Page 369 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

346 BIOMATERIALS

representative volume element of the composite. This approach results in the common rule of mixtures
equations for composites, where properties are relative to the volume fraction of the fibers and matrix.
Physical properties such as density are easily calculated by the following equations:
V + V + V = 1 (14.1)
f m v
r = r V + r V (14.2)
c f f m m
where V , V , and V are the volume fractions of the fiber, matrix, and voids, respectively, and sim-
f m v
ilarly, r , r , and r are densities of the composite, fiber, and matrix.
c f m
The rule of mixtures is useful in roughly estimating upper and lower bounds of mechanical prop-
erties of an oriented fibrous composite, where the matrix is istropic and the fiber orthotropic, with
coordinate 1 the principal fiber direction and coordinate 2 transverse to it. For the upper bound, the
Voight model is used (Fig. 14.3), where it is assumed that the strain is the same in the fiber and
matrix. For the lower bound, the Reuss model is used, where the stress is assumed to be the same.
This gives the following equations for composite moduli:
E = E V + E V (14.3)
1c 1f f m m
1 V f V m
= + (14.4)
E 2c E 2f E m
1 V f V m
= + (14.5)
G 12c G 12 f G m
where E and G are the Young’s modulus and shear modulus, respectively. The equations for trans-
verse modulus and shear modulus are known as the inverse law of mixtures. Some fibers, such as
carbon, have different properties along their longitudinal and transverse axes that the preceding
equations can take into account.

FIGURE 14.3 Composite stress models: (a) Voight or isostrain; (b) Reuss or isostress. Arrows indicate ten-
sion force direction.