Elastic fields due to dislocation arrays in anisotropic bimaterials

Based on the single-dislocation Green’s function, analytical solutions of the elastic fields due to dislocation arrays in an anisotropic bimaterial system are derived by virtue of the Cottrell summation formula. The singularity in the Peach–Koehler (P–K) force is removed by both rigorous mathematical approach and physical energy consideration. Numerical results for dislocation arrays in the Cu/Nb bimaterial with Kurdjumov–Sachs (K–S) orientation show that: (1) the traction continuity and periodic condition are both satisfied; (2) the maximum magnitude of the traction at the interface due to a mixed dislocation array is smaller than that due to a single mixed dislocation. In other words, the traction at the interface could be suppressed by the corresponding array with a relatively high density (L < 10 nm); however, the shear stress on the glide plane increases with increasing dislocation density; (3) the Cu/Nb interface attracts the mixed dislocation array in copper and repels the screw one there. This implies that the P–K force depends not only on the material properties, but also on the crystal orientation and the type of Burgers vector, among others.     

Vibration of beams with piezoelectric inclusions

A mathematical model for the vibration of beams with piezoelectric inclusions is presented. The piezoelectric inclusion in a non-piezoelectric matrix (host beam) is analyzed as two inhomogeneous inclusion problems, elastic and dielectric, by using Eshelby’s equivalent inclusion method. The natural frequency of the beam is determined from the variational principle in Rayleigh quotient form, which is expressed as functions of the elastic strain energy and dielectric energy of the piezoelectric inclusion. The Euler–Bernoulli beam theory and Rayleigh–Ritz approximation technique are used in the present analysis. In addition, a parametric study is conducted to investigate the influence of the energies due to piezoelectric coupling on the natural frequency of the beam.     

Surface and piezoelectric effects on the buckling of piezoelectric nanofilms due to mechanical loads

Piezoelectric nanofilms (PNFs) are often subject to compression in their applications. Bulking thus occurs for those with a thin thickness. In this paper we have conducted a comprehensive study of the bucking behaviors of PNFs which are treated as sandwich-plates to account for the effect of the surface elasticity and piezoelectricity. The results from the analytic formulae show that the surface and piezoelectric effects influence not only the critical buckling load but also the associated modes. Specifically the two effects depend strongly on the nature of the residual surface stress and the direction of the voltage applied on PNFs.     

Scattering attenuation of elastic waves due to low-contrast inclusions

Seismic scattering attenuation due to random lithospheric heterogeneity has been theoretically modeled using two approaches. One approach is the Born approximation theory (BAT), which is primarily used to treat weak continuous heterogeneity, and the other approach is the Foldy approximation theory (FAT), which deals with sparsely distributed discrete inclusions. We apply the BAT to elastic wave scattering due to inclusions having low contrast with the matrix, and compare the results with those predicted by the FAT. We thus investigate the valid wavenumber range of the BAT based on a reasonable assumption that the inclusions are distributed so sparsely that the FAT is effectively correct for any wavenumber. For simplicity, we consider a specific type of round inclusion, which is either two- or three-dimensional and has a two-valued wave velocity and/or mass density. Both theories are confirmed to yield essentially equivalent results below a certain wavenumber limit, depending on the contrast. This is known as the Rayleigh-Gans scattering regime. Beyond the wavenumber limit, the BAT overestimates the attenuation for common-mode scattering due to wave-velocity contrast, but remains valid with respect to the attenuation for scattering due to mass-density contrast and/or conversion scattering. These conclusions are independent of the spatial dimensions of the media as well as the modes of the elastic waves (P or S). Some advantages of the BAT over the FAT for application to low-contrast inclusions are discussed.     

Graded piezoelectric cylinders subjected to high electric fields and comparison of their frequency response with piezoelectric plates

In this paper we investigated a radially polarized piezoceramic cylinder with graded piezoelectric properties, and used a nonlinear model for piezoceramics subjected to high electric fields. We investigated the nonlinear behavior of this material by examining changes in its electric-field-dependent dielectric and piezoelectric coefficients caused by domain wall motion. The Galerkin finite-element method was used to solve the governing equations of the axisymmetrically loaded heterogeneous piezoceramic medium subjected to harmonic electrical loading. Stress, displacement, resonance, and frequency responses were compared for homogeneous and graded cylinders; additionally, we compared the results of linear and nonlinear studies. We showed that the effective stress was higher within the graded cylinder than within the homogeneous cylinder, and that the nonlinearity caused by domain wall motion was less pronounced for the graded cylinder than for the homogeneous cylinder. The frequency responses of homogeneous and heterogeneous piezocylinders were also compared with those of piezoelectric plates. We concluded that—unlike for graded plates, which have a more desirable frequency response than homogeneous plates—graded cylinders are not superior to homogeneous cylinders. The finite-element solution in this paper is verified by simulations using COMSOL Multiphysics software.     

Elastic vibrations due to a high-current arc discharge

The passage of transient electric currents through a gas into a solid occurs in high-voltage switches and in lightning strikes. The experiments reported in this paper were designed to examine the vibration of a cantilevered conductor under electromagnetic forces of the arc and structural currents. Arcs are created at low pressures in a vacuum facility and the induced vibrations of the beam are measured. It is shown that the deflections are proportional to the square of the arc currents. Further, 5-μs photographs of the 2-ms arc, using a Kerr cell, show that the arc is distorted by the magnetic field of the beam currents. A simple analytical model is shown to reproduce the basic results of the experiments.     

The electro-elastic field of the infinite piezoelectric medium with two piezoelectric circular cylindrical inclusions

The electro-elastic field of the infinite piezoelectric medium with two piezoelectric circular cylindrical inclusions is derived under the antiplane shear stresses and inplane electric fields. The analytical solution is obtained. The proposed method is based upon the use of conformal mapping and the theorem of analytic continuation. From the results obtained, it can be found that the electro-elastic field depends on the material constants of individual phases, the geometric parameters of the system and the applied antiplane shear stresses and electric fields at infinity. In addition, the specific cases when two circular cylindrical inclusions are tangent to each other and they are holes and/or rigid ones, are also studied in this paper. The project supported by the National Natural Science Foundation of China (19872023) and the Foundation of the Ministry of Education for trans-century outstanding scholars     

Elastic fields of interacting elliptic inhomogeneities

This paper presents a method for the calculation of two-dimensional elastic fields in a solid containing any number of inhomogeneities under arbitrary far field loadings. The method called pseudo-dislocations method, is illustrated for the solution of interacting elliptic inhomogeneities. It reduces the interacting inhomogeneities problem to a set of linear algebraic equations. Numerical results are presented for a variety of elliptic inhomogeneity arrangements, including the special cases of elliptic holes, cracks and circular inhomogeneities. All these complicated problems can be solved with high accuracy and efficiency.     

Eshelby problem of polygonal inclusions in anisotropic piezoelectric full- and half-planes

This paper presents an exact closed-form solution for the Eshelby problem of polygonal inclusion in anisotropic piezoelectric full- and half-planes. Based on the equivalent body-force concept of eigenstrain, the induced elastic and piezoelectric fields are first expressed in terms of line integral on the boundary of the inclusion with the integrand being the Green's function. Using the recently derived exact closed-form line-source Green's function, the line integral is then carried out analytically, with the final expression involving only elementary functions. The exact closed-form solution is applied to a square-shaped quantum wire within semiconductor GaAs full- and half-planes, with results clearly showing the importance of material orientation and piezoelectric coupling. While the elastic and piezoelectric fields within the square-shaped quantum wire could serve as benchmarks to other numerical methods, the exact closed-form solution should be useful to the analysis of nanoscale quantum-wire structures where large strain and electric fields could be induced by the misfit strain.     

Size-dependent electromechanical properties in piezoelectric superlattices due to flexoelectric effect

Piezoelectric superlattice is a potential component for nanoelectromechanical systems.Due to the strong nonlocal effect such as flexoelectric effect at interfaces,classical piezoelectric theory is unable to accurately describe the electromechanical response of piezoelectric superlattice at nanoscale scale.Based on the previous nonlocal thermodynamics theory with flexoelectric effect Liu et al.(2016),the sizedependent electromechanical properties of piezoelectric superlattices made of BaTiO_3(BTO)and PbTiO_3(PTO)layers are investigated systematically in the present work.Giant strain gradient is found near the interface between BTO and PTO layers,which leads to the significant enhancement of polarization in the superlattice due to the flexoelectric effect.For the piezoelectric BTO–PTO superlattices with different unitcell sizes,the thickness of interface with nontrivial strain gradient is almost constant.The influence of strain gradient at the interface becomes significant when the size of superlattice decreases.As a result,a strong size dependence of electromechanical properties is predicted for the piezoelectric BTO–PTO superlattices.In particular,for the superlattices with a specific thickness ratio of BTO and PTO layers,the piezoelectric response can be several times larger than that of bulk structure.The present work demonstrates a practical way to design the piezoelectric superlattices with high piezoelectric coefficient by using the nonlocal effect at nanoscale.     

On the modified Green operator integral for polygonal,polyhedral and other non-ellipsoidal inclusions

A “strange” particularity of polyhedral inclusions and of fibres of regular polygonal cross sections has been recently stressed in the literature. For respectively fully or transversally isotropic elasticity of the embedding material, they have a mean and a central Green operator integral both equal to the uniform one of respectively the sphere or the cylindrical fibre. In using the Radon transform (RT) method, this particularity is here shown to be shared by much larger shape types in the same limits of material elasticity symmetry. As a subcase, even more shape types fulfill the similar particularity for material linear properties of second-rank characteristic tensor, such as thermal conductivity, magnetic or dielectric properties. When calculated using the RT method, the modified Green operator integral at any interior point of a bounded domain (inclusion) takes the form of a weighted average over an angular distribution of a single elementary operator. The weight function is geometrically defined from the characteristic function of the domain, and the four-rank or second-rank elementary operator depends on the material linear property of concern. The RT method simply shows that the noticed particularity is due to matching symmetry between the inclusion shape (through the weight function) and the material property (through the elementary operator). The general geometrical characteristics of the inclusion shapes belonging to these sphere-class and cylindrical-fibre-class are specified, and some remarkable shapes of these classes are commented.     

Elastic compliances and mechanical and adjusting characteristics of composite piezoelectric transducers

We study the compression diagrams and elastic compliances of composite piezoelectric transducers. We find the typical points on the compression diagram which correspond to the mechanical stress of clearance cutting and smoothing the microroughnesses and to the ultimate compression strength with crack formation on the edges of piezoelectric crystal plates. We construct mechanical and adjusting characteristics of piezoelectric transducers and determine their static and dynamic characteristics.     

Debonded arc-shaped interface conducting rigid line inclusions in piezoelectric composites

We consider an arc-shaped conducting rigid line inclusion located at the interface between a circular piezoelectric inhomogeneity and an unbounded piezoelectric matrix subjected to remote uniform anti-plane shear stresses and in-plane electric fields. Moreover, one side of the rigid line inclusion has become fully debonded from the matrix or the inhomogeneity leading to the formation of an insulating crack. After the introduction of two sectionally holomorphic vector functions, the problem is reduced to a vector Riemann–Hilbert problem, which can be decoupled sequentially by repeated application of the orthogonality relations between the eigenvectors for two corresponding generalized eigenvalue problems.     

A unified model for piezocomposites with non-piezoelectric matrix and piezoelectric ellipsoidal inclusions

In this paper, the closed-form solutions of the electroelastic Eshelbys tensors of a piezoelectric ellipsoidal inclusion in an infinite non-piezoelectric matrix are obtained via the Greens function technique. Based on the generalized Budianskys energy-equivalence framework and the closed-form solutions of the electroelastic Eshelbys tensors, a unified model for multiphase piezocomposites with the non-piezoelectric matrix and piezoelectric inclusions is set up. The closed-form solutions of the effective electroelastic moduli of piezocomposites are also obtained. The unified model has a rigorous but simple form, which can describe the multiphase piezocomposites with different connectivities, such as 0–3, 1–3, 2–2, 2–3, 3–3 connectivities, etc. It can also describe the effects of non-interaction and interaction among the inclusions. As examples, the closed-form solutions of the effective electroelastic moduli are given by means of the dilute solution for the 0–3 piezocomposite with transversely isotropic piezoelectric spherical inclusions and by means of the dilute solution and the Mori–Tanakas method for the 1–3 piezocomposite with two kinds of transversely isotropic piezoelectric cylindrical inclusions. The predicted results are compared with experimental data, which shows that the theoretical curves calculated by means of the Mori–Tanakas method agree quite well with the experimental values, but the theoretical curves obtained by the dilute solution agree well with the experimental values only when the volume fraction of the ceramic inclusion is less than 0.3. The results in this paper can be used to analyze and design the multiphase piezocomposites.     

Elastic interaction of partially debonded circular inclusions. II. Application to fibrous composite

A complete analytical solution has been obtained of the elasticity problem for a plane containing periodically distributed, partially debonded circular inclusions, regarded as the representative unit cell model of fibrous composite with interface damage. The displacement solution is written in terms of periodic complex potentials and extends the approach recently developed by Kushch et al. (2010) to the cell type models. By analytical averaging the local strain and stress fields, the exact formulas for the effective transverse elastic moduli have been derived. A series of the test problems have been solved to check an accuracy and numerical efficiency of the method. An effect of interface crack density on the effective elastic moduli of periodic and random structure FRC with interface damage has been evaluated. The developed approach provides a detailed analysis of the progressive debonding phenomenon including the interface cracks cluster formation, overall stiffness reduction and damage-induced anisotropy of the effective elastic moduli of composite.     

Elastic fields in rotating transversely isotropic media

Representation of elastic fields in terms of scalar functions, which permit reducing the problems of determining these fields to determining scalar potentials, are generalized to the case of transversely isotropic media rotating at a constant angular velocity. Relations for calculating the parameters of surface acoustic waves (SAW) propagating in a rotating transversely isotropic halfspace with various directions of the medium material symmetry axis with respect to the half-space surface are given.