Strip electro-mechanical yielding model for piezoelectric plate cut along two equal collinear cracks

The multiple-crack problems for piezoelectric ceramics till now have not yet address the crack opening arrest problem. The present work addresses this paucity. A 2-D strip-electro-mechanical yielding model is proposed for a transversely isotropic piezoelectric media weakened by two internal equal collinear straight cracks. The infinite boundary is prescribed with combined uniform constant in-plane mechanical and electrical loads. Developed mechanical and electric strip zones are arrested by prescribing over their rims uniform, normal, cohesive yield point stress and saturation limit electric displacement. Two cases are considered when saturation zone is bigger than developed yield zone and vice versa. Stroh formulation together with complex variable technique is employed to obtain the solution. Closed form expressions are derived for saturation zone length, yield zone length, crack opening displacement (COD), crack opening potential jump (COP) and energy release rate (ERR). An illustrative numerical study is prescribed to determine the effect of various parameters on the crack growth arrest and presented graphically. The results reveal that the model is capable of crack arrest under small-scale mechanical and electric yielding.     

Strip-saturation model for piezoelectric plane weakened by two collinear cracks with coalesced interior zones

A strip-saturation model is proposed for a transversely isotropic piezoelectric plane weakened by two collinear equal cracks, when developed saturation zones at the interior tips of the cracks get coalesced. The plane is subjected to unidirectional, normal (to the crack length) in-plane tension and electric displacement. The developed saturation zones are arrested by distributing over their rims the normal, cohesive, unidirectional saturation-limit electrical displacement. The solution is obtained using Stroh formulation and complex variable technique. Closed form expressions are derived for crack opening displacement (COD), crack potential drop (COP), field intensity factors, length of saturation zone, energy release rate. Case study carried out for PZT-4 to show the effects of inter-crack distance on the stress intensity factor. The variations of energy release rates are plotted for PZT-4, PZT-5H and BaTiO₃ to study the effects of the geometry of the two cracks.     

Transient response of two collinear dielectric cracks in a piezoelectric solid under inplane impacts

The paper is focused on the dynamic analysis of two collinear dielectric cracks in a piezoelectric material under the action of in-plane electromechanical impacts. Considering the dielectric permeability of crack interior, the electric displacements at the crack surfaces are governed by the jumps of electric potential and crack opening displacement across the cracks. The permeable and impermeable crack models are the limiting cases of the general one. The Laplace and Fourier transform techniques are further utilized to solve the mixed initial-boundary-value problem, and then to obtain the singular integral equations with Cauchy kernel, which are solved numerically. Dynamic intensity factors of stress, electric displacement and crack opening displacement are determined in time domain by means of a numerical inversion of the Laplace transform. Numerical results for PZT-5H are calculated to show the effects of the dielectric permeability inside the cracks, applied electric loadings and the geometry of the cracks on the fracture parameters in graphics. The observations reveal that based on the COD intensity factor, a positive electric field enhances the dynamic dielectric crack growth and a negative one impedes the dynamic dielectric crack growth in a piezoelectric solid.     

Closed-form solutions for two collinear dielectric cracks in a magnetoelectroelastic solid

The problem of two collinear electromagnetically dielectric cracks in a magnetoelectroelastic material is investigated under in-plane electro-magneto-mechanical loadings. The semi-permeable crack-face boundary conditions are adopted to simulate the case of two collinear cracks full of a dielectric interior. Utilizing the Fourier transform technique, the boundary-value problem is reduced to solving singular integral equations with Cauchy kernel, which then are solved explicitly. The intensity factors of stress, electric displacement, magnetic induction, crack opening displacement (COD) and the energy release rates near the inner and outer crack tips are determined in closed forms for two cases of possible far-field electro-magneto-mechanical loadings respectively. Numerical results for a BaTiO₃–CoFe₂O₄ composite are carried out to show the effects of applied mechanical loadings on the crack-face electric displacement and magnetic induction, the stress intensity factor and the COD intensity factor, respectively. The obtained results reveal that when the applied mechanical loading is stress, applied electromagnetic loadings have no influences on the stress intensity factor. When the applied mechanical loadings is train, the applied positive electromagnetic loadings decrease the intensity factors of stress and COD, and the applied negative ones increase the intensity factors of stress and COD. The variations of energy release rates are also given to show the effects of the geometry of two collinear dielectric cracks.     

Basic solutions of a 3D rectangular limited-permeable crack or two 3D rectangular limited-permeable cracks in the piezoelectric/piezomagnetic composite materials

The solutions of a three-dimensional rectangular limited-permeable crack or two three-dimensional rectangular limited-permeable cracks in the piezoelectric/piezomagnetic composite materials were investigated by using the generalized Almansi’s theorem and the Schmidt method. Finally, the relations among the electric field, the magnetic flux field and the stress field near the crack tips were obtained and the effects of the electric permittivity, the magnetic permeability of the air inside the crack, the shape of the rectangular crack on the stress, the electric displacement and magnetic flux intensity factors in the piezoelectric/piezomagnetic composite materials were analyzed.     

Stress intensity factors for three cracks at the interfaces of a graded layer bonding two different materials

We consider two dissimilar elastic half-planes bonded by a nonhomogeneous elastic layer in which there is one crack at the lower interface between the elastic layer and the lower half-plane and two cracks at the upper interface between the elastic layer and the upper half-plane. The stress intensity factors for these three cracks are solved for when tension is applied perpendicular to the interface cracks. The material properties of the bonding layer vary continuously between those of the lower half-plane and those of the upper half-plane. The differences in the crack surface displacements are expanded in a series of functions that are zero outside the cracks. The unknown coefficients in the series are solved by the Schmidt method so as to satisfy the conditions inside the cracks. The stress intensity factors are calculated numerically for selected crack configurations.     

Electro-mechanical sliding frictional contact of a piezoelectric half-plane under a rigid conducting punch

This paper investigates the two-dimensional sliding frictional contact of a piezoelectric half-plane in the plane strain state under the action of a rigid flat or a triangular punch. It is assumed that the punch is a perfect electrical conductor with a constant electric potential. By using the Fourier integral transform technique and the superposition theorem, the problem is reduced to a pair of coupled Cauchy singular integral equations and then is numerically solved to determine the unknown contact pressure and surface electric charge distribution. The effects of the friction coefficient and electro-mechanical loads on the normal contact stress, normal electric displacement, in-plane stress and in-plane electric displacement are discussed in detail. It is found that the friction coefficient has a significant effect on the electro-mechanical sliding frictional contact behaviors of the piezoelectric materials.     

Two-dimensional solution for electro-mechanical behavior of functionally graded piezoelectric hollow cylinder

In this paper, the general theoretical analysis for a hollow cylinder made of functionally graded piezoelectric material subjected to two-dimensional electromechanical load, is developed. The material properties, except the Poisson’s ratio, are assumed to vary with the power law function through the thickness of the cylinder. The mechanical and electrical displacements are assumed to be a function of radial and circumferential directions. By using the separation of variables method and complex Fourier series, the Navier equations in terms of displacements are derived and solved.     

Approximation of the Stieltjes Integral and Applications in Numerical Integration

Some inequalities for the Stieltjes integral and applications in numerical integration are given. The Stieltjes integral is approximated by the product of the divided difference of the integrator and the Lebesgue integral of the integrand. Bounds on the approximation error are provided. Applications to the Fourier Sine and Cosine transforms on finite intervals are mentioned as well.     

傅里叶变换中一个含参变量广义积分的计算

本文利用含参变量广义积分计算中常用的求导和变量替换等技巧等给出了傅里叶变换中一个重要的含参变量广义积分的计算方法.     

Equality of two spectra arising in harmonic analysis and semigroup theory

We show that a new notion of a spectrum of a function ( is a Banach space), defined by B. Basit and the first author, coincides with the Arveson spectrum of some shift group, provided is uniformly continuous. We apply this result to prove a new version of a tauberian theorem.     

Effects of electric and mechanical loads on the morphological evolution of a void in piezoelectric films

This paper reports the result of an investigation into the effect of electric and mechanical loads on the morphological evolution of a void in piezoelectric materials based on a model for the morphological evolution of a void, the thermodynamics potential and energy principle. Thus, the path and the bifurcation condition of the morphological evolution of the void in piezoelectric materials are described, which gives some insight into the reliability of piezoelectric films under electric and mechanical loads.     

Boundary integral equation methods in the theory of elasticity of hemitropic materials: A brief review

The theory of elasticity of hemitropic materials has recently been the object of rigorous mathematical analysis. In particular, the potential method and the theory of pseudodifferential equations have been used in studying the solvability in various function spaces of the main boundary value and transmission problems, in smooth and in Lipschitz domains. The main features and results of this boundary integral equations approach are briefly reviewed here.     

EXISTENCE OF GLOBAL SMOOTH SOLUTION FOR SCALAR CONSERVATION LAWS WITH DEGENERATE VISCOSITY IN 2-DIMENSIONAL SPACE

This article concerns the existence of global smooth solution for scalar conservation laws with degenerate viscosity in 2-dimensional space. The analysis is based on successive approximation and maximum principle.     

Transmission problems in the theory of elastic hemitropic materials

The purpose of this article is to investigate mixed transmission-boundary value problems for the differential equations of the theory of hemitropic (chiral) elastic materials. We consider the differential equations corresponding to the time harmonic dependent case, the so called pseudo-oscillation equations. Applying the potential method and the theory of pseudodifferential equations we prove uniqueness and existence theorems of solutions to the Dirichlet, Neumann and mixed transmission-boundary value problems for piecewise homogeneous hemitropic composite bodies and analyze their regularity properties. We investigate also interface crack problems and establish almost best regularity results.     

夹层压电材料中垂直于界面的共线双裂纹动力学问题分析

采用Schmidt方法分析了在简谐反平面剪切波作用下,两个半空间夹层压电材料中的共线裂纹的动力学行为．压电材料层内裂纹垂直于界面,电边界条件假设为可导通．通过Fourier变换,使问题的求解转换为两对三重积分对偶方程．通过数值计算,给出了裂纹的几何尺寸、压电材料常数、入射波频率等对于应力强度因子的影响．结果表明,在不同的入射波频率范围,动力场将阻碍或促使压电材料内裂纹的扩展．与不可导通电边界条件相比,导通裂纹表面的电位移强度因子比不可导通裂纹的电位移强度因子要小许多．     

采用新方法研究加层压电材料中平行界面共线双裂纹的断裂问题

利用新的方法(Schmidt方法)研究加层压电材料中含共线并与材料界面平行的双裂纹在稳态弹性波作用下的动态问题,经富立叶变换使问题的求解转换为求解两对三重对偶积分方程.这些方程可以采用Schmidt方法来求解,这个方法不同与以前求解所利用的方法.结果表明应力强度因子不仅与裂纹的几何尺寸、入射波频率、加层厚度有关,而且与材料性质有关.     

Analytic solution for MHD Transient rotating flow of a second grade fluid in a porous space

This article looks into the unsteady rotating magnetohydrodynamic (MHD) flow of an incompressible second grade fluid in a porous half space. The flow is induced by a suddenly moved plate in its own plane. Both the fluid and plate rotate in unison with the same angular velocity. Analytic solution of the governing flow problem is obtained by using Fourier sine transform. Based on the modified Darcy's law, expression for velocity is obtained. The influence of pertinent parameters on the flow is delineated and appropriate conclusions are drawn. Several existing solutions of Newtonian fluid have been also deduced as limiting cases.     

Modeling of tsunami generation and propagation by a spreading curvilinear seismic faulting in linearized shallow-water wave theory

The processes of tsunami evolution during its generation in search for possible amplification mechanisms resulting from unilateral spreading of the sea floor uplift is investigated. We study the nature of the tsunami build up and propagation during and after realistic curvilinear source models represented by a slowly uplift faulting and a spreading slip-fault model. The models are used to study the tsunami amplitude amplification as a function of the spreading velocity and rise time. Tsunami waveforms within the frame of the linearized shallow-water theory for constant water depth are analyzed analytically by transform methods (Laplace in time and Fourier in space) for the movable source models. We analyzed the normalized peak amplitude as a function of the propagated uplift length, width and the average depth of the ocean along the propagation path.