Journal of Composite

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Longitudinal Shear Loading of a Unidirectional Composite

Donald F. Adams and Douglas R. Doner
Journal of Composite Materials 1967 1: 4
DOI: 10.1177/002199836700100102

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 Longitudinal Shear Loading of a

Unidirectional Composite
DONALD F. ADAMS AND DOUGLAS R. DONER
, Philco Corporation
Aeronutronic Division
Newport Beach
, California

The behavior of a unidirectional fiber-reinforced composite ma-

terial subjected to a shear loading in the direction of the filaments
is of fundamental importance in the understanding of composite
material behavior. The shear stiffness and strength values asso-
ciated with this type of loading can be considered as fundamental
properties characterizing the material.
The problem of a doubly periodic rectangular array of parallel
elastic filaments of doubly symmetric but otherwise arbitrary
shape contained in an elastic matrix material subjected to shear
stress components in the directions of the filament axes has been
formulated using a theory of elasticity analysis. A finite difference
representation of the governing partial differential equations of
equilibrium and stress-displacement equations has been utilized
in obtaining a solution by a systematic over-relaxation procedure.
Numerical results have been obtained for a variety of filament
cross-sectional shapes, various filament-to-matrix shear modulus
ratios (ranging from zero to infinity), and a range of filament
spacings varying from filaments nearly in contact to extremely
wide spacings in which interaction effects between filaments are
essentially zero. Theoretical results have been compared with the
limited experimental data presently available.
The results obtained permit a detailed description of the micro-
mechanical behavior of a composite material subjected to
longitudinal shear loading. A study of local stress distributions in
the composite provides a means of predicting shear strength as
well as stiffness as a function of the material properties of the
constituents and their geometry.

Presented at the Second Annual Symposium on High Performance Com-

posites Materials, Washington University, St. Louis, Missouri, October 20-21,
1966.
This work was sponsored in part by the Air Force Materials Laboratory
under Contract AF 33 (615)-2180 and the National Aeronautics and Space
Administration under Contract NAS 7-215.

INTRODUCTION
HE GOAL of mechanics analyses of composite materials is to predict
Tthe behavior of the material in actual structural applications. Such
structural analyses typically are based upon the assumption that the
fundamental material properties of the composite are known initially.
These material properties can be determined experimentally by testing
actual composite specimens. However, in recent years, more and more

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 attention has been given to methods of analytically predicting these
basic composite material properties as a function of the properties of the
constituent materials and their geometrical relationship to each other.
Such an approach to predicting composite behavior has become
known as &dquo;micromechanics,&dquo; as opposed to the analysis of members fab-
ricated from such composites, i.e., &dquo;macromechanics.&dquo;
It has been suggested that once the basic material properties of a
unidirectional fiber-reinforced composite are completely determined, the
behavior of a member constructed of such material can be predicted by
utilizing suitable analyses [1]. These basic material properties are asso-
ciated with the response of the unidirectional composite to axial, trans-
verse, and shear loadings, respectively. In the present discussion, the
prediction of composite material properties associated with shear loading
will be considered. The shear loading, assumed to be acting along the
filaments, will be referred to as longitudinal shear loading.
DESCRIPTION OF PROBLEM
To treat the are made as to filament
problem analytically, assumptions
packing arrangement and geometry of the individual filaments. The
method of solution to be used is based upon the existence of certain
symmetry conditions which require the assumption of a periodic array.
A rectangular filament packing array has been arbitrarily assumed, as
shown in Figure 1, although some other regular array such as hexagonal
packing could also have been utilized. It was felt, however, that a rec-
tangular array, which permits the variation of filament spacings in two
coordinate directions independently, offers more versatility than, for ex-
ample, a hexagonal array with only one geometric spacing variable. The
individual filament cross sections are assumed to be symmetrical about
each of the coordinate axes, x and y. Within this restriction, the filaments

Figure 1. Composite containing a rec-

tangular array of filaments imbedded
in an elastic matrix.

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can be ofarbitrary shape, i.e., circular, elliptical, diamond, square, rec-

tangular, hexagonal, etc.

Having established the assumptions of rectangular packing and
doubly symmetric filaments, the problem can be formulated exactly
(within the usual assumptions of the theory of linear elasticity). This
is perhaps the key point of this analysis.
Because of the assumed periodicity, a fundamental or repeating unit,
as indicated by the dashed lines of Figure 1, can be isolated, being
typi-
cal of the entire composite. Because of the assumed filament symmetry,
only the first quadrant of this fundamental unit need be considered, as
indicated in Figure 2. When the composite is subjected to longitudinal
shear loads applied at a distance from the element being analyzed, in
the directions indicated by the average values of flr and %zy in Figure 2,
a
complex shear stress distribution will be induced. This is the result of
the dissimilar material properties of the filaments and matrix and also
because of interactions between the filament being analyzed and adjacent
filaments.
However, because of the assumed symmetry, each longitudinal shear
component, Tzx and Tzy, when applied separately, will cause a uniform
axial displacement of the boundary of the fundamental region upon
which it acts. Thus, the problem can be formulated as a displacement
boundary value problem, interactions between adjacent filaments being
automatically and accurately taken into account.
METHOD OF ANALYSIS
The problem of longitudinal shear loading is defined by a displace-
ment field of the form

For such a system of displacements, the only nonvanishing stress com-

ponents are:

Figure 2. First quadrant of the funda-

mental region-longitudinal shear
loading.

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where G is the shear modulus of the constituent material, i.e., filament or
matrix, in which the shear stress is being calculated. Each constituent
will be assumed to be homogeneous, isotropic, and linearly elastic in the
present formulation although, as will be discussed, these assumptions can
be relaxed, if desired.
The equilibrium equations in the x and y directions are identically
satisfied, while equilibrium in the z direction requires that

Because of the assumed double periodicity of the filament geometry

and spacing, the displacement field must satisfy the requirement that

Itnormally is desired to solve the shear problem for a given set of

shear loading conditions, i.e., specifying Tzx and Tzy, rather than for given
boundary displacement conditions. However, it is much simpler to solve
the problem when expressed in terms of displacements as, for example,
in Equations (2) and (3). Thus, the procedure will be to first solve the
problem for a specified uniform displacement, w~, along the side x a =

of the fundamental region, the boundary condition on the other three

straight sides of the first quadrant of the fundamental region being,
from symmetry conditions:

Having solved this

problem, the average shear stress izr correspond-
ing to this
specified displacement, w~, is determined by first calculating
T*zx at points along the boundary x a and then calculating the average
=

value.
Thus, a solution for the case of an average shear loading Tzx along
the boundary x = a and zero shear along the boundary y = b has been
obtained. This same procedure can then be repeated to obtain a solution
for the case of a specified average shear loading ~czy along the boundary
y = b and zero shear along the boundary x = a, i.e., specifying a uni-
form displacement, w~~, along the boundary y b, and solving the dis- =

placement boundary problem satisfying the boundary conditions:

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 In solving the problems outlined, it is necessary to establish continuity
conditions the interface between the filament and the matrix. These
at
conditions, which are identical in both problems, are as follows:

( a ) continuity of axial displacement across the interface

(b) continuity of shear stress across the interface

The effective shear moduli of the composite material are determined

as follows:
’rxx loading only

zy loading only

Having obtained solution for each of the two problems outlined,

a

i.e., flrspecified, fly 0 and T~ specified, ~czx = 0, the solution of the

=

general problem of combined shear loading is obtained by superposition.

SOLUTION TECHNIQUE
The solution of the problem outlined in the previous section has been
formulated using a finite difference representation and a numerical re-
laxation procedure designed for high-speed digital computer operation.
The finite difference approximations of the partial derivatives contained
in Equations (2) and (3) make use of irregular grid spacings in both
coordinate directions. This is an important feature of the solution in
that it permits the use of close grid spacings in regions where it is de-
sired to determine stresses very accurately, e.g., in areas of high stress
concentration where stress gradients are very high, while permitting a
coarser spacing in less critical regions. This permits a given degree of

accuracy with a minimum amount of numerical computation and com-

puter storage capacity.
The finite difference representations of the partial derivatives con-
tained in Equations (2) and (3) are of the following forms:

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First Irregular Central Di$erences.

Second Irregular Central Differences.

First Irregular Forward Di$erences.

First Irregular Backward Differences.

The terms a, through al2 represent distances measured from the point
( i, ~ ) at which the difference form is being expressed (point 0 in Figure
3) to surrounding points (numbered 1 through 12 in Figure 3 ) . Node
points 5 through 8 are not actually used in the longitudinal shear prob-
lem since they are associated with partial derivatives of the form éJ2f I éJxéJy

Figure 3. Node identification number-

ing system.

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which do not appear in the formulation. The subscripts on each displace-
ment term, w, identify the grid coordinates of that displacement in terms
of the point ( i, ~ ) .
Central differences are used in representing the equilibrium equation,
Equation (3). In representing the boundary condition equations, Equa-
tions (5, 6), and the interface continuity equation, Equation (8), it be-
comes necessary to use either forward or backward differences in order
to remain within the first quadrant of the fundamental region. Typically,
an array size of approximately 20 x 20 has been found to be sufficiently
accurate for the problem being considered.
Having established a finite difference residual equation in terms of
displacements at each node point, a systematic over-relaxation procedure
has been utilized in obtaining the displacement field corresponding to
the specified stress and displacement boundary conditions. The stress-
displacement relations of Equation (2), expressed in difference form, can
then be utilized to calculate stresses at any specified node point.
A detailed description of the numerical solution technique, a complete
Fortran IV computer program listing, and a sample computer output
for a typical solution are given in Reference 2.

NUMERICAL RESULTS
A representative sample of the type of numerical results which can
be obtained by utilizing the solution outlined will be presented here. No
attempt will be made to present a complete parametric study, however.
In Figures 4 and 5 are presented values of stress concentration fac-
tors, SCF, and normalized composite shear stiffnesses, G/Gm, respec-
tively, for circular filaments in a square packing array subjected to a
shear stress component ’Czx. These results are plotted as functions of the
ratio of the shear moduli of the constituents, G¡/Gm, for various filament
volume contents, vf. The values of SCF plotted represent the maximum
shear stress occurring in the composite divided by the applied shear
stress ~czx. The maximum value occurs in the matrix at the intersection
of the filament-matrix interface with the x-axis (see Figure 1 or 2), i.e.,
at the point of closest proximity between adjacent filaments. The nor-
malized filament spacing, blr, is also indicated for each filament volume
content.
As can be seen in Figures 4 and 5, little stiffening is obtained for fila-
ment volume contents below 50%, even when very rigid filaments are
utilized, i.e., for high values of G¡/Gm. The stiffness of the composite
is essentially that of the matrix. Also, for low values of G¡/Gm, e.g., val-
ues less than 3 or 4, little stiffening is obtained even for high filament
volume contents. However, for combinations of high filament volume

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L-ongitudinal Shear Loading of a Unidirectional Composite

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content high values of GtlGm, very significant increases in shear
and
stiffness predicted. Unfortunately, local stress concentrations also
are

become extremely high, indicating the possibility of low composite

strengths. To indicate the significance of these predictions, glass-epoxy
composites have a ratio of shear moduli, GfIG,,,, of approximately 20 and
boron-epoxy composites a ratio of 120. Most composites presently being
utilized in structures have a filament volume content ranging from 65
to 70 percent. As Figures 4 and 5 indicate, at v f = 70 percent, a glass-
epoxy system will have a composite shear stiffness approximately 5.5
times greater than that of the epoxy matrix, while inducing a stress con-
centration factor of 2.5. Correspondingly, a boron-epoxy composite would
produce a value of GIG. of 7.0 and a SCF of 2.8. However, although
stiffness properties increase rapidly for higher filament volume contents,
local stress concentrations become prohibitively high.
It should be pointed out, however, that for high filament volume
contents (closely spaced filaments), the high stress concentrations are
very localized. This leads to the conclusion that if local yielding of the

Figure 6. Stress concentration factors

(SCF) for boron filaments of various
shapes in an epoxy matrix (Gf/G.~ =

120) subjected to longitudinal shear

loading (Tzx).

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matrix can occur, these high stresses will be relieved and high shear stiff-
nesses can be obtained while maintaining adequate composite shear
strengths.
In addition to the consideration of variable constituent material prop-
erties and filament volume contents, the present analysis also permits the
study of various filament cross-sectional shapes. Although highly irregu-
lar shapes can be analyzed, relatively regular cross sections will be con-
sidered here in order to permit systematic comparisons. As examples, the
SCF and G/Gm for filaments of various elliptical cross sections and one
square cross section are compared with values obtained for circular fila-
ments in Figures 6 and 7, respectively. A value of GFIG of 120 is as-
sumed. In all of these examples, the distances between adjacent filaments,
8, were assumed to be equal in both coordinate directions. Filament
spacing, normalized by the principal dimension ; of the filament, was
taken as the plotting variable, i.e., b/r was used. This is more meaning-
ful to the present study than filament volume content, v f, since, for two
different filament geometries, equal values of v could require a signifi-

Figure 7. Composite shear stiffness

(G/GtJ for boron filaments of various
shapes in an epoxy matrix (Gj/Gm =

120) subjected to longitudinal shear

loading (izx).

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cant difference in 8. And 6 is the significant parameter in dictating the
magnitude of the local stress concentration.
Composite stiffnesses, G/Gm, but not stress concentration factors,
SCF, are presented for square filaments since, theoretically, an infinite
stress will exist at each sharp corner. The stress at the midpoint of the
side of the square filament was found to be relatively low ( SCF = 1).
In practice it would be advantageous to use either square filaments with
rounded corners or to assume localized yielding or stress relaxation to
occur in the matrix material at the corners. Figures 6 and 7 also indicate
that significant increases in the composite shear stiffness can be obtained
when using noncircular filaments by orienting the filaments properly,
while at the same time not introducing large variations in stress concen-
tration factors.
Additional discussion of the practical implications of the present work
can be found in the reports
generated by Aeronutronic under contract
AF 33(615)-5198, entitled: Nonconservative Behavior of Composite
Materials.

ANALYTICAL AND EXPERIMENTAL

COMPARISONS OF RESULTS
The primary variables in the present analysis include matrix and fila-
ment material properties, filament volume content (or filament spacing),
and filament cross-sectional shape. However, few experimental results
are
presently available for variations in these parameters. Very little ex-
perimental work has been done with noncircular filaments and most test
data for circular filaments are for a relatively small range of filament
volume content, i.e., from 65 to 70 percent typically. Some data for vari-
ous composite systems are available, thus permitting a study of variations
of the ratio GfIG,,,. Typical experimental shear stiffness data along with
analytically predicted values are presented in Table I.
There are several other existing theoretical determinations of the
composite shear modulus. Hashin and Rosen [3] derived the following
equation for circular fibers arranged in a hexagonal array:

Chen and Cheng [4] used the theory of elasticity and derived the com-
posite elastic moduli of parallel fibers also in a hexagonal packing.
The comparison between the numerical results of this paper with
those derived from Equation (11) is shown in Table II. In all cases, the

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Table I. Theoretical and Experimental Values of Composite Shear Modulus (G)
for Various Composite Materials

hexagonal array gives a lower value than the square array for the same
filament volume content. The results of Reference 4 are very close to
those predicted by Equation (11). Based on the results listed in Table
I, the square array gives values closer to experimental measurement than
the hexagonal array. This may be explained by the fact that the square
array is a more realistic approximation of actual composites than the
hexagonal array.
A relationship between local stress concentrations and composite
strength has not been adequately established, and hence a correlation of
theory with experiment is not yet possible. However, the qualitative
observations previously made concerning variations in SCF with filament
volume content have been experimentally substantiated by observed
losses in composite shear strength for very dense filament packings. Ad-
ditional work remains to be done in this area, however.
Because of the limited experimental data available for verification of
the theory, comparisons have also been made with another, indepen-
dently derived, theoretical solution. The problem of two elastic circular
cylindrical inclusions of different diameters and material properties con-
tained in an infinite elastic matrix and subjected to longitudinal shear
stress components has been solved by Wilson and Goree [5] utilizing a
complex variables formulation. By assuming the two inclusions to be
identical and subjected to a loading identical to that acting on a doubly
periodic array of filaments having the same relatively small spacing in one
direction and a very large spacing in the other direction, the predicted
stresses in the matrix material between filaments should be very nearly
identical in the two cases. This has, in fact, been found to be true. For
example, for GFIG,, = 20 and blr = 0.8, the two-inclusion solution gives
SCF = 2.55, whereas the doubly periodic array, with blr in one direc-
tion equal to 0.8 and equal to 6.88 in the other direction, gives SCF =
2.53. This is typical of the close agreement obtained between the two

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Table II. Stress Concentration Factors and Composite Stiffnesses
for Circular Filaments

independent analytical solutions for comparable configurations and tends

to provide confidence in the accuracy of the results obtained.
A comparison of the two independent solutions also offers several
interesting observations concerning physical behavior. For example, for
a square packing array with 6/r 0.8, the predicted value of SCF is
=

1.61 as compared to 2.53 for a value of 8/r of 0.8 in one direction and
6.88 in the other. Thus, nearby surrounding filaments actually tend to
distribute the stress more uniformly and reduce the peak value signifi-
cantly. That this is generally true can be seen in Table II, where two-
inclusion and square-array stress concentration factors are presented for
a range of filament spacings and shear modulus ratios.

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 CONCLUSIONS
It be generally concluded that local stress concentrations in a
can

composite material subjected to longitudinal shear loading may be ex-

tremely high. The two independent analytical solutions obtained agree
closely and hence provide reasonable assurance that the values obtained
are correct within the assumptions made in the formulation of the prob-

lem.
However, one basic assumption made in the analysis is that both the
filaments and the matrix are linearly elastic and that no plastic or visco-
elastic behavior occurs. In an actual composite material, localized yield-
ing undoubtedly does occur in the typically weak matrix material, per-
mitting a redistribution of the high localized stresses. Depending upon
the type of matrix material being considered, either nonlinear elastic,
inelastic, or viscoelastic behavior may occur. Thus, a logical extension
of the present analysis will be to study this nonconservative material
behavior.
NOMENCLATURE
2a, 2b - Filament spacing in x and y directions
G = Shear modulus of composite or constituents
n = Normal direction
f = Dimension of filament (radius for circular filament)
SCF = Stress Concentration Factor
u, v, w - Displacements in x, y, z directions
w*, woo = Uniform displacement along x = a, y = b
vj = Filament volume content
Sub f =
Pertaining to Fiber or Filament
Sub m =
Pertaining to Matrix
Sub i, j = Grid indices in x and y directions
’Czx, LZY =
Shear stress components
’Czx, Tzy -
Components of average shear stress
8 = Filamentspacing
REFERENCES
1. S. W. Tsai, "Strength Characteristics of Composite Materials," NASA Contractor
Report NASA CR-224, April 1965.
2. S. W. Tsai, D. F. Adams, and D. R. Doner, "Effect of Constituent Material
Properties on the Strength of Fiber-Reinforced Composite Materials," Air Force
Materials Laboratory Contractor Report AFML-TR-66-190, August 1966.
3. Z. Hashin and B. W. Rosen, "The Elastic Moduli of Fiber-reinforced Materials,"
, Vol. 31E (1964), p. 223.
. Applied Mechanics
J
4. C. H. Chen and Shun Cheng, "Mechanical Properties of Fiber-reinforced Com-
posites and of Perforated Solids," J , Vol. 1 (1967), p. 30.
. Composite Materials
5. H. B. Wilson, Jr. and J. G. Goree, "Transverse Shear Loading in an Elastic
Matrix Containing Two Elastic Circular Cylindrical Inclusions," Mathematical
Studies of Composite Materials III, Rohm and Haas Company Report, Hunts-
ville, Alabama, June 1966.
(received September 2, 1966)

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