Plane problems in piezoelectric media with elliptic inclusion

I.IntroductionPiezoelectricmedia,asa"ex\'typeoffullctionalmaterial.arex'idel}'appliedtomanytechnologicalfieldsduetoitselectronlechallicalcouplillgeffect.Defects.likethatofothermaterials.arenotlimitedtocracks.x'oidsandinclusionsillpiezoelectricmaterialsorelements.Yet,stressconcentrationsornoll-ullitbrllldistl-ibutionsofelectricfieldillducedbythosedefectsareoneofthehe}l'filctorswllicllwouldleadpiezoelectricstructurestonon-normalfailure.Therel'ore.itisofgrealimportancetostudythepropertiesofthos…     

PLANE STRAIN PROBLEM OF PIEZOELECTRIC SOLID WITH ELLIPTIC INCLUSION

By using the complex variables function theory, a plane strain electro-elastic analysis was performed on a transversely isotropic piezoelectric material containing an elliptic elastic inclusion, which is subjected to a uniform stress field and a uniform electric displacement loads at infinity. Based on the present finite element results and some related theoretical solutions, an acceptable conjecture was found that the stress field is constant inside the elastic inclusion. The stress field solutions in the piezoelectric matrix and the elastic inclusion were obtained in the form of complex potentials based on the impermeable electric boundary conditions.     

含椭圆形刚性夹杂的压电材料平面问题

应用复变函数的Ｆａｂｅｒ级数展开方法，本文研究了含椭圆形刚性夹杂的压电材料平面问题，给出了问题的封闭解。解签表明，夹杂内的电场强度和电位移为常量。并通过算例分析，讨论了正，逆压电效应在基体孔周处的机电行为。     

On the electroelastic interaction of a piezoelectric screw dislocation with an elliptical inclusion in piezoelectric materials

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｉｎｔｅｒａｃｔｉｏｎｏｆｄｉｓｌｏｃａｔｉｏｎｓｗｉｔｈｉｎｃｌｕｓｉｏｎｓｉｓｏｆｃｏｎｓｉｄｅｒａｂｌｅｉｍｐｏｒｔａｎｃｅｆｏｒｕｎｄｅｒｓｔａｎｄｉｎｇｔｈｅｐｈｙｓｉｃａｌｂｅｈａｖｉｏｒｏｆｍａｔｅｒｉａｌｓ.Ｓｕｃｈｓｔｕｄｉｅｓｃａｎｐｒｏｖｉｄｅｄｉｎｆｏｒｍａｔｉｏｎｃｏｎｃｅｒｎｉｎｇｃｅｒｔａｉｎｓｔｒｅｎｇｔｈｅｎｉｎｇｏｒｈａｒｄｅｎｉｎｇｍｅｃｈａｎｉｓｍｓｉｎｎｕｍｂｅｒｏｆｔｒａｄｉｔｉｏｎａｌａｎｄｃｏｍｐｏｓｉｔｅｍａｔｅｒｉ…     

Anisotropic thermopiezoelectric solids with an elliptic inclusion or a hole under uniform heat flow

The two-dimensional problem of a thermopiezoelectric material containing an elliptic inclusion or a hole subjected to a remote uniform heat flow is studied. Based on the extended Lekhnitskii formulation for thermopiezoelectricity, conformal mapping and Laurent series expansion, the explicit and closed-form solutions are obtained both inside and outside the inclusion (or hole). For a hole problem, the exact electric boundary conditions on the hole surface are used. The results show that the electroelastic fields inside the inclusion or the electric field inside the hole are linear functions of the coordinates. When the elliptic hole degenerates into a slit crack, the electroelastic fields and the intensity factors are obtained. The effect of the heat flow direction and the dielectric constant of air inside the crack on the thermal electroelastic fields are discussed. Comparison is made with two special cases of which the closed solutions exist and it is shown that our results are valid.     

Coupled solution for anisotropic piezoelectric materials with cracks

The coupled elastic and electric fields for anisotropic piezoelectric materials with electrically permeable cracks are analyzed by using Stroh formula in anisotropic elasticity. It is shown from the solution that the tangent component of the electric field strength and the normal component of the electric displacement along the faces of cracks are all constants, and the electric field intensity and electric displacement have the singularity of type (1/2) at the crack tip. The energy release rate for crack propagation depends on both the stress intensity factor and material constants. The electric field intensity and electric displacement inside electrically permeable cracks are all constants.     

Two-dimensional electroelastic fundamental solutions for general anisotropic piezoelectric media

I.IntroductionDuetotheirintrinsiccouplingeffectbetweenmechanicalandelectricalfields,piezoelectricmaterialshavebeenwidelyusedintechnologyastransducersandsensorsand,morerecently,asactuatorsinsmartstructures.lnordertooptimizetheirmicrostructuresandunderstand…     

The effective properties of piezocomposites, part I: Single inclusion problem

The problem of a piezoelectric ellipsoidal inclusion in an infinite nonpiezoelectric matrix is very important in engineering. In this paper, it is solved via Green's function technique. The closed-form solutions of the electroelastic Eshelby's tensors for this kind of problem are obtained. The electroelastic Eshelby's tensors can be expressed by the Eshelby's tensors of the perfectly elastic inclusion problem and the perfectly dielectric inclusion problem. Since the closed-form solutions of the Eshelby's tensors of the perfectly elastic inclusion problem and the perfectly dielectric inclusion problem can be given by theory of elasticity and electrodynamics, respectively, the electroelastic Eshelby's tensors can be obtained conveniently. Using these results, the closed-form solutions of the constraint elastic fields and the constraint electric fields inside the piezoelectric ellipsoidal inclusion are also obtained. These expressions can be readily utilized in solutions of numerous problems in the micromechanics of piezoelectric solids, such as the deformation and energy analysis, damage evolution and fracture of the piezoelectric materials. The project supported by the National Natural Science Foundation of China     

Electro-elastic fundamental solutions of anisotropic piezoelectric materials with an elliptical hole

By using Stroh' complex formalism and Cauchy's integral method, the electro-elastic fundamental solutions of an infinite anisotropic piezoelectric solid containing an elliptic hole or a crack subjected to a line force and a line charge are presented in closed form. Particular attention is paid to analyzing the characteristics of the stress and electric displacement intensity factors. When a line force-charge acts on the crack surface, the real form expression of intensity factors is obtained. It is shown through a special example that the present work is correct. The project supported by the Fund of the State Education Commission of China for Excellent Young Teachers     

The effective properties of piezoelectric composite materials with transversely isotropic spherical inclusions

ＩｎｔｒｏｄｕｃｔｉｏｎＷｉｔｈｔｈｅｄｅｖｅｌｏｐｍｅｎｔｏｆｉｎｆｏｒｍａｔｉｏｎｉｎｄｕｓｔｒｙａｎｄｔｈｅａｐｅａｒａｎｃｅｏｆｓｍａｒｔｍａｔｅｒｉａｌｓａｎｄｓｍａｒｔｓｔｒｕｃｔｕｒｅｓ,ｉｔｂｅｃｏｍｅｓｍｏｒｅａｎｄｍｏｒｅｉｍｐｏ...     

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

ＩｎｔｒｏｄｕｃｔｉｏｎＤｕｅｔｏｔｈｅｉｎｔｒｉｎｓｉｃｃｏｕｐｌｉｎｇｃｈａｒａｃｔｅｒｉｓｔｉｃｓｂｅｔｗｅｅｎｅｌｅｃｔｒｉｃａｎｄｅｌａｓｔｉｃｂｅｈａｖｉｏｒｓ,ｔｈａｔｉｓ,ａｐｐｌｉｅｄｍｅｃｈａｎｉｃａｌｌｏａｄｉｎｇｐｒｏｄｕｃｅｓｅｌａｓｔｉｃｄｅｆｏｒｍａｔｉｏｎ ,ａｓｗｅｌｌａｓｅｌｅｃｔｒｉｃｆｉｅｌｄ ,ａｎｄｃｏｎｖｅｒｓｅｌｙｅｌｅｃｔｒｉｃｆｉｅｌｄｃａｎｇｉｖｅｒｉｓｅｔｏｅｌａｓｔｉｃｄｅｆｏｒｍａｔｉｏｎ ,ｐｉｅｚｏｅｌｅｃｔｒｉｃｍａｔｅｒｉａｌｓｈａｖｅ…     

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-)     

Coupled fields of two-dimensional anisotropic magneto-electro-elastic solids with an elliptical inclusion

ＩｎｔｒｏｄｕｃｔｉｏｎＤｅｔｅｒｍｉｎａｔｉｏｎｏｆｐｅｒｔｕｒｂｅｄｆｉｅｌｄｓｉｎｄｕｃｅｄｂｙｉｎｃｌｕｓｉｏｎｓｈａｓｂｅｅｎａｎｉｍｐｏｒｔａｎｔｔｏｐｉｃｆｏｒｓｔｕｄｙｉｎｇｔｈｅｐｈｙｓｉｃａｌｂｅｈａｖｉｏｒｓｏｆａｄｖａｎｃｅｄｃｏｍｐｏｓｉｔｅｍａｔｅｒｉａｌｓ.Ｆｒｏｍｔｈｅｐｏｉｎｔｏｆｖｉｅｗｏｆａｐｐｌｉｃａｔｉｏｎｓ,ｔｈｅａｄｖａｎｃｅｄｃｏｍｐｏｓｉｔｅｍａｔｅｒｉａｌｓｃａｎｂｅｄｉｖｉｄｅｄｉｎｔｏｔｗｏｃａｔｅｇｏｒｉｅｓ:ｏｎｅｉｓｔｈｅｓｔｒｕｃｔｕ…     

Basic solution of two parallel mode-I permeable cracks in functionally graded piezoelectric materials

The basic solution of two parallel mode-I permeable cracks in functionally graded piezoelectric materials was studied in this paper using the generalized Almansi’s theorem. To make the analysis tractable, it was assumed that the shear modulus varies exponentially along the horizontal axis parallel to the crack. The problem was formulated through a Fourier transform into two pairs of dual integral equations, in which unknown variables are jumps of displacements across the crack surface. To solve the dual integral equations, the jumps of displacements across the crack surfaces were directly expanded as a series of Jacobi polynomials. The solution of the present paper shows that the singular stresses and the singular electric displacements at the crack tips in functionally graded piezoelectric materials carry the same forms as those in homogeneous piezoelectric materials; however, the magnitudes of intensity factors depend on the gradient of functionally graded piezoelectric material properties. It was also revealed that the crack shielding effect is also present in functionally graded piezoelectric materials.     

Inclusion of an arbitrary polygon with graded eigenstrain in an anisotropic piezoelectric half plane

This paper presents an exact closed-form solution for the Eshelby problem of a polygonal inclusion with graded eigenstrains in an anisotropic piezoelectric half plane with traction-free on its surface. Using the line-source Green’s function, the line integral is carried out analytically for the linear eigenstrain case, with the final expression involving only elementary functions. The solutions are applied to the semiconductor quantum wire (QWR) of square, triangular, and rectangular shapes, with results clearly illustrating various influencing factors on the induced fields. The exact closed-form solution should be useful to the analysis of nanoscale QWR structures where large strain and electric fields could be induced by the non-uniform misfit strain.     

The effective properties of piezocomposites, part II: The effective electroelastic moduli

Since piezoelectric ceramic/polymer composites have been widely used as smart materials and smart structures, it is more and more important to obtain the closed-from solutions of the effective properties of piezocomposites with piezoelectric ellipsoidal inclusions. Based on the closed-from solutions of the electroelastic Eshelby's tensors obtained in the part I of this paper and the generalized Budiansky's energy-equivalence framework, the closed-form general relations of effective electroelastic moduli of the piezocomposites with piezoelectric ellipsoidal inclusions are given. The relations can be applicable for several micromechanics models, such as the dilute solution and the Mori-Tanaka's method. The difference among the various models is shown to be the way in which the average strain and the average electric field of the inclusion phase are evaluated. Comparison between predicted and experimental results shows that the theoretical values in this paper agree quite well with the experimental results. These expression can be readily utilized in analysis and design of piezocomposites. The project supported by the National Natural Science Foundation of China     

General treatment of mode III interfacial crack problems in piezoelectric materials

Summary  The anti-plane problem of N collinear interfacial cracks between dissimilar transversely isotropic piezoelectric media, which are subjected to piecewise uniform out-of-plane mechanical loading combined with in-plane electric loading at infinity, and also a line loading at an arbitrary point, is addressed by using the complex function method. In comparison with other relevant works, the present study has two features: one is that the analysis is based on the permeable crack model, i.e. the cracks are considered as permeable thin slits, and, thus, both the normal component of electric displacement and the tangential component of electric field are assumed to be continuous across these slits. The other feature is that explicit closed-form solutions are given not only in piezoelectric media, but also inside cracks when the media are subjected to the most general loading. It is shown that the singularities of electric displacement and electric field in the media are always dependent on that of stress for the general case of loading, and all the singularities of field variables are independent of the applied uniform electric loads at infinity. For the interfacial cracks the electric field is square-root singular at the crack tips and shows jumps across the interface, while the normal component of the electric field is linearly variable inside the crack, but the tangential component is square-root singular. However, for a homogeneous medium with collinear cracks, the electric field is always nonsingular in the medium while the electric displacement exhibits square-root singularity. Moreover, in this case, the electric field inside any crack is equal to a constant when uniform loads are applied at infinity. Received 22 November 1999; accepted for publication 20 July 2000     

Analysis of a partially debonded elliptic inhomogeneity in piezoelectric materials

A generalized solution was obtained for the partially debonded elliptic inhomogeneity problem in piezoelectric materials under antiplane shear and inplane electric loading using the complex variable method. It was assumed that the interfacial debonding induced an electrically impermeable crack at the interface. The principle of conformal transformation and analytical continuation were employed to reduce the formulation into two Riemann-Hilbert problems. This enabled the determination of the complex potentials in the inhomogeneity and the matrix by means of series of expressions. The resulting solution was then used to obtain the electroeiastic fields and the energy release rate involving the debonding at the inhomogeneity-matrix interface. The validity and versatility of the current general solution have been demonstrated through some specific examples such as the problems of perfectly bonded elliptic inhomogeneity , totally debonded elliptic inhomogeneity, partially debonded rigid and conducting elliptic inhomogeneity, and partially debonded circular inhomogeneity.     

压电体椭圆孔边的力学分析

基于复变函数的方法,以PZT-4材料为例,分别采用精确电边界条件和非导通电边界条件进行了远场均匀载荷作用下的横观各向同性压电体椭圆孔的力学分析并与相关结果进行对比。结果表明当椭圆孔退化为圆孔时,无论在远场作用力载荷或电载荷,两种电边界条件下的结果均能完全吻合。随着椭圆孔的愈加尖锐化,非导通电边界条件逐渐不能适用。     

Analysis of a plate containing an elliptic inclusion with eigencurvatures

Summary An infinite plate containing an elliptic subregion in which a uniform eigencurvature is prescribed is analyzed. The problem is formulated by using the classical plate theory. Employing the Maysel's relation, an integral-type solution to the equilibrium equation is expressed in terms of the eigencurvature. Closed-form solutions of the displacement and corresponding resultant moment are obtained for interior points as well as for exterior points of the ellipse. An infinite plate containing an elliptic inhomogeneity in which a uniform eigencurvature is prescribed is also considered. The disturbance of the displacement and corresponding resultant moment due to the inhomogeneity is determined by the equivalent eigencurvature method. Solutions of a circular finite plate with uniform eigencurvature in a circular zone are also obtained analytically. Received 30 September 1997; accepted for publication 3 February 1998