Biomedical Engineering Reference

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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.

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