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 anti-plane problems. First, the piezoelectric problem is written in the form of the vector-matrix $${\mathbb{R}}$$ -linear problem in a class of double periodic functions. In particular, application of the zeroth-order solution to the $${\mathbb{R}}$$ -linear problem yields a vector-matrix extension of the famous Clausius–Mossotti approximation. The vector-matrix problem is decomposed into two scalar $${\mathbb{R}}$$ -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.
Highlights
We study piezoelectric anti-plane shear of piezoelectric fibrous composites in a stationary electromagnetic field when the electric charge induces the elastic stresses and deformations and vice versa
It is worth noting that formulae obtained in the framework of the 2-dimensional theory of duality transformations [13,15,16] are exact, and formulae deduced by self-consistent methods [11,12,13,14] are approximate and hold only for dilute and weakly inhomogeneous composites
By closed-form analytical formulae, these authors mean the effective constants presented in a form that explicitly contains some parameters. These parameters can be numerically computed for fixed regular arrays by the truncation of infinite systems of linear algebraic equations following the method of Rayleigh
Summary
We study piezoelectric anti-plane shear of piezoelectric fibrous composites in a stationary electromagnetic field when the electric charge induces the elastic stresses and deformations and vice versa. It is worth noting that formulae obtained in the framework of the 2-dimensional theory of duality transformations [13,15,16] are exact, and formulae deduced by self-consistent methods [11,12,13,14] are approximate and hold only for dilute and weakly inhomogeneous composites. Guinovart et al [40] (see papers cited therein) extended the method of Rayleigh to piezoelectric fibrous materials It was declared in [40] that closed-form analytical formulae are given for the effective properties. By closed-form analytical formulae, these authors mean the effective constants presented in a form that explicitly contains some parameters These parameters can be numerically computed for fixed regular arrays by the truncation of infinite systems of linear algebraic equations following the method of Rayleigh. In the case ≤ 0, an extension of the methods [30,31,32,33,34,35,36,37,38,39] has to be made
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