THE UN IV ER SIT Y OF MI CHI GAN COLLEGE OF ENGINEERING Department of Engineering Mechanics Department of Mechanical Engineering Tire and Suspension Systems Research Group Technical Report No. 5 THE ELASTIC CONSTANTS OF CORD-RUBBER LAMINATES S. K. Clark Project Directors: S. K. Clark and R. A. Dodge UMRI Project 02957 administered by, THE UNIVERSITY OF MICHIGAN RESEARCH INSTITUTE ANN ARBOR October 1960

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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 ii

TABLE OF CONTENTS Page LIST OF FIGURES iv I. STATEMENT 1 IIo SUMMARY 2 IIIo DETERMINATION OF ELASTIC CONSTANTS 3 IV. ACKNOWLEDGMENTS 10 V. REFERENCES 11 iii

LIST OF FIGURES Figure Page 1 Eg/Gxy vs. cord half-angle a for various values of Gxy/Ex 2 FF/Gxy vs. cord half-angle a for various values of Gxy/Ex. 6 3 Gn,/G/xy vs. cord half-angle c for various values of Gxy/Ex. 7 4 Fg.F/Gxy vs. cord half-angle a for various values of 8 Gxy/Exy with Ex/Fxy = 1/3. iv

I. STATEMENT In Ref. 1 it was shown that for those cases where all plies of a multi-ply laminate were loaded in such a way that their cords were in tension, or alternately, if all plies were loaded in such a way that all their cords were in compression, then exact expressions for the moduli of the resulting orthotropic structures could be found. These moduli were expressed in terms of the cord half-angle a and the four elastic constants of a single sheet of the material making up the laminate. In Refo 2 it was shown from some simple approximations that the four elastic constants of a single sheet could be reduced to one ratio of two of these elastic constants. Under these conditions, it should be possible to calculate and plot the over-all orthotropic elastic constants of the complete laminate as a funcEx.~~~~~

IIo SUMMARY The elastic constants of a multi-plied orthotropic structure may be determined and plotted as a function of the cord half-angle a and one single numercial quantity representative of the degree of anisotropy of a single sheet of material used in the laminate. These calculations show that the cord stiffness and end count ply an important part in the elastic modulus of extension Ee at small cord angles. At larger cord angles these factors cease to be important and the over-all orthotropic modulus is determined primarily by the angle of the cords and the type of rubber used in the sheets. The cross modulus Fib may also be determined by similar techniques, as may the orthotropic shearing modulus Gi. Both of these are dependent upon cord stiffness and end count to some extent over the entire range of cord angles, particularly the latter term which is extremely sensitive to this variable. Graphical values of these functions are presented and should be useful in estimating the effects of design changes on the stiffness of cordrubber laminates. 2

III. o DETERMINATION OF ELASTIC CONSTANTS Equations (15) and (17) of Ref. 1 give expressions for the modulus of elasticity E~, the cross modulus Fan and the shearing modulus Gin associated with an orthotropic body made up of two plies with included cord half-angle Go. These moduli would hold equally well for multi-plied structures, such as 2, 4, or 6 plies, provided that the cords in all the plies are equally loaded. The equations relating these elastic constants to the aij terms discussed in Refs. 1 and 2 are a33 1 = as3 (al2 hala23 - aL2aL3 a3 i Fna a]la~2 - a's 2 /3 ai - (a32)al2a23 - a2a l2a + aa33 Multiplying each of these terms by the modulus Gxy gives E = aaGx)Gy) - 3(al_3() (G Gxy -2 (a(G a 11a223" a12a13 a3ll Gin 8 Ka3)(G'iy)a -a la22 - a2L a - (a32)(Gxy)a - a213 + (a33)(Gxy) (2) a1a22 - a12 E~~ a 3~~

It is seen from the form of Eqs. (2) that the aij terms enter either as products with Gxy or else as ratios with the other aij terms. Thus, each of these elastic constants ES, Fin, and Gal can be reduced to a function of two variables, the cord half-angle a and the degree of anisotropy GX, as previously shown in Ref. 2. These constants can be calculated and plotted as functions of the two variables. This has been done and the results are presented graphically in Figs. 1, 2, and 3. In Ref. 2 it is shown why some uncertainty still exists concerning the exact value of the ratio Ex/Fxy. In previous reports, such as Refs. 2 and 3, where numerical calculations had been performed to present information, the calculations were invariably made in two sets, one using a value of one-half for Ex/Fxy and the other using a value of one-third. In all previous calculations negligible differences were caused by the use of these two values of Ex/Fxy and the common data resulting from the use of either was presented. This situation prevails even here with respect to the data presented in Figso 1 and 3. However, the data of Fig. 2 concerning the quantity Fn/Gxy is the only example found where the use of these two different values gave slightly different results. Figure 4 shows the calculated values of FST/Gxy vs. cord halfangle for Ex/Fxy = 1/3; the resulting values differ but little from those of Fig. 2, as can be readily observed. It is seen that cord angles play a most dominant role in the variation of these elastic constants. In particular, the variation of extensional modulus ES with cord angle is extremely steep. This indicates that many applications of laminated structures might exist in other situations where the elastic characteristics must be controlled over wide rangeso 4

_ Ex /Fxy = 1/2 or 1/3 Gxy/Ey = 1/3 10000 Gxy/Ex =10-4 Gxy/Ex =103 1000 u 1 Gxy/Ex =10-2 100.. I I I I I I 20 40 60 80 100 CORD HALF-ANGLE - a Fig. 1. ES/Gxy vs. cord half-angle cx for various values of Gxy/Ex.

Ex /Fxy = 1/2 Gxy/Ey = 1/3 -10000 Gxy/Ex = 10-4 Gxy/Ex = 0-' -100OO0 LuL" Gxy/Ex= I0-2 -100.Gxy/Ex =-IO-I 20 40 60 80 100 CORD HALF-ANGLE - a Fig. 2. Fer/Gxy vs. cord half-angle a for various values of Gxy/Ex.

10000 I I I OI T I Ex /Fxy= 1/2 or 1/3 Gxy/Ey = I/3 1000 1/ Il... Gxy/Ex =10-4 Gxy/Ex =10,%:~~ IOCC~~Gxy/EIOCC 20 40 60 80 100 120 CORD HALF-ANGLE -a Fig. 3. Gerl/Gxy vs. cord half-angle us for various values of Gxy/Ex.

Ex /Fxy /= 3/ Gxy/Ey I/3 -.I 0000...... -_IQOOC Gxy/Ex = 10-4 Gxy~~~~~~~~~~~~~~/ Ex...0x ~ ~.IE.I~ I I -1000 GiE -to-Gxy/Ex I0-$ -100 //, Gxy/Ex:10-1 -I000,L ~G~/L LIO-I~ 20 40 60 80 100 CORD HALF-ANGLE - a Fig. 4. Ft/Gxy vs. cord half-angle G for various values of x with E~/x~ = 1/5.

An interesting deviation from the relatively insensitive reaction of E/Gy and FV/Gxy to changes in the ratio of Gxy/Ex is provided by Gen/Gxy, shown in Fig. 3. It may be seen that this quantity is highly dependent on both cord angle and the ratio Gxy/Ex, and further, that immense changes in shear stiffness are possible through design changes. A knowledge of both Gxy and Gxy/Ex is required before Figs. 1-4 can be used quantitatively. Both of these elastic constants can be estimated by Eqso (5) and (6) of Ref. 2.

IVo ACKNOWLEDGMENTS The calculations necessary for presenting this information were performed by Mr. Richard N. Dodge with assistance from Mr. Do H. Robbins, Mr. D. E. Zimmer, and Miss Gwendolynne Chang. Thanks are due to them for their care and patience in this work. 10

V. REFERENCES 1. So K. Clark, The Plane Elastic Characteristics of Cord-Rubber Laminates, The University of Michigan Research Institute, Technical Report 02957-3-T, Ann Arbor, Michigan. 2. So K. Clark, Interply Shear Stresses in Cord-Rubber Laminates, The University of Michigan Research Institute, Technical Report 02957-4-T, Ann Arbor, Mich. 3. So K. Clark, Cord Loads in Cord-Rubber Laminates, The University of Michigan Research Institute, Technical Report 02957h5-T, Ann Arbor, Michigan. 11

DISTRIBUTION LIST Name No. of Copies The General Tire and Rubber Co. 6 Akron, Ohio The Firestone Tire and Rubber Co. 6 Akron, Ohio B.F. Goodrich Tire Co. 6 Akron, Ohio Goodyear Tire and Rubber Co. 6 Akron, Ohio United States Rubber Co. 6 Detroit, Michigan S. So Attwood 1 R.o A. Dodge 1 G. Jo Van Wylen 1 The University of Michigan Research Institute File 1 S. Ko Clark 1 Project File 10 12

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