Page 370 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

BIOMEDICAL COMPOSITES 347

The rule of mixtures equations have several drawbacks. The isostrain assumption in the Voight
model implies strain compatibility between the phases, which is very unlikely because of different
Poisson’s contractions of the phases. The isostress assumption in the Reuss model is also unrealistic
since the fibers cannot be treated as a sheet. Despite this, these equations are often adequate to
predict experimental results in unidirectional composites. A basic limitation of the rule of mixtures
occurs when the matrix material yields, and the stress becomes constant in the matrix while continuing
to increase in the fiber.
∗
The ultimate tensile strength of a fibrous composite s depends on whether failure is fiber-
c
dominated or matrix-dominated. The latter is common when V is small. One result of such treatment is
f
σ *
*
*
σ = m EV + σ V (14.6)
c
f 1
m m
f
E
m
∗
where s is the fracture strength of the matrix.
m
Other results from this simple analytical approach for orthotropic composites are
α V
E 1f α V + E m m m
1f
f
α = (14.7)
1c
EV + E V
f
m m
1 f
1
C = ρ ( f C V + ρ m m m (14.8)
C V )
c
f
f
ρ
c
K = K V + K V (14.9)
c 1
m m
f
f
where a is the coefficient of thermal expansion, C is the specific heat, and K is the thermal conduc-
tivity. The coefficient of hygroscopic expansion b can be found by substituting a with b above.
These results are for the longitudinal directions only.
14.5.2 Theory of Elasticity Models
In this approach, no assumptions are made about the stress and strain distributions per unit volume.
The specific fiber-packing geometry is taken into account, as is the difference in Poisson’s ratio
between the fiber and matrix phases. The equations of elasticity are to be satisfied at every point in
the composite, and numerical solutions generally are required for the complex geometries of the
representative volume elements. Such a treatment provides for tighter upper and lower bounds on
the elastic properties than estimated by the rule of mixtures, as is described in the references used in
this section.
One illustrative result for the longitudinal Young’s modulus is
2 [( v − v ) 2 EVE V ]
f
m
f
f
m m
E = + E m + E ( f − E V ) f (14.10)
{ EV 1− ( v − v 2 ) + E ⎡ V 1− ( 1 v m − v m 2 1− v ) ⎤ ⎦ }
c 1
m
2
2 ) + (
f ⎣
mm
m
f
f
f
where v and v are the Poisson’s ratios of the fiber and matrix, respectively. Very small differences
f m
in the Poisson’s ratios of the phases cause such equations to be simplified to the rule of mixtures.
14.5.3 Semiempirical Models
Curve-fitting parameters are used in semiempirical and generalized equations to predict experimen-
tal results. The most common model was developed by Halpin and Tsai, and it has been modified for
aligned discontinuous fiber composites to produce such results as the following for the longitudinal
modulus: