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					                                                          15th ASCE Engineering Mechanics Conference

                                                      June 2-5, 2002, Columbia University, New York, NY


                                 X. Allan Zhong1 and Joe Padovan2


Dimensional analysis is used in this study to establish general relations between geometric features of
unidirectional fibrous lamina and the corresponding transverse responses, including transverse modulus and
stress distribution in the matrix. Prediction of transverse response of unidirectional fibrous lamina was made
long ago, e.g. the Halpin-Tsai equations. However, specific geometrical features of such composites are
typically lumped into the volume fraction of fibers. Using micromechanics via explicit representation of
fibers, it is demonstrated explicitly in this paper that the transverse response is dependent on two geometrical
parameters, one is the usual fiber volume fraction, and the other is the ratio of fiber diameter to fiber spacing.
A new formula for prediction of transverse response of the composite is established based on linear elastic
finite element analysis. The new formula is validated via lab test data. The formula is extended to nonlinear
analysis by integrating nonlinear matrix behavior in a separable way under certain restrictions. It is also
demonstrated that the stress distribution in the matrix and the location of peak stress changes with the
aforementioned ratio when fiber diameter and thickness are fixed. The effect of random fiber spacing is also

Keywords: composite geometry, unidirectional lamina, transverse response, dimensional analysis, linear
elasticity, and hyperelasticity.


Theories for the mechanical behavior of unidirectional fibrous lamina are of significance in fiber
reinforced engineering structures, since they form the foundation of theories for multidirectional
laminates in classical lamination theory. The micromechanical models for linear elastic composites
have been well established and are summarized extensively in many textbooks, e.g. [1 − 3]. Recent
efforts to include material and geometric nonlinearity in estimating effective properties of
composites can be found in, e.g. [4, 5]. In this paper we are interested in the geometric effect on the
transverse response of unidirectional fibrous lamina, because of their relevance in the design of
engineering components or structures, such as the plies in tires, conveyor belts, hoses, etc. In these
industrial applications, since the fiber, e.g. steel wire or wire bundle, is much stronger and tougher

    The Goodyear Tire & rubber company, Akron, OH 44316-001,
    Department of Mechanical Engineering, The University of Akron, Akron, OH 44325
than the (polymeric) matrix materials, the fatigue failure occurs in the matrix material only. In this
setting, the transverse responses of the lamina of interest include transverse modulus or stiffness,
matrix fatigue crack initiation and fatigue propagation of matrix cracks.

It has been established that the rule of mixtures can predict the longitudinal modulus of
unidirectional fibrous lamina pretty well in [6], and more recently in [5]. The prediction of
transverse modulus however is more problematic. In the design of unidirectional fibrous lamina,
such as plies in tires, fiber diameter, fiber spacing (usually called EPI, i.e. fiber ends per inch, in tire
industry) and lamina thickness are important geometrical parameters. These geometrical
parameters influence the constraint of the fiber imposed on the matrix material, and consequently
affect the apparent failure properties of the composites. Based on these considerations, it is
worthwhile to relate the transverse modulus of unidirectional fibrous lamina to these geometrical
parameters explicitly for easy application and interpretation. However, because of the
homogenization nature of typical approaches in composite mechanics, effective properties of fiber
reinforced composites are related to constituent properties and fiber volume fraction, so theories or
equations for the estimation of transverse modulus are not suitable for this type of application.
Another response of interest is the peak stress in the matrix between fibers because it is directly
related to fatigue crack initiation. The typical approaches of composite mechanics could not predict
this quantity due to the inherent averaging nature in these approaches. The same can be said
regarding transverse matrix cracking. So, instead of using composite mechanics approaches,
micromechanical finite element models are formulated to study the geometric effects on the
transverse response of unidirectional composites – transverse modulus (transverse stiffness), and
peak stress in the matrix. The matrix cracking related subject will be addressed in a future paper. A
2D-unit cell model is used in the following analysis.

As to be shown later, it is found via finite element analysis that the transverse modulus is dependent
on two geometric parameters, one is the usual fiber volume fraction, and the other is the ratio of
fiber diameter (D) to fiber spacing (d), i.e. D/d, in addition to constituent properties. A new and
simple formula for the transverse modulus is obtained under small deformation assumptions. This
formula is then extended to large deformation under certain restrictions. The peak stress in matrix
is found to be varying with the ratio nonlinearly. It is noted that these findings can not be predicted
from composite mechanics approaches, they can not be predicted even from the more sophisticated
rebar representation [7] of fibers.

Transverse modulus or stiffness represents one aspect of the structural function of the
unidirectional fibrous lamina. The simplest calculation of the transverse modulus is obtained by
considering the fiber and matrix to act as springs in series [1,2],

               E f Em
E2 =                                                                                                   (1)
       E f (1 − v f ) + E m v f

where E f , E m are moduli of fiber and matrix, and v f is the volume fraction of fiber. This is one
of the commonly used formulas for estimation of transverse modulus. But it does not account for
fiber geometry and spacing. The Halpin-Tsai Equation [6] is frequently used to account for these

geometrical features of the lamina. The equation for the transverse modulus is [6],

       E m [ E f (1 + ξv f ) + ξE m (1 − v f )]
E2 =                                                                                             (2)
        E f (1 − v f ) + ξE m (1 + v f / ξ )

where ξ is a factor depending on fiber geometry and spacing. Similar equations for other moduli
are also available. Application of equation (2) is contingent upon the determination of ξ with
respect to different fiber spacing and fiber shape. For fibers of nominal circular cross section and
 E f >> E m , which is the typical case in tire reinforcements, ξ = 2. So equation (2) is reduced to

       E m (1 + 2v f )
E2 =                     .                                                                       (3)
         (1 − v f )

There are other estimations of transverse modulus of unidirectional fibrous lamina, e.g. [5], which
are similar to equation (2) or its refinements. The method of cells [9, 10] is a rather general
approach, except that the method requires regular fiber packing. However it is shown in [11], that
predictions of transverse response, based on any regular fiber packing pattern considered, do not
correlate well with experimental results. Based on these considerations, we will compare our
prediction of transverse response to that of the Halpin-Tsai equation, i.e. equation (2) or (3) only.

For a lamina with dimension W > > T, under transverse loading (Figure 1), the plane strain
assumption of deformation in the lamina is a good approximation of the problem. In a plane strain
formulation, only the cross section perpendicular to fiber direction needs to be considered as
shown in Figure 2, where D is the effective fiber diameter, d is the spacing between fibers, and T is
the lamina thickness. The plane strain problem can be further simplified if there are many fibers in
the loading direction (i.e. L >> D + d), which is the case under consideration.

  F                                                                                          F


                   Figure 1 Schematics of a unidirectional fibrous lamina

                    T                D

                                               Figure 2: A unit cell

So with the assumptions that W >>T and L >> D + d, a plane strain problem for a unit in the loading
direction is appropriate for prediction of the lamina transverse response. The unit used here is the
portion bonded by dashed lines in Figure 2. Considering symmetry conditions, only a quarter of the
unit is analyzed in a FEA calculation. The fiber is assumed to be linear elastic material and it has
Young’s modulus E f and Poisson’s ratio γ f . The matrix can be either a linear elastic material
(Young’s modulus E m and Poisson’s ratio γ m = 0.49 (almost incompressible, as is the case for
elastomers)) or a hyperelastic material of Neo-Hookean type .

The quantities of interest here include transverse modulus E2 of the lamina, and the stiffness S of
the lamina. In linear elasticity, the relation between the two quantities is approximated as follow,

      F σWT     composite WT
S=      =    = E2                                                                               (4)
      δ   εL               L

where F is the force applied on the lamina, δ is the corresponding elongation, and W, L and T are
the dimensions of the lamina.

For the plane strain problem under consideration, we have

 composite          S
E2         =                                                                                   (5)
               T /( D + d )

Here S is the stiffness of the unit in Figure 2.

Apparently the transverse modulus of the unidirectional composite is a function of material
properties as well as its geometrical features, in general one can write

E2         = f ( E f , γ f , E m , γ m , D, d , T )                                            (6)

for linear elastic matrix material, and

E2         (λ ) = f ( E f , γ f , C10 , D, d , T , λ )                                                                                                                      (7)

for Neo-Hookean matrix material with coefficient C10. The secant transverse modulus or
normalized transverse stiffness is dependent on the deformation in the composite. In equation (7),
the quantity λ is the stretch ratio in the lamina.

Dimensional analysis reduces the number of variables the transverse modulus depends upon. The
relations (6) and (7) can be rewritten into the following forms,

E2              Ef D D
           = f(   , , ,γ f ,γ m )                                                                                                                                                (8)
   Em           Em d T

for linear elastic matrix, and

E2         (λ )      Ef D D
                = f(    , , , λ,γ f )                                                                                                                                            (9)
    C10              C10 d T

for hyperelastic matrix material.

For easy conversion of FEA results into a concise formula, equation (5), which relates stiffness to
transverse modulus, is integrated into equation (8) or (9). From linear elastic analysis of this unit
cell model, it is found that for given fiber properties, the relations as given in equation (1) and (3)
do not match FEA results, e.g. see Figure 3(a).
                                       Normalized transverse modulus of                                                    Normalized effective transverse modulus as a
                                            unidirectional lamina                                                                          function of p

                                                                                              Normalized transverse
      Normalized transverse

                              10                                                                                      10                        y = 0.3366x + 1.3339
                              8                                                                                                                      R2 = 0.9992


                              6                                                                                       6
                              4                                                                                       4
                              2                                                                                       2
                              0                                                                                       0
                                   0       1         2        3           4                                                0                5          10         15   20   25         30
                                                   p_2                                                                                                            p

(a)                                                                                     (b)

 Figure 3: a) Comparison between predictions from the Halpin-Tsai equation and
                       those from FEA, where     (1 + 2v ) ,                                                                           f
                                                                                                              p2 =
                                                                                                                               (1 − v f )
                                               b) comparison of (10) with FEA results,                                                      p=
                                                                                                                                                  1+ 2⋅v f D
                                                                                                                                                   1− v f d

    Based on dimensional analysis and the Halpin-Tsai equation, it turns out that the following
    equation can describe the transverse modulus very well

           E2                        S                       1+ 2⋅vf D
                      = S' =                    = 0.3366 ⋅ (        ) + 1.339                                                                                                                     (10)
              Em             E m (T /( D + d ))               1− vf  d
    The equation is applied to predict the stiffness change of composites studied in [13]. The
    predictions compare with test results very well, see Figure 4. A similar relation for nonlinear
    Neo-Hookean matrix material has been obtained.

                                                               Lamina transverse stiffness: model prediction vs. test
                                                                           results (Gent and Park, 1986)

                                       transverse stiffness
                                          per unit length


                                                                   0               5               10             15                               20
                                                                                         fiber spacing (mm)

                    Figure 4: Effect of fiber spacing on transverse stiffness and illustration of test
                                                    geometry in [13]

    As shown in Figure 5, as d/D decreases, the location of peak stress moves from very close to the
    fiber/matrix interface to the center between the fibers. As a matter of fact when d/D ≤ 0.4, the peak
    normal stress is always located at the center between adjacent fibers. It is also found that peak
    stress in the fiber is amplified for both load and displacement control, see Figure 6.

                               Normal stress along loading direction, d/D=0.1                                                                     Normal stress along loading direction, d/D=1

                    1.05                                                                                                               1.07
                                                                                                                   Normalized stress
Normalzied stress

                    1.03                                                                                                               1.04
                    1.02                                                                                                               1.03
                      1                                                                                                                   1
                    0.99                                                                                                               0.99
                           0           0.2                          0.4          0.6         0.8         1                                    0          0.2           0.4          0.6     0.8          1
                                                                   normalzied distance                                                                                normalized distance

                                       Figure 5: location of peak stress in the matrix between fibers

                                                 Variation of peak stress between adjacent fibers

               Normalzied peak stress
                                             0         0.2      0.4             0.6       0.8            1   1.2

                                                             load control         displacement control

                                         Figure 6. Normalized peak stress between fibers

Sensitivity of the new formula with respect to variation in                                     and in γ f and γ m , 3D effect,
effect of out of plane boundary conditions and Effect of non-uniform fiber spacing, are also studied.
Details of this study will be addressed in the full-length paper.


Via micromechanical finite element analysis we have demonstrated that

1) The transverse modulus of unidirectional fibrous lamina is dependent on constituent properties,
   fiber volume fraction and relative fiber spacing (d/D). A simple scaling law is established and
   the range of applicability is identified.
2) The location and magnitude of the peak stress in the matrix of a unidirectional fibrous lamina
   under transverse load is shown to be dependent on only one geometric parameter, the relative
   fiber spacing (d/D) irrespective if fibers are uniformly spaced or not. This is because the peak
   stress is primarily determined by fiber interaction. The peak stress approaches to a constant
   with respect to d/D for given load, when d/D is large.


The authors benefited from many suggestions made by Prof. Alan Gent. X.A.Z. also appreciates

discussions with S. Kim and Z. Bo. The authors are grateful to The Goodyear Tire & Rubber
Company for permission to publish this paper.
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