Functionally Graded Materials in Engineering - Structural Interfaces and Attachments in Biology - page 26

In conclusion, we outlined fundamental relations involved in the analysis of
FGM structures and illustrated the major difference between cases of in-surface and
through-the-thickness grading. Two approaches to the micromechanical problem
have been identified: a direct approach where the properties, such as the tensor of
stiffness, are determined based on the prescribed volume fraction, orientation, and
shape of the constituent phases and the reverse approach where the latter volume
fractions and geometry are specified based on the prescribed tensor of stiffness.
2.4 Representative Recent Studies of FGM in Engineering
In this section we refer to recent studies in various areas of FGM oriented toward
engineering applications. While several comprehensive reviews are cited above, we
concentrate here on relatively recent representative additions to the field. The
problems anticipated in biomedical and biological applications being mostly lim-
ited to a narrow range of temperatures, we eliminate the reference to heat transfer
problems and to the effects of temperature on stresses and properties of FGM.
2.4.1 Homogenization of FGM
The homogenization bamboo, a natural FGM, was undertaken using graded finite
elements and accounting for a continuous variation of properties through the bam-
boo wall [ 21 ]. A stochastic micromechanical solution utilizing a Mori-Tanaka
approach to the evaluation of probabilistic properties of an isotropic FGM was
considered in [ 22 ] accounting for uncertainties in constituent material properties
and their volume fraction. The outcome included the mean and standard deviation of
the elastic modulus and Poisson ratio as well as their probability density functions.
The finite-volume theory was developed for modeling of a graded material
microstructure with arbitrary shaped cross sections and subsequently applied to
the analysis of the conductivity and stiffness matrices [ 23 , 24 ]. It was found that the
proposed approach was competitive with a finite element method.
An additional consideration that may become essential in the micromechanics of
FGM with small size RVE or with steep gradients is nonlocal elasticity. Such issues
are important if a classical continuum model is not well suited to modeling of the
material because of small-scale effects [ 25 ]. The concept of nonlocal elasticity is
based on the presumption that the stress at a point depends both on the strain at the
same point as well as on the strains at other points within the body. As a result, the
internal characteristic length, such as the lattice parameter, can be incorporated into
constitutive equations. Nonlocal elastic models have been proposed and applied to
modeling composite structures incorporating nanotubes, but they may also be useful
for the characterization of FGM. These models incorporate the features of both the
classical continuum model as well as the internal small-scale length effects.