Framework Of Linear Piezoelectricity Research Articles

Exact piezoelectric solution for misfitted inclusion in finite spherical matrix: Applications to quantum dot core/shell crystals

Piezoelectric behavior of quantum dot core/shell structures in zinc-blende and wurtzite crystals are investigated in the framework of linear piezoelectricity. Strains arising from lattice mismatch in the core/shell structure are modeled as eigenstrains resulting from size-mismatched inclusion embedded in a finite spherical piezoelectric medium. Assuming that the core/shell piezoelectric structure exhibits spherically-hexagonal anisotropy, an exact solution is obtained through the eigenvalue decomposition method and the analytical expressions of the electroelastic fields are found using appropriate boundary and continuity conditions. It is found that the electroelastic fields can become singular or vanish at the center of the core for certain cases of spherical anisotropy. The analytical solutions are verified with the finite element analysis (FEA) and found to be in excellent agreement. FEA for the case of rectilinear anisotropy was also performed which showed considerable difference from the spherically anisotropic analytical solution. The closed-form solution obtained in this work can be used for any two-layer piezoelectric structures having spherical geometry subjected to various electromechanical loads.

Electroelastic Fields Induced by Two Collinear and Energetically Consistent Cracks in a Piezoelectric Layer

AbstractWithin the framework of linear piezoelectricity, the problem of two collinear electrically dielectric cracks in a piezoelectric layer is investigated under inplane electro-mechanical loadings. The energetically consistent crack-face boundary conditions are utilized to address the effects of a dielectric inside the cracks on the crack growth. The Fourier transform technique is applied to solve the boundary-value problem. Under the consideration of two-case electromechanical loadings, the electroelastic fields near the inner and outer crack tips are obtained through the Lobatto-Chebyshev collocation method. The special case of two collinear energetically consistent cracks in an infinite piezoelectric solid is analyzed and the closed-form solutions of the crack-tip electroelastic fields are further determined. Numerical results show the variations of stress intensity factors and energy release rates near the inner and outer crack tips on the applied electric fields, the geometry of cracks and the width of the piezoelectric layer in graphics. The observations reveal that the stress intensity factors are dependent not only on the adopted crack-face boundary conditions, but also on the applied mechanical loading.

Closed-form solution for elliptical inclusion problem in antiplane piezoelectricity with far-field loading at an arbitrary angle

Abstract An elliptical piezoelectric inclusion embedded in an infinite piezoelectric matrix is analyzed in the framework of linear piezoelectricity. Using the conformal mapping technique, a closed-form solution is obtained for the case of a far-field antiplane mechanical load, τ 0 , and an inplane electrical load, E 0 , at an arbitrary angle β . The stress and electric field distribution patterns for different defect shapes, loading angles, and material constants are studied. The energy release rates of self-similarly expanding and rotating defects in the presence of an electric field are obtained using the generalized M - and L -integrals as a function of the loading angle. The physical significance of these results is discussed in terms of the stress and electric field distributions as well as the energy release rates.

A piezoelectric screw dislocation interacting with a half-plane trimaterial composite

The electro-elastic stress investigation on the interaction between a screw dislocation and a half-plane trimaterial composite composed of three bonded dissimilar transversely isotropic piezoelectric materials is analyzed in the framework of linear piezoelectricity. Each layer is assumed to have the same material orientation with x3 in the poling direction. The dislocations are characterized by a discontinuous displacement and electric potential across the slip plane and are subjected to a line force and a line charge at the core. Based on the complex variable and the method of alternating technique, the solution of electric field and displacement field is expressed in terms of explicit series form. The solutions derived here can be applied to a variety of problems, for example, a half-plane bimaterial, a quarter-plane bimaterial, a quarter-plane material and a rectangular strip etc. Numerical results are provided to show the influences of the material combinations and geometric configurations on the electro-elastic fields and image force calculated through the generalized Peach-Koehler formula. The solutions proposed here can be served as Green’s functions for the analyses corresponding piezoelectric cracking problems.

Elliptical inclusion problem in antiplane piezoelectricity: Implications for fracture mechanics

An elliptical piezoelectric inclusion embedded in an infinite piezoelectric matrix is analyzed in the framework of linear piezoelectricity. Using the conformal mapping technique, a closed-form solution is obtained for the case of a far-field antiplane mechanical load and an inplane electrical load. The solution to a permeable elliptical hole problem is obtained as a limiting case of vanishing elastic modulus of the inclusion. This enables the study of the nature of crack tip electric field singularity which is shown to depend on the electrical boundary condition imposed on the crack faces. The energy release rate of a self-similarly expanding slender crack in the presence of electric fields is obtained by using the generalized M-integral. The energy release rate expression indicates that the electric field has a crack-arresting influence. This effect is inferred to have a more fundamental physical origin in the interaction between the applied electric field and the induced surface charges on the crack faces. An experimental result contradicting the theoretical prediction on the crack-arresting effect is also discussed.

Electroelastic field induced by thin interface electrodes between two bonded dissimilar piezoelectric ceramics

The electroelastic analysis of multiple collinear electrodes embedded at the interface of two bonded dissimilar piezoelectric ceramics is made. Within the framework of linear piezoelectricity, the Fourier transform technique is applied to reducing the problem to a singular integral equation with Cauchy kernel. Two particular cases are especially emphasized. For a single interface electrode, the electroelastic field is obtained in the entire plane of a two-phase piezoelectric composite in terms of elementary functions. For two collinear interface electrodes of equal length, a closed-form solution is determined along the interface. Obtained results reveal that near the electrode edges, the induced electroelastic field exhibits a square-root singularity, and the oscillatory singularity does not occur for arbitrary two piezoelectric ceramics poled in the same or opposite directions normal to the interface. Across the electrode, the normal component of stress is continuous, while that of strain exhibits a jump, implying strain incompatibility due to the mismatch of the material properties of two dissimilar poled ceramics.

Anti-plane piezoelastic study on plane layered media

Singularity problems in a multilayered piezoelectric body composed of arbitrary number of dissimilar plane layers perfectly bonded are analyzed in the framework of linear piezoelectricity. Each layer of the composite is assumed to posses the symmetry of a hexagonal crystal in the 6 mm class and subject to electromechanical loadings, which produce in-plane electric fields and out-of-plane displacement. Based on the complex variable theory and the method of alternating technique, the solution of electric and displacement fields is expressed in terms of explicit series form. The boundary shear stress and electric field for a practical case are discussed graphically. Some certain piezoelectric constant combination to make the composite perform the largest response of electric field is found and can be used to build a relatively effective sensor in a piezoelectric composite material system.

Piezoelectric study on confocally multicoated elliptical inclusion

Within the framework of linear piezoelectricity, an effective method is developed and used to derive an analytical solution for the problem of a confocally multicoated elliptical inclusion embedded in an unbounded matrix which is subjected to arbitrary electromechanical loadings. Each layer of the composite is assumed to have different material constants, but has the same material orientation with x 3 being the poling direction. The alternating technique in conjunction with the method of analytical continuation is applied to derive the general series solution of electric field and displacement field for each layer of the composite. This approach has a clear advantage in deriving the solution to the heterogeneous problem in terms of the solution to the corresponding homogeneous problem. Several specific solutions are provided in closed form, which are verified by comparison with the existing ones. Some numerical results are provided to investigate the influence of material combinations and different electromechanical loadings on the shear stress and electric field. Besides, they also show that the derived solutions are well satisfied with the boundary conditions.

Piezoelectric study on circularly cylindrical layered media

In this paper, an analytical solution in series form for the problem of a circularly cylindrical layered piezoelectric composite consisting of N dissimilar layers is presented within the framework of linear piezoelectricity. Each layer of the composite is assumed to be transversely isotropic with respect to the longitudinal direction ( x 3 direction), and the composite is subject to arbitrary electromechanical singularities infinitely extended in a direction perpendicular to the x 1– x 2 plane such that only in-plane electric fields and out-of-plane displacement are produced. The alternating technique in conjunction with the method of analytical continuation is applied to derive the general multilayered media solution in an explicit series form, whose convergence is guaranteed numerically. The distributions of the shear stress and electric field are found to be dependent on the material combinations and the magnitude and position of the electromechanical singularities. An exactly closed form solution is obtained and discussed graphically for a practical example.

Anti-plane piezoelastic study on singularities interacting with interfaces

Singularity problems in a tripiezoelectric body composed of three dissimilar materials bonded along two parallel interface are analyzed in the framework of linear piezoelectricity. Each layer of the trimaterial is assumed to possess the symmetry of a hexagonal crystal in the 6 mm class and subject to electromechanical loadings which produce in-plane electric fields and out-of-plane displacement. Based on the complex variable theory and the method of alternating technique, the solution of electric field and displacement field is expressed in terms of explicit series form. Some piezoelectric constant combinations to make the composite perform the largest response of electric field have been found, and can be used to build a relatively effective sensor in a piezoelectric composite material system.

An asymptotic theory for vibrations of inhomogeneous/laminated piezoelectric plates.

An asymptotic theory for the vibration analysis of inhomogeneous monoclinic piezoelectric plates is developed by using small parameter expansion. The theory includes the important special case of a laminated plate in which each layer is homogeneous and orthotropic, but distinct from the other layers by having a different material or a different orientation. A hierarchy of two-dimensional equations of the same homogeneous operator for each order is reduced from the three-dimensional framework of linear piezoelectricity. The elasticity version of the leading-order equation is the same as that of the classical Kirchhoff inhomogeneous plate theory and, therefore, is easily solvable. By contrast, it is not straightforward to find solutions of the higher-order equations. The solvability condition is thus established for this purpose, by which higher-order frequency parameters are derived. The present theoretical formulation is examined by comparing the present asymptotic results with an exact three-dimensional solution for a piezoelectric bimorph strip, and excellent agreement is reached. Some new results are presented.

Effect of an interphase layer on the electroelastic stresses within a three-phase elliptic inclusion

Electroelastic stresses induced by electromechanical loadings and lattice mismatch between components and surrounding materials are found to significantly influence the electronic performance of devices and, in some cases, are identified as a major cause of failure and degradation. To reduce electromechanical failure an effective method is to apply an intermediate layer, with appropriate geometry and material properties, between the components of dissimilar piezoelectric materials. In this paper, the effect of an intermediate layer on the electroelastic stresses within an elliptical inhomogeneity is examined within the framework of linear piezoelectricity. Exact closed-form solutions are obtained for the electroelastic stresses in the inclusion, the interphase layer and the matrix, respectively, under remote mechanical antiplane shear and inplane electric field, by means of the complex variable method. It is shown that the electroelastic stresses depend on only two complex coefficients. Simple formulae and numerical examples are used to illustrate the effects of the interphase layer on the electroelastic stresses within the inclusion, and the dependency of this effect on the aspect ratio of the elliptical inclusion.

Impact response of an interface crack in a hybrid piezoelectric laminate

In a hybrid laminate containing an interfacial crack between piezoelectric and orthotropic layers, the dynamic field intensity factors and energy release rates are obtained for electro-mechanical impact loading. The analysis is performed within the framework of linear piezoelectricity. By using integral transform techniques, the problem is reduced to the solution of a Fredholm integral equation of the second kind, which is obtained from one pair of dual integral equations. Numerical results for the dynamic stress intensity factor show the influence of the geometry and electric field.

Mode I field intensity factors of infinitely long strip in piezoelectric media

We consider the problem of determining the singular stresses and electric fields in a piezoelectric ceramic strip containing a Griffith crack under in-plane normal loading within the framework of linear piezoelectricity. The potential theory method and Fourier transforms are used to reduce the problem to the solution of dual integral equations, which are then expressed to a Fredholm integral equation of the second kind. Numerical values on the field intensity factors are obtained, and the influences of the electric fields for PZT-6B piezoelectric ceramic are discussed.

Interacting circular inclusions in antiplanepiezoelectricity

The problem of multiple piezoelectric circular inclusions, which are perfectly bondedto a piezoelectric matrix, is analyzed in the framework of linear piezoelectricity. Both the matrixand the inclusions are assumed to possess the symmetry of a hexagonal crystal in the 6 mm classand subject to electromechanical loadings (singularities) which produce in-plane electric fieldsand out-of-plane displacement. Based upon the complex variable theory and the method ofsuccessive approximations, the solution of electric field and displacement field either in theinclusions or in the matrix is expressed in terms of explicit series form. Stress and electric fieldconcentrations are studied in detail which are dependent on the mismatch in the materialconstants, the distance between two circular inclusions, and the magnitude of electromechanicalloadings. It is shown that, when the two inclusions approach each other, the oscillatory behaviorof the stress and electric field can be induced in the inclusion as the matrix and the inclusions arepoled in the opposite directions. This important phenomenon can be utilized to build a verysensitive sensor in a piezoelectric composite material system. The present derived solution canalso be applied to the inclusion problem with straight boundaries. The problem associated withthree-material media under electromechanical sources is also considered.

Circular inclusion problem in antiplane piezoelectricity

A circular piezoelectric inclusion embedded in an infinite piezoelectric matrix is analysed in the framework of linear piezoelectricity. A closed-form solution is obtained for the case of a far-field antiplane mechanical load and a far-field inplane electrical load. It is shown that in modeling cavities, imposing an impermeable boundary condition is a good approximation provided that the piezoelectric material has high dielectric constant and strong electro-elastic coupling. Stress and electric field concentrations are also studied. It is shown that a high electric field can be induced in the inclusion under a mechanical load when the matrix and the inclusion are poled in the opposite directions. The path-independent M-integral of elastostatics, generalized to take piezoelectric effect into account, is used to study the energetics of a self-similarly expanding piezoelectric inclusion.