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METALL-FORSCHUNG
β, γ and δ are material non-homogeneous
parameters that present the continuous
variation of the material properties
along the x-direction.
Exponential functions for representation
of the material properties usually
facilitate the analytical solution but do
not give real representation for material
properties, except for the upper and lower
surfaces of the FGM, where the effect
of the of volume fractions distributions
are not taken into account.
The most realistic way for representation
of the continuous gradation of the
material properties of FGM is the volume
fraction and rules of mixture. The
use of volume fraction and rules of mixture
overcome the drawbacks of exponential
functions for representation of
the material properties. Unfortunately,
the use of the volume fraction and rules
of mixture make the analytical solution
of the FGM problems very complicated
and may be impossible. The use of the
finite element method in such problems
is the most effective tools and overcomes
such difficulties. Many authors, see [7,
8, 9], analyzed FGM problems, in different
cases, using the volume fractions and
rules of mixture.
Volume fraction and rule of
mixtures of FGM
Consider a plate of FGM with porosity
that functionally graded from ceramic
and metal. The volume fraction of the
metal, the volume fraction of ceramic
and the porosity are V m , V c and p respectively.
The volume fractions of FGM
with porosity can be represented as:
V m = (z/t) m , (1)
238
V c = (1-V m ) (2)
Where z and t are Cartesian coordinate
and plate thickness, respectively. In addition,
m is a constant that represent
the composition variations between ceramic
and metal along z-axis. If m=1 the
variation of the composition of ceramics
and metal is linear. The composition is
ceramic-rich when m>1 and ceramicpoor
when m