Biomedical Engineering Reference
In-Depth Information
that is referred to as secondary pyroelectricity and the magnitude of the effects observed
in the two experiments would be quite different. Indeed, what is observed in the latter
condition is the primary effect plus the additional component of the secondary effect. This
phenomenon is not only limited to pyroelectricity, so a similar discussion could equally
be given for secondary thermal expansion, or secondary piezoelectricity, if the magnitude
of these effects is warranted. Normally the set of
σ
(stress),
E
(electric field), and
T
(temperature) are considered as independent variables, while
ε
(strain),
D
(electric dis-
placement), and
S
(entropy) are dependent variables. However, other alternatives are also
possible. Therefore, the general form of linear (first-order effects) constitutive equations
for a piezoelectric crystal are given in the following form:
=
s
E,T
ijkl
σ
kl
+
d
kij
E
k
+
α
ij
T
ε
ij
(3.9)
D
i
=
d
ijk
σ
jk
+
k
σ,T
E
j
+
p
i
T
(3.10)
ij
S
=
α
ij
σ
ij
+
p
i
E
i
+
C
σ,E
/T
T
(3.11)
where:
ε
ij
= second-rank strain tensor
s
ijkl
= fourth-rank elastic compliance tensor
σ
ij
= second-rank stress tensor
d
ijk
= third-rank piezoelectric coefficient tensor (direct and converse effects)
E
k
= first-rank electric field tensor
α
= second-rank thermal coefficients tensor (thermal expansion and piezocaloric effects)
T
= zero-rank temperature tensor
D
i
= first-rank electrical displacement
k
ij
= second-rank permittivity tensor
p
i
= first-rank pyroelectric coefficients tensor (pyroelectricity and electrocaloric effects)
(
C
/
T
) = zero-rank heat capacity tensor.
The subscripts
i, j, k
,and
l
each have a value between 1 and 3. Superscript
E
means that
the material property, in this case the elastic compliance, is measured in a constant electric
field (short-circuit condition), and
T
represents the material property (dielectric constant)
measured under constant stress. In general, there are 3
n
independent components for each
tensor of rank
n
. For instance, the piezoelectric coefficient matrix (rank 3) and elastic com-
pliance matrix (rank 4) have 27 and 81 independent components, respectively. However,
since
d
ijk
are symmetrical in
j
and
k, s
ijkl
are symmetrical in
i, j, k
,and
l
, respectively,
independent components can be reduced. For a given piezoelectric material, the number of
independent parameters can be further reduced by using symmetry relations in the mate-
rial. For instance, the piezoelectric PVDF, which is the focus of this section, is classified
under an orthorhombic system, class
mm
2(or
C
2
V
) among 32 defined crystal classes. The
number of independent variables which are required for this class is depicted in Figure 3.3.
