Functionally Graded Materials in Engineering - Structural Interfaces and Attachments in Biology - page 21

compliance. The concentration tensors depend on such objective factors as the
shape, volume fraction, and orientation of the inclusions as well as on the subjective
aspect of the analysis, i.e., the choice of the micromechanical theory. In particular,
if the volume fractions of all constitutive materials, except for the locally dominant
material, are small enough to neglect their interaction, the tensor of stiffness
obtained by the effective field method is [ 18 ]
LðxÞ¼L i ðxÞþ X
j

c j ðxÞ½L j ðxÞL i ðxÞT j ðxÞ½c i ðxÞI þ c j ðxÞT j ðxÞ (2.5)

where tensors T j ðxÞ relate the average strain in the i -phase to the average strain to
the j -phase inclusions. While the previous micromechanical formulation is three-
dimensional, not being constrained by geometry of the structure or solid, the following
discussion concentrates on relatively thin-walled structures that are typical in engi-
neering applications.
The stiffness tensor is the outcome of the homogenization procedure. This tensor
can be employed to determine the extensional, coupling, and bending stiffness
matrices that are necessary for the macromechancial analysis of FGM structures.
This procedure requires us to transform a local stiffness tensor to the global
coordinate system employed in the analysis. If a relatively thin structure consists
of a number of layers, each of them with its unique material grading, the transfor-
mation equation for the local stiffness tensor L k that governs the response of the k th
layer located at z k < z < z kþ 1 where z is the coordinate counted from the middle
surface of the structure is

Q k ðx 1 ; x 2 ; zÞ¼Pðy k ÞL k ðx 1 k ; x 2 k ; zÞ (2.6)

In ( 2.6 ), x 1 and x 2 are in-surface coordinates and Pðy k Þ is the transformation tensor
that depends on the angle y k between the global ðx 1 ; x 2 ; zÞ and local ðx 1 k ; x 2 k ; zÞ
coordinate systems. Examples of in-surface global coordinates are the length and
width coordinates in the Cartesian system or the axial and circumferential
coordinates in the cylindrical coordinate system.
The matrices of extensional, coupling, and bending stiffness are now evaluated
following the standard mechanics of composite materials approach:

ð

Q k ðx 1 ; x 2 ; zÞf 1 z 2

fAðx 1 ; x 2 Þ Bðx 1 ; x 2 Þ Dðx 1 ; x 2 Þg ¼ g d z (2.7)

z

where the integration is conducted over the entire thickness of the structure. Note
that the stiffness matrices in the left side of ( 2.7 ) are dependent on in-surface
coordinates. In “conventional” composite materials such dependence is absent,
but if the volume fraction, shape, or orientation of constituent phases of an FGM
structure vary with the x 1 and x 2 coordinates, the stiffness of the structure is variable
over the surface.