Functionally Graded Materials in Engineering - Structural Interfaces and Attachments in Biology - page 23

Equation ( 2.10 ) can often be simplified, since the matrices of extensional, coupling,
and bending stiffnesses are not fully populated (in other words, some of the
elements of these matrices are equal to zero). The matrices are presented
for isotropic, specialty orthotropic and generally orthotropic composite laminates
in every textbook on composite materials (e.g., [ 19 ]). In the case of FGM the
properties can vary in three mutually perpendicular directions oriented along the
coordinate axes. If the variations of properties within each RVE are negligible, i.e.,
we account for such variations on the macroscale, rather than on the microscale, the
FGM material has three planes of elastic symmetry, each of them perpendicular to
the corresponding coordinate axis.
Several possible grading schemes listed below correspond to differently
populated matrices of stiffnesses. While the mathematical details are omitted for
brevity, we can classify the following cases:

1. Properties vary in three coordinate directions. At the RVE level, the material is
generally orthotropic.
2. Properties vary in three coordinate directions. At the RVE level, the material is
specially orthotropic.
3. Properties vary in three coordinate directions. At the RVE level, material is
isotropic.
4. Properties vary in the thickness direction only. At the RVE level, material is
generally orthotropic.
5. Properties vary in the thickness direction only. At the RVE level, the material is
specially orthotropic.
6. Properties vary in the thickness direction. At the RVE level, the material is
isotropic.
While the first three cases correspond to in-surface variable stiffness, i.e., AðxÞ;
BðxÞ; DðxÞ , in the latter three cases the corresponding matrices are constant

resulting in a simpler analysis.
Equations of equilibrium and boundary conditions can be derived from the
energy principles (e.g., Hamilton's principle) or from the analysis of equilibrium
of an infinitesimal element of the thin-walled structure. In both cases, the thickness
z -coordinate is eliminated from the analysis, i.e., the equilibrium of an element of the
structure is analyzed using a two-dimensional formulation where the stresses that
may vary with all three coordinates are replaced with the equivalent system of stress
resultants and stress couples via ( 2.8 ). The form of equations of equilibrium depends
on geometry of the structure, i.e., a doubly curved shell has different equations than a
flat plate, etc. For simplicity, we show here static equations for a flat plate utilizing
the Cartesian coordinate system, x and y being in-surface coordinates:

N x ; x þ N xy ; y ¼ 0
N xy ; x þ N y ; y ¼ 0
M x ; xx þ 2 M xy ; xy þ M y ; yy þ N x w; xx þ 2 N xy w; xy þ N y w; yy ¼ pðx; yÞ (2.11)

where w ¼ wðx; yÞ is a deflection and pðx; yÞ is a pressure applied to the plate.