Electroelastic behavior of piezoelectric composites with coated reinforcements: Micromechanical approach and applications

In this work, a micromechanical model for the estimate of the electroelastic behavior of the piezoelectric composites with coated reinforcements is proposed. The piezoelectric coating is considered as a thin layer with active electroelastic properties different from those of the inclusion and the matrix. The micromechanical approach based on the Green’s functions technique and on the interfacial operators is designed for solving the electroelasticity inhomogeneous coated inclusion problem. The effective properties of a piezoelectric composite containing thinly coated inclusions are obtained through the Mori–Tanaka’s model. Numerical investigations into electroelastic moduli responsible for the electromechanical coupling are presented as functions of the volume fraction and characteristics of the coated inclusions. Comparisons with existing analytical and numerical results are presented for cylindrical and elliptic coated inclusions.     

Moving antiplane shear crack in hexagonal piezoelectric crystals

Closed-form solutions are obtained and discussed for the stress and electric displacement fields around a loaded Griffith-type antiplane shear strip crack moving in hexagonal piezoelectric crystals. Representative numerical results are presented for ZnO and PZT-4.     

Singular electro-mechanical fields near the apex of a piezoelectric bonded wedge under antiplane shear

This paper presents the explicit forms of singular electro-mechanical field in a piezoelectric bonded wedge subjected to antiplane shear loads. Based on the complex potential function associated with eigenfunction expansion method, the eigenvalue equations are derived analytically. Contrary to the anisotropic elastic bonded wedge, the results of this problem show that the singularity orders are single-root and may be complex. The stress intensity factors of electrical and mechanical fields are dependent. However, when the wedge angles are equal (α=β), the orders become real and double-root. The real stress intensity factors of electrical and mechanical fields are then independent. The angular functions have been validated when they are compared with the results of several degenerated cases in open literatures.     

Electroelastic intensification near anti-plane shear crack in orthotropic piezoelectric ceramic strip

The theory of linear piezoelectricity is applied to solve the antiplane electroelastic problem of an orthotropic piezoelectric ceramic strip with a finite crack, which is situated symmetrically and oriented in a direction normal to the edges of the strip. Fourier transforms are used to reduce the problem to the solution of a pair of dual integral equations. They are then reduced to a Fredholm integral equation of the second kind. Numerical values on the stress intensity factor and the energy release rate for some piezoelectric ceramics are obtained and the results are displayed numerically to exhibit the electroelastic interactions.     

Electroelastic behavior modeling of piezoelectric composite materials containing spatially oriented reinforcements

In this work, a modeling of electroelastic composite materials is proposed. The extension of the heterogeneous inclusion problem of Eshelby for elastic to electroelastic behavior is formulated in terms of four interaction tensors related to Eshelby’s electroelastic tensors. Analytical formulations of interaction tensors are presented for ellipsoidal inclusions. These tensors are basically used to derive the self-consistent model, Mori–Tanaka and dilute approaches. Numerical solutions are based on numerical computations of these tensors for various types of inclusions. Using the obtained results, effective electroelastic moduli of piezoelectric multiphase composites are investigated by an iterative procedure in the context of self-consistent scheme. Generalised Mori–Tanaka’s model and dilute approach are re-formulated and the three models are deeply analysed. Concentration tensors corresponding to each model are presented and relationships of effective coefficients are given. Numerical results of effective electroelastic moduli are presented for various types of piezoelectric inclusions and for various orientations and compared to existing experimental and theoretical ones.     

ELECTROELASTIC FIELD FOR AN IMPERMEABLE ANTI-PLANE SHEAR CRACK IN A PIEZOELECTRIC CERAMICS PLATE

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Electroelastic field for an impermeable anti-plane shear crack in a piezoelectric ceramics plate

Electroelastic behavior of a cracked piezoelectric ceramics plate subjected to four cases of combined mechanical-electrical loads is analyzed. The integral transform method is applied to convert the problem involving an impermeable anti-plane crack to dual integral equations. Solving the resulting equations, the explicit analytic expressions for electroelastic field along the crack line and the intensity factors of relevant quantities near the crack tip and the mechanical strain energy release rate are obtained. The known results for an infinite piezoelectric ceramics plane containing an impermeable anti-plane crack are recovered from the present results only if the thickness of the plate h → ∞. Biography: LI Xian-fang (1964-)     

The asymptotic behavior of doubly periodic strain states

The asymptotic behavior of the doubly periodic strain state on a three-dimensional body is discussed. The components of the elastic state are initially assumed to be bounded by an arbitrary polynomial. It is then shown that the components can be approximated by second degree polynomials whose coefficients can be readily computed from data generally available in such problems. The error in using the approximation at different points in the elastic body is on the order of the reciprocal of any polynomial of the distance to the boundary of the body.     

Failure analysis of unidirectional fiber composites under combined axial tension and shear

A method of analysis is proposed for investigating the mechanics of failure of unidirectional, fiber composite materials subjected to axial tension and shear. The mechanisms of failure are assumed to result from the interaction of the applied shear stress and local matrix shear stress concentrations which arise as a result of the scattered fiber breaks that occur throughout the material. Two modes of failure are identified. One is associated with the unstable growth of shear failure regions in the matrix. The other is primarily a tensile failure mode which is influenced by the applied shear. The analysis predicts that a variety of shear-tensile stress failure surfaces are possible, depending on material properties. The results suggest that radical changes in the shape of failure surfaces can occur as the result of environmental effects. This has significant implications for reliability.     

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 Hashin's 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 Schwarts's microstructural parameters, which have a physical meaning, and provide formulae for them. The equations easily lead to Hill's universal relations.     

On the plastic zone size of solids containing doubly periodic rectangular-shaped arrays of cracks under longitudinal shear

Multiple cracks interaction plays an important role in fracture behavior of materials. A number of studies have been devoted to analytical and numerical analyses of the doubly periodic arrays of cracks. A very natural and highly accurate solution procedure is proposed to describe the interaction effect among the doubly periodic rectangular-shaped arrays of cracks. The proposed solution is implemented in the framework of continuously distributed dislocation model and singular integral equation approach. The accuracy of this solution is proved through a comparison of results from the present simulation and known closed form solutions. Further, the interaction effects among the periodic cracks on the plastic zone size and crack tip opening displacement are studied. It is found that the interaction distance among the vertical and horizontal periodic cracks is quite different.     

Electroelastic analysis of a piezoelectric cylindrical fiber with a penny-shaped crack embedded in a matrix

The electroelastic response of a penny-shaped crack in a piezoelectric cylindrical fiber embedded in an elastic matrix is investigated in this study. Fourier and Hankel transforms are used to reduce the problem to the solution of a pair of dual integral equations. They are then reduced to a Fredholm integral equation of the second kind. Numerical values on the stress intensity factor, energy release rate and energy density factor for piezoelectric composites are obtained to show the influence of applied electric fields.     

Oblique edge crack in an anisotropic material under antiplane shear

An oblique edge crack in an anisotropic material under antiplane shear loadings is investigated. The antiplane problems are formulated based on a linear transformation method. An anisotropic solid containing an edge crack subjected to concentrated forces is first considered. The stress intensity factor for the edge crack with concentrated forces is obtained from the solution of the transformed edge crack in an isotropic material which is solved by using conformal mapping technique and complex function theory. The solution of the edge crack under concentrated loads is used to construct the stress intensity factor for the oblique edge crack in the anisotropic material subjected to antiplane distributed loads. Some numerical computations are carried out to calculate the stress intensity factors for the edge crack in inclined orthotropic materials subjected to point forces as well as distributed tractions.     

Electroelastic analysis of a piezoelectric finite wedge with mixed type boundary conditions under a pair of concentrated shear forces and free charges

The antiplane problem of a piezoelectric finite wedge with mixed type boundary conditions (i.e. traction free-electrically grounded or clamped-electrically insulated) assumed on the circular edge is studied in this paper. Using the first and second kinds of the finite Mellin transforms, the stress and electrical displacement in the piezoelectric finite wedge subjected to a pair of concentrated shear forces and free charges are derived analytically. The singularity orders and all generalized intensity factors can also be obtained. After being reduced to the crack problem, the energy density theory is applied to study the effects of the boundary conditions on the circular edge and the wedge radius on the fracture behaviors of the cracked piezoelectric medium. The problem can be further degenerated to several simple problems and the results agree well with those of previous studies.     

A unified treatment of the elastic elliptical inclusion under antiplane shear

Summary A generalized and unified treatment is presented for the antiplane problem of an elastic elliptical inclusion undergoing uniform eigenstrains and subjected to arbitrary loading in the surrounding matrix. The general solution to the problem is obtained through the use of conformal mapping technique and Laurent series expansion of the associated complex potentials. The resulting elastic fields are derived explicitly in both transformed and physical planes for the inclusion and the surrounding matrix. These relations are universal in the sense of being independent of any particular loading as well as the geometry of the matrix. The complete field solutions are provided for an elliptical inclusion under uniform loading at inifinity, and for a screw dislocation interacting with the elastic elliptical inclusion.     

Influence of imperfect elastic contact condition on the antiplane effective properties of piezoelectric fibrous composites

A fiber-reinforced periodic piezoelectric composite, where the constituents exhibit transverse isotropic properties, is considered. The fiber cross-section is circular and the periodicity is the same in two orthogonal directions. Imperfect mechanic contact conditions at the interphase between the matrix and fibers are represented in parametric form. In order to analyze the influence of the imperfect interface effect over the behavior of the composite, the effective axial piezoelectric moduli are obtained by means of the Asymptotic Homogenization Method. Some numerical examples are given.     

Electroelastic field of a piezoelectric annular finite cylinder

This paper presents a theoretical study of a piezoelectric annular cylinder under axisymmteric electromechanical loading. The piezoelectric material is assumed to be transversely isotropic and the general solutions of the governing equations are obtained in terms of a Fourier–Bessel series containing Bessel functions of the first and second kind. The boundary-value problems for vertical pressure and an electric charge loading applied to the ends of an annular cylinder are solved by expanding the applied loading in terms of a Fourier–Bessel series. Selected numerical results for the electroelastic field of an annular cylinder are presented for different aspect ratios of a cylinder and material properties.     

The total creep of viscoelastic composites under hydrostatic or antiplane loading

The problem of bounding the total creep (or total stress relaxation) of a composite made of two linear viscoelastic materials and subjected to a constant hydrostatic or antiplane loading is considered. It is done by coupling the immediate and the relaxed responses of the composite, which are pure elastic. The coupled bounds provide the possible range of the total deformation at infinite time as a function of the initial deformation of the composite. For antiplane shear existing bounds for coupled two-dimensional conductivity yield the required coupled bounds, and these are attained by doubly coated cylinder assemblages. The translation method is used to couple the effective bulk moduli of a viscoelastic composite at zero and infinite time. A number of microgeometries are found to attain the bulk modulus bounds. It is shown that the Hashin's composite sphere assemblage does not necessarily correspond to the maximum or minimum overall creep, although it necessarily attains the bounds for effective bulk moduli. For instance, there are cases when the doubly coated sphere microstructure or some special polycrystal arrangements attain the bounds on the total creep.