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. Ekvall, J.C. (1961) Elastic Properties of Orthotropic Monofilament Laminates, ASME
Aviation Conference, Los Angeles, California, 61-AV-56.
2. Ekvall, J.C. (1966) Structural Behavior of Monofilament Composites, AIAA/ASME
Structures, Structural Dynamics and Materials Conference, Palm Springs,
California, pp. 250.
3. 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.
4. Hill, R. (1965) A Self-Consistent Mechanics of Composite Materials, Journal of
Mechanics and Physics of Solids, Vol. 13, pp. 213.
5. 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.
6. Whitney, J.M. and Riley, M.B. (1966) Elastic Properties of Fiber Reinforced
Composite Materials, Journal of AIAA, Vol. 4, pp. 1537.
7. Hashin, Z. (1968) Assessment of the Self-Consistent Scheme Approximation –
Conductivity of Particulate Composites, Journal of Composite Materials, Vol. 2,
pp. 284.
8. 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. Paul, B. (1960) Prediction of Elastic Constants of Multiphase Materials, Transactions
of the Metallurgy Society of AIME, Vol. 218, pp. 36.
10. 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.
11. 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.
12. Schapery, R.A. (1968) Thermal Expansion Coefficients of Composite Materials
Based on Energy Principle, Journal of Composite Materials, Vol. 2, No. 3, pp.
380.
13. 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.
14. Adams. D.F. and Doner, D.R. (1967) Longitudinal Shear Loading of a Unidirectional
Composite, Journal of Composite Materials, Vol. 1, pp. 4.
15. Adams. D.F. and Doner, D.R. (1967) Longitudinal Shear Loading of a Unidirectional
Composite, Journal of Composite Materials, Vol. 1, pp. 152.
16. Chen, C.H. and Cheng, S. (1967) Mechanical Properties of Fiber-Reinforced
Composites, Journal of Composite Materials, Vol. 1, pp. 30.
17. Behrens, E. (1968) Thermal Conductivity of Composite Materials, Journal of
Composite Materials, Vol. 2, pp. 2.
18. Behrens, E. (1967) Elastic Constants of Filamentary Composite with Rectangular
Symmetry, Journal of Acoustical Society of America, Vol. 47, pp. 367.
19. Foye, R.L. (1966) An Evaluation of Various Engineering Estimates of the Transverse
Properties of Unidirectional Composites, SAMPE, Vol. 10, pp. 31.
20. Tsai, S.W. (1964) Structural Behavior of Composite Materials, NASA CR-71, July,
National Aeronautical and Space Administration CR-71.
21. 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.
22. 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-TR66-190, Air Force Materials Laboratory.
23. 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. 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.
B. 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.
C. 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