Composite Materials Research Progress

Eletromechanical Field Concentrations and Polarization Switching... 261
For 90 ◦ switching in the x2x3 plane,
U = 1
2 (σ22ε22 + σ33ε33 +2σ32ε32 + D2E2) (29)
The critical value of internal energy density is assumed in the following form:
where ɛ T 33
Uc = 1
2 ɛT 33(Ec) 2
is the dielectric permittivity at constant stress.
The constitutive equations (10) and (11) during polarization switching are
The new piezoelectric constant e ′ ikl
by
(30)
σij = cijklε l kl − e ′ kijEk (31)
D l i = e ′ iklε l kl + ɛikEk (32)
is related to the elastic and direct piezoelectric constants
e ′ 111 = d′ 111 c11 + d ′ 122 c12 + d ′ 133 c13
e ′ 122 = d ′ 111c12 + d ′ 122c11 + d ′ 133c13
e ′ 133 = d′ 111 c13 + d ′ 122 c13 + d ′ 133 c33
e ′ 123 =2d ′ 123c44
e ′ 131 =2d′ 131 c44
e ′ 112 =2d ′ 112c66
e ′ 211 = d′ 211 c11 + d ′ 222 c12 + d ′ 233 c13
e ′ 222 = d ′ 211c12 + d ′ 222c11 + d ′ 233c13
e ′ 233 = d′ 211 c13 + d ′ 222 c13 + d ′ 233 c33
e ′ 223 =2d ′ 223c44
e ′ 231 =2d′ 231 c44
e ′ 212 =2d′ 212 c66
e ′ 311 = d ′ 311c11 + d ′ 322c12 + d ′ 333c13
e ′ 322 = d′ 311 c12 + d ′ 322 c11 + d ′ 333 c13
e ′ 333 = d ′ 311c13 + d ′ 322c13 + d ′ 333c33
e ′ 323 =2d′ 323 c44
e ′ 331 =2d ′ 331c44
e ′ 312 =2d′ 312 c66
The components of the piezoelectricity tensor d ′ ikl are
d ′
�
ikl = d33ninknl + d31(niδil − ninknl)+ 1
2 d15(δiknl
�
− 2ninknl + δilnk)
where ni is the unit vector in the poling direction, δij is the Kroneker delta, and d33 = d333,
d31 = d311, d15 =2d131 are the direct piezoelectric constants.
(33)
(34)