Fundamentals of Materials Science and Engineering: An Integrated Approach, 3e

GTBL042-15 GTBL042-Callister-v2 August 29, 2007 8:52

15.5 Influence of Fiber Orientation and Concentration • 629

whereAm/AcandAf/Acare the area fractions of the matrix and fiber phases, re-

spectively. If the composite, matrix, and fiber phase lengths are all equal,Am/Acis

equivalent to the volume fraction of the matrix,Vm, and likewise for the fibers,Vf=

Af/Ac. Equation 15.6 now becomes

σc=σmVm+σfVf (15.7)

The previous assumption of an isostrain state means that

c=m=f (15.8)

and when each term in Equation 15.7 is divided by its respective strain,

σc

c

=

σm

m

Vm+

σf

f

Vf (15.9)

Furthermore, if composite, matrix, and fiber deformations are all elastic, thenσc/c=

Ec,σm/m=Em, andσf/f=Ef, theE’s being the moduli of elasticity for the respec-

tive phases. Substitution into Equation 15.9 yields an expression for the modulus of

elasticity of a continuous and aligned fibrous compositein the direction of alignment

(orlongitudinal direction),Ecl,as

For a continuous and Ecl=EmVm+EfVf (15.10a)

aligned fiber-

reinforced composite,

modulus of elasticity

in the longitudinal

direction

or

Ecl=Em(1−Vf)+EfVf (15.10b)

since the composite consists of only matrix and fiber phases; that is,Vm+Vf=1.

Thus,Eclis equal to the volume-fraction weighted average of the moduli of

elasticity of the fiber and matrix phases. Other properties, including density, also

have this dependence on volume fractions. Equation 15.10a is the fiber analogue of

Equation 15.1, the upper bound for particle-reinforced composites.

It can also be shown, for longitudinal loading, that the ratio of the load carried

by the fibers to that carried by the matrix is

Ff

Fm

=

EfVf

EmVm

(15.11)

Ratio of load carried

by fibers and the

matrix phase, for

longitudinal loading

The demonstration is left as a homework problem.

EXAMPLE PROBLEM 15.1

Property Determinations for a Glass Fiber-Reinforced

Composite—Longitudinal Direction

A continuous and aligned glass fiber-reinforced composite consists of 40 vol%

of glass fibers having a modulus of elasticity of 69 GPa (10× 106 psi) and 60

vol% of a polyester resin that, when hardened, displays a modulus of 3.4 GPa

(0.5× 106 psi).