Biomedical Engineering Reference
In-Depth Information
Solution : The formula
PL / AE will be applied in two ways to the bar with
axially aligned cylindrical cavities illustrated in Fig. 7.3 and subjected to an axial
force P . First apply the formula to the matrix material imagining a reduced cross-
sectional area A m not containing the voids, thus E m ¼
d ¼
PL / A m d
. Next apply the
formula to the entire bar containing the voids, thus E eff
3
. The relationship
between the cross-sectional total area A and the cross-sectional area A m occupied
by the matrix material is A m ¼
¼
PL
=
A
d
A (1 -
f c ). The desired result is established by
eliminating P , L ,
d
, A , and A m between these three formulas.
Problems
7.4.1 Calculate the effective bulk modulus, K eff , shear modulus, G eff , and Young's
modulus, E eff , for a composite material consisting of a steel matrix material
with spherical inclusions that contain water. The spherical inclusions, which
have a radius r , are contained within unit cells that are cubes with a dimension
of 4 r . The Young's modulus of steel is 200 GPa and the shear modulus of
steel is 77 GPa. The bulk modulus of water is 2.3 GPa.
7.4.2 Calculate the effective bulk modulus, K eff , shear modulus, G eff , and Young's
modulus, E eff , for a composite material consisting of a magnesium matrix
material and spherical inclusions. The spherical inclusions are made of steel,
have a radius r , and are contained within unit cells that are cubes with a
dimension of 5 r . The Young's modulus of steel (magnesium) is 200 GPa
(45 GPa) and the shear modulus of steel (magnesium) is 77 GPa (16 GPa).
7.4.3 Calculate the effective moduli for a composite material consisting of a steel
matrix material and cylindrical voids of radius r contained in unit squares 8 r
by 8 r . The Young's modulus of steel is 77 GPa and its Poisson's ratio is 0.33.
7.4.4 Considering the effective moduli for a composite material consisting of a
matrix material and aligned cylindrical voids, show that for small values of
the volume fraction the in-plane effective shear moduli and the in-plane
effective Young's moduli decrease more rapidly with increasing porosity
than the out-of-plane effective Young's moduli. For simplicity consider the
case when
n m is 1/3.
7.5 Effective Permeability
In this section the effective axial permeability of the bar with axially aligned
cylindrical cavities illustrated in Fig. 7.3 is calculated. The method used is the
simplest one available to show that Darcy's law is a consequence of the application
of the Newtonian law of viscosity to a porous medium with interconnected pores.
The Navier-Stokes equations (6.37) are a combination of the Newtonian law of
viscosity (5.36 N) and the stress equations of motion (3.37), as was shown in the
previous chapter. Thus one can say that Darcy's law is a consequence of the
application of either the Newtonian law of viscosity, or the Navier-Stokes
equations, to a porous medium with interconnected pores. There are a number of
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