Theoretical approach to effective electrostriction in inhomogeneous materials

Abstract

An analytical approach is developed for the effective electrostriction, a nonlinearly coupled electromechanical effect, in inhomogeneous materials based on Green's-function method. For an isotropic composite containing randomly oriented ferroelectric crystallites with cubic symmetry, we derive the first effective-medium-like formulas for calculating its effective electrostrictive coefficients. The effects of microstructural features ͑such as volume fraction, crystallite shape and orientation, and connectivity of phases͒ on the effective electrostrictive coefficients are illustrated numerically and discussed. The calculations show that it is possible to develop new electrostrictors combining large electrostriction with mechanical flexibility by choosing the best combination of ferroelectric ceramics and polymer.

Microstructural evolution and overall response of an initially isotropic ferroelectric polycrystal under an applied electric field

“…In light of the lamellar morphology of the switched pattern commonly observed in ferroelectric crystals (Merz, 1954;Hooton and Merz, 1955), we invoke the Eshelby (1957) and Mori and Tanaka (1973) approach with aligned ellipsoidal inclusions. This approach is similar to the Maxwell-Garnett approximation in transport phenomena (Nan, 1993); it has also been applied to study the electrostriction of composite materials (Nan and Weng, 2000). As pointed out by Weng (1984Weng ( , 1990Weng ( , 1992 in the pure elastic context, this approach has the virtues that it is closely related to the Hashin-Shtrikman (1963) and Willis' bounds (1977) and, with the lamellar morphology, further coincides with Walpole's (1969) exact solution.…”

The bounds of electrostrictive coefficients of relaxor-based ferroelectric ceramics

“…Similarly, at the other extreme, by assuming that the strain and electric ® eld are the same in all crystallites, that is " …i †ˆ-" and E …i †ˆ-E, which means that each crystallite is mechanically clamped but electrically free, we obtain the commonly used Voigt approximations for the e ective dielectric and elastic constants (Nan 1993):…”

“…In order to estimate the e ective electrostriction behaviour, we solve the average problem by using the simple Reuss-and Voigt-type average approximations. First, by assuming that the stress and electric displacement are the same in all crystallites, that is ¼ …i †ˆ-¼ and D …i †ˆ-D, which means each crystallite may expand freely but is clamped electrically, we obtain the commonly used Reuss approximations for the e ective dielectric and elastic constants (Nan 1993) ;…”

“…An exact derivation of the electrostrictive properties of a polycrystalline ceramic is apparently not possible (Nan and Weng 2000), as the electrostrictive strains developed in neighbouring crystallites are normally incompatible because of their mutual misorientation. By analogy to the elastic moduli of polycrystals, Devonshire (1951) calculated the electrostrictive coe cients Q i j of an isotropic ceramic by directly averaging the coe cients for randomly oriented cubic crystallites.…”

“…As most of the experimental research is on electrostrictive ceramics, it is highly desirable to relate the electrostrictive properties to the single-crystal properties and texture of the ceramics. Although several successful approaches have been developed to link the e ective properties with the microstructure of ceramics, resulting in many expressions for calculating the e ective properties (for example Nan (1993) ), there are no theoretical expressions for the electrostriction, owing to the complexity of this nonlinear coupled mechanical± electric e ect.…”