mechanical properties of composites

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Materials

Journal of Composite

http://jcm.sagepub.com/content/1/1/30The online version of this article can be found at:

DOI: 10.1177/002199836700100104

1967 1: 30Journal of Composite MaterialsC.H. Chen and Shun Cheng

Mechanical Properties of Fiber Reinforced Composites

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Mechanical Properties ofFiber Reinforced Composites

Theoretical expressions are obtained by means of classicaltheory of elasticity for determining the composite elastic con-stants for fiber reinforced plastics in terms of the elastic moduliand the geometric parameters of the constituents. Infinite seriesare used for the solution of the differential

equationsand a com-

bination of least square method and Fourier method is employed.Although the investigation is made for hexagonal array, theprinciples are also valid for other fiber arrangements. Numericalexamples have been worked out for hexagonal array and comparedwith an existing analysis.

C. H. CHEN

The B. F. Goodrich Company

Akron, Ohio

AND

SHUN CHENG

The University of WisconsinMadison, Wisconsin

INTRODUCTION

N RECENT YEARS fiber reinforced materials have been paid considerableattentions due to the search for materials of lightweight, great strength

and stiffness. Consequently the determination of their mechanical prop-erties becomes important, especially that of unidirectional composite. The

problem in general may be stated as follows. Given the elastic constantsof constituents, the proportion of each and a particular geometrical ar-rangement, find the composite elastic constants on the macroscopic level.

Although in reality the fibers are arranged at random, for the purposeof analysis some idealized arrangements of fibers are assumed. The worksof Cutler [1] and Bodine [2] are based on the idealized model in which

both the fiber and resin are in the form of rectangular bands which fillout the full layer. Shaffer [3] also used the same model and a strength ofmaterial

approachin his

study.Hashin and Shtrikman [4]

developeda

method for obtaining bounds on the effective moduli of composite elasticmaterials by variational approach. Later, again using variational method,Hashin and Rosen [5] derived bounds and i for the effective


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array and what-they-called random array. This paper is concerned withthe hexagonal array. Classical linear theory of elasticity is employed andinfinite series solutions are used.

The analysis is made on the assumptions: (1) that fibers are circu-lar and have equal and uniform size, (2) that the composites are homo-geneous in macroscopic sense, (3) that both resin and fibers are homo-

geneous and isotropic on the microscopic level, and (4) that there is

perfect bonding between fiber and resin. The microscopic elements whichare geometrically similar and experience the same mechanical behaviorwill be called the repeating element. The present method is to imposeto a

repeating

element

arbitrarycomposite strains.

By usingstress func-

tions or displacement functions and by requiring the continuities ofstresses and displacements at the boundaries between fiber and resin andbetween repeating elements, the composite stresses can be related tothose imposed composite strains. Thus composite elastic constants can beobtained from these relationships.

Due to the actual physical limitations of the electronic computersavailable, it is impossible to meet all of the requirements of the mathe-matical theory of elasticity rigorously, especially if infinite series solutionsare

used. In this paper approximationsare

made only in the boundaryconditions between resin and fibers and between repeating elements.There are no approximations in the equations of equilibrium or compati-bility other than those in the linear theory of elasticity.

In making the approximations in the boundary conditions, two differ-ent techniques related to &dquo;Least Square Method&dquo; are used. One is theFourier method whereby the square of a particular error over a particu-lar segment of the boundary is minimized with respect to a particularparameter in a given series.Another technique is to minimize the sum of

the squares of weighted errors by variation of the parameters. Combina-tion of the two methods, which is named as the combined least squaremethod is used in this paper.

ANALYSIS OF HEXAGONALARRAY REINFORCEMENT

Aportion of a cross-section perpendicular to the reinforcement isshown in Figure 1. The rhombus such asABCD represents a repeatingelement.Acoordinate

systemwill be set for each fiber area and resin

area with origin at the center of the area. Since the geometry is unaltered

by each rotation of ~r/3 about the z-axis passing through the center of afiber, the body is said to have an axis of elastic symmetry of the sixthorder [6]. Accordingly it has a type of anisotropy called transverselyisotropic, for which there are five independent elastic constants and thestress-strain relations become:

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1. Determination of Constants Cil, C12, CI3W33

Letthere

be normal strainsex, ey and ez of

constantvalues given

to

the composite under consideration. Under these composite strains thestrain and stress distributions for each resin and each fiber will be sym-metrical with respect to two geometrically symmetrical axes at 0 = 0and 0 = ~r/2 of each area. Hence it is sufficient to consider only a quarterportion of the rhombus such as CID. Since the macroscopic strains arethe same everywhere and with the assumption of ui = 0, v, = 0 the

rigid body displacements of the various areas will be as follows:

where the subscripts I, c, d, o refer to the area under consideration. Dueto the symmetry theAiry stress functions for each fiber may be writtenas:

and for each resin as:

The fiber stresses which are the same for all fibers are:


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Under the assumption of plane strain ( yxz = Yy. = 0) the fiber displace-ments are:

Similar expression for stresses and displacements in the resin can be ob-tained by replacingAm, 8~,, R fm, Sfm, Vf, G by Cm, Dm, R2m, Sjm, vj, Girespectively in the foregoing and using the appropriate pair of coordi-nates

for each resinarea.

The normal and tangential components of the stresses and displace-ments for fibers C and D at the boundaries are

where Un rro, u and v are given by Equations (3) and (4) with r = 1and appropriate subscripts. The normal and tangential components of thestresses and displacements for resin area can be obtained by following

where a gives the direction of the outer normal to the boundary. It should

be noted here that for resin 0, thereare

displacements dueto

rigid bodymotion, which are ex -B13b cos ao + eyb sin ao for ( un ) o and - ~x V-3 b sinao + -ey b cos a~ for (u,),,.

The conditions of continuity of stresses and displacements in the nor-mal and tangential directions across the boundaries Ll, L2 and L3 be-tween areas enables us to find the arbitrary constantsAm, Bm, Cm andDm in the series. To be specific it is required that



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The above requirements can be satisfied approximately if only a finitenumber of terms are used in the series but the errors can be minimized

in the sense of least square. The

necessary equationsfor

solving arbitrarycoefficients will be set up as following. Using the orthogonal propertiesof the circular function two equations are obtained by Fourier method

along the boundaries L, and L,.

The rest of equations are obtained by least square method. If let V~ _f lids, where ds is a differential element of length along the appropriateboundary segment, and let U = 4GiG(V3 + V4 + V11 + VIZ ) -f- V5 +V6 -f- (2Gi )2 2 ( V~ + V8 ) , these equations are then obtained by

Hence there are four equations for four unknownsAn, Bn, Cn and Dn-Since U contains the

rigid body displacements expressedin terms

of ex,ey and ez, Equations (9) will also have terms of ex, ey and ez. Thus whenthe set of Equations (8) and (9) are solved, the unknowns coefficients

ill b d i f h


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strains ex, ey and ez. The total composite forces are obtained by integrad-ing the normal stresses an and az along the lines IC, ID and over thearea ICD respectively. Thus the average composite stresses are given as:

where rib is the radius of resin I at the boundary. SinceAn, Bn, Cn, Dunare now all expressed in terms of ex, -ey, and ~, so will be Sx, Sy and S,whenAn, Bn, Cn, Dn are substituted into Equations (10).After terms of

ex, ey, ez are collected, we get

Asufficient number of terms in the series should be taken to make the

differences between gl2 and gzl as well as that between g13, gz3~ g3I~ g32within certain limit.

2. Determination of Constant C44Under the various conditions assumed the displacements w for both

resin and fiber due to shear distortion will be functions of r and 6 but not

of z. If let u = 0, v = 0 then the equation of elasticity for both fiber andresin reduce to: 172w = 0. Given the composite shear strains of ~yxz and

yyz the rigid body displacements can be written as follows:



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By nature of the deformation the displacement can be written for fiberas:

and for resin as:

each with subscript of particular area in concern. Notice that A~, and

Cm depend on firz and Bm and Dm depend on yyz.

The shear stresses are obtained by

The shear stress of the fiber at the boundary will be:

and that of resin will be:

The conditions of continuity of w and ~ynz across the boundary betweenareas require that

Using the combined least square method the following equations are

set up forYxz


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where U - Gj2 (V1 + V5 + V3 ) + V4, Vj == f I/ds, and ds is the dif-ferential elements along the appropriate line segment. The compositestresses are:

A suf$cient number of terms should be taken to make the difference be-

tween C44 and C55 within certain limits.

Figure 1. Hexagonal array.

NUMERICAL EXAMPLES

Typical numerical results are shown for two cases. The first case isa

homogeneous material where the properties of the fibers and resin are

equal. The input and output are:



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Note that for the present case:

These relations are necessary for isotropic materials. Thus the resultsshow that the program produces accurate results for this limiting case.

The second case is a nonhomogeneous material, with the followinginput and output:

The input corresponds to E-glass fiber and epoxy resin as the constituentmaterial. The fiber volume content is approximately 63%.

Similarly results for other fiber volume contents were run. The resultsin terms of engineering constants are shown as solid lines in Figure 2.

Typical relations between engineering constant and the components ofCz, matrix are:

- - .- .- - . -,

where Gij -1 is the inverse of Cij. In the present paper the transverse

isotropic plane is the 1-2 plane.Also shown in Figure 2 as dashed linesare the results of Reference 5 and its subsequent correction presented byDow and Rosen [7].

) MM is the largest number of m in the series; Ll and L2 are the number of divisionsinto which the boundaries Ll and L2 in Figure 1 are divided for the numerical integration.


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Not shown graphically in Figure 2 are the longitudinal stiffness ( E33 )and major Poissons ratio ( v23 ) because they are linear functions of thefiber volume content.

Figure 2. Numerical results.

DISCUSSIONS

The computational results for hexagonal array using the combinedleast square method are very satisfactory for the set of data used. Theerrors in those composite elastic constants, which are expectedly to beequal, range from .21 percents to 1.12 percents.

In general, the Fourier method has the advantages that it leads towell conditioned equations because of the orthogonality of the seriesalong the particular boundary, and that the required integration alongthe boundary may be obtained readily whereas with the combined least

square method the values are found by numerical integration. But theFourier method has a disadvantage that it can be used satisfactorily foronly a few boundary shapes. Moreover, as many different series are re-

quired for fulfilling the multiboundary conditions at the same boundary,the number of unknowns and

consequentlythe size of matrix will be

greatly increased, thus making the method less favorable.On the contrary the combined least square method has the advantage

that its application is not limited to the shape of the boundary and there-fore in principle to any array of fibers.

In performing the numerical integrations, trapezoidal rule is usedmainly because of its simplicity in programming. This is adequate as long



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as there are no singularities in the integrands. Good approximations are

expected if sufficient number of divisions along the integration segmentis used.In using the combined least square method, all portions of every

boundary and all errors due to the approximations in continuity ofstresses are treated equal. The errors due to the approximations in con-

tinuity of displacement was multiplied by 2Gi or 2VGiGf to obtain theequivalent weighted errors depending upon whether the boundary isbetween resins or between resin and fiber. The more the number of terms

is used the more the solution converges to the correct solution and themanner of weighting becomes of less importance.

It may be concluded that the combination of Fourier method and the

least square method is probably the best approach for these problems.The results reported here were obtained in the course of research

supported by theAir Force Materials Laboratory under ContractAF33( 615 ) -5070.

ACKNOWLEDGMENT

The authors are grateful to Prof. G. Pickett for the helpful comments and dis-cussions received during this work presented here. The authors would also like toextend their appreciation to Professor S. W. Tsai for his helpful suggestion andassistance concerning this problem and its presentation.

NOMENCLATURE

sx, Sy, sz = Composite normal stressesSyz, Sxx, Sxy - Composite shear stressesex, ey, e; - Composite normal strains

Yyz~ Yxz, qzy = Composite shear strainsC2~ = Composite elastic constants~M, V, w - Rigid body displacements in r, 0 and z directionsu, v, w - Total displacements in r, 0 aid z directionsux, uy = Displacements in x, y directionsern, u.~ = Stress and displacement normal to boundaryrut, rnz, ut - Stresses and displacement tangent to boundary0 - Stress functionW - Displacement functionvj,Gf - Poissons ratio and shear modulus of fiber

vi, Gi ._ Poissons ratio and shear modulus of resin

Rfm= m - 2 + 4 vf

Rim = m - ~ + 4 v;Sfm = - Rfm + 2( m + 1)Sim = - Rim + 2 (m + 1)b - 1/z distance between fibers

REFERENCES

1. V. C. Cutler, "Bending Analysis of Directionally Reinforced Pipe." Ph.D. Dis-sertation, Mechanics Dept., The University of Wisconsin, 1960.

2. R. Y. Bodine, "Some Elasticity Problems forAnAnisotropic Solid." Ph.D. Dis-sertation, Mechanics Dept., The University of Wisconsin, 1961.

3. B. W. Shaffer, "Stress-strain Relations of Reinforced Plastic Parallel and Normalto Their Internal Filaments."AIAA Journal, Vol. 2 (1964), p. 348.


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4. Z. Hashin and S. Shtrikman, "Note on a VariationalApproach to the Theory ofComposite Elastic Materials." J. Franklin Institute, Vol. 271 (1961), p. 336.

5. Z. Hashin andB. W.

Rosen,"The

ElasticModuli of

Fiber-Reinforced Materials."J.Applied Mechanics

,

Vol. 31 (1964), p. 223.6. S. G. Lekhnitskii, Theory ofElasticity ofanAnisotropic Elastic Body, (English

translation by P. Pern), Holden-Day, Inc., 1963.7. N. F. Dow and B. W. Rosen, "Evaluations of Filament-reinforced Composites

forAerospace StructuralApplications," NASAReport CR-207,April 1965.(receivedAugust 2, 1966)

Summer Workshop

PHYSICALASPECTS OF COMPOSITE MATERIALS

sponsored by

ARPAMonsanto/Washington UniversityAssociation

at

WASHINGTON UNIVERSITY-ST. LOUIS, MO.

JULY 13-21,1967

This workshop consists of two sessions: Introduction, andAd-vanced Topics. It is intended for technical people interested in themodern techniques in composite materials.

The 3-day introductory session will include the elementary for-mulation of the theories of elasticity, plates and shells, and linear

viscoelasticity.The 5-day advanced session will include micromechanics, aniso-

tropic fracture mechanics, dynamics of composite materials, visco-elastic stress analysis, plasticity, nonlinear viscoelasticity, metalliccomposites, structural synthesis, and problems of design and testing.

Complete program will be published in the near future.


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