Composite Materials Research Progress

260 Yasuhide Shindo and Fumio Narita
c11 = c1111 = c2222, c12 = c1122, c13 = c1133 = c2233, c33 = c3333
c44 = c2323 = c3131, c66 = c1212 = 1
2 (c11 − c12)
e15 = e131 = e223, e31 = e311 = e322, e33 = e333
The direction of the spontaneous polarization P s of each grain can change by 90 ◦ or
180 ◦ for ferroelectric switching induced by a sufficiently large electric field. In order to
develop a non-linear model incorporating the polarization switching mechanisms with the
electromechanical fields calculations, two criteria are used. The first criterion for polarization
switching is based on work down, and the second is internal energy density switching
criterion.
The first criterion [6] states that a polarization switches when the electrical and mechanical
work exceeds a critical value
σij∆εij + Ei∆Pi ≥ 2P s Ec
where ∆εij and ∆Pi are the changes in the spontaneous strain and polarization during
switching, respectively, and Ec is a coercive electric field. The changes in ∆εij = ε r ij and
∆Pi = P r
i for 180◦ switching can be expressed as
⎫
⎬
⎭
(17)
(18)
(19)
∆ε11 =0, ∆ε22 =0, ∆ε33 =0, ∆ε12 =0, ∆ε23 =0, ∆ε31 =0 (20)
∆P1 =0, ∆P2 =0, ∆P3 = −2P s (21)
For 90 ◦ switching in the x3x1 plane, there results
∆ε11 = γ s , ∆ε22 =0, ∆ε33 = −γ s , ∆ε12 =0, ∆ε23 =0, ∆ε31 =0 (22)
∆P1 = ±P s , ∆P2 =0, ∆P3 = −P s
For 90 ◦ switching in the x2x3 plane,
(23)
∆ε11 =0, ∆ε22 = γ s , ∆ε33 = −γ s , ∆ε12 =0, ∆ε23 =0, ∆ε31 =0 (24)
∆P1 =0, ∆P2 = ±P s , ∆P3 = −P s
The polarization switching criterion based on internal energy density (second criterion)
[7] is defined as
U = Uc
where U is the internal energy density and Uc is a critical value of internal energy density
corresponding to the switching mode. The internal energy density associated with 180 ◦
switching can be written as
U = 1
2 D3E3
In the case of 90 ◦ switching in the x3x1 plane, the internal energy density is
(25)
(26)
(27)
U = 1
2 (σ11ε11 + σ33ε33 +2σ31ε31 + D1E1) (28)