Page 371 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

348 BIOMATERIALS

E c 1 = 1+ξη V f
E m 1−η V f (14.11)

EE / m −1
f 1
where η = (14.12)
EE / m + η
f 1
and the Halpin-Tsai curve-fitting parameter is assumed to be x = 2L/d, where L is the length and d
the diameter of the fiber. The expression can be simply substituted to also obtain E and G using
2c 12
experimental values for x.
For two-dimensional randomly oriented fibers in a composite, approximating theory of elasticity
equations with experimental results yielded this equation for the planar isotropic composite stiffness
and shear modulus in terms of the longitudinal and transverse moduli of an identical but aligned
composite system with fibers of the same aspect ratio:
(14.13)
E = 3 / 8 E + 5 / 8 E
c 1 2
(14.14)
G = 1 / 8 E + 1 / 4 E
c 1 2
For a three-dimensional random orientation of fibers, a slightly different equation is proposed
for the isotropic tensile modulus:
(14.15)
E = 1 / 5 E + 4 / 5 E
c 1 2
The stiffness of particulate composites can be predicted depending on the shape of the particles. 6
For a dilute concentration of rigid spherical particles, the composite stiffness is approximated by
5( E − E V
)
E = p m p + E m (14.16)
c
+
32 EE m
/
p
where E and V are the stiffness and volume fraction of particles, respectively.
p p
It is important to note that any model for composite behavior requires experimental validation
and may prove to be quite inaccurate for not taking into account many irregularities typical in
composite design and processing. In addition, these results are usually valid for static and short-term
loading.
14.6 FRACTURE AND FATIGUE FAILURE

Failure of fiber-reinforced composites is generally preceded by an accumulation of different types of
internal damage that slowly or catastrophically renders a composite structure unsafe. The damage
can be process-induced, such as nonuniform curing control of a dental resin, or it can be service-
induced, such as undesired water absorption. Fiber breaking, fiber bridging, fiber pullout, matrix
cracking, and interface debonding are the failure mechanisms common in all composites, but the
sequence and interaction of these mechanisms depend on the type of loading and the properties of
the constituents, as well as on the interfacial shear strength (Fig. 14.4). Various fracture mechanics
theories are available for failure analysis of composites, among them the maximum stress theory
and the maximum strain theory described in more detail in Ref. 5.
Energy absorption and crack deflection during fracture lead to increased toughness of the
composite. The two most important energy-absorbing failure mechanisms in a fiber-reinforced
composite are debonding at the fiber-matrix interface and fiber pullout. If the interface bonds
relatively easily, the crack propagation is interrupted by the debonding process, and instead of moving
through the fiber, the crack is deflected along the fiber surface, allowing the fiber to carry higher