THE UNIVERSITY OF MI CHIGAN COLLEGE OF ENGINEERING Department of Engineering Mechanics Department of Mechanical Engineering Tire and Suspension Systems Research Group Technical Report No. 17 THE THREE-DIMENSIONAL ELASTIC CHARACTERISTICS OF CORD-RUBBER LAMINATES R. N. Dodge Project Director: S. K. Clark ORA Project 02957 administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR January 1964

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The Tire and Suspension Systems Research~ Group at The University of Michigan is sponsored by: FIRESTONE TIRE AND RUBBER COMPANY GENERAL TIRE AND RUBBER COMPANY B. F. GOODRICH TIRE COMPANY GOODYEAR TIRE AND RUBBER COMPANY UNITED STATES RUBBER COMPANY iii

TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS vii NOMENCLATURE ix I. FOREWORD 1 II. SUMMARY 2 III. THEORETICAL ANALYSIS 4 IV. CALCULATION OF ELASTIC PROPERTIES 26 V. ACKNOWLEDGMENTS 36 VI. REFERENCES 37 VII. DISTRIBUTION LIST 38

I-. i. I

LIST OF ILLUSTRATIONS Table Page I. Direction Cosines 11 II. Numerical Values of Elastic Properties and Interply Stress 29 Figure 1. Schematic view of a small section of a two-ply laminate. 4 2. Schematic view of a single ply of cord imbedded in rubber. 10 3. Single ply of cord imbedded in rubber at angle a to the vertical. - axis. 19 4. Comparison of interply shear stress cse of this report and that of Ref. 1. 30 5. Interply stress a' as calculated from Eq. (23). 31 6. Cross-moduli Fez and Fry as calculated from Eqo (21). 33 7. Shear moduli Get and G.T; as calculated from Eqs. (27) and (30). 34 8. Interply shear stresses acl and aCl calculated from Eqso (26) and (29). 35 vii

I

NOMENCLATURE English Letters: aij, cij Constants associated with generalized Hooke's law, using properties based on cord tension. a!. Constants associated with generalized Hooke's law, using properties based on cord compression. E,F,G Elastic constants for orthotropic laminates with cords in tension. E',F',G' Elastic constants for orthotropic laminates with cords in compression. I Direction cosine. u,vw Displacements in the x, y, z directions, respectively. W Elastic strain energy. x,y,z Orthogonal coordinates aligned along and normal to the cord direction. Greek Letters: ~a One-half the included angle between cords in adjoining plies in a two-ply laminate. C Strain. Poisson's ratio. Orthogonal coordinates aligned along and normal to the principal axes of elasticity, or orthotropic axes, in an orthotropic laminate. a Stress. a' Interply stress. ix

I. FOREWORD It has been known for some time that the stress-strain relationships perpendicular to the plane of a tire carcass could be of some importance in cases where the carcass becomes thick such as in aircraft or industrial tires. Such an analysis will require knowledge of the elastic properties of cord-rubber combinations perpendicular to the thickness of a number of plies, but will yield information concerning the possible interaction between forces in the plane of the carcass and ground pressure forces perpendicular to the plane of the carcass. Previous work on the elastic characteristics of cord rubber laminates has ignored stresses in this thickness direction. Thus, this report will serve two purposes: one is to give some indication of the size of the elastic constants perpendicular to the carcass of a tire, the second is to investigate what effects an inclusion of stresses in this direction might have on the previous development of plane elastic characteristics. This report is an extension of the development in Ref. 1; hence, much of the detail of Ref. 1 will be utilized in this development.

II. SUMMARY A three-dimensional analysis of the elastic characteristics of cordrubber laminates is presented in this report. The analysis is a continuation of Ref. 1i which was based on the principals of orthotropic materials. The investigation is confined to those str-uctures which have the cords of all plies in tension, This report shows that the extension moduli and cross-moduli, relating stress and strain in the plane of the cords, are not affected by the stress and strain in the thickness direction; hence, there are no coupling effects between the normal strains in the plane of the cords and the stresses in the direction of the thickness$.. The shear modulus is affected, but the magnitude of this change is very small for- tthe nwrierical calculations presented in this report,. In addition, the extension modui and cross-moduli representing the relationships between normal stresses and strains in the thickness direction are presented, as well as the addi-tional shearing effee-ts, which are shown to be completely independent of the stresses in the plane of the cords. Interply shear stresses again enter into the analysis,. They are used only insofar as they influence the elastic characteristics considered in this report. However, these stresses are shown to be different from those previously obtained by ignoring effects in the thickness direction, It may generally be concluded from the work reported here that the inclusion of the third direction is entirely possible when investigating the

elastic characteristics of orthotropic cord-rubber laminates. The only additional information needed is that relating stress and strain in the thickness direction of a single sheet of the constituent material.

III. THEORETICAL ANALYSIS A two-ply laminated structure is considered in which the cords are separated angularly in the two plies by an included angle 2a, as shown in Fig. 1. In that illustration, the heavy diagonal lines representing typical textile cords are shown being bisected by coordinate axes t and i, two arms of the orthogonal 5, i, 5 system. The elastic properties of this type of structure will be obtained and expressed in terms of the elastic properties of a single sheet of parallel textile fibers embedded in rubber and forming one lamina. Fig. 1. Schematic view of a small sect-ion of a two-ply laminate. This report will extend Ref. 1 by including the effects of stress in the G direction. The details of Ref. 1 will be referred to often in this analysis. Although discussion in this report is limited to the characteristics

of a two-ply system, the ideas and techniques developed allow fairly easy extension to any number of plies, provided that the structure remains orthotropic, that is, that it maintains three planes of elastic symmetry. Observation of the physical nature of the structure under investigation, shows that the usual form of Hooke's law can be applied directly, so long as the relationships between stress and strain are written in the A, i, and q directions, since these are the only directions in which coupling between normal stresses and shearing strains vanishes. In other words, these are the only directions in which we can apply normal stresses and have no shearing strain. The equations relating stress and strain for this condition are: E: En T E: en E no- aG ET; E~ EE = rl = PCrAnd a~~~ a~~rl ~115 C~i ls 5

Some simplification of Eq. (1) results when one recalls (Ref. 2) that the stresses and strains are components of symmetric tensors and, hence, cij = aji and eij = Eji. With this in mind, it can be seen that Ge] = GqI, G = GA, Ga = GMT. Note however, that By and [ie, pil and [l, etc. are not necessarily equal. Therefore, from this viewpoint one has twelve elastic constants: Eg, Er, E, A, LF I, } G,1 G~, Gg,. Equations (1) can be obtained by a somewhat more general approach, which will also yield additional information. Several authors, eg., Ref. 2, have shown that for an orthotropic material the relationship between stress and strain can be expressed in the following manner: ~ = C21 EG + C12 c+ Cz13 e + 0 + 0 + 0 C = Ca21 + C22 C, a3 + 0 + 0 + 0 E = _ C31 3E + C32 E + C33 6 + 0 + 0 + 0 (2) = 0 + 0 + O + 2C44 AEn, + + O 'C O0 + 0 + 0 + 0 + 2C55 E:. + O 0 O= + O + + 0 + + 2C66 1 when Cij represent elastic constants of the material. In addition, there is a function which represents the strain energy per unit volume stored in the material; this function is of the tomL

W = LCll Et + C22 e +C33 03 + (C13 + C31) E E + 4C44 -4 (3) + 4C:55 C~ + 4Cee 6E Since this function is quadratic in form, interchanging the subscripts on the C's will not alter its value; hence, Cij = Cji. A physical consequence of the unchanging nature of W is discussed in Ref. 1. With this in mind, one can rewrite Eq, (2) in this form: o' = C11" e + C12 6 + C13 E: n =- C12 e. + C22 6E + C23 6~ C: 0 C13 6e + C23 e- + C33 66 (4) T(;; = 2C44 CE% yt~; = 2C55 C~; arl - 2C66 6E Rewriting Eq. (4) for strain in terms of stress, we have (C22 C33 - C23) + (C13 C23 - C12 C33) + (C12 C23 - C13 C22) 22 C22 (C13 C23 - C12 C33) (011 C33 - C13) + (C12 C03 - Cll C23) C022 022 C22 (5) 7

(C12 C23 - C13 C22) a~ + (cC C13 - C1 C23) ~. + (Ci C22 - c2 C,lC2 12) C22 C22 C22 \: 2C44 2C55 2Ce Comparing Eqs. (1) and Eqs. (5), observe that (C13 C23- C12 C33) _ -t2_ = - 1 C22 E, E~ Fg (C12 C23 - C13 C22) _ - i C22 E E Fg (C12 C13 - C1 ) CT1 = 1 (6) C22 E E~ Er (C22 C33 -C-; = 1 o-;1C3 = __ C2 C22 2 C22 EC22 2C 66= Gg; 2C55 Ge; 2C4 " GC 4

The substitution of Eqs. (6) into Eqs. (5) yields the stress-strain relations for any orthotropic body in the form _ +-_ E -= Flt E F C l=,l P:15 GiG The form and terminology of Eqs. (7) will be used throughout the following discussion involving stress-stress relations. Note that Eqs. (7) contain nine independent elastic constants: the extension moduli Et, E., and E:; the cross-moduli FTi, FS:, and FTi; and the shear moduli GTi, Gig, and Gig. Leaving the general two-ply structure, we will now investigate a single ply which is considered to be constructed by imbedding a series of parallel, straight cords lying in a plane into a sheet-like matrix of elastic material. This construction is illustrated in Fig. 2. This sheet of material is also an orthotropic body if the axes of elastic symmetry are taken to be the x, y, and z axes. Since it is orthotropic it will have elastic constants similar to those expressed in Eqs. (7) except that the subscripts will now be x, y,

/ z, Fig. 2. Schematic view of a single ply of cord imbedded in rubber. and z instead of i, r and ~. It will be assumed that Ex, Ey Ez Fxz, Fyz Fxy, Gxy, and Gyzare known quantities. Some previous work has been done in obtaining these properties. Again, note that these are the properties of a single ply in the orthotropic directions only when all of the cords are in tension. Again referring to Fig. 2, consider a known stress ax, ay, %z' Cxy, axz, and ayz with a corresponding strain state cxe Cye' e~z xy' Gxz' and Eyz' It is desired to determine the corresponding stress and strain state referred to the orthogonal axes i, I, and ~, obtained by rotating the x, y, z system about the z arm through an angle of a. The direction cosines of the different axes with respect to each other due to this rotation are shown in Table I. 10

TABLE I DIRECTION COSINES x y z cos c -sin Q 0 sin a cos a O 0 0 5 o o ] It has been shown that the stress components referred to the, r, ~ system, written interms of the known components of the x, y, z system, take the form.2 2 cos2a cx + sin2a 5y - 2 sin a cos a~ axy T sin a x +cosa cy +2siny + 2 sin cos a G xy aC = aZ (8) (TSr = sin aC cos c rx - sin oe cos a cy + (cos2oa - sin2c)~xy = cos Ca yz + sin a axz ots = -sin a c + cos a xz Similarly, for the strain components referred to the, r,; system written in-termsof the known components of the x, y, z system, LI1

= cos2 Ex + sin2a c - sin a cos a exy y xy Ec = sin2a Ex + cos2a Ey + sin a cos a Cxy (9) - 2 sin c cos a ex - a sin c cos a ~y + (cos2o - sin2')EXy cos a cyz + sin a exz = - sin a cEyz + cos a xz In like manner, a similar set of equations could be written for the components of stress and strain in the x, y, z system in terms of the components in the ~,, g system. These will take the following form: ax = cos2c a + sin2a ca + 2 sin a cos a ac 2 2 o-r= sinO a c + cosc a - 2 sin a cos al a az = z ( (8)1 Cxy = - sin a cos a cr + sin a cos a ao + (cos2a - sin2a)a~n ayz = cosa - sin crxz = sin a c~ + cos a c~ Similarly, for the strain components referred to the x, y, z system written in terms of the known components of the ~ rj, system, 12

=X = costZa Ce + sin20e e + sin aG cos a Ec 2a 2 sin E + COsGE sin + cosin a E Cz = E (9) Exy = - 2 sin a cos o cE + 2 sin a cos a CE + (cos 2 - sin2a)e, Cyz = cos cz Eg - sin a S Exz = sin a cn + cos a c. The generalized form of Hooke's law may be written using the stresses as independent variables with respect to any desired coordinate axes; for example, for the ~,, g system = all ac + al2 Uo + ala +-,ali4 a'14 +-a15 d~'; +'"ale q'[ T - a2l a5 + a22 a, + a23 a1 + a24 U,] + a25 9g + a26 U. = a31 6C + a3s2 n + a33 + a34 + a35 Un + a36 a 6 (10) e =. l a + a42 a. + a43 a + a + a + a45 o +a46 ao = a51 a + a52 a s + a53 a + a54 Ue, + a55 UE + a56 a lC.....= a, -+= a62 )1'+C a63 0+ +.a6E4"<,' + a65 "a66 o. The values of the values of the various ai coefficients of Eqs. (10) will be determined by using the procedure outlined in Ref. 1. The coefficient al4 will be 13

determined to illustrate the procedure. The stress ar may be considered known, and cr, ari, r, cax, and al are set equal to zero. Then it follows from Eqs. (L0) that al -; (11) however, from Eq. (9) = cos2o Ex + sin y - sin G cos exy (12) Next, recalling that the single ply under consideration is.orthotropic with the x, y, z axes as the axes of orthotropy, the equations relating strain and stress can be written by referring to Eqs. (7). ax y Cz X x Fx y Fxz 4X ~y %z C=X + -- + Z Fxy Ey Fyz (13) z x + y + a x Fz Fyz Ez yXY G Gx G xz' z Gyz Now, referring to Eqs. (8)1, ax, y az, 'xy arf, and yz can be expressed in terms of the non-zero ': 14

x = 2 sin cos a, y = - 2 sin a cos 0c r (14) xy = (cos2s - sin2)an z yz, Cxz = 0. Substituting Eqs. (14) into Eqs. (13) gives Cy = 2 sin a cos a-E F+yS c~~ = 2 sin C cos (F F - (15) (cos2a - sin2) xy XZ yz Now, substituting Eqs. (15) into Eqs. (12), 2 cos3cs 2 n c a x cos sa os c- sin2ac = r Ex Ey in a Gxy +t _ cos3c sin ax + sin3x cos a.. FxyP

Therefore, a14 becomes '5 2 cos32 sin a 2 sin3a cos a a14 = - E=..,, Crl Ex Ey - cos C cos c (cosoac - sin2a)(G- + — By identical procedures one may obtain al5 = ale = a2e = a35 = a36 = a45 = a46 = as1 = a52 = a53 a25:- = a- = a61 = a62 = a63 = a64 = 0 a4l = a14 4.4 all = a+ + cos2 sin + x Ey y x 4 4 N7J..sin c cos a 2 a22 = i + + o si + 1 a33 Ez 2 2 1. 2 cos sin2oa) a44 = 4 cos2Qa sin2 x 1 + (cos - sin2 +E Ey Fx Gxy COS2~ sin2dl a 55 + G x r 16

cos2O sin2a 2. 221 sin2(z coS2 1 + 1 1 + (sin4a + cos'a) Ex E G Fxy cos2Q sin2ca a 13 =a31 = + Fxz Fyz sin 2c cos2~C a23 = a32 = F F F XZ yz 2 sin3x cos a 2 cos3 sin a 2(o2 a24 = a42 = Sf E + sin c cos oa(cos2oa - sin2Co) + I> a34 = a43 = 2 cos sin aF F a56 a65 = cos a sin (16) riXz Gyz Equations (10) and Eqs. (16) now allow the properties of an orthotropic sheet to be predicted in any direction q, ~, ~ at an angle a with the cord direction, If the results of Eqs. (16) are used, Eqs. (10) take the form a= aljCY + a12 crl + a13 cr + aL4 at, (a) (17) a 2 Ce + 822 C+ a23 M + a24 o (b) 17

C: a13 cr + a23 a + a33 C: + a34 Cr (c) (17) se, - a55 a + a5e c1r (e) Ea C a665 6 i (f) Note that Eqs. (17e:,f:)l are independent of Eqs. (17a-d). One concludes from this that there are no coupling effects between the stresses ac, crh, ca, and alp and the shearing strains in the e - ~ plane and the j - ~ plane. In like manner, there are no coupling effects between the shearing stresses a,; and ant and the strains cE, en, E:, and al. To determine the relationship between the stresses and strains in Eqs. (17a-d) consider now a sheet of the type shown in Fig. 2 but inclined at some angle a to the orthotropic sheet axes x and y. This is illustrated in Fig. 3. Imagine that is is desired to extend this sheet in the ~, i, and g directions by means of normal stresses only. Because the orthotropic axes x and y do not coincide with the 0, r axes, this is not possible; distortions cGn will inevitably accompany the application of any set of normal stresses a, CT, and ac. However, because the g axis coincides with the third orthotropic axis, the z axis, there are no accompanying shear strains eg or El. This is also seen in (17e,f), Because there are shearing distortions in the - plane, one must admit the existence of shearing stresses ~ as necessary for the distortionless extension of an element such as that represented by Fig. 5,

a /x Fig. 3. Single ply of cord imbedded in rubber at angle a to the vertical 5 - axis. where the reference axes do not coincide with the orthotropic axes. With this provision in:mind, one may go directly to (17a-d) and presume no shear strain acl, that is, a fi will exist to prevent the shear distortion when the normal stresses are applied to the structure. This will result in having only normal strains ce, en, and e:. For this condition Eqs. (17) become =c - all c0 + al2 a0 + al3 C: + a14 0C, e = a12 at + a22 a + a23 oa + a24 aid (18) = al3 0a + a23 r] + a33 ao; + a34 'er1 O = al4 C0a + a24 0, + a34 as + a44 con We now have four equations and four unknowns (%E, en, e:, aid). Solving the

last of Eqs. (18) for,r one obtains = _ a2 _ a43sa (19) a44 a44 a44 This is the stress that must be supplied from some external source to obtain distortionless extension. The use of this stress in the first three of Eqs. (18) yields f. -C 1 a4 a14> + 12 a a14 a12. + 13 14 a34 a44 / a44 a44 / _a. +cy a22- a 24> a24 4. a 12 + a 22.. + C 23 (20) - a44 a44 o a44 -a ax-e 3-14 a34 + a(2 24 a34 a34a a44'/,, a44 a44 Comparison of Eqs. (20) with Eqs. (7) shows that a2 a a34> E E E a4 E4 a24 E 'a (21) a1- a4 2 ( a14 a34 1 13 a24 a34 Fa a44 a44 F a44 Now consider the shear moduli Grl G~, and GT? which relate the shearing strains and stresses. Note again that this entire development is based 20

on the assumption that all cords are in a state of tension. This is a necessary assumption to assure that the sheet properties Ex, Ey, etc., which are assumed to be known, are constants for all plies involved. To obtain Gig associated with all cords in tension, one must postulate an extension-free distortion or a pure strain c4a, which occurs as the result of the proper application of the stresses A, Y a and o. It has already been mentioned that the application of either or both of the stresses aid and oC, does not affect the strain cop. Equations (17) become, when applied to this case, o = all t + a12 ai a + ad + al4 Ca O = al2 t + a22 a + a23 Ad + a24 stu (22) o = a13 Co + a23 a + a33ss a + a34 a~,or = aL4 Ca + a24 0a + a34 at + a44 en Solving the first three of Eqs. (22) for a~, an, and ad gives =,,i-al4(a22a33-a23) - a2(a34a23-a24a33) + a13(a22a34-a24a23j D an = adF1 ll(a23a3,4 —"a24a33) + ai4(a2ia33-"a23a31) + al3 (a24a31,3. a21a34))/D Y =, l11(a23a24-a22a34) - a12(al3a24-a12a34) + a14(a12a23-a22a13j D, Equations (23) represent the normal stress necessary for an extensionless 21

distortion. Substituting these into the last of Eqs. (22) gives G = 9 = ~)a14(a23-a22a33) + a24(a13-a1la33) + a34(a12-a1la22) + 2a14a24(a12a33-a13a24) + 2a24a34(aa23-a 12a13] /a33 (a la22-a12) + a23(a12a13-ajja23) + a13(al2a23-al3a22)] + a44 (24) The last two of Eqs. (17) will now be considered in an effort to obtain GSS and G. In the first case, a known alp will be applied and it will be assumed that the stresses will so distribute themselves that c.~ will be zero. For this case Eqs. (17) become cE5 = a55 a55 + a56 a] (25) 0 = a56 6Cr + a66 6C ~ Solving the second of Eqs. (25) for o~, one obtains the shear stress e, which must be supplied from some external source in order to have a pure shear strain ail resulting from the application of only the sheer stress ail' = a65 Substituting Eq. (26) into the first of Eqs. (25) gives6) 22=,

1 =.. = a55 - a56 (27) Ga5 ail a66/ For a second case, a known ant is applied and it is assumed that the stresses will be so distributed that cGl is zero. For this case Eqs. (17) become = a55ss + a56 C (28) CEl as56 + a66 Uc Solving the second of Eqs. (28) for alp, one obtains the shear stress a~t which must be supplied from some external source in order to have a pure shear strain c.5 resulting from the application of the shear stress any = 56, (29) Substituting.Eq. (29) into the second of Eqs. (28) gives = 66 - a6 (30) G: oC a55 Equations (21), (24), (27), and (30) now represent a set of equations from which one can calculate the elastic properties of an orthotropic sheet in any direction i, i, at an angle a with the cord direction x, yo It should be made clear at this point that the development of these equations has been obtained directly from the analysis of a single ply of material and, it has been observed by comparing these with Eqs. (7), that they are the same elas

tic properties that relate the stress and strain of the two-ply composite body. Hence, the moduli of the two-ply combination of Fig. 2 are the same as those of the single sheet of Fig. 3 when referred to the 5, i, ~ axis. Of course, these are the same only so long as the two plies are exactly the same in physical make-up and so long as the single sheet has been provided with the "extra" stresses expressed by Eqs. (19), (23), (26), and (29). It is appropriate at this point to compare the form of the elastic properties as developed in this report to those of Ref. 1. First of all, Et, E., and Fin are exactly the same for the two developments. This implies that the relationships between normal stresses and strains in the - r plane are unaffected by stresses and strains in the S direction. However, note that the "extra" stresses a0lp (Eq. (19) of this report and Eqs. (14) of Ref. 1) are not the same. Hence, the magnitude of the additional stress necessary for distortionless extension is different for the two developments. The magnitude of this difference is discussed in a subsequent portion of this report. The next comparison to be made is that of the shear modulus, Gno. A comparison of Eq. (17) of Ref. 1 to Eqs. (24) of this development shows that there is a considerable difference in the two expressions for Gin. The magnitude of this difference is discussed in a later part of this report. Note, also, in conjunction with G, that the "extra" stress a and cA necessary for extensionLess distortion are different in the two developments. These differences imply that the "extra" stresses necessary for pure distortion are affected by the stresses and strains in the third direction, These "extra" 24

stresses are the so-called "interply shear stresses" that have been discussed by this group in previous reports (Refs. 1, 4, etc.). In addition to the changes noted in the relationships of stress and strain in the ~ - r plane, one can now predict the relationships of stress and strain in the n - 5 and 5 - ~ planes. Also, note that additional interply stresses are present when one includes the stresses and strains in the C direction.

IV. CALCULATION OF ELASTIC PROPERTIES Two topics are discussed in this section, supported by evidence from actual calculations. First a comparison is made between those elastic properties which are common to this report and to Ref. 1. Second, typical calculations are presented for those additional properties that have been developed in this report. Equations (21), (24), etc., show that in order to calculate the elastic properties of the orthotropic sheet in a non-orthotropic direction, one must first calculate the aij coefficients. However, in order to calculate the aij's one must have some knowledge of the basic sheet properties Ex, Ey, Fxy, Gxy, Ez, Fxz. Fyz G Xz and Gyzo Previous work has been done in obtaining the first four of these quantities, and some typical values from Refs. 2 and 6 are used in the calculations which follow, Because the remaining five sheet properties have not been measured previously, there is no experimental information on their magnitudes. However, the approximate value of these quantities can be estimated from the geometry of construction of a single-)ply. By referring to Fig. 2, one can reasonably assume that Ez will be similar in magnitude to Ey, because, like Ey, Ez is a ratio of stress to strain in a direction perpendicular to the cords. In a like manner, one can reasonably assume that Fxz will. be similar in magnitude to Fxyj because, like Fxy, Fxz is a ratio of either stress in the cord direction and strain in the direction perpendicaul.ar to the cords or vice-versa. 26

On the other hand, Fyz is assumed to be similar in magnitude to 2Ey. This assumption is based on the fact that Fyz is a ratio of stress in a direction perpendicular to the cords to strain in a direction perpendicular to the cords but also perpendicular to the stress. Hence it will behave like the quantity El/1, where E1 refers to the extension modulus in a direction perpendicular to the cords, while 1j refers to the Poisson's ratio in a direction perpendicular to the cords. This in turn implies that Fyz is similar in magnitude to the ratio of Ey/IR - 2Ey. It is also reasonable to assume that Gxz and Gyz are similar in magnitude to Gxy, since they are all ratios of shear stress and shear strain taking place primarily in the material in which the cords are imbedded. These sheet properties are only estimates, with no experimental confirmation. However, they should serve adequately in the following calculations, since the calculations are only intended to illustrate some of the differences in the development of this report as compared to that in Ref. 1. With this in mind, the following typical values of sheet properties used in the calculations of this section are listed below: Ex = 223500 psi Fxy = - 405000 psi 'Ey = 970 psi Fxz - 400000 psi Ez = 1000 psi Fyz = - 2000 psi GX = 308 psi Gxz 300 psi Gyz = 350 psi 27

a = o0 150, 300, 45~, 600, 750, 90~ As mentioned previously, there is no difference between the Eg, E, and Fin of this report and those of Ref. 1. Therefore no discussion of these properties is necessary except to compare the interply shear stresses required for the distortionless extension necessary in these three properties. To do this a comparison must be made between the ae obtained from Eq. (19) of this report and that obtained from Eq. (14) of Ref. 1. The results of this comparison are shown in Table II and illustrated in Fig. 4. The difference in the two formulations can be attributed to the significant effect of a%, represented by the third term on the right-hand side of Eq. (19). In comparing the Ggn of this report (Eq. (24)) and that of Ref. 1 (Eq. (17)) observe that there is a considerable difference in the two expressions. This is because the interply stresses necessary for extensionless distortion are different in the two developments. However, Table I shows that the numerical value of GSn is altered very little by the inclusion of the effects of the: direction. This is further borne out by observing that the normal interply stresses are unaltered numerically. The additional normal interply stress ca, illustrated in Fig. 5, is small and contributes very little to the value of GSR. Figure 5 is a graphical representation of FS and Fig, and it is interesting to note that these values change sign over the range of cord angles used in the calculations. This implies that it is possible, with the correct combination of cord angle and sheet properties, to apply an extensional stress 28

TABLE II NUMERICAL VALUES OF ELASTIC PROPERTIES AND INTERPLY STRESSES Property 0Property 15 30 4 60 75 90 cAi-Eq. (19); this report* 0 9.81 28.8 50.8 28.8 9.81 o at -Eqs. (14); Ref. 1* 0 19.0 55.8 98.6 55.8 19.0 0 a//a~-Eqs. (23); this report 0 3.66 1.72 1.00.58.27 0 cr//a -Eqs. (16); Ref. 1 0 3.65 1.72 1.00.58.26 0 c/cr n-Eqs. (23); this report 0.27.58 1.00 1.72 3.66 o ca/a T-Eqs. (16); Ref. 1 0.26.58 1.00 1.72 3.65 0 c//a T-Eqs. (23); this report 0.0079.0047.0040.0046.0079 0 a n/ca- Eq. (26); this report 0 -.0412 -.0693 -.0769 -.0642 -.0361 0 a -/aq - Eq. (29); this report o -.0361 -.0642 -.0769 -.0693 -.0412 0 Et —Eqs. (21); this report and Eq. (15) of Ref. 1 223500 90380 7112 1225 842 942 970 En —Eqs. (21); this report and Eq. (15) of Ref. 1 970 924 842 1225 7112 90380 223500 Eu-Eqs. (21); this report 1000 1023 1131 1312 1131 1023 1000 Fn — Eqs. (21); this report and Eq. (15) of Ref. 1 -405000 -12480 -2519 - 1239 -2519 -12480 -405000 Fi -Eqs. (21); this report -400000 +31300 +8149 -172416 -2600 - 2053 - 2000 FTe -Eqs. (21); this report - 2000 - 2053 -2600 -172416 +8149 +31300 -400000 Ga T-Eq. (24); this report 308 14221 42048 55962 42048 14221 308 GtT1-Eq. (17); Ref. 1 308 14210 42020 55920 42020 14210 308 Gab-Eq. (27); this report 300 303 312 325 337 347 350 Gdm-Eq. (30); this report 350 347 337 325 312 303 300 *Calculated with normal stresses of 100.

o-o Eq. (19) of this report x-x Eq. (14) of this report Resulting From Application of Normal Stresses of 100 psi 100 x 80 60 X X 40 20 x x 0 15 30 45 60 75 90 a0 Fig. 4. Comparison of interply shear stress r'~ of this report and that of Ref. 1,

.0080.0070.0060.0050.0040.0030.0020 0 15 30 45 60 75 90 00o Fig. 5. Interply stress a: as calculated from Eq. (23). 31

in the 5 or i direction and have the thickness increase. In a similar manner, one may apply an extensional stress in the ~ direction and have an increase in length in either the ~ or r direction. Figures 6, 7, and 8 illustrate the remaining additional elastic characteristics obtainable from the analysis presented in this report. It is emphasized again that the results presented here are those of a typical example. There are many special combinations of geometry and basic sheet properties that would probably result in slightly different conclusions being drawn concerning the influence of various parameters on the elastic characteristics. 32

.5 x 10 F - / I ~\ -105 I f \ fI I -3 x 105 I 4 x 105 -4x105 X 0 15 30 45 60 75 90 ao Fig. 6. Cross-moduli Fe and Fe as calculated from Eq. (21). 33

350 ' W 340 G 330 320 - 310 300 290 280 I I 0 15 30 45 60 75 90 ao Fig. 7. Shear moduli GeG and G.l as calculated from Eqs. (27) and (30). 534

-.08 -.07 -.06 / -.05 I 1/ R: A\ -.04 -.03 //.01 0 15 30 45 60 75 90 0 a Fig. 8. Interply shear stresses ao' and at calculated from Eqs. (26) and (29). 35

V. ACKNOWLEDGMENTS The author wishes to express his gratitude to Professor S. K. Clark for his comments and criticisms regarding the contents of this report. Thanks are also expressed to Mr. F. Martin, Mr. A. Turner, and Mr. B. Riggs for their assistance in obtaining the calculated results presented.

VI. REFERENCES 1. S. K. Clark, The Plane Elastic Characteristics of Cord-Rubber Laminates, The University of Michigan, Office of Research Administration, Technical Report 02957-3-T, Ann Arbor, Michigan, October 1960. 2. I. S. Sokolnikoff, Mathematical Theory of Elasticity, 2nd ed., McGrawHill Book Co., Inc., New York 1956. 3. Chi-Teh Wang, Sc. D, Applied Elasticity, McGraw-Hill Book Co., Inc., New York 1953. 4. S. K. Clark, Interply Shear Stresses in Cord-Rubber Laminates, The University of Michigan, ORA Technical Report 02957-4-T, Ann Arbor, Michigan, October 1960. 5. S. K. Clark, N. L. Field, Tables of Elastic Constants of Orthotropic Laminates, The University of Michigan, ORA Technical Report 02957-16-T, Ann Arbor, Michigan, November 1962. 6. S. K. Clark, R. N. Dodge, N. L. Field, B. Herzog, Digital Computation of Two-Ply Elastic Characteristics, The University of Michigan, ORA Technical Report 02957-12-T, Ann Arbor, Michigan, October 1961. 37

VII. DISTRIBUTION LIST No. of Copies The General Tire and Rubber Company Akron, Ohio The Firestone Tire and Rubber Company Akron, Ohio 6 B. F. Goodrich Tire Company Akron, Ohio 6 Goodyear Tire and Rubber Company Akron, Ohio 6 United States Rubber Company Detroit, Michigan 6 S. S. Attwood 1 R. A. Dodge 1 G. J. VanWylen 1 The University of Michigan ORA File 1 S. Ko Clark 1 Project File 10 38

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