Hybrid fundamental-solution-based FEM for piezoelectric materials

In this paper, a new type of hybrid finite element method (FEM), hybrid fundamental-solution-based FEM (HFS-FEM), is developed for analyzing plane piezoelectric problems by employing fundamental solutions (Green’s functions) as internal interpolation functions. A modified variational functional used in the proposed model is first constructed, and then the assumed intra-element displacement fields satisfying a priori the governing equations of the problem are constructed by using a linear combination of fundamental solutions at a number of source points located outside the element domain. To ensure continuity of fields over inter-element boundaries, conventional shape functions are employed to construct the independent element frame displacement fields defined over the element boundary. The proposed methodology is assessed by several examples with different boundary conditions and is also used to investigate the phenomenon of stress concentration in infinite piezoelectric medium containing a hole under remote loading. The numerical results show that the proposed algorithm has good performance in numerical accuracy and mesh distortion insensitivity compared with analytical solutions and those from ABAQUS. In addition, some new insights on the stress concentration have been clarified and presented in the paper.     

Fracture mechanics of piezoelectric materials

This paper presents an analysis of crack problems in homogeneous piezoelectrics or on the interfaces between two dissimilar piezoelectric materials based on the continuity of normal electric displacement and electric potential across the crack faces. The explicit analytic solutions are obtained for a single crack in an infinite piezoelectric or on the interface of piezoelectric bimaterials. For homogeneous materials it is found that the normal electric displacement D₂, induced by the crack, is constant along the crack faces which depends only on the remote applied stress fields. Within the crack slit, the perturbed electric fields induced by the crack are also constant and not affected by the applied electric displacement fields. For bimaterials, generally speaking, an interface crack exhibits oscillatory behavior and the normal electric displacement D₂ is a complex function along the crack faces. However, for bimaterials, having certain symmetry, in which an interface crack displays no oscillatory behavior, it is observed that the normal electric displacement D₂ is also constant along the crack faces and the electric field E₂ has the singularity ahead of the crack tip and has a jump across the interface. Energy release rates are established for homogeneous materials and bimaterials having certain symmetry. Both the crack front parallel to the poling axis and perpendicular to the poling axis are discussed. It is revealed that the energy release rates are always positive for stable materials and the applied electric displacements have no contribution to the energy release rates. This revised version was published online in July 2006 with corrections to the Cover Date.     

Interface corners in piezoelectric materials

Through proper arrangement, the constitutive law, strain-displacement relation and equilibrium equation of piezoelectric materials can be written in the same mathematical form as those of elastic materials and hence Stroh formalism can be extended for piezoelectric analysis. Based on this viewpoint, the authors’ previous works for fracture analysis of anisotropic elastic materials, e.g. the eigen-relation for determining singular orders, the near-tip solutions, and the unified definition of stress intensity factors for interface corners, can also be applied to piezoelectric materials. In this paper, the theoretical framework of our previous works is briefly introduced, and then an efficient and accurate computing method (H-integral) and its required auxiliary solutions are derived for extracting the stress/electric intensity factors of interface corners made up of piezoelectric materials. This theoretical framework and H-integral form a universal solution technique that is valid for the fracture analysis of cracks, corners, interface cracks, and interface corners. Besides, the special cases that suggest how we simulate elastic insulators/conductors from piezoelectric materials are discussed. Several numerical examples are dealt with to display the feasibility and applicability of the proposed approaches, and finally, a numerical example which exhibits how the electric load influences the fracture behavior is also studied.     

Electric equivalent circuit for flexural vibrations in piezoelectric materials

General expressions for the values of the components of an electric equivalent circuit are derived for flexural vibrations. These expressions are applied to some interesting electrode configurations. The obtained values are in good agreement with experimental values and values specified by manufacturers of watch crystals. It is noted that solving Laplace's equation for the dielectric field is sufficient in order to obtain the values of the vibration amplitude, the piezoelectric current, and the equivalent components. The piezoelectric part of the electric field need only be considered if very accurate values of the resonance frequencies are desired. It is shown how accurate estimates can be obtained without the need of advanced calculation tools.     

Evaluation of materials for artificial heart valves

The heart of a normal human being beats about 38 million cycles every year. An artificial heart valve, to perform at this rate in the adverse conditions inside the heart for 20 years or more, should be highly wear-resistant with excellent fatigue strength. Thus, the study of mechanical and physical properties of the materials intended for use in artificial valves becomes an inseparable part of the valve development process itself. The physical and mechanical requirements of the materials used in the Chitra heart valve have been evaluated by studying their water absorption, adhesive wear and abrasive properties. The mechanical durability of the device has been assessed by accelerated life cycle testing. The test systems developed for the above are described here. The results show UHMW-PE to be a highly wear-resistant material suitable for the occluder. The accelerated wear tests show that the valve with Haynes-25 alloy cage and UHMW-PE disc has durability in excess of 50 years.     

Cross-property connections for fiber reinforced piezoelectric materials with anisotropic constituents

The paper addresses the problem of the connection between effective elastic stiffnesses, piezoelectric coefficients and dielectric permeabilities of a fiber reinforced piezoelectric composite with both phases (the matrix and the fibers) being transversely-isotropic. These connections allow one to predict the entire set of macroscopic elastic stiffnesses through one or two measurements of dielectric permeability. The solutions for square and hexagonal arrays of fibers and for randomly located parallel fibers are constructed and compared. The analytical results show very good agreement with available experimental data. As a side result, it is shown that the mutual positions of inhomogeneities produce only a minor effect and that applicability of the non-interaction approximation is much wider than expected.     

Weight functions for interface cracks in dissimilar anisotropic piezoelectric materials

Weight functions proposed for interface cracks in dissimilar isotropic materials (Gao, 1991; Chen and Hasebe, 1994) are extended to treat those in piezoelectric materials. The difficulties in separating the eight distinct complex arguments are overcome. The pseudo-orthogonal properties of the eigenfunction expansion form found in isotropic dissimilar cases(Chen and Hasebe, 1994) are proved to be valid in the present cases although the mathematical manipulations performed here seem much more complicated than those in isotropic dissimilar materials. Several path-independent integrals are obtained and all the coefficients in the eigenfunction expansion form, including the K _I, K _II, K _III and K _e, could be calculated by the weight functions introduced in this paper. It is concluded that the weight functions presented here provide a powerful tool to calculate the dominant parameters at the interface crack tip without any special treatment to the singular stress field of the near-tip region.     

Towards a fracture mechanics for brittle piezoelectric and dielectric materials

A theoretical fracture mechanics for brittle piezoelectric and dielectric materials is developed consistent with standard features of elasticity and dielectricity. The influence of electric field and mechanical loading is considered in this approach and a Griffith style energy balance is used to establish the relevant energy release rates. Results are given for a finite crack in an infinite isotropic dielectric and for steady state cracking in a piezoelectric strip. In the latter problem, the effect of charge separation in the material and discharge in the crack are considered. Observations of crack behavior in piezoelectrics under combined mechanical and electrical load are discussed to assess which features of the theory are useful.     

A new thermoelectroelastic solution for piezoelectric materials with various openings

Summary A new solution is obtained for thermoelectroelastic analysis of an insulated hole of various shapes embedded in an infinite piezoelectric plate. Based on the exact electric boundary conditions on the hole boundnary, Lekhnitskii's formulation and conformal mapping, the solution for elastic and electric fields has been obtained in closed form in terms of complex potential. The solution has a simple unified form for various holes such as ellipse, circle, triangle and square openings. As an application of the solution, the hoop stress and electric displacement (SED) and the solution for crack problems are discussed. Using the above results, the SED intensity factor and strain energy release rate can be obtained analytically. One numerical example is considered to illustrate the application of the proposed formulation and compared with those obtained from impermeable model.     

A one-dimensional model for non-linear behaviour of piezoelectric composite materials

In this paper, a one-dimensional analytical model is proposed to investigate the non-linear behaviour for piezoelectric and piezoelectric fibre reinforced composite (PFRC) materials in the fibre direction. The required linear and non-linear constants for purely piezoelectric materials are obtained using the curve-fitting method and the measured S₃–E₃ non-linear loops of the corresponding piezoelectric materials, whereas those for PFRC materials are determined by employing the quadratic non-linear constitutive equations for a purely piezoelectric material, the iso-field assumptions and linear and non-linear constants of the composite constituents. A numerical study is conducted. The numerical results reveal a significant effect of stress T₃ on S₃–E₃ non-linear behaviour for both soft PZT–5H ceramics and PZN–4.5%PT crystals. It is also found that the piezoelectric fibre volume fraction V_f and strain S₃ can significantly affect the T₃–E₃ non-linear behaviour for both PZT–5H/piezo-polymer polyvinylidene fluoride (PVDF) and PZN–4.5%PT/PVDF PFRC materials. A good correlation is noted between the piezoelectric constants d₃₃ and e₃₃ predicted using the present method and those recommended by manufacturer for PZT–5H ceramics and PZN–4.5%PT crystals.     

Complex material coefficients and energy ratios for lossy piezoelectric materials

This correspondence reviews complex material coefficients of piezoelectric materials and their influence on the energy ratios in coupled systems. In lossless systems, it is shown for the length extensional (LE 33) mode in a C_∞ material that there are at least 4 energy ratios that will produce the same coupling value. In addition, in the (LE 33) mode there are at least 2 different experimental conditions to define an energy ratio that produces the same coupling. With the introduction of loss, these 2 experiments and the 4 energy ratios diverge and no longer produce the same coupling. It is shown that the instantaneous ratio of coupled to input energy or the time average of this ratio, if a numerical value is desired, is the appropriate energy ratio in a dissipating system.     

Interaction integrals for thermal fracture of functionally graded piezoelectric materials

This paper presents domain form of the interaction integrals based on three independent formulations for computation of the stress intensity factors and electric displacement intensity factor for cracks in functionally graded piezoelectric materials subjected to steady-state thermal loading. Each of the formulation differs in the way auxiliary fields are imposed in the evaluation of interaction integral and each of them results in a consistent form of the interaction integral in the sense that extra terms naturally appear in their derivation to compensate for the difference in the chosen crack tip asymptotic fields of homogeneous and functionally graded piezoelectric medium.     

Interface waves in functionally graded piezoelectric materials

We analyze the propagation of antiplane or shear-horizontal waves near the interface between two half-spaces of piezoelectric ceramics. The material properties vary in the direction perpendicular to the interface. Both electroded and unelectroded interfaces are considered. Transcendental equations that determine the dispersion relations of the waves are obtained. They reduce to a few known results in the literature as special cases. Different from similar waves in homogeneous materials, the waves obtained are dispersive. The equations for the dispersion relations are solved numerically. It is found that the wave speeds are sensitive to the variation of material properties. This suggests the possibility of manipulating the wave propagation behavior through proper design of materials.     

Review: environmental friendly lead-free piezoelectric materials

Lead zirconate titanate (PZT) based piezoelectric materials are well known for their excellent piezoelectric properties. However, considering the toxicity of lead and its compounds, there is a general awareness for the development of environmental friendly lead-free materials as evidenced from the legislation passed by the European Union in this effect. Several classes of materials are now being considered as potentially attractive alternatives to PZTs for specific applications. In this paper, attempts have been made to review the recent developments on lead-free piezo materials emphasizing on their preparation, structure–property correlation, etc. In this context, perovskite systems such as bismuth sodium titanate, alkali niobates (ANbO₃), etc. and non-perovskites such as bismuth layer-structured ferroelectrics are reviewed in detail. From the above study, it is concluded that some lead-free compositions show stable piezoelectric responses even though they do not match the overall performance of PZT. This has been the stimulant for growing research on this subject. This topic is of current interest to the researchers worldwide as evidenced from the large number of research publications. This has motivated us to come out with a review article with a view that it would give further impetus to the researchers already working in this area and also draw the attention of the others.     

Linear electro-elastic fracture mechanics of piezoelectric materials

The concepts of linear elastic fracture mechanics, generalized to treat piezoelectric effects, are employed to study the influence of the electrical fields on the fracture behavior of piezoelectric materials. The method of distributed dislocations and electric dipoles, already existing in the literature, is used to calculate the electro-elastic fields and the energy-release rate for a finite crack embedded in an infinite piezoelectric medium which is subjected to both mechanical and electric loads. The energy-release rate expressions show that the electric fields generally tend to slow the crack growth. It is shown that the stress intensity factor criterion and the energy-release rate criterion differ when the energetics of the electric field is taken into account. The study of crack tip singular stress field yields a possible explanation for experimentally observed crack skewing in the presence of a strong electric field.     

Closed form solutions for partially debonded circular inclusion in piezoelectric materials

Summary This paper deals with the problem of a partially debonded piezoelectric circular inclusion in a piezoelectric matrix. This boundary value problem is reduced to two Riemann-Hilbert problems through the use of the analytical continuation theory.Closed form solutions are obtained by considering the behavior of the complex field potentials at origin and infinity. The formulae for the electro-elastic field intensity factors of the interfacial crack are derivedexplicitly. Several particular cases are provided to show the effect of the crack angle, the mechanical and electrical properties and the loads on the electroelastic field singularities.