Voigt-Reuss-Hill Moduli for Ferroelectric Aggregates

Classical means of computing "average" isotropic elastic moduli of polycrystalline aggregates began with Voigt who performed a space averaging of monocrystal elastic stiffnesses to arrive at aggregate values <c>. Reuss performed a complementary averaging of the elastic compliances to produce values <s>. The two sets represent extremes. Hill used an arithmetic average of these results to yield values that often agree well with experiment. Voigt-Reuss-Hill procedures apply to averaging moduli subject solely to mechanical boundary conditions. Piezoelectrics in general, and ferroelectrics in particular, are required to be characterized by both mechanical and electrical boundary conditions. How the proper averaging is to be performed is described. Lithium niobate, a strong piezocoupling ferroelectric, is used as a numerical example to demonstrate that the piezoelectric effect must be taken into account. A self-consistent averaging method is also disclosed.