The Behavior of Structures Composed of Composite Materials Part 14 pdf

384 The factor represents a measure of the constituent element packing geometry and loading conditions. For example, for the transverse modulus is used while for calculation of the in-plane shear modulus a value of is used. It should be noted that the values of as given above provide reasonable predictions for the elastic constants up to certain volume fractions of fiber packing density and also for reasonable bounds on certain fiber geometries. For predicting the fourth technical engineering constant, the major Poisson’s ratio the rule of mixtures can again be used. Thus, When considering anisotropic and twisted fibers, such as yarns, a modification of the above formulae is necessary. B. Physical properties An important factor in determining the elastic properties of composites is knowledge concerning the proportion of constituent materials used in the respective lamina/laminates. These proportions can be given in terms of either weight fractions of volume fractions. From an experimental viewpoint, a measure of the weight fractions is easier to obtain than is the corresponding volume fractions of constituent elements. There is however, an analytical connection between these proportioning factors which allows conversion from weight to volume fraction and vice versa. Since volume fractions are key to elastic properties calculations, this connection remains important. The expressions necessary for this development follow. Definitions Volume Fractions Weight Fraction f, m, c refer to fiber, matrix, composite respectively In order to interrelate the above quantities analytically, we make use of familiar density-volume relations. Thus, 385 refers to density The above equation can be rewritten in terms of volume fractions by dividing thru by . Thus, Equation (1) can be couched alternately in terms of constituent weights so that, Dividing the above equation by we obtain Introducing now the relationships between weight, volume and density, we have, The relationship for and in terms of and can now be easily established by inverting the above relations. Further, while the current derivation has been limited by to two constituent elements, the extension to and the inclusion of multiple elements can be easily made. A relation between weight and volume fractions of fiber or matrix can thus be analytically expressed in terms of the following equations 386 where, Equations (2) has been plotted in Figure 11 for the following fiber types and corresponding fiber densities as shown in Table 1. This figure is useful for converting between weight and volume fraction of fibers for respective materials. Other fiber types, with defined densities can be added to the plotted data as inferred by interpolation. It should be noted that for volume fractions of fiber greater than 75% care in the use of Figure 1 should be exercised. This is due to the fact that there are theoretical as well as practical limits which can be attached to the maximum allowable packing densities oriented with different fiber arrays. As specific examples for the most common arrays encountered the square and hexagonal, the maximum fiber volume fractions allowed would be 78% and 91% respectively. These results can be obtained from simple analysis which is included below for the two most common fiber packing geometries. 387 Fiber Packing Geometry 1. Hexagonal Array: Consider triangle ABC (Area occupied by the fibers) Volume Fraction 388 2. Square Packing: Consider a square ABCE References 1. 2. 3. 4. 5. 6. 7. 8. Ekvall, J.C. (1961) Elastic Properties of Orthotropic Monofilament Laminates, ASME Aviation Conference, Los Angeles, California, 61-AV-56. Ekvall, J.C. (1966) Structural Behavior of Monofilament Composites, AIAA/ASME Structures, Structural Dynamics and Materials Conference, Palm Springs, California, pp. 250. Hill, R. (1965) Theory of Mechanical Properties of Fiber-Strengthened Materials – Self Consistent Model, Journal of Mechanics and Physics of Solids, Vol. 13, pp. 189. Hill, R. (1965) A Self-Consistent Mechanics of Composite Materials, Journal of Mechanics and Physics of Solids, Vol. 13, pp. 213. Whitney, J.M. (1966) Geometrical Effects of Filament Twist on the Modulus and Strength of Graphite Fiber-Reinforce Composite, Textile Research Journal, September, pp. 765. Whitney, J.M. and Riley, M.B. (1966) Elastic Properties of Fiber Reinforced Composite Materials, Journal of AIAA, Vol. 4, pp. 1537. Hashin, Z. (1968) Assessment of the Self-Consistent Scheme Approximation – Conductivity of Particulate Composites, Journal of Composite Materials, Vol. 2, pp. 284. Hashin, Z. (1965) On Elastic Behavior of Fiber-Reinforced Materials of Arbitrary Transverse Phase Geometry, Journal of Mechanism and Physics of Solids, Vol. 13, pp. 119. 389 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. Paul, B. (1960) Prediction of Elastic Constants of Multiphase Materials, Transactions of the Metallurgy Society of AIME, Vol. 218, pp. 36. Hashin, Z. and Rosen, W. (1964) The Elastic Moduli of Fiber-Reinforced Materials, Journal of Applied Mechanism, Vol. 31, June, pp. 223, Errate, Vol. 32, 1965, pp. 219. Hashin, Z. and Shtrikman, S. (1963) A Variational Approach to the Theory of the Elastic Behavior of Multiphase Materials, Journal Mechanics and Physics of Solids, pp. 127. Schapery, R.A. (1968) Thermal Expansion Coefficients of Composite Materials Based on Energy Principle, Journal of Composite Materials, Vol. 2, No. 3, pp. 380. Adams, D.F. and Tsai, S.W. (1969) The Influence of Random Filament Packing on the Transverse Stiffness of Unidirectional Composites, Journal of Composite Materials, Vol. 3, pp. 368. Adams. D.F. and Doner, D.R. (1967) Longitudinal Shear Loading of a Unidirectional Composite, Journal of Composite Materials, Vol. 1, pp. 4. Adams. D.F. and Doner, D.R. (1967) Longitudinal Shear Loading of a Unidirectional Composite, Journal of Composite Materials, Vol. 1, pp. 152. Chen, C.H. and Cheng, S. (1967) Mechanical Properties of Fiber-Reinforced Composites, Journal of Composite Materials, Vol. 1, pp. 30. Behrens, E. (1968) Thermal Conductivity of Composite Materials, Journal of Composite Materials, Vol. 2, pp. 2. Behrens, E. (1967) Elastic Constants of Filamentary Composite with Rectangular Symmetry, Journal of Acoustical Society of America, Vol. 47, pp. 367. Foye, R.L. (1966) An Evaluation of Various Engineering Estimates of the Transverse Properties of Unidirectional Composites, SAMPE, Vol. 10, pp. 31. Tsai, S.W. (1964) Structural Behavior of Composite Materials, NASA CR-71, July, National Aeronautical and Space Administration CR-71. Halpin, J.C. and Tsai, S.W. (1967) Environmental Factors in Composite Materials Design, AFML-TR-67-423 Air Force Materials Laboratory, Wright-Patterson Air Force Base, Ohio. Tsai, S.W., Adams, D.F. and Doner, D.R. (1966) Effect of Constituent Material Properties on the Strength of Fiber-Reinforced Composite Materials, AFML-TR- 66-190, Air Force Materials Laboratory. Ashton, J.E., Halpin, J.C. and Petit, P.H. (1969) Primer on Composite Materials: Analysis, Technonic Publishing Co., Inc., Stanford, Conn., pp. 113. Appendix 2. Test Standards for Polymer Matrix Composites. As can be discerned from the test material, the role of the engineer in controlling the design process using composite materials requires considerable expertise beyond traditional levels for establishing design criteria. A fundamental input into any design process is the requirement for obtaining the necessary materials properties data as well as establishing the overall material response in order to identify the types of failure events that can occur. Thus the data base for composites is an evolutionary process in which current accepted test standards are being reviewed and revisions adopted as well as composite modes of failure identified and tabulated. As a ready means of access and awareness to the test procedures in current practice, test standards have been included. It should be mentioned that in general the engineer executes tests of the following type: A. B. C. Interrogative, that is, those examining some aspect, or is seeking fundamental information on certain properties, relations, or physical constants of materials, those using unique test apparatus. Developmental, that is, those tests required to obtain additional data to ensure meeting performance specifications on a selected material. In such cases both standard and modified standard test equipment may be used by the engineer. Standardized, that is, those tests which utilize controlled test procedures which have been adapted from sanctioned test committee and professional engineering society recommendations. Such tests are almost universally run using commercially available test equipment and with specific geometry specimens. While all three of the aforementioned type of tests provide important data, it is the standardized test that we tend to rely upon when requiring data for materials. This is especially true since engineers in general wish to be able to duplicate specific tests using accessible equipment rather than designing totally unique test facilities. In view of these statements, the following standards given in Table 1 are provided which describe a number of common mechanical tests. Details concerning the test specimen geometry and procedures can be found in the appropriate standard. Appreciation is expressed to Dr. Gregg Schoeppner, AFRL/MLBCM for his contribution to Appendix 2. 392 Appendix 3. Properties of Various Polymer Composites. Using such tests as described in the standards of Appendix 2, a listing of selected material properties for continuous filament unidirectional composites is included as Table A3-1 below. The symbols used in Table A3-1 are: Modulus of elasticity in the fiber direction Modulus of elasticity perpendicular to the fiber direction Major Poisson’s ratio, i.e., In-plane shear stiffness Tensile strength in the fiber direction Compressive strength in the fiber direction Tensile strength normal to the fiber direction Compressive strength normal to the fiber direction In-plane shear strength Fiber volume fraction Coefficient of thermal expansion in the fiber direction Coefficient of thermal expansion perpendicular to the fiber direction Coefficient of moisture expansion in the fiber direction Coefficient of moisture expansion perpendicular to the fiber direction. For conversion from the psi units used in Table A3-1 for stress and modulus of elasticity, To determine the density of many of the composite materials given on the next page, use the Rule of Mixtures of Section 2.4 (pp. 51-52), along with the fiber densities given in Table 1 of Appendix 1 (pg. 387), and the polymer matrix densities given in Table 1.2 (pg. 8). 394 [...]... 341, 348, 350, 354, 355 Henderson, J 129, 142 Hill, R 311, 312, 326, 330, 331, 388 Hilton, H.H 77, 79 Hofer, K.E 334, 335, 348, 350, 355 Hoffman, O 313, 314, 330, 331 Hsu, T.M 335, 355 Hsu, Y.S 253, 254 Huang, N.N 54, 77 Hwu, C 133, 142 Inman, D.J 129, 142 Jen, M.M 354, 357 Jones, D.L.C 129, 140 Jones, R.M Jurf, R.A 57, 78 398 Kelly, A 310, 331 Kerr, A.D 135, 142 Kim, R.Y 353,356 Koya, T 260, 299 Kuno,... 351 load capacity 337 kayak paddles 31 Kevlar 4, 25 fibers 7, 28, 141 , 387 Kevlar 49/epoxy composite 81, 144 , 148 , 206, 394 specimen 338 Kevlar 49/E-glass hybrid polyester composite 395 Kevlar 49 woven epoxy composite 395 polyester composite 395 kinematic motion 304 relations 67 kinetic energy 119, 293, 294 Kirchoff edge condition 94, 95, 115 ladder side rail 20, 282 Lagrangian 293 lamina 3, 57, 58,... Wetherhold, R.C 339, 356 White, J.C 252, 253 White, S 32 Whitney, J.M 114, 141 , 184, 200, 252254, 353, 356 Wilson, D.W 58, 78, 141 , 142 Yi, S 77, 79 Yon, J 354, 357 Young, D 132, 142 , 183, 199, 200, 285, 299 Yu, Y.Y 76, 79 Subject Index A380 super jumbo jet 23 acetals 7, 13 acoustic signature 27 acryonitride – butadiene styrene (ABS) 7, 13 adaptive structures adhesive 281, 282, 333, 335-338, 340, 342, 343 bonded... coupling 73, 105, 113, 131, 138, 139, 180, 183, 185, 218, 276, 277 bending-twisting coupling 73, 106, 139, 140 , 276, 277 biaxial compressive load 133 stress state 236 bicycles 31 bi-harmonic operator 107 bi-modular materials 141 biodegradable composites 32 biological tissue 141 404 biomedical engineering 141 bird strike 179 blades 32 blast 27, 28 boat hulls 16, 28 body forces 88, 89, 260, 287 Boeing 23 bolt... 40, 66, 112 complete orthogonal set of functions 95 compliance matrix 46, 50, 316 Composite Worldwide Inc 28 compressive failure mode 348 failure strength 313 loads 131, 132, 135, 155, 158, 188, 199, 243, 244, 315 mechanical properties 140 modulus of elasticity 8, 184 stiffness 141 strength 305, 312, 315, 393 stresses 324 transverse loads 315 yield strength 306, 312, 314, 316 compression 320, 323, 325,... Zenkert, D 138, 142 Zukas, J.A 58, 78 Ulrich, D.R 78 Van Siclen, R.C 350-354, 356 Vinson, J.R 41, 57, 58, 76-79, 106, 109, 114, 138, 141 , 142 , 200, 230, 243, 253, 254, 276, 285, 299, 337, 338, 339, 346, 347, 356 Vizzini, A.J 253, 254 Volkersen, O 334, 354 von Mises, R 326, 331 Waddoups, M.E 331 Walker, W.J 78 Waltz, T 230, 253 Wang, D.Y 334, 354 Warburton, G 132, 142 , 183, 200, 285, 299 Wetherhold, R.C... 109, 110, 114, 160, 168-170, 222, 232, 233, 235-239, 264, 282, 285, 286 clamped-clamped beam 173, 184, 184, 267 clamped-free beam 170, 183, 184, 283 clamped-simply supported beam 172, 183, 267 Class A finish 16 406 classical beam 160, 182, 197, 294 beam theory 70, 94, 155, 156, 182, 290, 292, 295, 298 plate theory 70, 91, 93, 94, 96, 111-113, 115, 124, 139, 215, 272 shell theory 70, 215 theory 68,... coefficient of hygrothermal expansion 55, 59, 64, 276, 337, 393 moisture expansion 393 thermal expansion 1, 8, 53, 55, 59, 64, 192, 276, 334, 393 cold press molding 15, 16 collapse 35, 243 column 58, 87, 155, 160, 198, 199 combine 32 combined axial compression and bending 250 external pressure 251 hydrostatic pressure 251 torsion 252 combined loading 250, 316 failure theories 306 strength theories 306... interlock composite 4 ply laminate 34, 73, 338, 368 of twist 137 402 anisotropic 27, 34 compliance matrix 40, 41 elastic stiffness matrix 40, 41 elasticity 39 failure theory 309 fiber 52 laminate 330 materials 39, 40, 306, 309, 311 strength 309 tensor 40 anisotropy 40, 259, 274, 311 antenna 26, 27 applied lateral loads 135 load 243, 251 surface shear stresses 90, 91, 113 torque 243 aramid fibers 7, 51, 141 ... specimens 335 strength 335 stress 27 tests 335, 338, 350 fiber 4, 21, 25, 27, 29, 304, 365, 367, 375 buckling 140 cutting 348 dominated failure 305 failure 328 materials 140 optics 5 orientation 246, 305, 351 packing geometry 378, 389 placement 19 412 properties 51 reinforced composite 3, 5, 13, 14, 16, 39, 62, 309, 312 volume fraction 98, 372, 393 fiberglass 11, 26, 29 boats 28 bolted joints 348 laminates . Approach to the Theory of the Elastic Behavior of Multiphase Materials, Journal Mechanics and Physics of Solids, pp. 127. Schapery, R.A. (1968) Thermal Expansion Coefficients of Composite Materials Based. Reinforced Composite Materials, Journal of AIAA, Vol. 4, pp. 1537. Hashin, Z. (1968) Assessment of the Self-Consistent Scheme Approximation – Conductivity of Particulate Composites, Journal of Composite. Loading of a Unidirectional Composite, Journal of Composite Materials, Vol. 1, pp. 152. Chen, C.H. and Cheng, S. (1967) Mechanical Properties of Fiber-Reinforced Composites, Journal of Composite Materials,