The Behavior of Structures Composed of Composite Materials Part 14 pdf

Physical properties An important factor in determining the elastic properties of composites isknowledge concerning the proportion of constituent materials used in the respectivelamina/la

The factor represents a measure of the constituent element packing geometry andloading conditions For example, for the transverse modulus is used while forcalculation of the in-plane shear modulus a value of is used It should benoted that the values of as given above provide reasonable predictions for the elasticconstants up to certain volume fractions of fiber packing density and also for reasonablebounds on certain fiber geometries

For predicting the fourth technical engineering constant, the major Poisson’s ratiothe rule of mixtures can again be used Thus,

When considering anisotropic and twisted fibers, such as yarns, a modification of theabove formulae is necessary

B Physical properties

An important factor in determining the elastic properties of composites isknowledge concerning the proportion of constituent materials used in the respectivelamina/laminates These proportions can be given in terms of either weight fractions ofvolume fractions From an experimental viewpoint, a measure of the weight fractions iseasier to obtain than is the corresponding volume fractions of constituent elements.There is however, an analytical connection between these proportioning factors whichallows conversion from weight to volume fraction and vice versa Since volume fractionsare key to elastic properties calculations, this connection remains important Theexpressions necessary for this development follow

Definitions

f, m, c refer to fiber, matrix, composite respectively

In order to interrelate the above quantities analytically, we make use of familiardensity-volume relations Thus,

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 beanalytically expressed in terms of the following equations

Fiber Packing Geometry

1 Hexagonal Array:

Consider triangle ABC

(Area occupied by the fibers)

Volume Fraction

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

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, 66-190, Air Force Materials Laboratory.

AFML-TR-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 controllingthe design process using composite materials requires considerable expertise beyondtraditional levels for establishing design criteria A fundamental input into any designprocess is the requirement for obtaining the necessary materials properties data as well asestablishing the overall material response in order to identify the types of failure eventsthat can occur Thus the data base for composites is an evolutionary process in whichcurrent accepted test standards are being reviewed and revisions adopted as well ascomposite modes of failure identified and tabulated

As a ready means of access and awareness to the test procedures in currentpractice, test standards have been included It should be mentioned that in general theengineer 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, thoseusing 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 andmodified 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 societyrecommendations Such tests are almost universally run using commerciallyavailable test equipment and with specific geometry specimens

While all three of the aforementioned type of tests provide important data, it is thestandardized test that we tend to rely upon when requiring data for materials This isespecially true since engineers in general wish to be able to duplicate specific tests usingaccessible equipment rather than designing totally unique test facilities In view of thesestatements, the following standards given in Table 1 are provided which describe anumber of common mechanical tests Details concerning the test specimen geometry andprocedures can be found in the appropriate standard

Appreciation is expressed to Dr Gregg Schoeppner, AFRL/MLBCM for hiscontribution 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 selectedmaterial properties for continuous filament unidirectional composites is included as TableA3-1 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 ofelasticity,

To determine the density of many of the composite materials given on the nextpage, use the Rule of Mixtures of Section 2.4 (pp 51-52), along with the fiber densitiesgiven in Table 1 of Appendix 1 (pg 387), and the polymer matrix densities given inTable 1.2 (pg 8)

394

395

285, 299

Flaggs, D.L 141, 337, 339, 343, 355,

356

Foye, R.L 389Fujita, A 354, 356Gandhi, K.R 354Gellert, E 24, 36Gere, J 164, 200Ghosh, S.K 135, 142Goland, M 334, 339, 355Greenberg, J.B 253, 254Grimes, G.C 335, 336, 341, 355Hahn, H.T 50, 52, 77, 314, 331Halpin, J.C 50, 77, 389Harris, C.E 23, 36Hart-Smith, L.J 336, 337, 347, 355,

356

Hashin, Z 50, 77, 388, 389Hawley, A.V 335, 341, 348, 350, 354,

355

Henderson, J 129, 142Hill, R 311, 312, 326, 330, 331, 388Hilton, H.H 77, 79

Hofer, K.E 334, 335, 348, 350, 355Hoffman, O 313, 314, 330, 331Hsu, T.M 335, 355

Hsu, Y.S 253, 254Huang, N.N 54, 77Hwu, C 133, 142

Inman, D.J 129, 142Jen, M.M 354, 357

Jones, D.L.C 129, 140Jones, R.M

Jurf, R.A 57, 78

347, 354, 356

Osgood, W.R 336, 355Pagano, N.J 66, 79Pajerowski, J 355Paliwal, D.N 135, 142Paul, B 389

Petit, P.H 389Pipes, R.B 50, 77, 78Potter, P.C 36Preissner, E.C 237, 253Rajapakse, Y.D.S 58, 78Raju, B.B 354, 357Ramberg, W 336, 355Rankine, W.J.M 306, 331Reddy, J.N 141, 142, 253, 254Reddy, V.S 142

Reissner, E 75, 76, 79, 286, 289, 293,

299, 334, 339, 355

Renton, W.J 334, 337, 338, 346Riley, M.B 388

Ross, C.A 252, 254Roy, B.N 252, 253Running, D.M 354, 356Sandhu, R.S 331Sankar, B.V 253Schapery, R.A 389Sen, J.K 356

Shames, I 77Sharpe, W.N., Jr 339, 355Shaw, D 253, 254Sheinman, I 253, 254Shen, C 57

Sherman, I 253, 254Shuart, M.J 23, 36Sierakowski, R.L 36, 58, 78, 138, 252,

254

Simitses, G.J 253, 254Sloan, J.G 62, 68, 274, 299

Wu, C.I 299

Wu, E.M 314, 330, 331

Yi, S 77, 79Yon, J 354, 357Young, D 132, 142, 183, 199, 200,

285, 299

Yu, Y.Y 76, 79Zeng, Q.G 354, 357Zenkert, D 138, 142Zukas, J.A 58, 78

advanced beam theory 184, 193

Advanced Enclosed Mast/Sensor (AEM/S)

anisotropic 27, 34

compliance matrix 40, 41elastic stiffness matrix 40, 41elasticity 39

failure theory 309

fiber 52

laminate 330

materials 39, 40, 306, 309, 311strength 309

area moment of inertia 269

Arleigh Burke destroyers (DDG 51) 27armored vehicle 32

artificial intelligence 364

A54

3501-6 graphite/epoxy 394Peek (APC42) 394

cargo door assembly 24

carrier film sheets 21

Cartesian coordinate system 42, 43, 66, 217, 206, 293

Consolidated Vultee B-36 Bomber 333

constant amplitude load 338

constituent properties 367

constitutive equations 33, 34, 40, 42, 63, 66, 67, 76, 87, 92, 112, 132, 138, 155, 156,

178, 183, 185, 194, 195, 218, 239, 246, 274, 275, 371, 375contact molding 17

properties 51

reinforced composite 3, 5, 13, 14, 16, 39, 62, 309, 312volume fraction 98, 372, 393

finite difference methods 334

finite element methods 334, 339, 351, 365, 377

first ply failure 328, 330

free-free beam 183

Freedonia Group, Inc 21

Kirchoff edge condition 94, 95, 115

ladder side rail 20, 282

Lagrangian 293

lamina 3, 57, 58, 63, 66, 67, 69, 70, 74, 75, 87, 90, 97, 98, 105, 117, 125, 159, 218, 231,

246, 274, 280, 304, 305, 311, 316, 318, 322-324, 328, 329, 375, 384

failure 304, 305, 316