the direction of themaximum stress. (The maximum and minimum stressesare normal to each other, and with axes chosen in thisway, the shear stress, τxy, is zero.)

The equations for balanced angle ply laminates arerelatively simple, thus, for example, we have, for theYoung’s modulus in the axial direction, Ex = Q11 − νxyQ12,where the Poisson’s ratio, νxy = Q12/Q22. The stiffnessin the normal direction, Ey, is the complement of Ex

and is given by Ey = ExQ22/Q11, and the shear modu-lus, Gxy = Q66 where the Qij ’s have the usual mean-ings (e.g., Q11 = Q11 cos4 φ + 2(Q12 + 2Q66) cos2 φ sin2

φ +

160

120

80

40

00 15 30 45

Angle (degrees)60 75 90

Stif

fnes

s (G

Pa)

25 mm43 mm100 mm

ASTM (25 mm wide, 150 mm long)

width, 20 mm length

Expected resultfrom LPT theory

ASTM resultsshow edgesoftening effect

LPT theory with100% restraint

Figure 3. Young’s modulus results for carbon–epoxy angle plylaminates of various widths and gauge lengths, compared withlaminated plate theory.

Laminate angle (degrees)

35 30 25 20 15 10 5

1 10

Max

imum

Poi

sson

’s r

atio

100 1000

Modulus ratio (E1/E2)

10,000 100,000

100

10

1

0.1

Figure 4. Showing the very high Poisson’s ratios possible withangle ply laminates.

Q22 sin4φ and Q11 = E1/[1 − ν12ν21] etc.) The correspond-

ing strains are given by εx = σx/Ex − νxyσy/Ex and εy =σy/Ey − νxyσy/Ex.

We eliminate σx and σy between the last two equations,using σy = ασx and with εy = βεx where β is a parameterthat allows us to set the y strain independently of thex strain. This gives a very simple design equation thatenables us to evaluate the requisite ply angle, φ, given anyvalues of α and β:

αEx

Ey+ νxy(αβ − 1) − β = 0

The process thereafter appears to be straightforward[8], and an organic construction approach could be used.We could build up our wing or windmill blade, area byarea, based on the local loads expected and hence theideal value of the ply angle and laminate thickness at thislocation. (Nature does this, as can be seen by examining thecellulose fibers in trees.) This sort of thing can be readilyput into effect with modern methods of fiber placementconstruction.

ANGLE PLY LAMINATES 3

Carbon

Glass

[0, +45, 90, −45]s

150

100

50

15 30Ply angle (degrees)

Fra

ctur

e to

ughn

ess

(MP

a m

1/2 )

45 60 75 Figure 5. Fracture resistance of angle ply laminates com-pared with those from a quasi-isotropic laminate.

1

0.5

0 15Ply angle (degrees)

30 45

Str

ess

and

mod

ulus

rat

io

0

σ y /σ x Ey/Ex

Pressure vessel(Winding angle 54° 44′)

Figure 6. Stress ratio and modulus ratio plotted againstlaminate angle for balanced angle ply laminates.

REFERENCES

1. Hull D, Legge MT, Spencer B. Composites 1978;9:17–24 and263–271.

2. Tsai S. NASA CR224, 1965.3. Jackson PW, Cratchley D. J Mech Phys Solids 1966;14:49–64.4. Khatibzadeh M, Piggott MR. Compos Sci Technol

1996;56:1443–1451.5. Piggott MR, Liu K, Wang J. ASTM STP 1383, 2000, pp.

324–333.6. Piggott MR, Zhang W. In: Blackman BRK, Pavan A, Williams

JG, editors. ESIS 3 Proceedings. Switzerland. Rotterdam,Boston, Heidelberg, London : Elsevier; 2002. pp. 445–454.

7. Soden PD, Kitching R, Tse PC, et al. Compos Sci Technol1993;46:363–378.

8. Piggott MR. What engineers should know about fibre com-posites. Toronto: MERP Enhanced Composites Inc.; 2006. pp.49–50. (Now obtainable directly from the author)

FURTHER READING

Jones RM. Mechanics of composite materials. New York:McGraw-Hill; 1975. (2nd edn. Philadelphia (PA): Taylor &Francis; 1999).

Piggott MR. Polym Compos 2000;21:506–513.