1

Halpin-Tsai Equations

SE 253A Mechanics of Laminated Composite Structures

Lecture Supplemental Pack 3

Instructor: Prof. Hyonny Kim

Department of Structural Engineering

University of Califomia, San Diego

1

Halpin- Tsai Equations

Longitudinal Modulus

El == cfEf +cmEm

Major Poisson's Ratio

M

224

8 20 ••crx

E216

I For E2 e =2

-Em123

where:

M = composite values of E2' G12, and V23

Mf = fiber values of E2' G12, and V23

Mm = matrix values of E2' G12, and V23

~ = geometry factor for E2' G12, and v23

- function of fiber geometry, packing arrangement, and loading conditions

- found by Elasticity solutions, and by fitting 11 to other solutions (e.g., FEA of RVEs)

Original reference citation: John C. Halpin and Stephen W. Tsai, "Effects of Environmental Factors on Composite Materiais," AFRL-TR-67 -423, June 1969.

Additional reference: J.C. Halpin and J. L. Kardos, "The Halpin-Tsai Equations: A Review," Polymer Engineering and Science, 16(5), 1976, pp. 344-352.

Following figures in this package extracted from: Robert M. Jones, Mechanics of Composite MateriaIs, 2nd Ed., Taylor & Francis, 1999. 2

FIIER VOLUME (V l) [FIIER SPACING (8/r»

78%

[0.01

8

4

4

o~--~--~~~~--~--~~~~--~----~~~ 1 2 4 6 8 10 20 40 60 100 200 400 600 1000

E,/Em Figure 3-32 Halpin-Tsai Calculations (Circles) versus Adams and Doner's

Calculations for E2 of Circular Fibers in a Square Array (After Halpin and Tsai [3-17])

3

:ê Cf)o

~ ü05 'õ~ Cf)"-

.S2 Cf)-C; .(llÜ

-5 ~ Q) •....~LL

_06 Cf)

••••.Q) O§">-:~~ -0-t:l.i..JQ)-o..o C;

&C\i

E .!!2-

Figure 3-33 Halpin- Tsai Calculations (Circles) versus Adams and Doner's ~ j Calculations for G12 of Circular Fibers in a Square Array ~~(After Halpin and Tsai [3-17])

flBER VOLUME (Vt> p=IBER SPAC INO ( 8/r)]

5

24

8

2

4 6 8 10 40 60 100

~

Gm

400 600 1000 20 200

4

35~----------------------

6

25

E2 106 psi 20

15

10

5

I BORON-EPOXY ~

Ef = 60 X 106 psi (410 GPa) Em = .5 x 106 psi(35 GPa)

vf = .2 v m = .34

200

150 E2 GPa

100

50

'- a_1b- O O

O .2 .4 .6 .8 1 FIBER-VOLUME FRACTION, Vf

Figure 3-34 Halpin-Tsai Calculations (Circles) versus Foye's Calculations for E2 of Rectangular Cross-Section Fibers in a Diamond Array (After Halpin and Tsai [3-17])

~=2~ b

5

Figure 3-35 Halpin-Tsai Calculations (Circles) versus Foye's Calculations for G12 of Rectangular Cross-Section Fibers in a Diamond Array (After Halpin and Tsai [3-17])

6

Model

E1 (GPa) E2 (GPa) V12 G12 (GPa)

Voigt 41.8 41.8 0.22 17.0

Reuss 7.35 7.35 0.34 2.74

Hybrid VR 41.7 8.21 0.28 2.74

Sq. Fiber 41.7 10.8 0.26 3.64

Halpin-Tsai 41.7 13.1 0.28 3.93

(round, sq. array)

Test Oata* 39 8.6 0.28 3.8

Sample Calculation - Halpin- Tsai

Gf and Gm approximated by E

2(1 + v)

For round fibers is square array, geometry factor for E2 and G12 are: ~E2 = 2

~G12 = 1

• Glass Fibers:

Ef = 73 GPa, vf = 0.22, c, = 0.55

• Epoxy Matrix:

Ef = 3.5 GPa, Vf = 0.35, c, = 0.45

Elastic properties calculated to be:

E1 = 41.7 GPa

E2=13.1 GPa

G12 = 3.93 GPa

V12 = 0.28

7

Example Calculation - Comparison to Other Models

• Glass Fibers:

Ef = 73 GPa, vf = 0.22, c, = 0.55

• Epoxy Matrix:

Ef = 3.5 GPa, Vf = 0.35, c, = 0.45

Gf and Gm approximated by E

2(1 + v)

* test data from Daniel & Ishai textbook for c, = 0.55

8