Biomedical Engineering Reference
In-Depth Information
0
x
0
y
0
xy
where
e
and
e
are middle surface axial strains and
g
is the middle surface
shear strain.
The considerations discussed with regard to (
2.11
) remain valid if the problem
is solved utilizing the stress function. In the case of in-surface grading the substitu-
tion of the stress resultants and stress couples expressed in terms of the stress
function and deflection in the third equation (
2.11
) and in (
2.13
) results in exceed-
ingly complicated equations. If grading is limited to the through-the-thickness
z
-coordinate, the corresponding procedure is simplified, essentially coinciding
with such procedures for typical laminated composite structures. For example, if
an FGM panel is symmetric about the middle surface and the principal axes of
the material coincide with the coordinate system (specially orthotropic RVE), the
equation of equilibrium and the compatibility condition become:
2
w;
xx
w;
yy
D
11
w;
xxxx
þ
2
ðD
12
þ
2
D
66
Þw;
xxyy
þ D
22
w;
yyyy
¼ pðx; yÞþhð';
yy
w;
xx
2
a
22
';
xxxx
þð
2
a
12
þ a
66
Þ';
xxyy
þ a
11
';
yyyy
¼ðw;
xy
Þ
';
xy
w;
xy
þ ';
xx
w;
yy
Þ
(2.14)
1
where the elements of the extensional compliance matrix
½a
ij
¼½A
ij
and the
subscripts comply with standard notation in mechanics of composites.
As is evident from the present discussion, analytical solutions for FGM
structures may be possible if grading is limited to the thickness direction. In the
general case, the solution has to be numerical. The strain energy of an FGM
structure is introduced in the same manner as for any other material, i.e.,
ðð
ð
1
2
U ¼
ðs
x
e
x
þ s
y
e
y
þ s
z
e
z
þ t
yz
g
yz
þ t
xz
g
xz
þ t
xy
g
xy
Þ
d
(2.15)
O
O
where
is the volume occupied by the structure. The difference between a
conventional composite material and a heterogeneous FGM where grading can
vary with in-surface coordinates becomes apparent if we substitute the stress-strain
relations (
2.1
) into (
2.15
).
It is necessary to emphasize here that assumptions regarding analytical
expressions for the tensor of stiffness of FGM should be treated with caution.
This is because in realistic situations we can prescribe variations of the volume
fraction of constituent materials, i.e., analytical functions
c
n
ðxÞ
. However, analytical
expressions for the stiffness, i.e.,
AðxÞ;
O
DðxÞ
, should be derived based on the
adopted micromechanics, reflecting the local volume fraction, shape, and orienta-
tion of the inclusions (constituent phases), rather than being assumed arbitrarily. On
the other hand, it is possible to develop a reverse scheme starting with the tensor of
stiffness of FGM and using micromechanics to specify the corresponding volume
fraction and geometry of inclusions.
BðxÞ;
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