Poisson's ratio of ber-reinforced composites
Henrik Christiansson
Theoretical Physics, Royal Institute of Technology,
S-100 44 Stockholm, Sweden
Johan Helsing
Department of Mathematical Sciences, Rensselaer Polytechnic Institute,
Troy, New York 12180-3590
Abstract
Poisson's ratio ow diagrams, that is, the Poisson's ratio versus the ber fraction, are obtained numerically for hexagonal arrays of elastic circular bers
in an elastic matrix. High numerical accuracy is achieved through the use
of an interface integral equation method. Questions concerning xed point
theorems and the validity of existing asymptotic relations are investigated and
partially resolved. Our ndings for the transverse eective Poisson's ratio, together with earlier results for random systems by other authors, make it possible to formulate a general statement for Poisson's ratio ow diagrams: For
composites with circular bers and where the phase Poisson's ratios are equal
to 1/3, the system with the lowest stiness ratio has the highest Poisson's ratio. For other choices of the elastic moduli for the phases, no simple statement
can be made.
I. INTRODUCTION
Several authors have recently devoted their attention to Poisson's ratio ow diagrams,
that is, the eective Poisson's ratio as a function of the inclusion volume fraction, for composite materials. [1{14]. Some authors consider explicitly discrete materials such as discrete
fractals [1] and lattice networks of springs [2{5], but most work has been for continua. The
Poisson's ratio for two-dimensional elastic composites has been studied analytically in the
framework of eective medium approximations [6,16], and numerically with methods such as
nite element methods [13] and boundary element methods [11], and with discretized-spring
schemes applied to digital images of materials [8{10,12,13,15]. The studies undertaken with
the digital image based method of the Poisson's ratio ow diagrams have been for isotropic,
random or periodic structures of circular voids [9] and circular rigid inclusions [13]. Computations have also been done for composites with random structure, where the phases have
nite elastic moduli [10,14]. The error in the above computations typically is of the order of three per cent. For systems with strong inhomogeneities and smaller inter-inclusion
distances, the error is larger.
1
Two-dimensional eective medium approximations for random circular inclusions predict
so-called xed points for the eective Poisson's ratio of two-phase composites in dierent
ways. One xed point relates to composites where both phases have the same Poisson's ratio.
If the phase Poisson's ratio is equal to 1/3, then the eective Poisson's ratio should also be
1/3 for any matrix Young's modulus and any volume fraction of the inclusions [6,10] and this
xed point will be a straight line in the Poisson's ratio ow diagram. Numerical calculations
with the digital image based method on random ber-reinforced composites [14] suggest
that this prediction may be a good approximation for a wide class of composites. Another
xed point for the eective Poisson's ratio relates to composites where one phase consists
of either circular voids or circular rigid inclusions. For such materials, when the inclusions
form a connected path (percolation), the eective medium approximation predicts that the
eective Poisson's ratio is independent of both the matrix Young's modulus and its Poisson's
ratio. The existence of this type of Poisson's ratio xed point for percolating voids has been
proved analytically [8,17] and demonstrated numerically [8,9], for any suspension of voids.
The precise value of the xed point, however, depends on the geometrical conguration [8].
For example, the xed point for a hexagonal array of circular voids is 1. Recent numerical
calculations [13] suggest that systems with rigid inclusions do not exhibit a xed point of
this type.
Asymptotic relations for the eective Poisson's ratio have been derived for periodic systems of circular voids [9] or circular rigid bers [13]. Those asymptotic relations are in good
agreement with numerical results [9,13].
Recently, one of us [24] presented a numerical algorithm for solving the elastostatic
equation of a two-dimensional, locally isotropic, two-component composite. For a hexagonal
array of circular bers the implementation of this algorithm is particularly simple. Machine
accurate solutions can be computed in a few seconds for both high degree of inhomogeneity
and small inclusion separation distances.
The purpose of this paper is to test the validity of xed point theorems and asymptotic
expressions for the eective Poisson's ratio when the bers are near touching. This is
accomplished by using the above mentioned algorithm to calculate detailed Poisson's ratio
ow diagrams from near homogeneity to the regime where asymptotic expressions become
valid, for hexagonal arrays of elastic circular bers.
The paper is organized as follows. Section II summarizes some results from linear elasticity and gives relations between dierent moduli. Section III describes our numerical algorithm, which we refer to as the interface integral equation method. Section IV presents Poisson's ratio ow diagrams and compares those to earlier results and predictions [9,10,13,14].
II. LINEAR ELASTICITY
The stress-strain relations for a linear elastic and isotropic material can be written as
1 [(1 + (d)) (d) ];
=
(1)
ij
ij
ij
E (d)
kk ij
= ((d) d2 (d))kk ij + 2(d) ij :
2
(2)
Here ij and ij are the strain and stress tensors and d is the dimension, which can be
two or three. Eq. (1) denes the Young's modulus E (d) and Poisson's ratio (d). Eq. (2)
denes the bulk modulus (d) and the shear modulus (d). Two-dimensional elasticity can be
considered as an approximation to either of two important special cases in three-dimensional
elasticity: plane-stress and plane-strain elasticity. When a material is in the form of a thin
sheet and subjected to planar loads, stress across its thickness can often be neglected. The
resulting, two-dimensional, elastostatic problem is called plane-stress. When a very thick
material that is translationally invariant in the direction of its thickness is subjected to
planar loads, the strain along the thickness can often be neglected. The resulting, twodimensional, elastostatic problem is called plane-strain. Plane-strain elasticity is physically
relevant when considering ber-reinforced materials subjected to loads across the bers.
The mappings from three-dimensional elastic moduli to the relevant two-dimensional
elastic moduli can be retrieved from Eqs. (1,2), using the assumptions in the two special
cases. The results are summarized in Table 1 [12].
In the following we will consider two-dimensional elasticity for ber-reinforced composites. The eective elastic two-dimensional moduli will be denoted by e , e , e and
Ee .
III. THE INTERFACE INTEGRAL EQUATION METHOD
Integral equation methods are by now standard for solving simple boundary value problems in locally isotropic elasticity. See Chen and Zhou [18] for a theoretical discussions
and a literature survey. Reasons why integral equation methods are often preferred over
other numerical schemes are related to storage requirements, accuracy, and discretization.
A numerical example of particular interest is the paper of Greenbaum, Greengard, and
Mayo [19]. The authors solve the elastostatic equation rapidly and to high accuracy using
iterative techniques and a biharmonic extension of the Fast Multipole Method [20{22]. A
recent numerical example involving inclusions is the work on Ostwald ripening by Jou, Leo,
and Lowengrub [23].
For some reason, integral equation methods have not been used much for solving homogenization problems in locally isotropic elasticity. We know of only two papers in the
literature. The rst papers is by Eischen and Torquato [11]. The authors use a standard
boundary element method for nite body problems, match boundary conditions at interfaces, and compute three-digit accurate eective moduli for some hexagonal arrays of bers.
The ratio of ber separation distance to ber radius is larger than 0.06. The other paper
is by one of us [24]. Here an integral equation for a periodic composite is explicitly written
down and is solved with Fourier methods. A simple formula for extracting eective properties from the solution is derived. This leads to a fast and accurate algorithm which we refer
to as the interface integral equation method. For moderate inhomogeneity we can compute
eective moduli for arbitrarily dense arrays in a few seconds on a simple workstation. The
two main reasons that the interface integral equation method is orders of magnitudes better
than the boundary element method in this case are 1) We only discretize the interfaces,
not the boundary of the unit cell. The periodic boundary conditions are treated with lattice sums [25]. 2) We use a quadrature rule with superalgebraic convergence, not low-order
accurate quadrature.
3
IV. RESULTS AND DISCUSSION
We have computed the eective Poisson's ratio e for hexagonal arrays of bers in a
matrix. Most calcualtions were for systems where the Poisson's ratios of the matrix 1 and
of the bers 2 were equal and the Young's moduli of the matrix E1 and bers E2 were
dierent. It follows immediately from the integral equation [24], that the eective Poisson's
ratio depends only on the stiness ratio E2=E1 and the Poisson's ratios of the phases. The
resulting Poisson's ratio ow diagrams are shown in Figures 1-4 with E2=E1 and 1 = 2 as
variable parameters. We have considered non-overlapping bers in hexagonal arrays which
means that
p the calculations of Poisson's ratio do not extend beyond the percolation threshold
p
f2 = =2 3.
First we calculated the eective Poisson's ratio for systems with 1 = 2 = 0:5 and
stiness ratios (E2=E1) = 2000 and (E2=E1) = 0:0005 which is approximately equivalent to
rigid inclusions and voids, respectively. Those structures have been studied before with other
methods [9,13]. Eective Poisson's ratio ow diagrams for those systems are shown together
with ow diagrams for other stiness ratios in Figure 1. Our results are in agreement with
existing graphical data for voids [9] (open circles) and rigid inclusions [13] (lled circles).
It has been suggested [14] that for random suspensions of inclusions and 1 = 2 = 1=3,
the eective Poisson's ratio is e = 1=3 for all ber fractions of inclusions f2 and all stiness
ratios E2=E1. Figure 2 illustrates that this does not hold for hexagonal arrays of bers.
Figure 3 shows Poisson's ratio ow diagrams for 1 = 2 = 0. Previous calculations on
periodic structures with the digital image based method [9,13] have only been possible with
1 = 2 1=3. The graphs with (E2 =E1 ) = 2000 and (E2 =E1 ) = 500 are almost on top of
each other. From gure 3 it is obvious that even when the Poisson's ratio is zero, a large
range of the eective Poisson's ratios can be obtained.
Figure 4 shows the Poisson's ratio ow diagrams for stiness ratio (E2=E1) = 100 and
with 1 = 2 varying from -0.5 to 0.5 with the step 0.1. This could be compared to the result
from eective medium approximations [10] and numerical calculations [10,14] on random
suspension of bers. The conclusion in those studies was stated as follows: when 1 = 2 >
1=3, the eective Poisson's ratio always decreases and is bounded below by 1/3 when two
phases are mixed. If 1 = 2 < 1=3 the value of e always increases and is bounded above
by 1/3 when two phases are mixed. Figure 4 shows that for hexagonal arrays of bers this
statement is valid at low ber fractions of bers when the stiness ratio is larger than one,
but invalid for moderate and high ber fractions of bers. For instance the graph with
1 = 2 = 0:4 is not bounded by e = 1=3. A similar Poisson's ratio ow diagram for
systems with stiness ratios less than one would violate this statement even more. Our
conclusion is that the stiness ratio has a large inuence on the eective Poisson's ratio for
periodic structures.
The parameter choice 1 = 2 = 1=3, suggested [10,14] as a xed point for random
systems of bers, plays a certain role even for this periodic composite structure. We have
investigated Poisson's ratio ow diagrams with various parameter choices and reached the
following conclusions: For every f2, there is a lowest number up(f2) and a highest number
low (f2) such that e < 1 = 2 whenever 1 = 2 > up(f2 ) and such that e > 1 = 2
whenever 1 = 2 < low (f2). In general up(f2), and low (f2) are quite dierent. Only in
the dilute limit, f2 < 0:2, do they coincide to up(0) = low (0) = 1=3. This conclusion is a
4
generalization of a suggestion by Garboczi and Day [14] that the eective Poisson's ratio e
satises 1 = 2 < e < 1=3 and 1 = 2 > e > 1=3 for any random geometry involving
overlapping and nonoverlapping bers.
For 1 = 2 = 1=3 the system with lowest stiness ratio has the highest eective Poisson's
ratio as can be seen in Fig. 2. Numerical calculations using the digital image based method
on random systems of circular bers [10] and 1 = 2 = 1=3, showed that the system with
lowest stiness ratio had the highest eective Poisson's ratio although the deviation from 1/3
was small for all considered stiness ratios and not as large as in our calculations. A general
statement for circular bers, supported by our calculations, would be that for 1 = 2 = 1=3
and given a structure, periodic or random, the system with the lowest stiness ratio has the
highest eective Poisson's ratio. For other values of 1 = 2, no general statement for the
eective Poisson's ratio can be made.
p
Chen,
p Thorpe and Davis [13] derived an asymptotic formula, near percolation (f2 =
=2 3) for the eective Poisson's ratio of a hexagonal array of rigid bers;
1 + 1 :
p
(3)
e =
7 1
We calculated the eective Poisson's ratio with the integral equation method for two systems
with ber fractions near percolation and high stiness ratio, (E2=E1) = 2000. The rst
system had the phase Poisson's ratios 1 = 2 = 0:5 and the second had the phase Poisson's
ratios 1 = 0:5, 2 = 0:2. Fiber fraction f2 = 0:90689 gave e = 0:245 for the rst system
and e = 0:181 for the second system. Fiber fraction f2 = 0:906899 gave e = 0:21 for
the rst system and e = 0:14 for the second system. Equation (3) gives ep = 0:23 with
1 = 0:5. The dierence between our results and the asymptotic result has to do with the
order in which we approach innitely rigid bers and touching bers. In the derivation of
Eq. (3) the bers are rst assumed to be innitely rigid, then touching is approached. Our
calculations show that large but nite values of the ber Young's modulus could give results
for the eective Poisson's ratio near percolation that dier signicantly from the asymptotic
relation Eq. (3). To our knowledge, no asymptotic expressions of uniform validity exist
that include more material parameters. See McPhedran, Poladian and Milton [26] for an
uniformly valid asymptotic formula in the context of electrostatics. Our results show that no
xed point is reached for rigid inclusions when percolation is approached. This conclusion
has also been reached by Chen, Thorpe and Davis [13] using the digital image based method
on a system of rigid circular bers.
This paper demonstrates the possibility of accurate calculations of the eective Poisson's
ratio for two-phase ber reinforced composites over the whole parameter space (fi; i; Ei)
(i = 1; 2), using the interface integral equation method. Our calculations show that a hexagonal array of circular bers does not exhibit Poisson's ratio xed points in any of the ways
discussed in the introduction. The only exception is of course voids in any two-dimensional
structure, which have been shown analytically [8,17] to exhibit Poisson's ratio xed points.
We also conclude that the stiness ratio of the phases has a larger inuence on the eective
Poisson's ratio for periodic systems, than can be seen in the existing results [10,14] for random systems. A general statement about the eective Poisson's ratio for circular bers in
random or periodic structures is made. The validity of existing asymptotic relations for the
eective Poisson's ratio near percolation is investigated.
5
ACKNOWLEDGMENTS
We would like to acknowledge useful discussions with L. Gibiansky, G. W. Milton, A. B.
Movchan and M. F. Thorpe. HC gratefully acknowledges support from the Swedish research
councils NFR and NUTEK and the Goran Gustafsson foundation.
6
REFERENCES
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[5] L. M Schwartz, S Feng, M. F. Thorpe and P. N. Sen, Phys. Rev. B 32, 4607 (1985).
[6] M. F. Thorpe and P. N. Sen, J. Acoust. Soc. Am. 77, 1674 (1985).
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[8] M. F. Thorpe and I. Jasiuk, Proc. R. Soc Lond. A 438, 531 (1992).
[9] A. R. Day, K. A. Snyder, E. J. Garboczi and M. F. Thorpe, J. Mech. Phys. Solids 40,
1031 (1992).
[10] K. A. Snyder, E. Garboczi and A. R. Day, J. Appl. Phys. 72, 5948 (1992).
[11] J. W. Eischen and S. Torquato, J. Appl. Phys. 74, 159 (1993).
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[13] J. Chen, M. F. Thorpe and L. C. Davis, J. Appl. Phys. 77, 4349 (1995).
[14] E. J. Garboczi and A. R. Day, J. Mech. Phys. Solids 43, 1349 (1995).
[15] L. C. Davis, K. C. Hass, J. Chen and M. F. Thorpe, Appl. Mech. Rev. 47, 5 (1994).
[16] J. Berryman, J. Acoust. Soc. Am. 68, 1820 (1980).
[17] A. V. Cherkaev, K. A. Lurie and G. W. Milton, R. Proc. Soc. A 438, 519 (1992).
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[20] V. Rokhlin, J. Comput. Phys. 60, 187 (1985).
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[22] J. Carrier, L. Greengard and V. Rokhlin, SIAM J. Sci. and Stat. Comput. 9, 669 (1988).
[23] H. J. Jou, P. H. Leo and J. S. Lowengrub, submitted to J. Comp. Phys. (1995).
[24] J Helsing, J. Mech. Phys. Solids 43, 815 (1995).
[25] L. Berman and L. Greengard, J. Math. Phys. 35, 6036 (1994).
[26] R. C. McPhedran, L. Poladian and G. W. Milton, Proc. R. Soc. Lond. A 415, 185
(1988).
7
1.0
holes
rigid incl.
νeff
E2/E1=0.0005
E2/E1=0.002
E2/E1=0.01
E2/E1=0.05
0.5
E2/E1=20
E2/E1=100
E2/E1=500
E2/E1=2000
0.0
0.0
0.5
1.0
f2
FIG. 1 Eective Poisson's ratio e versus the ber fraction f2 for a hexagonal array of
circular bers. The phase Poisson's ratios are 1 = 2 = 0:5. The dashed lines have stiness
ratios (E2=E1) < 1. The exact stiness ratios are listed to the right and the top value in
the column is associated with the dashed curve with largest maximum value. The solid lines
have stiness ratios (E2=E1 ) > 1. The corresponding stiness ratios are listed to the right
and the top value in the column is associated with the solid curve with smallest minimum
value. Earlier results for holes [9] (open circles) and rigid inclusions [13] (lled circles) are
included.
8
1.0
E2/E1=0.0005
E2/E1=0.002
E2/E1=0.01
E2/E1=0.05
νeff
0.5
E2/E1=20
E2/E1=100
E2/E1=500
E2/E1=2000
0.0
0.0
0.5
1.0
f2
FIG. 2 Eective Poisson's ratio e versus the ber fraction f2 for a hexagonal array of
circular bers. The phase Poisson's ratios are 1 = 2 = 1=3. The dashed lines have stiness
ratios (E2=E1) < 1. The solid lines have stiness ratios (E2=E1) > 1. The corresponding
stiness ratios are listed to the right in the same way as in Fig. 1.
9
1.0
E2/E1=0.0005
E2/E1=0.002
E2/E1=0.01
E2/E1=0.05
νeff
0.5
E2/E1=2000
E2/E1=500
E2/E1=100
E2/E1=20
0.0
0.0
0.5
1.0
f2
FIG. 3 Eective Poisson's ratio e versus the ber fraction f2 for a hexagonal array of
circular bers. The phase Poisson's ratios are 1 = 2 = 0. The dashed lines have stiness
ratios (E2=E1) < 1. The solid lines have stiness ratios (E2=E1) > 1. The corresponding
stiness ratios are listed to the right in the same way as in Fig. 1.
10
0.5
νeff
0.0
-0.5
0.0
0.5
1.0
f2
FIG. 4 Eective Poisson's ratio e versus the ber fraction f2 for a hexagonal array of
circular bers. The phase Poisson's ratios are 1 = 2 and take the values -0.5 to 0.5 with
step 0.1. The stiness ratio is (E2=E1) = 100.
11
V. TABLES
TABLE I. The connections between the two-dimensional
elastic moduli and the three-dimensional elastic moduli
in the plane-strain condition and the plane-stress condition.
plane-strain elasticity
(2)
(2)
plane-stress elasticity
(3)
= 1 (3)
(3)
E (2)
= 1 E( (3))2
= (1
E (3)
(3))(1
(2)
= 2(1E+ (3))
(3)
2 (3))
(2)
= (3)
E (2)
= E (3)
(3)
(2)
= 2(1E
(2)
= 2(1E+ (3))
(3))
(3)
12