Effective anti-plane properties of piezoelectric fibrous composites

Abstract

Anti-plane shear of piezoelectric fibrous composites is theoretically investigated. The geometry of composites is described by the 2-dimensional geometry in a section perpendicular to the unidirectional fibers. The previous constructive results obtained for scalar conductivity problems are extended to piezoelectric antiplane problems. First, the piezoelectric problem is written in the form of the vector-matrix R-linear problem in a class of double periodic functions. In particular, application of the zeroth-order solution to the R-linear problem yields a vector-matrix extension of the famous Clausius-Mossotti approximation. The vector-matrix problem is decomposed into two scalar R-linear problems. This reduction allows us to directly apply all the known exact and approximate analytical results for scalar problems to establish high-order formulae for the effective piezoelectric constants. Special attention is paid to non-overlapping disks embedded in a two-dimensional background.

Mathematical Models of Elastic and Piezoelectric Fields in Two-Dimensional Composites

“…A constructive method of decompositions and reduction of the vector-matrix Rlinear problems to the scalar R-linear problems were proposed in [72]. Here, we do not discuss that method closely related to our approaches and refer to [72] for details and examples.…”

“…Here, we do not discuss that method closely related to our approaches and refer to [72] for details and examples.…”

Composites with invisible inclusions: Eigenvalues of ℝ-linear problem

“…Local fields in fibrous composites are described by solutions of the Riemann-Hilbert and the R-linear problems for multiply connected domains [13,21,22,26,27,28]. The physical properties of the components of traditional composites are expressed in terms of the positive constants, c.f., conductivity, permeability, permittivity etc.…”