Composite Biomaterials - Biomaterials - page 84

is only one nonzero strain value; there is no lateral strain. A modulus tensor with 21 independent ele-
ments describes a triclinic crystal, which is the least symmetric crystal form. The unit cell has three dif-
ferent oblique angles and three different side lengths. A triclinic composite could be made with groups
of fibers of three different spacings, oriented in three different oblique directions. Triclinic modulus
elements such as C 2311 , known as cross-coupling constants, have the effect of producing a shear stress in
response to a uniaxial strain; this is undesirable in many applications. An orthorhombic crystal or an
orthotropic composite has a unit cell with orthogonal angles. There are nine independent elastic moduli.
The associated engineering constants are three Young's moduli, three Poisson's ratios, and three shear
moduli; the cross-coupling constants are zero when stresses are aligned to the symmetry directions.
An example of such a composite is a unidirectional fibrous material with a rectangular pattern of fibers
in the cross-section. Bovine bone, which has a laminated structure, exhibits orthotropic symmetry, as
does wood. In a material with hexagonal symmetry, out of the nine C elements, there are five indepen-
dent elastic constants. For directions in the transverse plane, the elastic constants are the same, hence
the alternate name transverse isotropy. A unidirectional fiber composite with a hexagonal or random
fiber pattern has this symmetry, as does human Haversian bone. In cubic symmetry, there are three
independent elastic constants: a Young's modulus, E , a shear modulus, G , and an independent Poisson's
ratio, ν. Cross-weave fabrics have cubic symmetry. Finally, an isotropic material has the same material
properties in any direction. There are only two independent elastic constants, hence E , G , ν, and also
the bulk modulus B are related in an isotropic material. Isotropic materials include amorphous solids,
polycrystalline metals in which the grains are randomly oriented, and composite materials in which the
constituents are randomly oriented.
Anisotropic composites offer superior strength and stiffness in comparison with isotropic ones.
Material properties in one direction are gained at the expense of properties in other directions. It is sen-
sible, therefore, to use anisotropic composite materials only if the direction of application of the stress
is known in advance.

4.4 Particulate Composites

It is often convenient to stiffen or harden a material, commonly a polymer, by the incorporation of par-
ticulate inclusions. The shape of the particles is important (see Christensen, 1979). In isotropic systems,
stiff platelet (or flake) inclusions are the most effective in creating a stiff composite, followed by fibers, and
the least effective geometry for stiff inclusions is the spherical particle, as shown in Figure 4.3. A dilute
concentration of spherical particulate inclusions of stiffness E i and volume fraction V i , in a matrix (with
Poisson's ratio assumed to be 0.5) denoted by the subscript m, gives rise to a composite with a stiffness E :

5
32

( EEV

EE

− )

i mi

E = + E (4.4)

+ ( )

m

/

i m

The stiffness of such a composite is close to the Hashin-Shtrikman lower bound for isotropic com-
posites. Even if the spherical particles are perfectly rigid compared with the matrix, their stiffening
effect at low concentrations is modest. Conversely, when the inclusions are more compliant than the
matrix, spherical ones reduce the stiffness the least and platelet ones reduce it the most. Indeed, soft
platelets are suggestive of crack-like defects. Soft platelets therefore result not only in a compliant com-
posite, but also a weak one. Soft spherical inclusions are used intentionally as crack stoppers to enhance
the toughness of polymers such as polystyrene (high-impact polystyrene), with a small sacrifice in
stiffness.
Particle reinforcement has been used to improve the properties of bone cement. For example, inclu-
sion of bone particles in polymethyl-methacrylate (PMMA) cement somewhat improves the stiff-
ness and improves the fatigue life considerably (Park et al., 1986). Moreover, the bone particles at the