Functionally Graded Materials in Engineering - Structural Interfaces and Attachments in Biology - page 24

As is easily observed, the problem formulated in ( 2.11 ) is statically indetermi-
nate since there are six unknown stress resultants and stress couples in three
equilibrium equations. This is a common feature of mechanics of solids, fluids,
and gases. Nevertheless, the problem can be resolved by expressing six stress
resultants and couples in ( 2.11 ) in terms of three unknown displacements via
( 2.9 ), ( 2.10 ) and the strain-displacement relations omitted here for brevity. The
latter relations can be either linear or nonlinear leading to geometrically linear or
nonlinear problems, respectively. The boundary conditions are formulated in terms
of displacements and rotations (kinematic conditions) and stress resultants and
couples (static conditions).
The major difference between two types of FGM becomes apparent upon
the analysis of ( 2.11 ). In the case of grading confined to variations of the volume
fraction (and/or shapes and orientations of constituent phases) exclusively through
the thickness of the structure, the tensor of stiffness is independent of the in-surface
coordinates x and y. Accordingly, equations of equilibrium in terms of displace-
ments contain only derivatives of displacements, while the elements of matrices
A

B and D are constant. However, if the volume fractions (and/or other grading
variables) of constituent phases vary with the in-surface coordinates, i.e., Aðx; yÞ;
Bðx; yÞ;

;

Dðx; yÞ , derivatives of stress resultants and stress couples in ( 2.11 ) contain
derivatives of the corresponding stiffness coefficients. The analytical solution of the
corresponding equations of equilibrium becomes impossible, except for simple
superficial cases. However, a numerical solution utilizing finite element or finite
difference methods is feasible even in this case since in-surface gradients of the
properties are relatively small and it is possible to size a mesh where each element
can be assigned constant in-surface properties.
An alternative method that is used in the theory of plates and shells, including
those from FGM, is based on the introduction of the stress function. In the case of a
flat FGM plate, this function ð'Þ is introduced through the following relations:

N x ¼ '; yy ; N y ¼ '; xx ; N xy ¼'; xy (2.12)

The substitution of the stress function defined by ( 2.12 ) into equations of equilib-
rium ( 2.11 ) results in identities for the first two equations. The third equation can be
formulated in terms of the stress function and unknown deflection. This equation
has to be complemented with the equation expressing compatibility conditions.
These conditions guarantee single-valued displacements in case the problem is
solved in terms of stresses or stress functions. They do not have to be considered
if the problem is solved in terms of displacements since such solution yields
unique values of displacements throughout the domain occupied by the structure.
In the problem of plane stress the compatibility conditions are reduced to one
equation [ 20 ]:

0
x ; yy þ e

0
y ; xx g

0

xy ; xy ¼ w; xy 2

e w; xx w; yy (2.13)