Limits on Unidirectional Composites

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Journal of Composite

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DOI: 10.1177/002199836900300418

1969 3: 728Journal of Composite MaterialsH.E. Brandmaier

Limits on Unidirectional Composite Material Properties

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Limits on Unidirectional

CompositeMaterial

Properties

H. E. BRANDMAIER, Curtiss-Wright Corporation

,

Wood-Ridge

,

N. J. 07075

(ReceivedAugust 26, 1969)

This note presents some interesting properties of unidirectional composite

materials deduced from a study of the maxima, minima, and zeros of the trans-formation equations for the compliance matrix elements. Similar equations forthe stiffness matrix elements were given by Tsai and Pagano [1]. Equations for

the compliance matrix elements in tabular form are:

where

As an example of the use of this table

The possible variation of these elements with 0 is constrained by the invariants

where PI and P, are constants. Thus Si2 is directly related to S~,. The behavior of

S~6 and S._,., is similar to Sis and S~I and will not be considered further.The first derivative of equation (1) with respect to 0 is,

or, using equation (1),

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which is zero at 0, 7r/2, and 0 defined by,

The second derivative is

For most composite materials, E1, > E22 > G12 or S11 < S22 < S66- Therefore

Sz2- Sss+ 511 - 2512 < 0 and 0 < a < 1.As a result the second derivative is positive

and S11 is a minimum at 0 and 1T/2. Under these conditions 5~1 is a maximum at

4 < B ~ ~ since from equation (6), 0 > cos 2 ~ > -l. Near the lower 0 limit either511 ~ S22 which is not satisfied by composite materials or 566 ~ S22 + 511 - 2SI2,while at the upper limit See = 2 (S22 - S~2). Substituting these limits into equation

(2) shows that the maximum value of S11 varies between about 4 S66 and S22.Using the same approach, S~6 and 5~2 are maxima at 0 and ~r/2 and minima at

~r/4 for 0 < a < 1. The 7r/4 values are,

which is independent of 566 and

Differentiating S,6 with respect to 0 yields

and

Maxima or minima thus occur at 0 defined by,

Since 4 cos 2a+ a=+2 ~(~r +2 , equation (11) shows that the + in equation (12)Since 4 cos 2a+a=2 B/(~) +2, equation (11) shows that the + in equation (12)defines a maximum for which 0 < 7T/4 while the - defines a minimum for which

0 > 4 . Considering that 0 < a < 1, equation (12) further limits the location of the

maximum to the narrow range, g < 0 * 6 and the location of the minimum to the



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Figure 1. Variation of E/E22 with e and

E/Ezz.Figure 2. Variation of shear couplingconstant with 8 and Ell/E22.

somewhat broader range, 3 g < 8 ~ 2 . The corresponding S,, range is 4 S66 to

3B~3.

1..4522 for the maximum and - 4&dquo; 566 to 0 for the minimum. Except at the lower 0

limit the maximum will be greater in magnitude than the minimum. Comparingthe minima and maxima conditions for 5~1 and Sis further reveals that:

1. From equation (5), S16 is zero at the angles at which 5~1 is a maximum or aminimum.

2. From equation (10) inflection points (zero second derivative) occur in the

5~1 curve at the angles at which Sis is a minimum and/or maximum.

3. The magnitude of the peak values of 5~h Siz, and 5~6 are all about4 I S66 whichoccurs at very small values of G12.



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Figure 1 illustrates the effect of increasing Ell, due to the development of

stiffer filaments, on the behavior of Youngs modulus E = 1/Sil; the fact that

E= E11 at 0 = 0 identifies each curve. Typical values of Ell, E22, and v12 were

assumed. Over much of the 0 range E is relatively insensitive to Ell and at-

G1,/E2~ - G12 from equation (6) for E11 > E~2.As &dquo;a&dquo; increases the minimum

decreases and shifts to lower values of 0. Figure 2 gives corresponding data for

the shear coupling constant which relates the shear strain and the longitudinalstrain in the same manner that Poissons ratio relates the transverse and longi-tudinal strains. While Poissons ratio is less than 1 for the chosen parameters,

the peak shear coupling constant becomes very large as E11 increases.As E22/G12

increases, the magnitudes of the minimum and maximum increase for a givenvalue of E11/E22.

REFERENCE

1. S. W. Tsai and N. J. Pagano, "Invariant Properties of Composite Materials" CompositeMaterials Workshop, ed. by S. W. Tsai, J. C. Halpin, and N. J. Pagano, TechnomicPublishing Company, Inc., Stamford, Conn., (1968), p. 233.


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