4.4 Constitutive Relations for Macro-fiber Composites
71
where B φ denote the electric field matrix, and Φ is the electric voltage vector applied
on MFC patches.
4.5 Electroelastic Nonlinear Constitutive Relations
Concerning structures deforming in elastic range and under strong electric field, the
nonlinear constitutive equations including second-order of electroelastic terms are
adopted [19]
ε i j = s i jkl σ kl + d i jm E m +
1
2
β i jmn E m E n ,
(4.71)
D m = d mkl σ kl + mn E n +
1
2
χ mkn E k E n .
(4.72)
Here, the Latin indices, i, j, k, l, m, n, take the numbers 1, 2 or 3, while i j or
kl denote only 11, 22, 33, 12 or 21, 13 or 31, 23 or 32. In (4.71) and (4.72), ε i j
and σ kl , denote respectively the strain and stress components, D m and E n are the
electric displacement and electric field components. The coefficients s i jkl , d mkl and
mn represent, respectively, the tensors of elastic compliance constants, piezoelectric
constants and dielectric constants, β i jmn and χ mkn are the nonlinear electroelastic
constants and nonlinear electroelastic susceptibility constants, respectively.
Again using the Voigt notation, given in Table 4.1, Eqs. (4.71) and (4.72) can be
re-written as
ε p = s pq σ q + d pm E m +
1
2
β pmn E m E n ,
(4.73)
D m = d mq σ q + mn E n +
1
2
χ mkn E k E n .
(4.74)
Here, the elastic compliance constants s pq are calculated by the material elastic
properties as
s 11 =
1
Y 1
,
s 12 = −
ν 12
Y 1
= −
ν 21
Y 2
, s 22 =
1
Y 2
,
s 44 =
1
κG 23
, s 55 =
1
κG 13
,
s 66 =
1
G 12
,
(4.75)
where Y i , ν 12 and G i j are the Young’s moduli, the Poisson’s ratios and the shear
moduli, and κ = 5/6 is the shear correction factor.
Assuming each piezoelectric patch has only one pair of electrodes, electric field
can be applied in one polarization direction. If the polarization aligns along the
thickness direction, it leads to m = n = k = 3. Again, because of the characteristic
of plates and shells, the transverse normal strain assumes zero, σ 3 = 0. Therefore,
71
where B φ denote the electric field matrix, and Φ is the electric voltage vector applied
on MFC patches.
4.5 Electroelastic Nonlinear Constitutive Relations
Concerning structures deforming in elastic range and under strong electric field, the
nonlinear constitutive equations including second-order of electroelastic terms are
adopted [19]
ε i j = s i jkl σ kl + d i jm E m +
1
2
β i jmn E m E n ,
(4.71)
D m = d mkl σ kl + mn E n +
1
2
χ mkn E k E n .
(4.72)
Here, the Latin indices, i, j, k, l, m, n, take the numbers 1, 2 or 3, while i j or
kl denote only 11, 22, 33, 12 or 21, 13 or 31, 23 or 32. In (4.71) and (4.72), ε i j
and σ kl , denote respectively the strain and stress components, D m and E n are the
electric displacement and electric field components. The coefficients s i jkl , d mkl and
mn represent, respectively, the tensors of elastic compliance constants, piezoelectric
constants and dielectric constants, β i jmn and χ mkn are the nonlinear electroelastic
constants and nonlinear electroelastic susceptibility constants, respectively.
Again using the Voigt notation, given in Table 4.1, Eqs. (4.71) and (4.72) can be
re-written as
ε p = s pq σ q + d pm E m +
1
2
β pmn E m E n ,
(4.73)
D m = d mq σ q + mn E n +
1
2
χ mkn E k E n .
(4.74)
Here, the elastic compliance constants s pq are calculated by the material elastic
properties as
s 11 =
1
Y 1
,
s 12 = −
ν 12
Y 1
= −
ν 21
Y 2
, s 22 =
1
Y 2
,
s 44 =
1
κG 23
, s 55 =
1
κG 13
,
s 66 =
1
G 12
,
(4.75)
where Y i , ν 12 and G i j are the Young’s moduli, the Poisson’s ratios and the shear
moduli, and κ = 5/6 is the shear correction factor.
Assuming each piezoelectric patch has only one pair of electrodes, electric field
can be applied in one polarization direction. If the polarization aligns along the
thickness direction, it leads to m = n = k = 3. Again, because of the characteristic
of plates and shells, the transverse normal strain assumes zero, σ 3 = 0. Therefore,