On the Theory of Adaptive Composites
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(d)- •o
0
cid)=O
(b) (c)
(A
FAKA~
Figure 1. Formation of an adaptive composite.
Figure 2.
Deformation and polarization of an adaptive composite.
149
Mat. Res. Soc. Symp. Proc. Vol. 459 0 1997 Materials Research Society
If the mobility of interdomain interface is high, this thermodynamic effect results in additional contributions to the deformation and the polarization, increasing the compliance ("superelasticity") and the susceptibility of the composite [1,2]. We show below that the increase of compliance and susceptibility should have critical 2-type behavior if the thickness of the composite layer approaches to some critical values. THEORETICAL MODEL Consider a composite which consists of a periodic alternation of plane-parallel layers of an active ferroelectric phase and a passive phase. An elementary unit of the composite is shown in Fig. 1 c. /6=h/Hand 1-fr=I-h/H are relative thicknesses of the layers. The ferroelectric phase has a polytwin periodic microstructure, a is the fraction of domain 2. For simplicity, a 2-dimensional composite is considered [1]. The self strains of a para-ferroelectric transformation are:
Elc° =(
6
J
22rO =
0
(1) 1
_0)
with the twin strain: E0 = Co _ Eo =
20
(2)
The chosen form of the self-strain allow us to use the 2D model for the description of the 3D polydomain microstructure, formed by the two domains of a tetragonal phase with the c-axis in the plane of the ferroelectric layer with e0 e= 0 '-Eo" (Fig.3) [3]. For both cases the minimum misfit is obtained for equidomain microstructure if it is assumed that the minimum misfit equals zero for the 2D case.
[001]
Fr
Figure 3.
210
+0 12
/10101
pom
Two domains of a tetragonal phase formed from a cubic crystal.
150
ENERGY OF COMPOSITE The energy of the composite is a sum of the elastic energy of long-range internal stresses, the elastic energy of short-range microstresses near the interfaces between the composite layers and the energy of the interdomain interfaces [3]. The expression for the energy per unit area of the composite and the graphical presentation of each term in that expression is schematically illustrated in Table I. The first term describes energy of elastic interaction between the layers and has a minimum at a1-=/2. e12 is the energy of indirect interaction between domains [3]. e12=1/2E0 (2 0)2 , where E 0 is the Young's modulus, for the 2D model. eo is the misfit energy of single domain layer and is equal to 1/2E0 e0 2 for the 2D model. The second term describes approximately the energy of the microstresses [4]. The microstresses are localized within a thin layer near the interfaces between the composite phases; its thickness is approximately equal to the period D of the polydomain microstructure. The microstresses and their energy have been calculated in many papers [5-9]. However, the analysis of the results of the calculations shows that the simple expression above gives a fairly good estimation of microstress energy for polydomain microstructures with a period less than the thicknes
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