Composite Biomaterials - Biomaterials - page 83

250

Composite stiffness

200

Voigt,
aligned fiber

150

Dilute
random fiber,
3-D

HS
upper

100

Dilute
particle

50

HS lower

Dilute platelet

Reuss

0

0 0.2 0.4

Volume fraction

0.6 0.8 1

FIGURE 4.3 Stiffness versus volume fraction for Voigt and Reuss models, as well as for dilute isotropic suspen-
sions of platelets, fibers, and spherical particles embedded in a matrix. Phase moduli are 200 and 3 GPa.

The Reuss stiffness E , represented by

− 1

=+ −

V
E

1 V

E

i
i

i

(4.2)

E

m

is less than that of the Voigt model. The Voigt and Reuss models provide upper and lower bounds,
respectively, upon the stiffness of a composite of arbitrary phase geometry (Paul, 1960). The bounds are
far apart if, as is common, the phase moduli differ a great deal, as shown in Figure 4.3. For composite
materials which are isotropic, the more complex relations of Hashin and Shtrikman (1963) provide
tighter bounds upon the moduli (Figure 4.3); both the Young and shear moduli must be known for each
constituent to calculate these bounds.

4.3 Anisotropy of Composites

Observe that the Reuss laminate is identical to the Voigt laminate, except for a rotation with respect to
the direction of load. Therefore, the stiffness of the laminate is anisotropic , that is, dependent on direc-
tion (Lekhnitskii, 1963; Nye, 1976; Agarwal and Broutman, 1980). Anisotropy is characteristic of com-
posite materials. The relationship between stress σ ij and strain ε kl in anisotropic materials is given by the
tensorial form of Hooke's law as follows:

3 3

∑ 1

σ = C ε (4.3)

ij ijkl kl

k = 1 l =

Here C ijkl is the elastic modulus tensor. It has 3 4 = 81 elements; however, since the stress and strain are
represented by symmetric matrices with six independent elements each, the number of independent
modulus tensor elements is reduced to 36. An additional reduction to 21 is achieved by considering elas-
tic materials for which a strain energy function exists. Physically, C 2323 represents a shear modulus since
it couples a shear stress with a shear strain. C 1111 couples axial stress and strain in the 1 or x -direction,
but it is not the same as Young's modulus. The reason is that Young's modulus is measured with the lat-
eral strains free to occur via the Poisson effect, while C 1111 is the ratio of axial stress to strain when there