Modelling of Polydomain Smart Materials
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BaTiO 3, PbTiO 3, (Pbl.,Zrx)TiO 3 [PZT], ceramics and thin film to an external electric field have been presented by several authors. 59 The domain interface movement under mechanical stress and electric or magnetic fields enables an additional mode of deformation. In previous paper we showed the effect of a bias electric field on the mechanical deformation beyond the saturated domain under the electric field.'" The electric field creates a stable polarization and makes the mechanical deformation much more difficult. In this work we describes the deformation of layer composites due to the evolution of equilibrium domain structure under the electric field at different constraint condition. EQUILIBRIUM DOMAIN STRUCTURES OF POLYDOMAIN MATERIAL According to the symmetry relation between the parent cubic and the product tetragonal phase as a result of phase transformation, e.g., BaTiO 3, three different orientational variants (or domains) of the tetragonal phase can be formed in the active layer. Three domains are characterized by the self-strain (or spontaneous strain) as shown in Figure 1:
-1 0-i=eo ,0 0 1
2= o 0 0 0
3=eo -1)
-1 00
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Mat. Res. Soc. Symp. Proc. Vol. 360 @1995 Materials Research Society
(o)
a)
IC)
'I
b)
5 Figure 1.
Elementary unit of a layer composite before (a) and after (b) the transformation (c) 90* and 180a domains in the ferroelectric phase.
[001] P//[001]
i
P//[111]
[100] ,/ r-=1/2
P//[ 110] Figure 2.
Polarization and "striction" of polydomain phase under electric field.
152
where eo=(ao-a)/ao, and x=(c-ao)/(ao-a) is the parameter of transformation tetragonality, a, is a lattice parameter of the cubic phase, and a and c are lattice parameters of the tetragonal phase. The minimum of the elastic energy corresponds to equilibrium domain structure consisting of all three types of domains with self strain and the polarization as follows:
S=
a1
1 +a 2 92 +a 393 ,
(2)
P=(a 1.+a 2y+a 3Z)(1-2y)P]
The components of i are e .=co(K+l)(ccl°), cyy.-'o(lc+1)(a1,l°+C,3°-0ct,-ct 3), £zz=o 0 (--1
)(cL3-
ocr)where tcl0=l/(x+l), ct30=(c-l)/(K+l), a,, i=1,2,3, is a volume fraction of each domain, y is a volume fraction of polarization along electric field (E), and P. is saturated polarization of each domain. y is the same in all domains due to compatibility of electric field across domain boundaries as shown in Figure 2. So the direction of polarization (P) as well as total self strain is determined only by domain composition. EQUILIBRIUM DOMAIN STRUCTURES UNDER EXTERNAL ELECTRIC FIELD The enthalpy of the layer composite under the external electric field (E) is (3)
h(a,E) = p(1-P) e(a,) + PhE(aj,y)
The elastic energy is obtained as a result of constraint as shown in Fig. 1 with a simple approximation that the elastic moduli of both phases are the same and the phases are isotropic as follow:
e = PI(-0)e(a•) = P(1-0) G/(l+v) ( e2+ C2+2ve ey
(4)
where G= E/2(1-v) is effective modulus, E is Young's modulus, v is Poisson's ratio of the
is a fraction of the active phase in a layer comp
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