Federico J. Sabina

Closed-form expressions for the effective coefficients of a fiber-reinforced composite with transversely isotropic constituents – II. Piezoelectric and square symmetry

Abstract A two-phase parallel fiber-reinforced periodic elastic composite is considered wherein the constituents exhibit transverse isotropy. The fiber cross-section is circular and the periodicity is the same in two orthogonal directions. Simple closed-form formulae are obtained for the effective properties of this composite by means of the asymptotic homogenization method. Numerical computation of these is easy. The analytical solution of the required resulting plane- and antiplane-strain local problems, which turns out to be only three, makes use of potential methods of a complex variable and properties of Weierstrass elliptic and related functions with periods (1,0) and (0,1). Dvoraks universal type of relations for this composite are easily derived in an elementary new way without solving any local problem. This result also applies when the interface may be arbitrarily shaped, but compatible with the square symmetry. Comparison with experimental data is shown. The above results include the situation when one or both phases are isotropic.

A simple self-consistent analysis of wave propagation in particulate composites

Abstract A simple self-consistent embedding scheme is developed for the approximate analysis of waves in a composite consisting of a matrix containing inclusions. It employs an approximate solution for scattering from a single inclusion, which allows the development of simple explicit equations, comparable with those already known for elastostatics, which are easily solved by iteration. The performance of the scheme is discussed in relation to a variety of illustrative examples.

Closed-form expressions for the effective coefficients of fibre-reinforced composite with transversely isotropic constituents. I: Elastic and hexagonal symmetry

The purpose of this paper is to determine the effective elastic, piezoelectric and dielectric properties of reinforced piezoelectric composite materials with unidirectional cylindrical fibres periodically distributed in two directions at an angle π/3 by means of the asymptotic homogenization method. Each periodic cell of the medium is a binary piezoelectric composite wherein both constituents are homogeneous piezoelectric materials with transversely isotropic properties. This paper makes use of some results obtained in Part I. Relatively simple closed-form expressions for the overall properties are obtained by means of potential methods of a complex variable and Weierstrass elliptic and related functions. Schulgasser universal type of relations are derived in a simple new way by means of the homogenized asymptotic method. The number of local problems to get all coefficients is two. The numerical computation of these effective properties is simple.

Unit cell models of piezoelectric fiber composites for numerical and analytical calculation of effective properties

Numerical unit cell models of 1-3 periodic composites made of piezoceramic unidirectional cylindrical fibers embedded in a soft non-piezoelectric matrix are developed. The unit cell is used for prediction of the effective coefficients of the periodic transversely isotropic piezoelectric cylindrical fiber composite. Special emphasis is placed on a formulation of the boundary conditions that allows the simulation of all modes of the overall deformation arising from any arbitrary combination of mechanical and electrical loading. The numerical approach is based on the finite element method and it allows extension to composites with arbitrary geometrical inclusion configurations, providing a powerful tool for fast calculation of their effective properties. For verification, the effective coefficients are evaluated for square and hexagonal arrangements of unidirectional piezoelectric cylindrical fiber composites. The results obtained from the numerical technique are compared with those obtained by means of the analytical asymptotic homogenization method for different volume fractions. Furthermore, the results are compared with other analytical and numerical methods reported in the literature.

Overall properties of piezocomposite materials 1-3

Abstract The purpose of this paper is to present analytical expressions of the effective constants of reinforced piezoelectric composite materials with periodically distributed unidirectional cylindrical fibers for transducer applications. Each periodic cell of the medium is a binary piezoelectric composite wherein both constituents are homogeneous piezoelectric materials with hexagonal symmetry. The general formulae obtained allow the prediction of global properties of an important class of piezocomposites. Numerical calculations show an adequate concordance with other theoretical models. In the analysis, the periodicity of the structure is assumed to be much smaller than the elastic wavelength. Finally, we apply these results to a 1–3 material and obtain new piezocomposites with better global properties for biomedical imaging and hydrophone applications.

Scattering of SH waves by surface cavities of arbitrary shape using boundary methods

Abstract In this paper a boundary method is used to numerically solve the problem of scattering of SH waves by a bounded surface cavity or arbitrary shape in a half-space. This method reduces the dimension of the problem by one, but avoids the introduction of singular integral equations. A close connection is established between this method and least-squares collocation. Results are obtained using a multipole expansion in terms of Hankel functions about the origin. Comparison with some known exact solutions for SH wave motion yields very good agreement. It is observed that, in the case of a trench with steep walls, local amplification factors can sometimes significantly exceed 100%.

Self-consistent analysis of waves in a matrix-inclusion composite—I. Aligned spheroidal inclusions

Abstract A simple self-consistent scheme which has already been implemented for wave propagation through a matrix containing spherical inclusions is developed and applied to a composite comprising an isotropic matrix containing a random array of aligned spheroidal inclusions. Results are presented for a variety of cases, including a matrix containing cavities of very small aspect ratio, either fluid-filled or empty, to simulate a body containing an array of aligned penny-shaped cracks.

Overall behavior of two-dimensional periodic composites

The overall properties of a binary elastic periodic fiber-reinforced composite are studied here for a cell periodicity of square type. Exact formulae are obtained for the effective stiffnesses, which give closed-form expressions for composites with isotropic components including ones for empty and rigid fibers. The new formulae are simple and relatively easy to compute. Examples show the dependences of the stiffnesses as a function of fiber volume fraction up to the percolation limit. The specific example of glass fibers in epoxy yields new curves, which correct those displayed before by Meguid and Kalamkarov. Comparison with experimental data is very good. Bruno, Hill and Hashins bounds are compared with the exact solution. In most cases, the latter is very close to a bound in a given interval. A useful fact to know, where the easy formula afforded by the bound is advantageous. Plots of effective properties are also given for values of the shear moduli ratio of the two media. The overall parameters in the cases of empty and rigid fibers are also shown. The exact formulae explicitly display Avellaneda and Schwartss microstructural parameters, which have a physical meaning, and provide formulae for them. The equations easily lead to Hills universal relations.

Magnetic anomaly due to a vertical right circular cylinder with arbitrary polarization

A closed form solution for the total anomalous magnetic field due to a vertical right circular cylinder with arbitrary polarization is derived under the assumption that the magnetization is uniform. As expected, the computed field is similar to the field due to a “similar” prism‐shaped body.

CLOSED-FORM THERMOELASTIC MODULI OF A PERIODIC THREE-PHASE FIBER-REINFORCED COMPOSITE

ABSTRACT In previous works, using the asymptotic homogenization method (AHM), analytical formulae have been obtained for all global elastic constants of a binary fiber composite with perfect interfaces. In many cases of interest the perfect interphase is not an adequate model and it is necessary to include in the analytical models one or more interphases separating the reinforcement inclusion phase from the host matrix phase. In this article, an extension of AHM to thermoelastic heterogeneous problems is given. A simple closed form of effective properties for a three-phase unidirectional transversely isotropic composite is presented. By using homogenization schemes for periodic media, the local problems are solved and effective thermoelastic properties moduli are determined. The method is based on the assumption that the scale ratio between the periodic cell and the whole composite tends to zero. New universal relations for the three-phase thermoelastic composite are found from the AHM. In order to analyze the interphase effect, the effective thermoelastic moduli are compared with some theoretical approaches and experimental results reported in the literature.