PURE TENSION WITH OFF-AXIS TESTS FOR ORTHOTROPIC LAMINATES.

Transcription

1 PURE TENION WITH OFF-AXI TET FOR ORTHOTROPIC LAMINATE. G. Verchery, X.J. Gong. IAT - Insiu upérieur de l Auomobile e des Transpors, LRMA - Laboraoire de Recherche en Mécanique e Acousique, 49, Rue Mademoiselle Bourgeois - BP 3, 5827 Nevers Cedex, France. Georges.Verchery_isa@u-bourgogne.fr UMMARY : Orhoropic maerials can be classified in four caegories, wo of which possessing four direcions of exremum for he Young s modulus. Deformaion under axial load in hese direcions always occur wihou shear. I maes possible a complee measuremen of in-plane elasic consans by hree pure ension ess for maerials wih four direcion of exremum Young s modulus. This paper presens he analysis and experimenal evidence for laminaes. KEYWORD : off-axis es, ensile ess, orhoropy, elasic consans, laminaes. INTRODUCTION. Complee deerminaion of he mechanical properies of anisoropic maerials by simple ess is sill a challenge, even when limiing o elasic properies. This is mainly due o he many couplings exising in hese maerials, which do no allow simulaneously simple loading and simple deformaion excep for a limied number of cases, such as ensile ess in he direcions of he symmery axes. Generally such special cases are no enough o deermine all he elasic consans. The aim of he presen paper is o show ha, for cerain ypes of orhoropic maerials, i is possible o design oher simple ess o complee he deerminaion of he elasic properies. More precisely, for hese maerials, pure ension can be obained in he elasic range in four differen direcions, i.e. he wo axes of symmery and wo off-axis direcions (symmerical wih respec o he axes of orhoropy, all of which are direcions of exremum Young s modulus and zero coupling beween shear and longiudinal effecs. Many composie laminaes belong o his class of orhoropic maerials, and consequenly can be characerised compleely in heir elasic range by hree independen ensile ess. We derive hereafer he equaions and condiions of applicaion, and give experimenal evidence for such complee characerisaion of orhoropic maerials.

2 CLAIFICATION OF PLANE ORTHOTROPIC MATERIAL. The analysis hereafer is closely relaed o he ensorial naure of elasic consans (siffnesses or compliances, so use is made of he ensorial noaion, wih four subscrips for he Caresian componens of hese fourh order ensors. For convenience, he maching of he ensorial and usual conraced noaions for compliances is reminded in Appendix. As explained in Appendix 2, for an orhoropic maerial, he six ensorial componens of he elasic compliances, in axes x, y (wih axis x a he angle θ from he orhoropy axis of larger Young s modulus, can be expressed wih four invarian parameers,, r, r, and an ineger index, in he form : ( xxyy xyxy xyyy yyyy ( θ ( θ ( θ ( θ ( θ ( θ ( ( ( ( ( ( r sin 4θ r sin 4θ cos 2θ sin 2θ sin 2θ cos 2θ From hese Eqs., he following fundamenal relaion appears : ( θ (2 4 ( θ θ Consequenly, he number and direcions of he exremum values of Young s modulus E(θ or is inverse ( θ are deermined by he zero values of he coupling compliance ( θ. This coupling compliance, which conrols he ineracion beween longiudinal and shear effecs, can be expressed as : { cos θ} (3 sin 2θ r r ( ( θ 2 o, when r > r, here are only wo direcions of exremum, which are of course he orhoropy axes, where sin 2θ is zero. When r r, he las facor in Eq.(3 can be zero oo. Le : (4 r ( r cos2ω wih ω π 4 for even, and π 4 ω π 2 for odd. Then Eq.(3 wries : (5 θ 4( r sin 2θ sin( θ ω sin( θ ω ( which shows four direcions of zero coupling compliance and exremum Young s modulus, a relaive angles equal o, 9, and ± ω. Consequenly, orhoropic maerials can be classified wih respec o heir compliances ino four caegories :

3 i odd and r > r, wih only wo direcions of exremum values for he Young s modulus, i.e. for he maximum and 9 for he minimum (relaive angles, ii odd and r r, wih four direcions of exremum values, i.e. for he absolue maximum, ± ω for he wo absolue minima and 9 for a local minimum, iii even and r > r, wih only wo direcions of exremum values for he Young s modulus, i.e. for he maximum and 9 for he minimum, iv even and r r, wih four direcions of exremum values, i.e. for a local minimum, ± ω for he wo absolue maxima and 9 for he absolue minimum. For case ii (odd, he polar curve represening he Young s modulus wih he direcion is cross-shaped, wih is longer branches along he direcion, while for case iv (even, his curve is bone-shaped, wih is major exensions along he ± ω direcions. Furher, for boh cases, i can be shown ha he shear modulus in he (or 9 direcion is relaed o he Young s modulus and he Poisson s raio in he ω (or ω direcion, according o he formula : (6 ( ω 2G( E ν ( ω which is an exension of he well-nown relaion beween Young s modulus, Poisson s raio and shear modulus for isoropic maerials. This relaion should be disinguished from he similar formula : (7 ( π 4 2G( E ν ( π 4 which holds for any orhoropic maerial, bu does no relae properies in direcions of simple ess. uch heoreical classificaion is really meaningful. I can be checed easily from published daa for crysal elasic consans [] ha differen cases do exis for crysals. Furher, for laminaes, for which elasic properies can be ailored by conrol of he sacing sequence, all ypes can be obained. Using he classical laminaed plae heory and varying he basic laminas and he sacing sequences, i is easy o predic ha, for insance, laminaes reinforced wih unbalanced fabrics and cross-ply laminaes should belong generally o he i- and ii-ypes, while angle-ply laminaes should be generally of he iii- and iv-ypes. APPLICATION TO TENILE TET. I is well-nown ha, in he orhoropy axes, he compliance marix has he following form : ( which shows ha unidirecional load along any orhoropy axis produces longiudinal and ransverse srains, wihou shear deformaion.

4 Now le us consider Caresian axes in he off-axis direcion ω of exremum Young s modulus and is orhogonal direcion ζ. In hese axes, i resuls from Eq.(2 ha he coupling compliance ωωωζ, proporional o he derivaive of he longiudinal compliance ωωωω, is zero. o he compliance marix ges he special form : (9 ωωωω ωωζζ 2 ωωζζ ζζζζ ωζζζ 2 4 ωζζζ ωζωζ Comparison beween Eqs. (8 and (9 shows ha sress-srain behaviour in hese axes is no so simple han in he orhoropy axes. The non-zero compliance ωζζζ induces coupling beween shear and exension in he ζ direcion. However, no coupling occurs for unidirecional load in he ω direcion. In oher words, pure ension occurs for loading in he direcion of exremum Young s modulus. I resuls ha four pure ension ess can be applied o orhoropic maerials falling ino caegories ii and iv described in he previous secion, in he four differen direcions of maximum or minimum Young s modulus (, 9 and ± ω. Each es can lead o wo srain measuremens in he longiudinal and ransverse direcions, providing eigh values of he Young s moduli and he Poisson s raios in he corresponding direcions. Of course, all hese values are no independen. pecially, i is obvious ha measuremens a ω and ω should give he same resuls for Young s moduli and Poisson s raios, and i is well nown ha he Poisson s raios a and 9 saisfy a reciprociy relaion. However, four independen quaniies can be exraced from he measuremens, leading o a complee characerisaion of he elasic properies of he maerials, wih he Young s moduli a and 9, he Poisson s raio a and he shear modulus a. According o Eq.(6, he shear modulus can be obained simply by combining srain measuremens for ension in he ω direcion. Oher measuremens can be discarded, or used o chec he relaions occurring beween non independen quaniies, or also incorporaed wih he ohers in an opimisaion process for beer deerminaion of elasic consans, as described in Ref.[2]. Finally, alhough no sricly necessary, measuremens of shear srain during he off-axis ension ess could be done, in order o chec ha shear is null, so o confirm he sae of pure ension. EXPERIMENTAL IMPLEMENTATION. An experimenal research was conduced on various laminaes o illusrae his off-axis pure ension es. We manufacured glass-epoxy and carbon-epoxy laminaes from prepregs by hopressing. pecimens were cu in he, 9 and ω direcions and equipped wih hreedirecional rosees on boh faces (gauges a, 45, and 9 in he specimen axes. The classical laminaed plae heory was used o predic he value of he off-axis angle ω. For he off-axis es, we acually observed ha he shear srain in he specimen axes (obained by combining convenienly he signals of he hree gauges was zero wih an excellen precision all along he elasic range, demonsraing a pure ension, as in he orhoropy direcions. From hese hree pure ension ess, we go enough independen measuremens o deermine compleely he elasic properies of laminaes. We measured he Young s moduli and he

5 Poisson s raio direcly from he signals of he and 9 gauges wih he wo classical ension ess in he orhoropy direcions, and he shear modulus by combining he signals of he and 9 gauges in he off-axis pure ension es, in accordance wih Eq.(6. From hese in-plane engineering consans, we also compued compliances and siffnesses. Table gives resuls obained for a 7-ply carbon-epoxy laminae wih he following anisymmerical sacing sequence (angles in degrees, from he boom o he op : [ / 9 / 3 5 / 9 / / 9 / 3 2 ]. Experimenal values are followed by heir sandard deviaion in braces. The Young s and shear moduli are in GPa. Predicions obained from he classical laminaed plae heory (CLPT wih experimenal daa for he lamina properies are also presened for comparison purpose. The seleced off-axis angle ω, as obained by he classical laminaed plae heory, was 28. Table. E E2 ν 2 G2 Experimen (2.8 (.37 (.8 (.75 CLPT COMMENT AND CONCLUION. We have shown, by analysis and experimen, ha i is possible, for cerain ypes of orhoropic laminaes, o deermine compleely he in-plane elasic properies (membrane siffnesses, compliances and engineering consans from a oal of hree pure ension ess (an off-axis es in a special direcion and he wo classical ension ess in he axes of symmery. The presen wor can receive several ineresing exensions. Firs, we have shown in anoher wor ha similar heory and experimen apply o bending ess for some ypes of laminaes orhoropic in flexure, leading o a complee deerminaion of heir bending siffnesses. Furher, he basic relaion beween he derivaive of he inverse of he Young s modulus and he coupling compliance is no limied o orhoropy. In fac i is valid for complee anisoropy, and anisoropic maerials can be classified in four caegories wih respec o heir compliances, wih wo or four direcions of exremum Young s moduli. As in he orhoropic case, pure ension ess can be achieved in hese direcions. Finally, in he off-axis ensile ess, we observed ha he shear srains remained very small beyond he limi of lineariy, hus suggesing ha his off-axis es migh be exended in he non-elasic range. However, his needs furher experimenal confirmaion, which is presenly in progress. REFERENCE.. Nye J.F. : Physical properies of crysals, heir represenaion by ensors and marices, Oxford Universiy Press, Oxford, England, 985 (firs published in 957.

6 2. Grédiac M., Vaurin A., Verchery G. : A general mehod for daa averaging of anisoropic elasic consans, Journal of Applied Mechanics, Transacions of he AME, 993, Vol. 6, pp Jones R. M. : Mechanics of Composie Maerials, McGraw-Hill, Taylor & Francis, New Yor, Tsai. W., Hahn H. T. : Inroducion o composie maerials, Technomic, Lancaser, Pennsylvania, Tsai. W. : Composie design, hird ediion, Thin Composies, Dayon, Ohio, Verchery G. : Les invarians des enseurs d ordre quare du ype de l élasicié (Invarians of fourh ran ensors wih he symmery of he elasiciy ensors, Proceedings of he Euromech Colloquium 5 (Villard-de-Lans, 979, J.P. Boehler Ed., Ediions du CNR, Paris, 982, pp Verchery G.: Designing wih anisoropy, Texile Composies in Building Consrucion, Par 3 : Mechanical behaviour, design and applicaions, P. Hamelin & G. Verchery Eds., Pluralis, Paris, 99, pp APPENDIX. The relaion beween he so-called conraced noaion and he ensorial noaion for he fourh order compliance ensor inroduces facors 2 and 4, and is given in marix form as : (A Furher, for orhoropic maerials, in he symmery axes, he engineering consans and he componens of he compliance ensor are relaed by : (A E ν 2 E ν 2 E 2 E 2 G 2 Beware of differen definiions ha may be found for Poisson s raios (compare [3], [4], [5]. APPENDIX 2. The polar descripion of anisoropy was inroduced in a general form by Verchery [6], [7]. I represens any plane ensor wih scalars, moduli (which are posiive and polar angles, generalising nown conceps such as modulus and angle for a vecor, Mohr s circle consrucion for second order symmerical ensors, Tsai s circles for fourh order elasiciy ensors [4], [5]. Wih his polar represenaion, for a compleely anisoropic maerial, he six Caresian componens of he elasic compliances are expressed in any Caresian axes x and x 2 wih wo scalars,, wo moduli r, r, and wo polar angles ϕ, ϕ, as :

7 (A r sin 4ϕ r sin 4ϕ cos2ϕ sin 2ϕ sin 2ϕ cos2ϕ Whereas he ransformaion relaions, when he axes are roaed by an angle θ, are raher cumbersome for he Caresian componens, hey ge a very simple form for hese polar parameers. Namely, he polar angles are decreased by his same angle θ, while he scalars and he moduli are invarian, as well as he difference beween he polar angles. Coming bac o he Caresian componens gives in he new axes x and y (for general anisoropy and arbirary reference axes x and x 2 : (A-4 xxyy xyxy xyyy yyyy r sin 4( ϕ r sin 4( ϕ θ θ θ θ θ θ cos2( ϕ θ sin 2( ϕ θ sin 2( ϕ θ cos2( ϕ θ I can be shown ha orhoropy is obained when he following invarian relaion holds : (A-5 ϕ ϕ 4 π where is an ineger and he orhoropy axes are a angles ϕ and ϕ π 2. From Eqs.(A-3, i appears ha here are only wo disinc cases, i.e. for even and for odd values of. However, hese wo cases can be analysed simulaneously. Finally, i is convenien in he orhoropic case o choose he reference axes so ha ϕ π 2. Consequenly he axes x and x 2 are he orhoropy direcions, wih he Young s modulus being larger along he firs axis han in he second ( E > E2, or conversely <. Then, wih θ being he relaive angle (from he x axis o he curren x axis of he roaed frame, he Caresian componens of he compliance in axes x and y wrie as in Eq.(.